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3800	https://www.quora.com/What-are-some-good-examples-of-Newtons-third-law-to-every-action-there-is-an-equal-and-opposite-reaction-in-real-life	What are some good examples of Newton's third law, 'to every action, there is an equal and opposite reaction,' in real life? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In physics real life examples of kar... stimulus and response newton's third law of mot... science real-life cases practical examples actions and reactions science physics 5 What are some good examples of Newton's third law, "to every action, there is an equal and opposite reaction," in real life? All related (56) Sort Recommended Ron Brown Decades of teaching physics to undergrads · Author has 13.4K answers and 81.8M answer views ·5y Originally Answered: What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? · For one thing, one doesn’t “prove” Newton’s third law - or any other law of physics, for that matter. Laws in physics are just a result of verified observations and experiments. If an observation were made that was inconsistent with this law, then the law would be modified to account for the newly discovered inconsistency. In the case of Newton’s third law, it is simply the observation that all forces are interactions between two different objects - each exerting the same force on the other but in the opposite direction. You asked about an example in my life that I have experienced very recentl Continue Reading For one thing, one doesn’t “prove” Newton’s third law - or any other law of physics, for that matter. Laws in physics are just a result of verified observations and experiments. If an observation were made that was inconsistent with this law, then the law would be modified to account for the newly discovered inconsistency. In the case of Newton’s third law, it is simply the observation that all forces are interactions between two different objects - each exerting the same force on the other but in the opposite direction. You asked about an example in my life that I have experienced very recently. How about currently? I am sitting in a chair holding my laptop writing this post - right now. I am pushing downward on the chair as it is pushing upward on me. My computer is pushing downward on my lap as my lap is pushing upward on the computer with exactly the same force. My fingertips are pushing downward on the keys and I can feel the interaction, the pressure of the keys against my fingertips. And it doesn’t matter whether the objects are moving or not. You cannot exert a force against nothing. So any force your exert against s omething is an interaction that is simultaneously exerting the same force against you. Increase that force and the force against you increases. Push your right index finger against your left index finger. (Go ahead … no one is looking - or judging you.) See that? You felt the same pressure in both fingers. Both fingers compressed the same amount. Which finger is exerting the force? Both are. Which is the action force and which the reaction? There is no difference - it is just an interaction between your two index fingers. Now push harder - and both fingers feel the increased pressure. That is the essence of Newton’s third law. Upvote · 9 5 9 1 Sponsored by Complex Law Attention UK Drivers: Check PCP refund eligibility now. If you took out car finance between 2007–2021, use Complex Law to check your refund eligibility. Learn More 999 185 Assistant Bot · 10mo Newton's third law states that for every action, there is an equal and opposite reaction. Here are some practical examples from everyday life that illustrate this principle: Walking: When you walk, your foot pushes backward against the ground. The ground reacts by pushing your foot forward with an equal force, allowing you to move ahead. Swimming: A swimmer pushes water backward with their hands and feet. The water pushes them forward with an equal and opposite force, propelling them through the water. Rocket Launch: Rockets work on this principle by expelling gas downward at high speed. The reac Continue Reading Newton's third law states that for every action, there is an equal and opposite reaction. Here are some practical examples from everyday life that illustrate this principle: Walking: When you walk, your foot pushes backward against the ground. The ground reacts by pushing your foot forward with an equal force, allowing you to move ahead. Swimming: A swimmer pushes water backward with their hands and feet. The water pushes them forward with an equal and opposite force, propelling them through the water. Rocket Launch: Rockets work on this principle by expelling gas downward at high speed. The reaction to this action is that the rocket is pushed upward into the sky. Jumping: When you jump, you push down on the ground with your legs. The ground pushes back up with an equal force, lifting you into the air. Recoil of a Gun: When a bullet is fired, the gun exerts a forward force on the bullet. Simultaneously, the bullet exerts an equal and opposite force backward on the gun, causing it to recoil. Rowing a Boat: When a rower pushes the water backward with an oar, the water pushes the boat forward with an equal force. Bouncing a Ball: When a ball hits the ground, it exerts a downward force on the ground. The ground exerts an equal and opposite force upward, causing the ball to bounce back up. Sitting in a Chair: When you sit on a chair, your body exerts a downward force on the chair. The chair exerts an equal and opposite force upward, supporting your weight. These examples demonstrate how Newton's third law is evident in various physical activities and interactions in our daily lives. Upvote · Krish V Nair Studied at College of Engineering, Trivandrum · Author has 1.1K answers and 1.6M answer views ·5y Originally Answered: What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? · I hope you are ready for THIS answer! Science would not be so highly respected if this were not to be true. The simplest example you can think of is the recoil of a gun/rifle. If you are indeed asking for a personal experience, I shall tell you that, although it is embarrassing. It was an NCC camp while in college, using an old 303 rifle, I became over confident and ignored instructions after I fired a few accurate shots and held it without pressing thr butt against my shoulder. The recoil hit hard against my collar bone. I had to go to the hospital for an x-ray and lost the chance to win the f Continue Reading I hope you are ready for THIS answer! Science would not be so highly respected if this were not to be true. The simplest example you can think of is the recoil of a gun/rifle. If you are indeed asking for a personal experience, I shall tell you that, although it is embarrassing. It was an NCC camp while in college, using an old 303 rifle, I became over confident and ignored instructions after I fired a few accurate shots and held it without pressing thr butt against my shoulder. The recoil hit hard against my collar bone. I had to go to the hospital for an x-ray and lost the chance to win the first prize.When a rifle fires a pellet at tremendous speed, the higher mass of the rifle will translate this momentum into a much milder recoil onto our shoulder. No matter how many times you check the forces mass multiplied by the respective accelerations, (M×a). People who have held the gun with a loose grip have had broken collar bones. This is a corollary of the law of conservation of mass as well as energy. A guy who finds that the calorific value of fuel used in a vehicle is more than the mechanical work done will finally realise the frictional losses which gets converted to heat was ignored. However what is true in our field of observation is not easily proved at the macro level beyond it. Thus, subtle energy exchanges between humans, for example seem to remain unbalanced. But in the future there is little doubt that this will also be established, as this is such a profound truth that is not limited by time or locality.But until then we can comfort ourselves with the theory of Karma which says exactly the same! Upvote · 9 4 Related questions More answers below What is a real life example of Newton's third law? What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? People say Newton’s third law, “For every action, there is an equal and opposite reaction,” is not accurate. Is it true? How can I apply Newton's third law in real life? According to Newton's third law, for every reaction, there is an equal and opposite reaction. Do you observe this in your personal life? Cite examples of situations in your life when you did something that resulted in an equal and opposite reaction? Rankine Gajendra Studied Mathematics&Physics · Author has 103 answers and 708.7K answer views ·7y Originally Answered: What are the real life examples of 3 Newton Laws? · Newton’s third law of motion states that every action has an equal and opposite reaction. Action and reaction are in terms of force and are equal in magnitude and opposite in direction. The action-reaction pair are said to be inseparable, coexisting or not existing at all, for there has to be a reaction to every action. Imagine a universe without the third law, where you keep flapping your arms in the pool but could not swim? This law can be seen around us very frequently. Have a look! If any law has made it most to the general public’s references in conversations, it is the Newton’s third law Continue Reading Newton’s third law of motion states that every action has an equal and opposite reaction. Action and reaction are in terms of force and are equal in magnitude and opposite in direction. The action-reaction pair are said to be inseparable, coexisting or not existing at all, for there has to be a reaction to every action. Imagine a universe without the third law, where you keep flapping your arms in the pool but could not swim? This law can be seen around us very frequently. Have a look! If any law has made it most to the general public’s references in conversations, it is the Newton’s third law of motion. It is the most popular scientific law that appeals to the masses. It is because of the third law of motion that a slap hurts to both the individuals involved, the one who beats and the one who’s beaten. No wonder those high fives with your friends hurt you too! Have a seat How does it feel to sleep on a hard surface? The body feels the pain. It is because the surface exerts an equal and opposite force on you. Heavier the object, more is the opposite force applied. When you sit for too long on your chair, you start feeling pain. Now you know why this happens. Walk you through it You can’t walk on the slippery surface but you can easily do so on a rough surface. This is because the horizontal component of the force you exert on the floor for pushing it backwards gets a reaction force from the rough ground in terms of friction acting forwards on your feet, but slippery surface lacks this friction. Therefore, always take small steps on slippery grounds to minimize horizontal force you exert on it. Hit it hard Ever fired a gun? If you try one, please take care that it recoils. It means that the piston exerts a force on the bullet to propel it, but the reaction from the bullet causes the piston to move back, taking you along with it. Ever felt a jerk while batting in cricket, or pain in leg while kicking a football? It is because when you exert the force on the ball to hit it hard, the ball also exerts equal force on you and this causes the jerk or pain you feel. Slapping someone is not cool, more so, because this action also has equal and opposite reaction! Whatever force you exert on the cheek, the same force is exerted by the cheek on your palm. For that matter, hitting any surface will result in pain in your hand. Move it The birds use action and reaction pair while flying. The wings push the air downwards, and the air pushes the bird upwards. As stated above, the third law of motion helps us swim as we propel ourselves forward and push the water behind us. To the stars! Rocket propulsion is also a good example of Newton’s Third Law. The exhaust from the rocket pushes the ground and the ground pushes the rocket with equal and opposite force to cause the latter to move forward. If the third law of motion never existed, we would never have made it into the space! Source -Google search engine Happy Learning! Upvote · 99 15 William Hudson Studied at Jourdanton High School ·10y Originally Answered: What are some good examples of Newton's third law in real life? "To every action, there is an equal and opposite reaction" · I live in South Texas, and as such I have been raised to hunt my whole life. We shoot hogs, as they are pests and would ruin my grandfather's peanut farm, and because they were a good source of meat for the family (we would also donate extra meat to poor families in town). On my first hog hunt, I was very young and my uncle took me out to a hunting blind to see if any hogs would show up. He handed me the rifle, and I asked him if it would "kick," or have uncomfortable recoil. He said, "No, it's just like a BB gun." After sitting in the blind for a couple of hours, hogs came out and I chose the Continue Reading I live in South Texas, and as such I have been raised to hunt my whole life. We shoot hogs, as they are pests and would ruin my grandfather's peanut farm, and because they were a good source of meat for the family (we would also donate extra meat to poor families in town). On my first hog hunt, I was very young and my uncle took me out to a hunting blind to see if any hogs would show up. He handed me the rifle, and I asked him if it would "kick," or have uncomfortable recoil. He said, "No, it's just like a BB gun." After sitting in the blind for a couple of hours, hogs came out and I chose the biggest one to shoot. He said, "Pull the trigger when you're ready." I looked through the scope, put the cross hairs on the hog, took a deep breath to relax, and calmly pulled the trigger. When I did, I learned the hard way at a very young age about Newton's Third Law. As the bullet is propelled out the barrel of the gun, the expanding gases also push on the shoulder of the shooter. After the shot, the gun recoiled so violently, I ended up laying on my back on the floor of the hunting blind, and my uncle ended up laughing until he cried. Newton's Third Law, gittin' 'er done. Upvote · 9 5 9 1 Promoted by Almedia Marc Hammes Personal finance journalist @ Almedia ·Updated Apr 25 What is the best site to do online work for money? Dropshipping, freelancing, virtual assistant… These are only some “side gigs” I've tried to get extra cash. Ask me if I enjoyed them, and you'll see me frown. But eventually, my search paid off - I learned that there is a way to earn real money just by… playing games. I know, it sounds like one of those “too good to be true” pitches. I was dubious too. But I gave it a shot, and 30 days later, I had $955 sitting in my PayPal. No cap. That’s when I realized: Freecash isn’t a hustle. It’s a true lifesaver. … Freecash? What is that? I'll put it super simply: Freecash is a site that pays you for completi Continue Reading Dropshipping, freelancing, virtual assistant… These are only some “side gigs” I've tried to get extra cash. Ask me if I enjoyed them, and you'll see me frown. But eventually, my search paid off - I learned that there is a way to earn real money just by… playing games. I know, it sounds like one of those “too good to be true” pitches. I was dubious too. But I gave it a shot, and 30 days later, I had $955 sitting in my PayPal. No cap. That’s when I realized: Freecash isn’t a hustle. It’s a true lifesaver. … Freecash? What is that? I'll put it super simply: Freecash is a site that pays you for completing simple tasks in apps and games. You're basically a game tester — but way more fun. And without a CV 😁 So the process is easy: Register and complete your first task. Then, go through other tasks to see what’s available on the platform. There are higher and lower paying tasks. 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So, yes, playtime can be your payday! 🔝 Don't miss your chance to earn good money and have fun. Sign up on Freecash and enjoy all the perks right away. Upvote · 999 585 99 75 99 14 Peter Upton BA in Physics&Mathematics, The Open University · Author has 14.4K answers and 10.6M answer views ·5y Originally Answered: What is a real life example of Newton's third law? · Get two inflated rubber balloons. Charge them by rubbing them on your hair/ jumper. Suspend them by cotton or thin string. They repel each other. The angles suggest that bthe forces are equal in size and opposite directions. Both forces are electrical an they each act on s different object. Another expample would be to get two bar magnets, one in each hand. Hold them close together pole- pole. They either both experience an attractive force or both a repulsive force. You never get one magnet is attracted by the other feels nothing. What is not an example of Newton 3: This: where gravity pulls the Continue Reading Get two inflated rubber balloons. Charge them by rubbing them on your hair/ jumper. Suspend them by cotton or thin string. They repel each other. The angles suggest that bthe forces are equal in size and opposite directions. Both forces are electrical an they each act on s different object. Another expample would be to get two bar magnets, one in each hand. Hold them close together pole- pole. They either both experience an attractive force or both a repulsive force. You never get one magnet is attracted by the other feels nothing. What is not an example of Newton 3: This: where gravity pulls the mass down and the spring pulls the mass up. Yes the two forces are equal and opposite but Newton 3 also mrequires the two forces to act on different objects be of exactly the same type. Here one force is gravity and the other is an elastic force from the spring Upvote · 9 1 Related questions More answers below Does 'karma' describe Newton's third law of motion 'every action, there is an equal and opposite reaction'? How do you reflect on the third law of motion by Isaac Newton: “in every action there is always an equal or opposite reaction.”? Can you give some examples of where Newton's third law explains "for every action there is an equal and opposite reaction"? What are some examples of the third law of motion? According to Newton's third law of motion, every action has an equal and opposite reaction. But when we move an object, why does it move? Ashok Ghai B.E. from IIT Roorkee (mechanical engineering) (Graduated 1967) · Author has 993 answers and 293.3K answer views ·5y Originally Answered: What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? · I am conducting research on those cosmic powers that control the evolution of life on earth. On learning a method to contact those powers, i ventured to send them a message to know the future of mankind. I promptly received their reply in such a miraculous manner that it has startled the scientists all over the world. A noted professor of IIT Kanpur and many other scientists have certified it to be a cosmic message containing important knowledge about our future. Upvote · 9 2 Promoted by Betterbuck Anthony Madden Writer for Betterbuck ·Updated Mon What are the weirdest mistakes people make on the internet right now? Here are some of the worst mistakes I’ve seen people make: Not using an ad blocker If you aren’t using an ad blocker yet, you definitely should be. A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load. No more YouTube ads, no more banner ads, no more pop-up ads, etc. Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options. Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your privac Continue Reading Here are some of the worst mistakes I’ve seen people make: Not using an ad blocker If you aren’t using an ad blocker yet, you definitely should be. A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load. 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Upvote · 999 216 99 22 9 4 Peter Upton BA in Physics&Mathematics, The Open University · Author has 14.4K answers and 10.6M answer views ·Updated 5y Originally Answered: What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? · It was along time ago. I had an old 6 volt VW Beetle. In really cold weather, it did not start reliably. Sometime I had to push (bump) start it. I probably could not afford/did not want to afford a new battery. When pushing the car to get it moving, my wife being in the car ready to drop the clutch to start the engine, I noticed that in icy conditions, I could not push the car as normal. It was not related to my strength, I could push the car with increasing force until my feet slipped on the ice/snow. It was very obvious that what limited how hard I could push the car was the grip I could get Continue Reading It was along time ago. I had an old 6 volt VW Beetle. In really cold weather, it did not start reliably. Sometime I had to push (bump) start it. I probably could not afford/did not want to afford a new battery. When pushing the car to get it moving, my wife being in the car ready to drop the clutch to start the engine, I noticed that in icy conditions, I could not push the car as normal. It was not related to my strength, I could push the car with increasing force until my feet slipped on the ice/snow. It was very obvious that what limited how hard I could push the car was the grip I could get on the ground. Netwons 3 rd law- Forward force on car due to friction with the ground = Backwards force on road due to friction with the ground. Upvote · TJ Berens Retired Aerospace Defense Consultant at United States Armed Forces · Author has 24.8K answers and 12M answer views ·4y Originally Answered: According to Newton's third law, for every reaction, there is an equal and opposite reaction. Do you observe this in your personal life? Cite examples of situations in your life when you did something that resulted in an equal and opposite reaction? · According to Newton's third law, for every reaction, there is an equal and opposite reaction. Do you observe this in your personal life? Cite examples of situations in your life when you did something that resulted in an equal and opposite reaction? Newton’s context was physics. Things have contexts. :D Upvote · 9 2 Sponsored by Hedged Tired of overpaying for your business energy? The old business energy model is broken, and we have the solution. Learn More 99 28 Manuel Valdez Former IT & Telecomunications Director at Higher Education · Author has 1.8K answers and 315.4K answer views ·1y Originally Answered: Can you give examples of Newton’s third law in everyday life? · If two bodies exert forces on each other, these forces have the same magnitude but opposite directions: planes, rockets, boats, walking, gun shooting recoil, balloon blowing, boxing punching balls, rebound of throwing a flat stone over water surface, ball rebound, bouncing, propellers. and so on. Planes, rockets, ships move forward because fuel or air exhausted by turbine or propeller, they push with the same magnitude in the opposite direction. When a Ballon blows air out of it, it moves. Try bending a small thin pine tree, it will resist to move, because it is pushing back. When you stop push Continue Reading If two bodies exert forces on each other, these forces have the same magnitude but opposite directions: planes, rockets, boats, walking, gun shooting recoil, balloon blowing, boxing punching balls, rebound of throwing a flat stone over water surface, ball rebound, bouncing, propellers. and so on. Planes, rockets, ships move forward because fuel or air exhausted by turbine or propeller, they push with the same magnitude in the opposite direction. When a Ballon blows air out of it, it moves. Try bending a small thin pine tree, it will resist to move, because it is pushing back. When you stop pushing, it will come back to its original position. Upvote · Ron Brown Decades of teaching physics to undergrads · Author has 13.4K answers and 81.8M answer views ·4y Originally Answered: What exactly does Newton’s third law of motion "For every action, there is an equal and opposite reaction." mean, and are there any examples? · The essence of what the third law means is that all forces are interactions between two different objects. That when two objects interact, they exert equal forces on each other, but in opposite directions. Examples? Any two things that interact. You push on the wall, the wall pushes on you with the same force at the same time but in opposite directions. Earth’s gravitational pull on the Moon (or you) is identical to the the Moon’s (or your) gravitational pull on the Earth. But the forces act on the different objects and are in opposite directions. The bat hits the ball with exactly the same for Continue Reading The essence of what the third law means is that all forces are interactions between two different objects. That when two objects interact, they exert equal forces on each other, but in opposite directions. Examples? Any two things that interact. You push on the wall, the wall pushes on you with the same force at the same time but in opposite directions. Earth’s gravitational pull on the Moon (or you) is identical to the the Moon’s (or your) gravitational pull on the Earth. But the forces act on the different objects and are in opposite directions. The bat hits the ball with exactly the same force that the ball hits the bat. And so on. Upvote · Martin Brilliant Former MTS at Bell Labs (1966–1989) · Author has 13.6K answers and 12.3M answer views ·4y Originally Answered: What exactly does Newton’s third law of motion "For every action, there is an equal and opposite reaction." mean, and are there any examples? · Newton’s third law means that If object A exerts a force on object B, then object B exerts an equal force on object A in the opposite direction. A good example is in billiards, or pool. If the cue ball strikes the 2 ball exactly in the middle, the cue ball stops and the 2 ball starts moving in the same direction. That’s because the 2 ball exerts enough force on the cue ball to stop it, and the cue ball exerts an equal and opposite force on the 2 ball. What exactly does Newton’s third law of motion "For every action, there is an equal and opposite reaction." mean, and are there any examples? Upvote · 9 1 Pax Sreekantan I've been alive. · Author has 4K answers and 2.6M answer views ·Updated 3y Cars, planes and rockets. If there was no reaction from the road, the car wouldn't move forward. The action is the car's wheels pushing back on the road, and via friction with the road surface, the road pushes the car forward. Think of what happens on an icy road when you're starting from rest - the wheel doesn't push on the ice because it's slippery, it just spins in place - so no reaction, and so the car doesn't move forward. Jet engines are similar - the engine rams air out the back at a high momentum, and in return the aircraft gets propelled forward. A jet engine would not work in outer s Continue Reading Cars, planes and rockets. If there was no reaction from the road, the car wouldn't move forward. The action is the car's wheels pushing back on the road, and via friction with the road surface, the road pushes the car forward. Think of what happens on an icy road when you're starting from rest - the wheel doesn't push on the ice because it's slippery, it just spins in place - so no reaction, and so the car doesn't move forward. Jet engines are similar - the engine rams air out the back at a high momentum, and in return the aircraft gets propelled forward. A jet engine would not work in outer space, since there is no air to push back. Rockets work because they actually eject material (spent rocket fuel) out the back at a high momentum. Upvote · 9 1 Related questions What is a real life example of Newton's third law? What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? People say Newton’s third law, “For every action, there is an equal and opposite reaction,” is not accurate. Is it true? How can I apply Newton's third law in real life? According to Newton's third law, for every reaction, there is an equal and opposite reaction. Do you observe this in your personal life? Cite examples of situations in your life when you did something that resulted in an equal and opposite reaction? Does 'karma' describe Newton's third law of motion 'every action, there is an equal and opposite reaction'? How do you reflect on the third law of motion by Isaac Newton: “in every action there is always an equal or opposite reaction.”? Can you give some examples of where Newton's third law explains "for every action there is an equal and opposite reaction"? What are some examples of the third law of motion? According to Newton's third law of motion, every action has an equal and opposite reaction. But when we move an object, why does it move? What is not an example of Newton's third law? What are the real life examples of 3 Newton Laws? Every action has an equal and opposite reaction. What is the law? Just Asking Is Karma considered as Newtons Third law (i.e For every action, there is an equal and opposite reaction)? What are Newton's third law daily examples with action and reaction? Related questions What is a real life example of Newton's third law? What is the best example of your life that you experienced very recently that proved Newton’s third law that every action has an equal and opposite reaction? People say Newton’s third law, “For every action, there is an equal and opposite reaction,” is not accurate. Is it true? How can I apply Newton's third law in real life? According to Newton's third law, for every reaction, there is an equal and opposite reaction. Do you observe this in your personal life? Cite examples of situations in your life when you did something that resulted in an equal and opposite reaction? Does 'karma' describe Newton's third law of motion 'every action, there is an equal and opposite reaction'? 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3801	https://virtualnerd.com/algebra-1/linear-equation-analysis/slope-rate-of-change/understanding-slope/slope-definition	What Does the Slope of a Line Mean? \| Virtual Nerd Real math help. Algebra 1 Switch to: Middle Grades Math Pre-Algebra Algebra 2 Geometry Common Core SAT Math ACT Math Texas Programs Texas Standards Analyzing Linear Equations Rate of Change and Slope Defining Slope What Does the Slope of a Line Mean? What Does the Slope of a Line Mean? Note: You can't learn about linear equations without learning about slope. The slope of a line is the steepness of the line. There are many ways to think about slope. Slope is the rise over the run, the change in 'y' over the change in 'x', or the gradient of a line. Check out this tutorial to learn about slope! Keywords: definition slope meaning steepness steeper compare slopes Algebra 1 Switch to: Middle Grades Math Pre-Algebra Algebra 2 Geometry Common Core SAT Math ACT Math Texas Programs Texas Standards Analyzing Linear Equations Rate of Change and Slope Defining Slope Background Tutorials Defining Slope Image 2: What Does Negative Slope Mean?Image 3: What Does Negative Slope Mean? What Does Negative Slope Mean? What does a negative slope mean? What does the graph of a negative slope look like? Find the answers to these questions by watching this tutorial! Image 4: What Does Positive Slope Mean?Image 5: What Does Positive Slope Mean? What Does Positive Slope Mean? What does a positive slope mean? What does the graph of a positive slope look like? Find the answers to these questions by watching this tutorial! Image 6: What is Rate of Change?Image 7: What is Rate of Change? What is Rate of Change? Trying to describe the how something changes in relation to something else? Use rate of change! In this tutorial, learn about rate of change and see the difference between positive and negative rates of change! Horizontal and Vertical Lines Image 8: What Are Vertical Lines?Image 9: What Are Vertical Lines? What Are Vertical Lines? You may be able to guess that vertical lines are lines that go straight up and down, but did you know that all vertical lines have the same slope? In this tutorial, learn all about vertical lines including their slope and what the equation of a vertical line looks like! Image 10: What Are Horizontal Lines?Image 11: What Are Horizontal Lines? What Are Horizontal Lines? Ever look at the horizon when the sun is rising or setting? Know why it's called the horizon? It's a horizontal line! And just like the horizon, horizontal lines go straight left and right. In this tutorial, you'll learn all about horizontal lines including their slope and what the equation of a horizontal line looks like. Further Exploration Defining Slope Image 12: What's the Formula for Slope?Image 13: What's the Formula for Slope? What's the Formula for Slope? When you're dealing with linear equations, you may be asked to find the slope of a line. That's when knowing the slope formula really comes in handy! Learn the formula to find the slope of a line by watching this tutorial. Finding Slopes Image 14: How Do You Find the Slope of a Line from a Graph?Image 15: How Do You Find the Slope of a Line from a Graph? How Do You Find the Slope of a Line from a Graph? Trying to find the slope of a graphed line? First, identify two points on the line. Then, you could use these points to figure out the slope. In this tutorial, you'll see how to use two points on the line to find the change in 'y' and the change in 'x'. Then, you'll see how to take these values and calculate the slope. Check it out! About Terms of Use Privacy Contact
3802	https://testbook.com/question-answer/the-maximum-value-of-sin-costheta--617272e255e829f74956fba3	[Solved] The maximum value of sin θ - cos θ is Get Started ExamsSuperCoachingTest SeriesSkill Academy More Pass Skill Academy Free Live Classes Free Live Tests & Quizzes Previous Year Papers Doubts Practice Refer & Earn All Exams Our Selections Careers English Hindi Home Quantitative Aptitude Trigonometry Question Download Solution PDF The maximum value of sin θ - cos θ is This question was previously asked in UP Junior Aided Teacher (UPJASE) 2021 Science & Mathematics Official Paper Download PDFAttempt Online View all SUPER TET Papers > 2 3+1 2 √2 1 Answer (Detailed Solution Below) Option 3 : √2 Crack Super Pass Live with India's Super Teachers FREE Demo Classes Available Explore Supercoaching For FREE Free Tests View all Free tests > Free SUPER TET Full Test 1 53.9 K Users 150 Questions 150 Marks 150 Mins Start Now Detailed Solution Download Solution PDF Concept: The maximum value of a sin⁡x+b cos⁡x=√a 2+b 2 Calculation: The maximum value of sin θ - cos θ =1 2+(−1)2=2 Hence, the maximum value is √2. Download Solution PDFShare on Whatsapp Latest SUPER TET Updates Last updated on May 29, 2024 SUPER TET 2024 Notification is to be released soon. The SUPER TET Exam is conducted by the Uttar Pradesh Basic Education Board (UPBEB) for recruitment to the posts of Principals and Assistant Teachers in Junior High Schools across the state.The vacancy rate is expected to be around 17000. The minimum eligibility criteria for Assistant Teacher positions is graduation, whereas the minimum eligibility criteria for Principal positions is graduation combined with 5 years of teaching experience. Interested candidates can review the UP SUPER TET Syllabus and Exam Pattern from this page. India’s #1 Learning Platform Start Complete Exam Preparation Daily Live MasterClasses Practice Question Bank Mock Tests & Quizzes Get Started for Free Trusted by 7.6 Crore+ Students More Trigonometry Questions Q1.A tower stands vertically on the ground. From a point on the ground which is 27.6 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 45°. Find the height of the tower. Q2.From a point P on a level ground, the angle of elevation of the top of the tower is 30°. If the tower is 50 m high, the distance of point P from the foot of the tower is: Q3.Find the value of sin 2⁡68∘+sin 2⁡22∘2(cos 2⁡17∘+cos 2⁡73∘). Q4.If cot⁡θ=7 8,then evaluate 8(1+sin⁡θ)(1−sin⁡θ)7(1+cos⁡θ)(1−cos⁡θ). Q5.From the top of a 60 m high building, the angles of depression of the top and bottom of a lamp post are 30° and 60°, respectively. Find the height of the lamp post. Q6.If sin5A = cos(A – 42°), where 5A is an acute angle, find the value of A. Q7.A person standing 30 metres away from a tower observes the angle of elevation to the top of the tower as 30°. Find the height of the tower. Q8.Simplify: (sec⁡θ−1 sec⁡θ)×(c o s e c θ−1 c o s e c θ)×(sin⁡θ−1 sin⁡θ). Q9.If sin 4⁡θ−cos 4⁡θ=1 2,find the value of 2 sin 2⁡θ−1. Q10.The angle of elevation of a ladder leaning against a wall is 60°, and the foot of the ladder is 4.5 m away from the wall. What is the length of the ladder? Suggested Test Series View All > UP LT Grade 2025 Mock Test Series 590 Total Tests with 2 Free Tests Start Free Test Pedagogy for All Teaching Exams (Paper 1 & 2) - Let's Crack TET! 345 Total Tests with 3 Free Tests Start Free Test More Quantitative Aptitude Questions Q1.A, B, and C can complete a work alone in 20, 24, and 30 days respectively. They start working together. After 2 days, A leaves the work. Two days after A leaves, C also leaves the work. In how many total days will the work be completed? Q2.A and B start a business with Rs. 16000 and Rs. 20000. After x months B left the business. After one year total profit is Rs. 3300 and share of A is Rs. 1800. Find the value of x? Q3.Total surface area of sphere is 5544 m2. The volume of sphere is 8t. Find the value of t? Q4.Ratio of age of A and B is 5:3. Ratio of age of A after 5 years and Age of B 5 years ago is 3:1. Find age of C if age of C is 10 years older than A? Q5.Average marks of 25 students are 60. Marks of two students taken as 72 and 52 instead off 75 and 50. Find difference of old and new average? Q6.A can finish the work alone in [x + 20] days. B can finish the work alone in [x + 10] days. A is 25% less efficient than B. If A works for 10 consecutive days, then takes a 1-day break, repeating this cycle until the work is done. Under this pattern, A finishes the task in [n + 3] days. Similarly, B also works 6 days continuously, then takes a 2-days break, and completes the task in [m + 2] days. D completes the same task in (m - 12) days C completes it in [n + 20] days. Based on this information, determine which of the following statements is true: I)B and C together complete the work in [x + 4] days. II) A and D together complete the work in [(n/2) – 3] days III) Sum of m and n is perfect cube number. Q7.On a particular day, each of Alok, Sameer, and Amit sold three types of gadgets in their respective shops. Alok and Sameer sold an identical number of Type X gadgets while Sameer sold twice as many of Type Y gadgets as Alok. The ratio of the number of Type Z gadgets sold by Alok, Sameer, and Amit was 4:6:5, respectively. Each of the three sellers sold an identical number of Type X gadgets at different prices per unit. Assertion (A): It is possible that Alok sold each Type X gadget at a profit of Rs 5, each Type Y gadget at a profit of Rs 3, and each Type Z gadget at a loss of Rs 2, and made an overall profit of Rs 150; Sameer sold each Type X gadget at a profit of Rs 4, each Type Y gadget at a profit of Rs 2, and each Type Z gadget at a loss of Rs 1, and made an overall profit of Rs 200; and Amit sold each Type X gadget at a profit of Rs 3, each Type Y gadget at a loss of Rs 2, and each Type Z gadget at a profit of Rs 5, and made an overall profit of Rs 180. Reason (R): Framing and s Q8.A car and a bike start from the same point and travel along a straight road. The car travels at a constant speed of 60 km/h and the bike at 40 km/h. After how much time will the car be 50 km ahead of the bike? Consider the following statements: Statement 1: The car starts 2 hours before the bike. Statement 2: The bike travels at a constant speed of 40 km/h. Statement 3: The car travels at a constant speed of 60 km/h. Q9.Ravi and Rohan decided to run a 1200 m long race on a track as long as the length of the race. Which of the statement(s) below is (are) sufficient to conclude that Rohan caught up with Ravi when 1/4 of the track length still remained? Consider the following statements: Statement 1: Ravi is allowed to start 10 seconds before Rohan started running. Statement 2: Ravi ran at 9 m per second. Statement 3: Rohan ran at a speed of 12 m/s. Q10.A Question is given below followed by two Statements 1 and 11. Question: If the average marks obtained by 110 students is 20, how many students have failed? Statement I: The average marks of failed students are 12. Statement II: The average marks of passed students are 22. Which one of the following is correct in respect of the above Question and the Statements? 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3803	https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_1e_(OpenStax)/05%3A_Systems_of_Linear_Equations	Skip to main content 5: Systems of Linear Equations Last updated : Feb 13, 2022 Save as PDF Chapter 4 Review Exercises 5.1: Solve Systems of Equations by Graphing Buy Print CopyView on Commons Donate Page ID : 15150 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) 5.1: Solve Systems of Equations by Graphing : + 5.1E: Exercises 5.2: Solve Systems of Equations by Substitution : Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph. : + 5.2E: Exercises 5.3: Solve Systems of Equations by Elimination : We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression. The third method of solving systems of linear equations is called the Elimination Method. : + 5.3E: Exercises 5.4: Solve Applications with Systems of Equations : + 5.4E: Exercises 5.5: Solve Mixture Applications with Systems of Equations : + 5.5E: Exercises 5.6: Graphing Systems of Linear Inequalities : + 5.6E: Exercises Chapter 5 Review Exercises Chapter 4 Review Exercises 5.1: Solve Systems of Equations by Graphing
3804	http://physics.bu.edu/~duffy/semester2/c02_field_ring.html	\| \| \| --- \| \| \| \| Field from a Ring of Charge What is the magnitude of the electric field at the center of a ring of charge of radius a? Assume there is a charge Q uniformly distributed over the ring. zero- not zero but less than kQ/a2- kQ/a2- larger than kQ/a2 The field from one side of the ring cancels the field from the other, so the net field at the center is zero. What is the magnitude of the electric field a distance x from the center of the ring, along the axis of the ring? zero- not zero but less than kQ/r2- kQ/r2- larger than kQ/r2 The field from one part of the ring is partly cancelled by the field from the opposite side of the ring. The field is non-zero but has a magnitude less than kQ/r2. The field can be determined exactly by integrating. The strategy is to split the ring up into tiny pieces and give each piece a tiny charge dQ. Each of these charges sets up a tiny field dE at our point, where: \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. The field from one side of the ring cancels the field from the other, so the net field at the center is zero. What is the magnitude of the electric field a distance x from the center of the ring, along the axis of the ring? zero- not zero but less than kQ/r2- kQ/r2- larger than kQ/r2 The field from one part of the ring is partly cancelled by the field from the opposite side of the ring. The field is non-zero but has a magnitude less than kQ/r2. The field can be determined exactly by integrating. The strategy is to split the ring up into tiny pieces and give each piece a tiny charge dQ. Each of these charges sets up a tiny field dE at our point, where: \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. What is the magnitude of the electric field a distance x from the center of the ring, along the axis of the ring? zero- not zero but less than kQ/r2- kQ/r2- larger than kQ/r2 The field from one part of the ring is partly cancelled by the field from the opposite side of the ring. The field is non-zero but has a magnitude less than kQ/r2. The field can be determined exactly by integrating. The strategy is to split the ring up into tiny pieces and give each piece a tiny charge dQ. Each of these charges sets up a tiny field dE at our point, where: \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. The field from one part of the ring is partly cancelled by the field from the opposite side of the ring. The field is non-zero but has a magnitude less than kQ/r2. The field can be determined exactly by integrating. The strategy is to split the ring up into tiny pieces and give each piece a tiny charge dQ. Each of these charges sets up a tiny field dE at our point, where: \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. The strategy is to split the ring up into tiny pieces and give each piece a tiny charge dQ. Each of these charges sets up a tiny field dE at our point, where: \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| --- --- --- \| \| dE \| = \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| \| \| \| \| \| --- \| k dQ\| \| \| \| \| r2 \| \| Add these fields as vectors to get the net field: E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. E = Σ dE When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. When we integrate to find the net field all the components along the axis add and all the components perpendicular to the axis cancel. We put a factor of: cos(θ) = x/r in the integral so we're only including the components along the axis. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| ∫ \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| = \| ∫ \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| \| \| \| \| \| --- \| k dQ cos(θ)\| \| \| \| \| r2 \| \| \| \| \| \| --- \| k x dQ\| \| \| \| \| r3 \| \| For every piece of the ring r2 = x2 + a2 \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge. \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| E = Ex \| = \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| ò \| dQ \| = \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| \| \| \| \| \| --- \| k x\| \| \| \| \| (x2 + a2)3/2 \| \| \| \| \| \| --- \| k x Q\| \| \| \| \| (x2 + a2)3/2 \| \| One way to check your answer is to plug in the extremes. When x is zero, which means the point is at the center of the ring, the equation gives E = 0, which is good. When x is much larger than a the expression reduces to the point charge equation. This also makes sense, seeing as when you're very far away from the ring it looks like a point charge.
3805	https://onlinemicrobiolab.ir/wp-content/uploads/2025/02/CLSI-2025-Streptococcus-spp.-%CE%B2-Hemolytic-Group.pdf	CLSI M100-Ed35 For Use With CLSI M02 and CLSI M07 128 © Clinical and Laboratory Standards Institute. All rights reserved. Table 2H-1 Streptococcus spp. -Hemolytic Group CLSI M02 and CLSI M07 Testing Conditions Medium: Disk diﬀusion: MHA with 5% sheep blood Broth dilution: CAMHB with LHB (2.5% to 5% v/v); the CAMHB should be supplemented to 50 μg/mL calcium for daptomycin (see CLSI M071 for instructions for preparation of LHB). Agar dilution: MHA with sheep blood (5% v/v); recent studies using the agar dilution method have not been performed and reviewed by the subcommittee. Inoculum: Colony suspension, equivalent to a 0.5 McFarland standard, using colonies from an overnight (18- to 20-hour) sheep blood agar plate Incubation: 35°C ± 2°C Disk diﬀusion: 5% CO2; 20–24 hours Dilution methods: ambient air; 20–24 hours (CO2 if necessary, for growth with agar dilution) QC Recommendations Refer to the following: • Tables 4B and 5B that list acceptable QC ranges applicable for each method • Appendix I to develop a QC plan When a commercial test system is used for antimicrobial susceptibility testing, refer to the manufacturer’s instructions for QC strains and QC ranges. Table 2H-1. Zone Diameter and MIC Breakpoints for Streptococcus spp. -Hemolytic Group Refer to Table 3J for additional testing recommendations, reporting suggestions, and QC. General Comments (1) Refer to Table 1H-1 for antimicrobial agents that should be considered for testing and reporting by microbiology laboratories. (2) For disk diﬀusion, test a maximum of 9 disks on a 150-mm plate and 4 disks on a 100-mm plate. Measure the diameter of the zones of complete inhibition (as judged by the unaided eye), including the diameter of the disk (see CLSI M02QG2). The zone margin should be considered the area showing no obvious, visible growth that can be detected with the unaided eye. Do not measure the zone of inhibition of hemolysis. Measure the zones from the upper surface of the agar illuminated with reflected light, with the cover removed. Ignore faint growth of tiny colonies that can be detected only with a magnifying lens at the edge of the zone of inhibited growth. (3) For -hemolytic streptococci when testing chloramphenicol, clindamycin, erythromycin, linezolid, tedizolid, and tetracycline by broth microdilution MIC, trailing growth can make end-point determination diﬃcult. In such cases, read the MIC at the lowest concentration where the trailing begins. Tiny buttons of growth should be ignored (see CLSI M071). This document is protected by copyright. CLSI order #Ord-1294879, Downloaded on 1/14/2025. Licensed to: vikash ranjan CLSI M100-Ed35 For Use With CLSI M02 and CLSI M07 129 © Clinical and Laboratory Standards Institute. All rights reserved. Table 2H-1 Streptococcus spp. -Hemolytic Group CLSI M02 and CLSI M07 Table 2H-1. Streptococcus spp. -Hemolytic Group (Continued) Antimicrobial Agent Disk Content Interpretive Categories and Zone Diameter Breakpoints, nearest whole mm Interpretive Categories and MIC Breakpoints, μg/mL Comments S I R S I R PENICILLINS (7) An organism that is susceptible to penicillin can be considered susceptible to antimicrobial agents listed here when used for approved indications and does not need to be tested against those agents. For groups A, B, C, and G -hemolytic streptococci, penicillin is tested as a surrogate for ampicillin, amoxicillin, amoxicillin-clavulanate, ampicillin-sulbactam, cefazolin, cefepime, ceftaroline, cephradine, cephalothin, cefotaxime, ceftriaxone, ceftizoxime, imipenem, ertapenem, and meropenem. For group A -hemolytic streptococci, penicillin is also a surrogate for cefaclor, cefdinir, cefprozil, ceftibuten, cefuroxime, and cefpodoxime. Penicillin or ampicillin 10 units 10 μg ≥ 24 ≥ 24 – – – – ≤ 0.12 ≤ 0.25 – – – – See general comment (5). CEPHEMS (PARENTERAL) (Including cephalosporins I, II, III, and IV. Please refer to Glossary I.) See comment (7). Cefepime or cefotaxime or ceftriaxone 30 μg 30 μg 30 μg ≥ 24 ≥ 24 ≥ 24 – – – – – – ≤ 0.5 ≤ 0.5 ≤ 0.5 – – – – – – Ceftaroline 30 μg ≥ 26 – – ≤ 0.5 – – (4) For this table, the -hemolytic group includes the large colony–forming pyogenic strains of streptococci with group A (S. pyogenes), C, or G antigens and strains with Group B (S. agalactiae) antigen. Small colony–forming -hemolytic strains with group A, C, F, or G antigens (S. anginosus group, previously S. milleri) are considered part of the viridans group, and breakpoints for the viridans group should be used (see Table 2H-2). (5) Penicillin and ampicillin are drugs of choice for treating -hemolytic streptococcal infections. Susceptibility testing of penicillins and other -lactams approved by the FDA for treatment of -hemolytic streptococcal infections does not need to be performed routinely, because nonsusceptible isolates (ie, penicillin MICs > 0.12 and ampicillin MICs > 0.25 μg/mL) are extremely rare in any -hemolytic streptococci and have not been reported for S. pyogenes. If testing is performed, any -hemolytic streptococcal isolate found to be nonsusceptible should be re-identified, retested, and, if confirmed, submitted to a public health laboratory. See Appendix A for additional instructions. (6) Breakpoints for Streptococcus spp. -hemolytic group are proposed based on population distributions of various species, pharmacokinetics of the antimicrobial agents, previously published literature, and the clinical experience of subcommittee members. Systematically collected clinical data were not available for review with many of the antimicrobial agents in this table. NOTE: Information in boldface type is new or modified since the previous edition. This document is protected by copyright. CLSI order #Ord-1294879, Downloaded on 1/14/2025. Licensed to: vikash ranjan CLSI M100-Ed35 For Use With CLSI M02 and CLSI M07 130 © Clinical and Laboratory Standards Institute. All rights reserved. Table 2H-1. Streptococcus spp. -Hemolytic Group (Continued) Antimicrobial Agent Disk Content Interpretive Categories and Zone Diameter Breakpoints, nearest whole mm Interpretive Categories and MIC Breakpoints, μg/mL Comments S I R S I R CARBAPENEMS See comment (7). Doripenem – – – – ≤ 0.12 – – Ertapenem – – – – ≤ 1 – – Meropenem – – – – ≤ 0.5 – – GLYCOPEPTIDES Vancomycin 30 μg ≥ 17 – – ≤ 1 – – LIPOGLYCOPEPTIDES Dalbavancin – – – – ≤ 0.25 – – (8) Report only on S. pyogenes, S. agalactiae, and S. dysgalactiae. Oritavancin – – – – ≤ 0.25 – – Telavancin – – – – ≤ 0.12 – – LIPOPEPTIDES Daptomycin – – – – ≤ 1 – – (9) Not routinely reported on organisms isolated from the lower respiratory tract. MACROLIDES (10) Susceptibility and resistance to azithromycin, clarithromycin, and dirithromycin can be predicted by testing erythromycin. (11) Not routinely reported on organisms isolated from the urinary tract. Erythromycin 15 μg ≥ 21 16–20 ≤ 15 ≤ 0.25 0.5 ≥ 1 (12) Rx: Recommendations for intrapartum prophylaxis for group B streptococci are penicillin or ampicillin. Although cefazolin is recommended for penicillin-allergic women at low risk for anaphylaxis, those at high risk for anaphylaxis may receive clindamycin or vancomycin (if the isolate is not susceptible to clindamycin).3 Group B streptococci are susceptible to ampicillin, penicillin, and cefazolin but may be resistant to erythromycin and clindamycin. When clindamycin is being considered for intrapartum prophylaxis (eg, pregnant woman with severe penicillin allergy), erythromycin and clindamycin (including ICR) should be tested, but only clindamycin should be reported. See Table 3J. Table 2H-1 Streptococcus spp. -Hemolytic Group CLSI M02 and CLSI M07 This document is protected by copyright. CLSI order #Ord-1294879, Downloaded on 1/14/2025. Licensed to: vikash ranjan CLSI M100-Ed35 For Use With CLSI M02 and CLSI M07 131 © Clinical and Laboratory Standards Institute. All rights reserved. Table 2H-1. Streptococcus spp. -Hemolytic Group (Continued) Antimicrobial Agent Disk Content Interpretive Categories and Zone Diameter Breakpoints, nearest whole mm Interpretive Categories and MIC Breakpoints, μg/mL Comments S I R S I R MACROLIDES (Continued) Azithromycin 15 μg ≥ 18 14–17 ≤ 13 ≤ 0.5 1 ≥ 2 Clarithromycin 15 μg ≥ 21 17–20 ≤ 16 ≤ 0.25 0.5 ≥ 1 Dirithromycin 15 μg ≥ 18 14–17 ≤ 13 ≤ 0.5 1 ≥ 2 TETRACYCLINES (13) Isolates that test susceptible to tetracycline are considered susceptible to doxycycline and minocycline. Tetracycline 30 μg ≥ 23 19–22 ≤ 18 ≤ 2 4 ≥ 8 FLUOROQUINOLONES Levofloxacin 5 μg ≥ 17 14–16 ≤ 13 ≤ 2 4 ≥ 8 Gatifloxacin 5 μg ≥ 21 18–20 ≤ 17 ≤ 1 2 ≥ 4 Grepafloxacin 5 μg ≥ 19 16–18 ≤ 15 ≤ 0.5 1 ≥ 2 Ofloxacin 5 μg ≥ 16 13–15 ≤ 12 ≤ 2 4 ≥ 8 Trovafloxacin 10 μg ≥ 19 16–18 ≤ 15 ≤ 1 2 ≥ 4 PHENICOLS Chloramphenicol 30 μg ≥ 21 18–20 ≤ 17 ≤ 4 8 ≥ 16 See comment (11). LINCOSAMIDES Clindamycin 2 μg ≥ 19 16–18 ≤ 15 ≤ 0.25 0.5 ≥ 1 See comments (11) and (12). (14) For isolates that test erythromycin resistant and clindamycin susceptible or intermediate, testing for ICR by disk diﬀusion using the D-zone test or by broth microdilution is required before reporting clindamycin. See Table 3J, CLSI M02,4 and CLSI M07.1 STREPTOGRAMINS Quinupristin-dalfopristin 15 μg ≥ 19 16–18 ≤ 15 ≤ 1 2 ≥ 4 (15) Report only on S. pyogenes. Table 2H-1 Streptococcus spp. -Hemolytic Group CLSI M02 and CLSI M07 This document is protected by copyright. CLSI order #Ord-1294879, Downloaded on 1/14/2025. Licensed to: vikash ranjan CLSI M100-Ed35 For Use With CLSI M02 and CLSI M07 132 © Clinical and Laboratory Standards Institute. All rights reserved. Table 2H-1. Streptococcus spp. -Hemolytic Group (Continued) Antimicrobial Agent Disk Content Interpretive Categories and Zone Diameter Breakpoints, nearest whole mm Interpretive Categories and MIC Breakpoints, μg/mL Comments S I R S I R OXAZOLIDINONES (16) S. agalactiae and S. pyogenes that test susceptible to linezolid are considered susceptible to tedizolid. Isolates that test nonsusceptible to linezolid should be tested against tedizolid if that result is needed for treatment. Linezolid 30 μg ≥ 21 – – ≤ 2 – – Tedizolid 2 μg ≥ 15 – – ≤ 0.5 – – (17) Report only on S. pyogenes and S. agalactiae. Abbreviations: CAMHB, cation-adjusted Mueller-Hinton broth; CO2, carbon dioxide; FDA, US Food and Drug Administration; I, intermediate; ICR, inducible clindamycin resistance; LHB, lysed horse blood; MHA, Mueller-Hinton agar; MIC, minimal inhibitory concentration; QC, quality control; R, resistant; S, susceptible. Symbol: , designation for “Other” agents not included in Tables 1 but have established clinical breakpoints. Table 2H-1 Streptococcus spp. -Hemolytic Group CLSI M02 and CLSI M07 References for Table 2H-1 1 CLSI. Methods for Dilution Antimicrobial Susceptibility Tests for Bacteria That Grow Aerobically. 12th ed. CLSI standard M07. Clinical and Laboratory Standards Institute; 2024. 2 CLSI. M02 Disk Diﬀusion Reading Guide. 2nd ed. CLSI quick guide M02-Ed14-QG. Clinical and Laboratory Standards Institute; 2024. 3 American College of Obstetricians and Gynecologists. Prevention of group B streptococcal early-onset disease in newborns: ACOG Committee Opinion, Number 797. Obstet Gynecol. 2020;135(2):e51-e72. doi:10.1097/AOG.0000000000003668 4 CLSI. Performance Standards for Antimicrobial Disk Susceptibility Tests. 14th ed. CLSI standard M02. Clinical and Laboratory Standards Institute; 2024. This document is protected by copyright. CLSI order #Ord-1294879, Downloaded on 1/14/2025. Licensed to: vikash ranjan
3806	https://en.wikipedia.org/wiki/Nitric_acid	Jump to content Nitric acid Afrikaans العربية Asturianu Azərbaycanca تۆرکجه বাংলা 閩南語 / Bn-lm-gí Беларуская Беларуская (тарашкевіца) Български Bosanski Brezhoneg Català Чӑвашла Čeština Cymraeg Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Galego 한국어 Հայերեն हिन्दी Hrvatski Ido Bahasa Indonesia Interlingua Íslenska Italiano עברית Jawa ಕನ್ನಡ ქართული Қазақша Кыргызча ລາວ Latina Latviešu Lietuvių Magyar Македонски മലയാളം Bahasa Melayu 閩東語 / Mìng-dĕ̤ng-ngṳ̄ Монгол မြန်မာဘာသာ Nederlands 日本語 Nordfriisk Norsk bokmål Norsk nynorsk Occitan Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ پنجابی پښتو Polski Português Romnă Русский Scots Shqip සිංහල Simple English سنڌي Slovenčina Slovenščina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் తెలుగు ไทย Türkçe Українська اردو Tiếng Việt 文言 Winaray 吴语粵語中文 Edit links From Wikipedia, the free encyclopedia Highly corrosive mineral acid Not to be confused with nitrous acid. Nitric acid \| Pure nitric acid \| \| \| \| \| \| \| \| \| --- \| \| \| \| \| \| \| Names \| \| \| IUPAC name Nitric acid \| \| \| Other names Aqua fortis Spirit of niter Eau forte Hydrogen nitrate Acidum nitricum \| \| \| Identifiers \| \| \| CAS Number \| 7697-37-2Y \| \| 3D model (JSmol) \| Interactive image Interactive image \| \| ChEBI \| CHEBI:48107Y \| \| ChEMBL \| ChEMBL1352Y \| \| ChemSpider \| 919Y \| \| ECHA InfoCard \| 100.028.832 \| \| EC Number \| 231-714-2 \| \| Gmelin Reference \| 1576 \| \| KEGG \| D02313Y \| \| MeSH \| Nitric+acid \| \| PubChem CID \| 944 \| \| RTECS number \| QU5775000 \| \| UNII \| 411VRN1TV4Y \| \| UN number \| 2031 \| \| CompTox Dashboard (EPA) \| DTXSID5029685 \| \| InChI InChI=1S/HNO3/c2-1(3)4/h(H,2,3,4)Y Key: GRYLNZFGIOXLOG-UHFFFAOYSA-NY InChI=1/HNO3/c2-1(3)4/h(H,2,3,4) Key: GRYLNZFGIOXLOG-UHFFFAOYAO \| \| \| SMILES N+(O)[O-] ON(=O)=O \| \| \| Properties \| \| \| Chemical formula \| HNO3 \| \| Molar mass \| 63.012 g·mol−1 \| \| Appearance \| Colorless liquid \| \| Odor \| Acrid, suffocating \| \| Density \| 1.51 g/cm3, 1.41 g/cm3 [68% w/w] \| \| Melting point \| −42 °C (−44 °F; 231 K) \| \| Boiling point \| 83 °C (181 °F; 356 K) 68% solution boils at 121 °C (250 °F; 394 K) \| \| Solubility in water \| Miscible \| \| log P \| −0.13 \| \| Vapor pressure \| 48 mmHg (20 °C) \| \| Acidity (pKa) \| −1.4 \| \| Conjugate base \| Nitrate \| \| Magnetic susceptibility (χ) \| −1.99×10−5 cm3/mol \| \| Refractive index (nD) \| 1.397 (16.5 °C) \| \| Dipole moment \| 2.17 ± 0.02 D \| \| Thermochemistry \| \| \| Std molar entropy (S⦵298) \| 146 J/(mol·K) \| \| Std enthalpy of formation (ΔfH⦵298) \| −207 kJ/mol \| \| Hazards \| \| \| GHS labelling: \| \| \| Pictograms \| \| \| Signal word \| Danger \| \| Hazard statements \| H272, H290, H314, H331 \| \| Precautionary statements \| P210, P220, P280, P303+P361+P353, P304+P340+P310, P305+P351+P338 \| \| NFPA 704 (fire diamond) \| 3 0 2 OX \| \| Flash point \| Non-flammable \| \| Lethal dose or concentration (LD, LC): \| \| \| LC50 (median concentration) \| 138 ppm (rat, 30 min) \| \| NIOSH (US health exposure limits): \| \| \| PEL (Permissible) \| TWA 2 ppm (5 mg/m3) \| \| REL (Recommended) \| TWA 2 ppm (5 mg/m3) ST 4 ppm (10 mg/m3) \| \| IDLH (Immediate danger) \| 25 ppm \| \| Safety data sheet (SDS) \| ICSC 0183 \| \| Related compounds \| \| \| Other anions \| Nitrous acid \| \| Other cations \| Sodium nitrate Potassium nitrate Ammonium nitrate \| \| Related compounds \| Dinitrogen trioxide Dinitrogen tetroxide Dinitrogen pentoxide Nitrogen oxide Nitrogen monoxide Nitrogen dioxide \| \| Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa). Y verify (what is YN ?) Infobox references \| \| Chemical compound Nitric acid is an inorganic compound with the formula HNO3. It is a highly corrosive mineral acid. The compound is colorless, but samples tend to acquire a yellow cast over time due to decomposition into oxides of nitrogen. Most commercially available nitric acid has a concentration of 68% in water. When the solution contains more than 86% HNO3, it is referred to as fuming nitric acid. Depending on the amount of nitrogen dioxide present, fuming nitric acid is further characterized as red fuming nitric acid at concentrations above 86%, or white fuming nitric acid at concentrations above 95%. Nitric acid is the primary reagent used for nitration – the addition of a nitro group, typically to an organic molecule. While some resulting nitro compounds are shock- and thermally-sensitive explosives, a few are stable enough to be used in munitions and demolition, while others are still more stable and used as synthetic dyes and medicines (e.g. metronidazole). Nitric acid is also commonly used as a strong oxidizing agent. History Medieval alchemy The discovery of mineral acids such as nitric acid is generally believed to go back to 13th-century European alchemy. The conventional view is that nitric acid was first described in pseudo-Geber's De inventione veritatis ("On the Discovery of Truth", after c. 1300). However, according to Eric John Holmyard and Ahmad Y. al-Hassan, nitric acid was also referenced in various earlier Arabic works such as the Ṣundūq al-ḥikma ("Chest of Wisdom") attributed to Jabir ibn Hayyan (8th century) or the Taʿwīdh al-Ḥākim attributed to the Fatimid caliph al-Hakim bi-Amr Allah (985–1021). The recipe in the Ṣundūq al-ḥikma attributed to Jabir has been translated as follows: Take five parts of pure flowers of nitre, three parts of Cyprus vitriol and two parts of Yemen alum. Powder them well, separately, until they are like dust and then place them in a flask. Plug the latter with a palm fibre and attach a glass receiver to it. Then invert the apparatus and heat the upper portion (i.e. the flask containing the mixture) with a gentle fire. There will flow down by reason of the heat an oil like cow's butter. Nitric acid is also found in post-1300 works falsely attributed to Albert the Great and Ramon Llull (both 13th century). These works describe the distillation of a mixture containing niter and green vitriol, which they call eau forte (aqua fortis). Modern era In the 17th century, Johann Rudolf Glauber devised a process to obtain nitric acid by distilling potassium nitrate with sulfuric acid. In 1776 Antoine Lavoisier cited Joseph Priestley's work to point out that it can be converted from nitric oxide (which he calls "nitrous air"), "combined with an approximately equal volume of the purest part of common air, and with a considerable quantity of water."[a] In 1785 Henry Cavendish determined its precise composition and showed that it could be synthesized by passing a stream of electric sparks through moist air. In 1806, Humphry Davy reported the results of extensive distilled water electrolysis experiments concluding that nitric acid was produced at the anode from dissolved atmospheric nitrogen gas. He used a high voltage battery and non-reactive electrodes and vessels such as gold electrode cones that doubled as vessels bridged by damp asbestos. The industrial production of nitric acid from atmospheric air began in 1905 with the Birkeland–Eyde process, also known as the arc process. This process is based upon the oxidation of atmospheric nitrogen by atmospheric oxygen to nitric oxide with a very high temperature electric arc. Yields of up to approximately 4–5% nitric oxide were obtained at 3000 °C, and less at lower temperatures. The nitric oxide was cooled and oxidized by the remaining atmospheric oxygen to nitrogen dioxide, and this was subsequently absorbed in water in a series of packed column or plate column absorption towers to produce dilute nitric acid. The first towers bubbled the nitrogen dioxide through water and non-reactive quartz fragments. About 20% of the produced oxides of nitrogen remained unreacted so the final towers contained an alkali solution to neutralize the rest. The process was very energy intensive and was rapidly displaced by the Ostwald process once cheap ammonia became available. Another early production method was invented by French engineer Albert Nodon around 1913. His method produced nitric acid from electrolysis of calcium nitrate converted by bacteria from nitrogenous matter in peat bogs. An earthenware pot surrounded by limestone was sunk into the peat and staked with tarred lumber to make a compartment for the carbon anode around which the nitric acid is formed. Nitric acid was pumped out from an earthenware pipe that was sunk down to the bottom of the pot. Fresh water was pumped into the top through another earthenware pipe to replace the fluid removed. The interior was filled with coke. Cast iron cathodes were sunk into the peat surrounding it. Resistance was about 3 ohms per cubic meter and the power supplied was around 10 volts. Production from one deposit was 800 tons per year. Once the Haber process for the efficient production of ammonia was introduced in 1913, nitric acid production from ammonia using the Ostwald process overtook production from the Birkeland–Eyde process. This method of production is still in use today. Physical and chemical properties Commercially available nitric acid is an azeotrope with water at a concentration of 68% HNO3. This solution has a boiling temperature of 120.5 °C (248.9 °F) at 1 atm (100 kPa; 15 psi). It is known as "concentrated nitric acid". The azeotrope of nitric acid and water is a colourless liquid at room temperature. Two solid hydrates are known: the monohydrate HNO3·H2O or oxonium nitrate [H3O]+[NO3]− and the trihydrate HNO3·3H2O. An older density scale is occasionally seen, with concentrated nitric acid specified as 42 Baumé. Contamination with nitrogen dioxide Nitric acid is subject to thermal or light decomposition and for this reason it was often stored in brown glass bottles: 4 HNO3 → 2 H2O + 4 NO2 + O2 This reaction may give rise to some non-negligible variations in the vapor pressure above the liquid because the nitrogen oxides produced dissolve partly or completely in the acid. The nitrogen dioxide (NO2) and/or dinitrogen tetroxide (N2O4) remains dissolved in the nitric acid coloring it yellow or even red at higher temperatures. While the pure acid tends to give off white fumes when exposed to air, acid with dissolved nitrogen dioxide gives off reddish-brown vapors, leading to the common names "red fuming nitric acid" and "white fuming nitric acid". Nitrogen oxides (NOx) are soluble in nitric acid. Fuming nitric acid Main article: Red fuming nitric acid Commercial-grade fuming nitric acid contains 98% HNO3 and has a density of 1.50 g/cm3. This grade is often used in the explosives industry. It is not as volatile nor as corrosive as the anhydrous acid and has the approximate concentration of 21.4 M. Red fuming nitric acid, or RFNA, contains substantial quantities of dissolved nitrogen dioxide (NO2) leaving the solution with a reddish-brown color. Due to the dissolved nitrogen dioxide, the density of red fuming nitric acid is lower at 1.490 g/cm3. An inhibited fuming nitric acid, either white inhibited fuming nitric acid (IWFNA), or red inhibited fuming nitric acid (IRFNA), can be made by the addition of 0.6 to 0.7% hydrogen fluoride (HF). This fluoride is added for corrosion resistance in metal tanks. The fluoride creates a metal fluoride layer that protects the metal. Anhydrous nitric acid White fuming nitric acid, pure nitric acid or WFNA, is very close to anhydrous nitric acid. It is available as 99.9% nitric acid by assay, or about 24 molar. One specification for white fuming nitric acid is that it has a maximum of 2% water and a maximum of 0.5% dissolved NO2. Anhydrous nitric acid is a colorless, low-viscosity (mobile) liquid with a density of 1.512–3 g/cm3 that solidifies at −42 °C (−44 °F) to form white crystals. Its dynamic viscosity under standard conditions is 0.76 mPa·s. As it decomposes to NO2 and water, it obtains a yellow tint. It boils at 83 °C (181 °F). It is usually stored in a glass shatterproof amber bottle with twice the volume of head space to allow for pressure build up, but even with those precautions the bottle must be vented monthly to release pressure. Structure and bonding The two terminal N–O bonds are nearly equivalent and relatively short, at 1.20 and 1.21 Å. This can be explained by theories of resonance; the two major canonical forms show some double bond character in these two bonds, causing them to be shorter than N–O single bonds. The third N–O bond is elongated because its O atom is bonded to H atom, with a bond length of 1.41 Å in the gas phase. The molecule is slightly aplanar (the NO2 and NOH planes are tilted away from each other by 2°) and there is restricted rotation about the N–OH single bond. Reactions Acid-base properties Nitric acid is normally considered to be a strong acid at ambient temperatures. There is some disagreement over the value of the acid dissociation constant, though the pKa value is usually reported as less than −1. This means that the nitric acid in diluted solution is fully dissociated except in extremely acidic solutions. The pKa value rises to 1 at a temperature of 250 °C. Nitric acid can act as a base with respect to an acid such as sulfuric acid: HNO3 + 2 H2SO4 ⇌ [NO2]+ + [H3O]+ + 2 HSO−4; Equilibrium constant: K ≈ 22 The nitronium ion, [NO2]+, is the active reagent in aromatic nitration reactions. Since nitric acid has both acidic and basic properties, it can undergo an autoprotolysis reaction, similar to the self-ionization of water: 2 HNO3 ⇌ [NO2]+ + NO−3 + H2O Reactions with metals Nitric acid reacts with most metals, but the details depend on the concentration of the acid and the nature of the metal. Dilute nitric acid behaves as a typical acid in its reaction with most metals. Magnesium, manganese, and zinc liberate H2: Mg + 2 HNO3 → Mg(NO3)2 + H2 Mn + 2 HNO3 → Mn(NO3)2 + H2 Zn + 2 HNO3 → Zn(NO3)2 + H2 Nitric acid can oxidize non-active metals such as copper and silver. With these non-active or less electropositive metals the products depend on temperature and the acid concentration. For example, copper reacts with dilute nitric acid at ambient temperatures with a 3:8 stoichiometry: 3 Cu + 8 HNO3 → 3 Cu(NO3)2 + 2 NO + 4 H2O The nitric oxide produced may react with atmospheric oxygen to give nitrogen dioxide. With more concentrated nitric acid, nitrogen dioxide is produced directly in a reaction with 1:4 stoichiometry: Cu + 4 H+ + 2 NO−3 → Cu2+ + 2 NO2 + 2 H2O Upon reaction with nitric acid, most metals give the corresponding nitrates. Some metalloids and metals give the oxides; for instance, Sn, As, Sb, and Ti are oxidized into SnO2, As2O5, Sb2O5, and TiO2 respectively. Some precious metals, such as pure gold and platinum-group metals do not react with nitric acid, though pure gold does react with aqua regia, a mixture of concentrated nitric acid and hydrochloric acid. However, some less noble metals (Ag, Cu, ...) present in some gold alloys relatively poor in gold such as colored gold can be easily oxidized and dissolved by nitric acid, leading to colour changes of the gold-alloy surface. Nitric acid is used as a cheap means in jewelry shops to quickly spot low-gold alloys (< 14 karats) and to rapidly assess the gold purity. Being a powerful oxidizing agent, nitric acid reacts with many non-metallic compounds, sometimes explosively. Depending on the acid concentration, temperature and the reducing agent involved, the end products can be variable. Reaction takes place with all metals except the noble metals series and certain alloys. As a general rule, oxidizing reactions occur primarily with the concentrated acid, favoring the formation of nitrogen dioxide (NO2). However, the powerful oxidizing properties of nitric acid are thermodynamic in nature, but sometimes its oxidation reactions are rather kinetically non-favored. The presence of small amounts of nitrous acid (HNO2) greatly increases the rate of reaction. Although chromium (Cr), iron (Fe), and aluminium (Al) readily dissolve in dilute nitric acid, the concentrated acid forms a metal-oxide layer that protects the bulk of the metal from further oxidation. The formation of this protective layer is called passivation. Typical passivation concentrations range from 20% to 50% by volume.[full citation needed] Metals that are passivated by concentrated nitric acid are iron, cobalt, chromium, nickel, and aluminium. Reactions with non-metals Being a powerful oxidizing acid, nitric acid reacts with many organic materials, and the reactions may be explosive. The hydroxyl group will typically strip a hydrogen from the organic molecule to form water, and the remaining nitro group takes the hydrogen's place. Nitration of organic compounds with nitric acid is the primary method of synthesis of many common explosives, such as nitroglycerin and trinitrotoluene (TNT). As very many less stable byproducts are possible, these reactions must be carefully thermally controlled, and the byproducts removed to isolate the desired product. Reaction with non-metallic elements, with the exceptions of nitrogen, oxygen, noble gases, silicon, and halogens other than iodine, usually oxidizes them to their highest oxidation states as acids with the formation of nitrogen dioxide for concentrated acid and nitric oxide for dilute acid. C (graphite) + 4 HNO3 → CO2 + 4 NO2 + 2 H2O 3 C (graphite) + 4 HNO3 → 3 CO2 + 4 NO + 2 H2O Concentrated nitric acid oxidizes I2, P4, and S8 into HIO3, H3PO4, and H2SO4, respectively. Although it reacts with graphite and amorphous carbon, it does not react with diamond; it can separate diamond from the graphite that it oxidizes. Xanthoproteic test Nitric acid reacts with proteins to form yellow nitrated products. This reaction is known as the xanthoproteic reaction. This test is carried out by adding concentrated nitric acid to the substance being tested, and then heating the mixture. If proteins that contain amino acids with aromatic rings are present, the mixture turns yellow. Upon adding a base such as ammonia, the color turns orange. These color changes are caused by nitrated aromatic rings in the protein. Xanthoproteic acid is formed when the acid contacts epithelial cells. Respective local skin color changes are indicative of inadequate safety precautions when handling nitric acid. Production Industrial nitric acid production uses the Ostwald process. The combined Ostwald and Haber processes are extremely efficient, requiring only air and natural gas feedstocks. The Ostwald process' technical innovation is the proper conditions under which anhydrous ammonia burns to nitric oxide (NO) instead of dinitrogen (N2). The nitric oxide is then oxidized, often with atmospheric oxygen, to nitrogen dioxide (NO2): 2 NO + O2 → 2 NO2 The dioxide then disproportionates in water to nitric acid and the nitric oxide feedstock: 3 NO2 + H2O → 2 HNO3 + NO The net reaction is maximal oxidation of ammonia: NH3 + 2 O2 → HNO3 + H2O Dissolved nitrogen oxides are either stripped (in the case of white fuming nitric acid) or remain in solution to form red fuming nitric acid. Commercial grade nitric acid solutions are usually between 52% and 68% nitric acid by mass, the maximum distillable concentration. Further dehydration to 98% can be achieved with concentrated H2SO4. Historically, higher acid concentrations were also produced by dissolving additional nitrogen dioxide in the acid, but the last plant in the United States ceased using that process in 2012. More recently, electrochemical means have been developed to produce anhydrous acid from concentrated nitric acid feedstock. Laboratory synthesis Laboratory-scale nitric acid syntheses abound. Most take inspiration from the industrial techniques. A wide variety of nitrate salts metathesize with sulfuric acid (H2SO4) – for example, sodium nitrate: NaNO3 + H2SO4 → HNO3 + NaHSO4 Distillation at nitric acid's 83 °C boiling point then separates the solid metal-salt residue. The resulting acid solution is the 68.5% azeotrope, and can be further concentrated (as in industry) with either sulfuric acid or magnesium nitrate. Alternatively, thermal decomposition of copper(II) nitrate gives nitrogen dioxide and oxygen gases; these are then passed through water or hydrogen peroxide as in the Ostwald process: 2 Cu(NO3)2 → 2 CuO + 4 NO2 + O2 2 NO2 + H2O → HNO2 + HNO3 or 2 NO2 + H2O2 → 2 HNO3 Uses The main industrial use of nitric acid is for the production of fertilizers. Nitric acid is neutralized with ammonia to give ammonium nitrate. This application consumes 75–80% of the 26 million tonnes produced annually (1987). The other main applications are for the production of explosives, nylon precursors, and specialty organic compounds. Precursor to organic nitrogen compounds See also: Nitration In organic synthesis, industrial and otherwise, the nitro group is a versatile functional group. A mixture of nitric and sulfuric acids introduces a nitro substituent onto various aromatic compounds by electrophilic aromatic substitution. Many explosives, such as TNT, are prepared this way: C6H5CH3 + 3 HNO3 → C6H2(NO2)3CH3 + 3 H2O Either concentrated sulfuric acid or oleum absorbs the excess water. H2S2O7 + H2O → 2 H2SO4 The nitro group can be reduced to give an amine group, allowing synthesis of aniline compounds from various nitrobenzenes: Use as an oxidant The precursor to nylon, adipic acid, is produced on a large scale by oxidation of "KA oil"—a mixture of cyclohexanone and cyclohexanol—with nitric acid. Rocket propellant Nitric acid has been used in various forms as the oxidizer in liquid-fueled rockets. These forms include red fuming nitric acid, white fuming nitric acid, mixtures with sulfuric acid, and these forms with HF inhibitor. IRFNA (inhibited red fuming nitric acid) was one of three liquid fuel components for the BOMARC missile.[unreliable source?] Niche uses Metal processing Nitric acid can be used to convert metals to oxidized forms, such as converting copper metal to cupric nitrate. It can also be used in combination with hydrochloric acid as aqua regia to dissolve noble metals such as gold (as chloroauric acid). These salts can be used to purify gold and other metals beyond 99.9% purity by processes of recrystallization and selective precipitation. Its ability to dissolve certain metals selectively or be a solvent for many metal salts makes it useful in gold parting processes. Analytical reagent In elemental analysis by ICP-MS, ICP-AES, GFAA, and Flame AA, dilute nitric acid (0.5–5.0%) is used as a matrix compound for determining metal traces in solutions. Ultrapure trace metal grade acid is required for such determination, because small amounts of metal ions could affect the result of the analysis. It is also typically used in the digestion process of turbid water samples, sludge samples, solid samples as well as other types of unique samples which require elemental analysis via ICP-MS, ICP-OES, ICP-AES, GFAA and flame atomic absorption spectroscopy. Typically these digestions use a 50% solution of the purchased HNO3 mixed with Type 1 DI Water. In electrochemistry, nitric acid is used as a chemical doping agent for organic semiconductors, and in purification processes for raw carbon nanotubes. Woodworking In a low concentration (approximately 10%), nitric acid is often used to artificially age pine and maple. The color produced is a grey-gold very much like very old wax- or oil-finished wood (wood finishing). Etchant and cleaning agent The corrosive effects of nitric acid are exploited for some specialty applications, such as etching in printmaking, pickling stainless steel or cleaning silicon wafers in electronics. A solution of nitric acid, water and alcohol, nital, is used for etching metals to reveal the microstructure. ISO 14104 is one of the standards detailing this well known procedure. Nitric acid is used either in combination with hydrochloric acid or alone to clean glass cover slips and glass slides for high-end microscopy applications. It is also used to clean glass before silvering when making silver mirrors. Commercially available aqueous blends of 5–30% nitric acid and 15–40% phosphoric acid are commonly used for cleaning food and dairy equipment primarily to remove precipitated calcium and magnesium compounds (either deposited from the process stream or resulting from the use of hard water during production and cleaning). The phosphoric acid content helps to passivate ferrous alloys against corrosion by the dilute nitric acid.[citation needed] Nitric acid can be used as a spot test for alkaloids like LSD, giving a variety of colours depending on the alkaloid. Nuclear fuel reprocessing Nitric acid plays a key role in PUREX and other nuclear fuel reprocessing methods, where it can dissolve many different actinides. The resulting nitrates are converted to various complexes that can be reacted and extracted selectively in order to separate the metals from each other. Nitric acid is a corrosive acid and a powerful oxidizing agent. The major hazard posed by it is chemical burns, as it carries out acid hydrolysis with proteins (amide) and fats (ester), which consequently decomposes living tissue (e.g. skin and flesh). Concentrated nitric acid stains human skin yellow due to its reaction with the keratin. These yellow stains turn orange when neutralized. Systemic effects are unlikely, and the substance is not considered a carcinogen or mutagen. The standard first-aid treatment for acid spills on the skin is, as for other corrosive agents, irrigation with large quantities of water. Washing is continued for at least 10–15 minutes to cool the tissue surrounding the acid burn and to prevent secondary damage. Contaminated clothing is removed immediately and the underlying skin washed thoroughly. Being a strong oxidizing agent, nitric acid can react violently with many compounds. Use in acid attacks Nitric acid is one of the most common types of acid used in acid attacks. ^ He goes on to point out that "nitrous air" is the reverse, or "nitric acid deprived of air and water." ^ a b c d e f g NIOSH Pocket Guide to Chemical Hazards. "#0447". National Institute for Occupational Safety and Health (NIOSH). ^ "nitric acid_msds". ^ Bell, R. P. (1973), The Proton in Chemistry (2nd ed.), Ithaca, NY: Cornell University Press ^ a b Zumdahl, Steven S. (2009). Chemical Principles 6th Ed. Houghton Mifflin Company. p. A22. ISBN 978-0-618-94690-7. ^ "Safety Data Sheet" (PDF). fishersci.com. Fisher Scientific International. 23 March 2015. p. 2. Archived (PDF) from the original on 10 September 2022. Retrieved 4 October 2022. ^ a b Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. pp. 465–471. doi:10.1016/C2009-0-30414-6. ISBN 978-0-08-037941-8. ^ Examples: Multhauf, Robert P. (1966). The Origins of Chemistry. London: Oldbourne. pp. 140–141. But among them we find the rudiments of processes which were finally to lead to the discovery of the mineral acids, sulphuric, hydrochloric and nitric. The mineral acids manifest themselves clearly only about three centuries after al-Razi, in the works of Europeans [...] Needham, Joseph; Ping-Yü, Ho; Gwei-Djen, Lu; Sivin, Nathan (1980). Science and Civilisation in China. Vol. 5: Chemistry and Chemical Technology. Cambridge: Cambridge University Press. Part IV: Spagyrical Discovery and Invention: Apparatus, Theories and Gifts, p. 195. ISBN 978-0-521-08573-1. It is generally accepted that mineral acids were quite unknown both to the ancients in the West and to the Arabic alchemists. Al-Hassan, Ahmad Y. (2001). Science and Technology in Islam: Technology and applied sciences. UNESCO. p. 59. ISBN 978-92-3-103831-0. The text is given here in full because of the prevailing notion that Islamic chemists did not produce mineral acids. Karpenko, Vladimír; Norris, John A. (2002). "Vitriol in the History of Chemistry". Chemické listy. 96 (12): 997–1005. [...] dating the discovery of nitric acid is likewise uncertain. It is estimated that this discovery took place after 1300 [...] A passage from the second part of Pseudo-Geber's Summa perfectionis [...] was long considered to be the earliest known recipe for sulfuric acid [...] Newman, William R. (2006). Atoms and Alchemy: Chymistry and the Experimental Origins of the Scientific Revolution. Chicago: University of Chicago Press. p. 98. ISBN 978-0226576961. [...] between the time when the Summa perfectionis was written and the seventeenth century, the mineral acids–sulfuric, hydrochloric, nitric, and the mixture of the latter two, called aqua regia, had been discovered. ^ Karpenko & Norris 2002, p. 1002. As Karpenko & Norris note, the uncertain dating of the pseudo-Geber corpus (which was probably written by more than one author) renders the date of its description of nitric acid equally uncertain. According to Al-Hassan 2001, p. 62, recipes for the preparation of nitric acid also occur in the Liber Luminis luminum, a Latin treatise usually attributed to Michael Scot (died before 1236) but perhaps translated by him from the Arabic. One of the manuscripts of the Liber Luminis luminum mentions that it was translated by Michael Scot; see Moureau, Sébastien (2020). "Min al-kīmiyāʾ ad alchimiam. The Transmission of Alchemy from the Arab-Muslim World to the Latin West in the Middle Ages". Micrologus. 28 (22): 87–141. hdl:2078.1/211340. 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The Manufacture of Chemicals by Electrolysis. D. Van Nostrand Co. pp. 30–32. Retrieved 2019-09-15. ^ Dean, John (1992). Lange's Handbook of Chemistry (14 ed.). McGraw-Hill. pp. 2.79–2.80. ISBN 978-0-07-016194-8. ^ "Nitric acid fuming 100%". Sigma-Aldrich. Retrieved May 15, 2025. ^ "nitric acid viscosity". Wolfram Alpha Knowledgebase (2002). Champaign, Illinois. ^ a b Cox, A. P.; Ellis, M. C.; Attfield, C. J.; Ferris, A. C. (1994). "Microwave spectrum of DNO3, and average structures of nitric and nitrous acids". Journal of Molecular Structure. 320 (1–2): 91–106. Bibcode:1994JMoSt.320...91C. doi:10.1016/0022-2860(93)08008-R. ^ Luzzati, V. (1951). "Structure cristalline de l'acide nitrique anhydre". Acta Crystallographica (in French). 4 (2): 120–131. Bibcode:1951AcCry...4..120L. doi:10.1107/S0365110X51000404. ^ a b Allan, D. R.; Marshall, W. G.; Francis, D. J.; Oswald, I. D. H.; Pulham, C. R.; Spanswick, C. (2010). "The crystal structures of the low-temperature and high-pressure polymorphs of nitric acid" (PDF). Dalton Transactions (Submitted manuscript). 39 (15): 3736–3743. doi:10.1039/B923975H. PMID 20354626. ^ Cox, A. P.; Riveros, J. M. (1965). "Microwave Spectrum and Structure of Nitric Acid". The Journal of Chemical Physics. 42 (9): 3106. Bibcode:1965JChPh..42.3106C. doi:10.1063/1.1696387. ^ IUPAC SC-Database A comprehensive database of published data on equilibrium constants of metal complexes and ligands ^ a b c d e Housecroft, Catherine E.; Sharpe, Alan G. (2008). "Chapter 15: The group 15 elements". Inorganic Chemistry (3rd ed.). Pearson. ISBN 978-0-13-175553-6. ^ ASTM standard A967-05 ^ Ōsawa, Eiji (December 2007). "Recent progress and perspectives in single-digit nanodiamond". Diamond and Related Materials. 16 (12): 2018–2022. Bibcode:2007DRM....16.2018O. doi:10.1016/j.diamond.2007.08.008. ^ Sherman, Henry Clapp (2007). Methods of organic analysis. Read Books. p. 315. 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Nature. ^ a b Thiemann, Michael; Scheibler, Erich; Wiegand, Karl Wilhelm (2000). "Nitric Acid, Nitrous Acid, and Nitrogen Oxides". Ullmann's Encyclopedia of Industrial Chemistry. Weinheim: Wiley-VCH. doi:10.1002/14356007.a17_293. ISBN 978-3527306732. ^ Clark, John D. (1972). Ignition!. Rutgers University Press. ISBN 978-0-8135-0725-5. ^ "BOMARC Summary". TheMilitaryStandard. Retrieved 2025-04-07. ^ Eaton, Andrew D.; Greenberg, Arnold E.; Rice, Eugene W.; Clesceri, Lenore S.; Franson, Mary Ann H., eds. (2005). Standard Methods For the Examination of Water and Wastewater (21 ed.). American Public Health Association. ISBN 978-0-87553-047-5. Also available on CD-ROM and online by subscription.[page needed] ^ Jewitt, Jeff (1997). Hand-applied finishes. Taunton Press. ISBN 978-1-56158-154-2. Retrieved 2009-05-28. ^ Muraoka, Hisashi (1995). "Silicon wafer cleaning fluid with HNO3, HF, HCl, surfactant, and water" U.S. patent 5,635,463 ^ ISO 14104:2017 – Gears – Surface temper etch inspection after grinding, chemical method. ^ Fischer, A. H.; Jacobson, K. A.; Rose, J.; Zeller, R. (1 May 2008). "Preparation of Slides and Coverslips for Microscopy". Cold Spring Harbor Protocols. 2008 (6): pdb.prot4988. doi:10.1101/pdb.prot4988. PMID 21356831.{{cite journal}}: CS1 maint: article number as page number (link) ^ Curtis, Heber D. (February 1911). "Methods of Silvering Mirrors". Publications of the Astronomical Society of the Pacific. 23 (135): 13. Bibcode:1911PASP...23...13C. doi:10.1086/122040. hdl:2027/mdp.39015018047608. S2CID 120665136. ^ O’Neal, Carol L; Crouch, Dennis J; Fatah, Alim A (April 2000). "Validation of twelve chemical spot tests for the detection of drugs of abuse". Forensic Science International. 109 (3): 189–201. doi:10.1016/S0379-0738(99)00235-2. PMID 10725655. ^ May, Paul (November 2007). "Nitric acid". Retrieved 2009-05-28. ^ "Nitric acid: Toxicological overview". Health Protection Agency. Retrieved 2011-12-07. ^ Rees, Anna (1 October 2013). "Freeze mob to highlight the issue of acid attacks". RESET.to. Retrieved 25 June 2021. External links NIOSH Pocket Guide to Chemical Hazards National Pollutant Inventory – Nitric Acid Fact Sheet Calculators: surface tensions Archived 2020-02-22 at the Wayback Machine, and densities, molarities and molalities Archived 2020-02-22 at the Wayback Machine of aqueous nitric acid \| Hydrogen compounds \| \| --- \| \| H3AsO3 H3AsO4 HArF HAt HSO3F H[BF4] HBr HBrO HBrO2 HBrO3 HBrO4 HCl HClO HClO2 HClO3 HClO4 HCN HCNO H2CrO4/H2Cr2O7 H2CO3 H2CS3 HF HFO HI HIO HIO2 HIO3 HIO4 HMnO4 H2MnO4 H2MoO4 HNC NaHCO3 HNCO HNO HNO2 HNO3 H2N2O2 HNO5S H3NSO3 H2O H2O2 H2O3 H2O4 H2O5 H3PO2 H3PO3 H3PO4 H4P2O7 H5P3O10 H[AuCl4] H2[PtCl6] H2OsCl6 H2S H2S2 H2Se H2SeO3 H2SeO4 H4SiO4 H2[SiF6] HSCN HNCS H2SO3 H2SO4 H2SO5 H2S2O3 H3O H2S2O6 H2S2O7 H2S2O8 CF3SO3H H2Te H2TeO3 H6TeO6 H4TiO4 H2Po H[Co(CO)4] \| \| \| v t e Nitrogen species \| \| --- \| \| Hydrides \| NH3 NH4+ NH2− N3− NH2OH N2H4 HN3 N3− NH5 (?) \| \| Organic \| NR3 >C=NR −CONR2 −CN HCN CN− (CN)2 H2NCN HOCN HNCO HNCS CH2N2 –NO –NO2 \| \| Oxides \| NO / (NO)2 N2O3 HNO2 / NO− 2 / NO+ NO2 / (NO2)2 N2O5 HNO3 / NO− 3 / NO+ 2 NO3 HNO / (HON)2 / N2O2− 2 / N2O H2NNO2 HO2NO / ONOO− HO2NO2 / O2NOO− NO3− 4 H4N2O4 / N2O2− 3 \| \| Halides \| NF NF2 NF3 NF5 (?) NCl3 NBr3 NI3 FN3 ClN3 BrN3 IN3 NH2F N2F2 NH2Cl NHF2 NHCl2 NHBr2 NHI2 \| \| Oxidation states \| −3, −2, −1, 0, +1, +2, +3, +4, +5 (a strongly acidic oxide) \| \| v t e Salts and covalent derivatives of the nitrate ion \| \| --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- \| HNO3 \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| He \| \| LiNO3 \| Be(NO3)2 \| B(NO3)−4 \| RONO2 +CO3 +C2O4 \| NO3− NH4NO3 \| HOONO2 \| FNO3 +F \| Ne \| \| NaNO3 \| Mg(NO3)2 \| Al(NO3)3 Al(NO3)−4 \| Si \| P \| +SO4 \| ClONO2 +Cl \| Ar \| \| KNO3 \| Ca(NO3)2 \| \| Sc(NO3)3 \| Ti(NO3)4 \| VO(NO3)3 \| Cr(NO3)3 \| Mn(NO3)2 \| Fe(NO3)2 Fe(NO3)3 \| Co(NO3)2 Co(NO3)3 \| Ni(NO3)2 \| CuNO3 Cu(NO3)2 \| Zn(NO3)2 \| Ga(NO3)3 \| Ge \| As \| +SeO3 \| BrNO3 +Br \| Kr \| \| RbNO3 \| Sr(NO3)2 \| \| Y(NO3)3 \| Zr(NO3)4 \| NbO(NO3)3 \| MoO2(NO3)2 \| Tc \| Ru(NO3)3 \| Rh(NO3)3 \| Pd(NO3)2 \| AgNO3 \| Cd(NO3)2 \| In(NO3)3 \| Sn(NO3)4 \| Sb4O4(OH)2(NO3)2 \| Te \| INO3 +IO3 \| Xe(NO3)2 \| \| CsNO3 \| Ba(NO3)2 \| \| Lu(NO3)3 \| Hf(NO3)4 \| TaO(NO3)3 \| WO2(NO3)2 \| ReO3NO3 \| Os \| Ir3O(NO3)10 \| Pt(NO3)2 Pt(NO3)4 \| Au(NO3)−4 \| Hg2(NO3)2 Hg(NO3)2 \| TlNO3 Tl(NO3)3 \| Pb(NO3)2 \| Bi(NO3)3 BiO(NO3) \| Po(NO3)4 \| At \| Rn \| \| FrNO3 \| Ra(NO3)2 \| \| Lr \| Rf \| Db \| Sg \| Bh \| Hs \| Mt \| Ds \| Rg \| Cn \| Nh \| Fl \| Mc \| Lv \| Ts \| Og \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| La(NO3)3 \| Ce(NO3)3 Ce(NO3)4 \| Pr(NO3)3 \| Nd(NO3)3 \| Pm(NO3)3 \| Sm(NO3)3 \| Eu(NO3)3 \| Gd(NO3)3 \| Tb(NO3)3 \| Dy(NO3)3 \| Ho(NO3)3 \| Er(NO3)3 \| Tm(NO3)3 \| Yb(NO3)3 \| \| \| Ac(NO3)3 \| Th(NO3)4 \| PaO(NO3)3 \| U(NO3)4 UO2(NO3)2 \| Np(NO3)4 NpO(NO3)3 NpO2NO3 NpO2(NO3)2 \| Pu(NO3)3 Pu(NO3)4 PuO2(NO3)2 \| Am(NO3)3 AmO2(NO3)2 \| Cm(NO3)3 \| Bk(NO3)3 \| Cf(NO3)3 \| Es(NO3)3 \| Fm \| Md \| No \| \| \| \| Authority control databases \| \| --- \| \| International \| GND FAST \| \| National \| United States France BnF data Japan Czech Republic Israel \| \| Other \| Yale LUX \| Retrieved from " Categories: Pnictogen oxoacids Nitrogen oxoacids Mineral acids Photographic chemicals Drug testing reagents Oxidizing acids Nitrogen(V) compounds Hidden categories: CS1: long volume value Articles containing Latin-language text Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference CS1 errors: ISBN date CS1 French-language sources (fr) Wikipedia articles needing page number citations from November 2022 CS1 maint: article number as page number Articles with short description Short description is different from Wikidata Wikipedia indefinitely semi-protected pages Chemical articles with multiple compound IDs Multiple chemicals in an infobox that need indexing ECHA InfoCard ID from Wikidata Chembox having GHS data Articles containing unverified chemical infoboxes Chembox image size set Articles containing French-language text All articles with incomplete citations Articles with incomplete citations from October 2022 Unclassified articles missing image alternative text All articles lacking reliable references Articles lacking reliable references from April 2025 All articles with unsourced statements Articles with unsourced statements from September 2011 Webarchive template wayback links
3807	https://www.microbehunter.com/microscopy-forum/viewtopic.php?t=5535	Dark field difficulties - MicrobeHunter.com Microscopy Forum MicrobeHunter.com Microscopy Forum You can also access this page with: www.microscopy-forum.com Skip to content SearchAdvanced search Quick links Unanswered topics Active topics Search FAQ Login Register Board index Search Unanswered topics Active topics Dark field difficulties Here you can discuss different microscopic techniques and illumination methods, such as Brightfield, Darkfield, Phase Contrast, DIC, Oblique illumination, etc. Post Reply Print view SearchAdvanced search 24 posts • Page 1 of 1 Message Author matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Dark field difficulties Quote #1Postby matt123 » Thu Dec 21, 2017 2:34 pm Hello, I'm relatively new to microscopy and am using an Amscope T490B compound microscope. I'm an experienced macro photographer ( and so my main interest is in photo microscopy but I've got a lot to learn when it comes to microscopy. I've been experimenting with different lighting techniques since getting my microscope. One difficulty I'm having is creating a dark field that is truly dark. I'm using a glass filter with a black circular stopper made out of insulation tape which sits beneath the condenser and I've experimented with a variety of different sized stoppers. To keep things simple I've been mainly using the 10x objective. It seems that no matter what sized stopper I use I can never get the background to be darker than a dark gray colour. Yet I regularly see photos and videos taken using dark field in which the background is jet black. Is this jet black a result of post processing in Photoshop or should I be able to create a jet black background using the microscope alone? My backgrounds are also full of specks and dots of dust and imperfections within the slides/cover slips etc but I realise that is a separate problem that's difficult to avoid. I think I understand the basic theory behind dark field so my gray background obviously suggests the stop is not effectively blocking the light but no matter what I try I can't seem to solve it. Does anyone have any suggestions? Many thanks. Matt www.mattcolephotography.co.uk Top mrsonchusPosts:4175Joined: Tue Feb 03, 2015 9:42 pmLocation: Cumbria, UK Re: Dark field difficulties Quote #2Postby mrsonchus » Thu Dec 21, 2017 3:45 pm Hello Matt, I've just tried darkfield too, and the result was quite good, with a jet-black background. Have a look at my post from yesterday here -->rough & ready darkfield Here I used a ground-glass 'filter' in my condenser's filter-holder with the stop made from a piece of 'Blu-Tack' simply squashed to the right diameter (i.e. the stop) to block all direct light. Field-iris full open, move the condenser up & down until two conditions are met; 1) the edgesof the 'stop' whatever you use to make it, are in focus at the same time as the specimen is.. 2) the stop is of a size that exactly (within reason of course) matches the field of view of the objective - that is to say blocks all light coming 'straight through' the condenser - the rest of the light will 'come from around' the stop and illuminate the specimen 'sideways' whilst the background will be black (as no light is getting through from below) - the so-called 'dark-field'. Your too-light background is very likely due to the stop being insufficiently opaque and not stopping all light completely - the 'Blu-Tack' stops all light and is easy to improvise to the right size to get started with... My first result seen in the post is with a x2.5 objective, I've yet to try it out with the others such as the x10 for starters... John B. p.s. welcome. John B Top matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Re: Dark field difficulties Quote #3Postby matt123 » Thu Dec 21, 2017 7:19 pm Many thanks John and also for the welcome. I was fairly confident that my stops were blocking the light but I've just tried using blu tack and I still have exactly the same problem - a grey background. The dark field effect is working i.e. the subject is illuminated by side lighting but I cannot get a black background. I've varied the stop size and on each occasion the only point I get the dark field effect is when the condenser is in its highest position almost touching the underneath of the slide. The only exception is if I make the stop too large and then the subject is still in darkness even when the condenser is in its highest position. So I'm not actually sure how to do your points (1) and (2) above as there is only 1 point at which I get the dark field effect (and that is when the condenser is in its highest position). I don't know what else to try or what I could be doing wrong... Matt www.mattcolephotography.co.uk Top mrsonchusPosts:4175Joined: Tue Feb 03, 2015 9:42 pmLocation: Cumbria, UK Re: Dark field difficulties Quote #4Postby mrsonchus » Thu Dec 21, 2017 7:31 pm Hi Matt, hmm, which objective are you using, a x4 or x10 maybe? A picture of your 'scope's stage and condenser may be useful. In the meantime I'll run through it again and take some images to try to give a far better idea of what I did - back hopefully later tonight with an update.... John B. John B Top matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Re: Dark field difficulties Quote #5Postby matt123 » Thu Dec 21, 2017 8:08 pm thanks John. I have the same problem whether I use the 4x or the 10x objectives. I'm not yet sure how you post photos but the following page on Amazon shows a number of images of my microscope including the stage and condenser ... ords=t490b Matt www.mattcolephotography.co.uk Top MicroBobPosts:3155Joined: Sun Dec 25, 2016 9:11 amLocation: Northern Germany Re: Dark field difficulties Quote #6Postby MicroBob » Thu Dec 21, 2017 8:45 pm Hi Matt, your grey background - does it occur together with a verly light object? The metering system of a camera is adjusted to give images of medium brightness. It tends to get exposure wrong in dark field microscopy. Dark field enhances all dirt in the slide and in the light path. When you see a really clean dark field image it usually has been cleaned with an image editing program. Does your microscope have a field diaphragm and is the lamp movable? When you have an immersion condensor (n.A. over 1) you might try to put a drop of demineralized water between Condensor to lens and slide. Getting really good micro images is not easy and first you have to know enough about microscopy itself. So don't let these difficulties disencourage you. Bob Top apochronautPosts:6994Joined: Fri May 15, 2015 12:15 am Re: Dark field difficulties Quote #7Postby apochronaut » Thu Dec 21, 2017 8:57 pm The condenser does usually need to be at it's highest possible N.A. to get the best DF. Is there some reason why you don't want to use the condenser at it's highest elevation? If the condenser is too low, you will need a very large stop in order to ensure that you are only getting light that originates outside the N.A. of the objective. If any light originates from an N.A. lower than that of the objective, it will cause a gray image. Top MicroBobPosts:3155Joined: Sun Dec 25, 2016 9:11 amLocation: Northern Germany Re: Dark field difficulties Quote #8Postby MicroBob » Thu Dec 21, 2017 9:09 pm Many microscopes have a stop that limits the movement of the condensor. It might be adjusted a bit conservative so that you can't turn the condensor high enough. Normaly it should be able to just touch the slide. Top matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Re: Dark field difficulties Quote #9Postby matt123 » Thu Dec 21, 2017 10:35 pm thanks for the further help and advice. Microbob - the grey background isn't an exposure issue unfortunately as I'm see it through the eyepieces as well as in the photo. I'm quite experienced in terms of photography and post processing so can create a darker background using the camera/software but I suspect I should be able to generate a darker background using the microscope alone. My microscope doesn't have a field diaphragm (only a diaphragm below the condenser) and the light isn't moveable. I won't let this discourage me. I must admit I'm already quite pleased with some of the images I've got so far. apochronaut - it's not a problem that the condenser is always at its highest point but I mentioned it as it seemed to contrast with John's advice to move the condenser up and down until 2 conditions are met. Thanks again. I'll keep experimenting. Matt www.mattcolephotography.co.uk Top mrsonchusPosts:4175Joined: Tue Feb 03, 2015 9:42 pmLocation: Cumbria, UK Re: Dark field difficulties Quote #10Postby mrsonchus » Thu Dec 21, 2017 11:08 pm Hi Matt, yes that's the way I managed to do it with my 'scope and a x2.5 objective, which isn't a common objective either. I tried to 'do it' with my x10 objective and failed miserably - hence the lack of a follow-up image.... I would definitely suggest that Apo's advice will be the more valuable to you as mine is very much geared to my particular setup whereas Apo's will be very sound advice 'from the ground up' as it were. My experience and knowledge of microscopy and in particular the methods re many types of 'scope comes nowhere near Apo's level of expertise, knowledge & experience. It was very useful to me also to read the comment re the attainment of maximum n.a. (of the condenser) as I'm very new indeed to darkfield of any type also. For now though I'm really pleased with the x2.5 objective's result - I'll work-on from there too. John B. Last edited by mrsonchus on Thu Dec 21, 2017 11:34 pm, edited 1 time in total. John B Top apochronautPosts:6994Joined: Fri May 15, 2015 12:15 am Re: Dark field difficulties Quote #11Postby apochronaut » Thu Dec 21, 2017 11:17 pm The stop is smaller , the farther from the objective it gets, so both the maximum and minimum N.A.'s of the condenser reduce, when compared to the same stop, raised higher. It sounds like you are getting some direct light that is creeping past the stop into the objective. Raising the stop, will effectively make it larger, relative to the aperture of the objective and ensure that the only light entering the system, is outside the aperture of the objective and thus ensuring a dark background. There is also a requirement that the DF stop be absolutely centered, otherwise direct light can creep in from one side only and foul the works. DF tends to work best, if you maximize the aperture of the condenser. John; are you using an aux. condensr lens with the 2.5X and not with the 10X? Top mrsonchusPosts:4175Joined: Tue Feb 03, 2015 9:42 pmLocation: Cumbria, UK Re: Dark field difficulties Quote #12Postby mrsonchus » Thu Dec 21, 2017 11:33 pm Hi Apo' I used the condenser with the top element swung-out for the x2.5, with the condenser well below it's maximum travel. The bottom lens of the 600 series is I think n.a. 0.25 or thereabouts. I tried both with the top lens (1.25 n.a.) in and out with the x10 and with the condenser up in it's usual (almost max height) position as when focusing the field-iris leaves in the image plane with the specimen. It really was a rushed job though and I'll take another look tomorrow with a far more careful and accurate approach and see what I can get with the x10. The x2.5 gives me the nice darkfield rather easily, unlike the x10 so far at least. Thanks for the pointers, they really are useful snippets that help a lot. John B. p.s. perhaps I should try the 0.90 top-lens condenser with the x10 - I've both the 1.25 and the 0.90 versions to choose from. John B Top JimTPosts:3247Joined: Fri Oct 24, 2014 1:57 pm Re: Dark field difficulties Quote #13Postby JimT » Thu Dec 21, 2017 11:45 pm Matt123, here is info about stop sizes based on the obj. being used. Hope this is a help. Magnification Numerical Aperture Stop Size (mm) 1X 0.03 25-30 2x 0.05 8-11 4X 0.10 8-14 10x 0.25 16-18 20X 0.40 18-20 20x 0.65 20-22 40X 0.65 22-24 I must admit I'm already quite pleased with some of the images I've got so far. We'll be looking forward to seeing some. JimT Top lorezPosts:735Joined: Wed Dec 17, 2014 1:48 am Re: Dark field difficulties Quote #14Postby lorez » Fri Dec 22, 2017 3:07 am One thing that is often overlooked is the quality of the lenses in the system. I have conducted workshops where each participant provided their own microscope and even though the accessories being used were identical the results were not. All the technique in the world cannot produce a quality image from a "budget" microscope. lorez Top SuphotPosts:100Joined: Wed Mar 11, 2015 4:35 am Re: Dark field difficulties Quote #15Postby Suphot » Fri Dec 22, 2017 9:05 am Hi matt123 May I suggest you to test your set up in the Dark field like this; Cut a normal photocopy paper in the size of standard microscope slide (around 1 x 3 inch I think). Use your Prepared Microscope slide and set up your Dark field to the best result as you can. (Cross section of any wood stem is a good to try). Take Prepared Microscope slide out of the microscope stage, replace it with normal photocopy paper that you have cut. In this step, when you look at the microscope stage, you should see Bright dot of illumination from the condenser focus on photocopy paper. The focus of bright dot should be very small. When you look into microscope eyepieces, you can see the magnified image of photocopy paper. It will not show Dark field image because all of the photocopy paper was illuminated no dark background can be show. Now use your hand to move the photocopy paper out of the stage completely. Looking into eyepieces, your microscope should show complete black image. If it show grey image, take eyepiece out and looking into the microscope. You should see the back of microscope objective, it should be dark, no illumination from the condenser can be seen. If the back of Objective was illuminated, the Dark field can not be done. I hope you can see by yourself that where the light came from. You may see that the field stop is not large enough, or the field stop is not exactly center or adjusting condenser up or down. You may try to fixed this until the back of Objective lens is dark. Please note that, material that you use to make a Dark filed stop should be completely opaque. The illumination from condenser was very high intensity, I usually stack multiple sheet of black tape together to make it completely black. I hope this may help Suphot Top photomicroPosts:207Joined: Thu Aug 03, 2017 10:28 amLocation: UK Re: Dark field difficulties Quote #16Postby photomicro » Fri Dec 22, 2017 10:16 am Isn't this model supposed to come with a dedicated dark-field condenser? Presumably this comes with ready made stops off the right size... Not sure therefore why you are making your own. Top Hobbyst46Posts:4393Joined: Mon Aug 21, 2017 9:02 pm Re: Dark field difficulties Quote #17Postby Hobbyst46 » Fri Dec 22, 2017 10:36 am Hi Matt You already got many expert advices that in total explain the optics. Let me add an example. I use a Zeiss GFL with very good optics (original Zeiss and Olympus, they all yield tge same DF moreor less). I have a phase condenser. Zeiss instructs to use the Ph2 or Ph3 phase stops of the condenser for DF wth low mag objectives. So I do and get good DF with 6.3x-16x-25x. It is seldom pitch black, often very dark gray, but the diatoms are brightly illuminated and shine against the background. So do the dust particles of course. Reading your post I took a CD plastic package box, you know it consists of a black plate and a transparent cover. Took it apart. The center of the black plate has three annular section openings that form a non-continuous annular "window". Surrounded by a "ridge" - a raised part, like a ring. I cut it out with a sharp knife. The improvised stop, diameter about 30mm, is shown below. I placed it inside the filter holder that resides below the condenser. Focused on a diatom slide (home made and fairly poor quality). Created Kohler illumination with the 25x/0.45 objective. The condenser is near the slide, raised as far as it goes. Brightfield position. Then open the iris to maximum. Use strong illumination - I have a 10W LED lamp. Then lowered the condenser slowly until a DF is obtained - and the stop is fairly in focus. Again, not pitch black but dark. I believe that there is nothing wrong with your optics and the insulation black band is also adequate. The mechanical and optical optimization is the challenge here and it is trial and error. Incidentally, I think that AmScope markets DF condensers as well. Happy Holidays. Top Hobbyst46Posts:4393Joined: Mon Aug 21, 2017 9:02 pm Re: Dark field difficulties Quote #18Postby Hobbyst46 » Fri Dec 22, 2017 10:38 am Sorry - here is the picture of my improvized darkfield stop. Attachments DF stop from CD box-1.jpg (17.68 KiB) Viewed 17513 times Top matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Re: Dark field difficulties Quote #19Postby matt123 » Fri Dec 22, 2017 10:49 am many thanks for the further help. Suphot - I will definitely try going through those steps to work out where the apparent light leak is coming from. Amscope do produce a dark field condenser and I did actually purchase one. However, frustratingly it is slightly too wide to fit into the condenser ring. The condenser that the scope came with is already very tight and the DF condenser is very slightly wider. I got onto Amscope who apologised and said it was a result of a slight redesign! Hence why I've been making home made stops. Matt www.mattcolephotography.co.uk Top Hobbyst46Posts:4393Joined: Mon Aug 21, 2017 9:02 pm Re: Dark field difficulties Quote #20Postby Hobbyst46 » Fri Dec 22, 2017 10:53 am Just to note: Of course the diameter of the active part of this piece is that inside the three windows, 11-12mm. A Common CD box. Opaque black plastic. Top Micro-BobPosts:30Joined: Mon Dec 05, 2016 12:18 pm Re: Dark field difficulties Quote #21Postby Micro-Bob » Fri Dec 22, 2017 11:13 am Hi Matt, I think Suphots method will help to show here problems are. Microscope illumination with low power objectives is often only improvised to get along with the standard condenser, so dark field methods will be different in this case. If you are interested in the theory behind it you might habve a look at Köhler illumination. Köhler worked out a theory for setting up the microscope for optimized resolution and contrast. True Köhler illumination is not so often found in microscopes. In many cases there is a frosted lens or filter somewhere in the light path that makes setting up easier an the light more uniform, but giving away a little contrast. You microscope is not made to offer this adjustability, but you may be able to see wheter there is room for optimisation. You could e.g. use a diy field diaphragm to close out unneeded light. Your objectives are no plan apos for shure, but I think that contrast won't be their weak point. Whith many microscopes it is fairly easy to get a good image with the 4x and 10x objective. With the 40x and 100x it is more demanding, a good slide, good optics and an experienced user is needed. So it is probably best to get it right for the lower magnifications first. Bob Top matt123Posts:6Joined: Thu Dec 21, 2017 2:16 pm Re: Dark field difficulties Quote #22Postby matt123 » Fri Dec 22, 2017 11:37 am thanks Bob. As you note, I'm not able to use Kohler illumination because of a lack of a field diaphragm but I have ordered a cheap iris diaphragm to see if I can improvise a little with the light source. Last night I did try something that seemed to darken the background slightly. Although I don't have a field aperture I artificially reduced the diameter of my light source by placing a black ring of metal on the lamp which reduced the 'aperture' of the lamp to around 15-20mm. I then reduced the size of my blu tack stop accordingly. As I say, this did seem to help. I don't really see why it did help as I presumably should be able to use the full aperture of the lamp with a larger stop and create the same dark background. But I wonder if my lack of a field diaphragm (and my light source being too wide?) is part of the problem. I have upgraded my 10x objective with an Amscope plan achromat but I must admit I can't see a great deal of difference, if any. It was something of an experiment but maybe I'd need to spend more to see an improvement. I'm very familiar with macro lenses for cameras where that is certainly true. However, I must say that I have been quite impressed with the quality of the Amscope objectives but that may reflect my lack of experience of different objectives. Matt www.mattcolephotography.co.uk Top Micro-BobPosts:30Joined: Mon Dec 05, 2016 12:18 pm Re: Dark field difficulties Quote #23Postby Micro-Bob » Fri Dec 22, 2017 1:33 pm For dark field you want a hollow cone of light. The outer angle so that the back lens of the objective is fully used. Your dark field spot has to fit to the objective, it is not influenced by the field diaphragm. One part of setting up a Köhler illumination is focusing the condenser on the field iris and closing the field iris so it just moves outside the visible image circle. You might be able to do something like this with a diaphragm or cardboard stop on top of the light opening. Top manu de hanoiPosts:20Joined: Wed May 02, 2018 2:04 pm Re: Dark field difficulties Quote #24Postby manu de hanoi » Thu May 03, 2018 9:19 am here are some last resort tips: -lower the iso if you're on a digital camera -make sure the iris, the stop, output of the condenser, objective are properly aligned (there are some tutorials and tips above) -make sure you dont have ambiant light parasiting ! -a good way to debug darkfield is to insert a sheet of white paper instead of the slide, you should see the dark core and bright ring when moving the condenser up/down (and possibly light leaks), but you want them to be nearly merged for observation Top Post Reply Print view Display: Sort by: Direction: 24 posts • Page 1 of 1 Return to “Illumination Techniques” Jump to Important links and infos ↳ Welcome to the new forum! ↳ Important announcements ↳ Some interesting links General ↳ Introduce yourself ↳ My microscope ↳ Sandbox ↳ About the website, the magazine and this forum ↳ Articles and images to be published ↳ Off Topic Microscopy Forum ↳ Pictures and Videos ↳ Beginner's corner ↳ Illumination Techniques ↳ Digital processing ↳ Microscopes and optics ↳ Microscopy accessories ↳ Camera systems and imaging ↳ Specimens, samples and slides ↳ Morphology and Behavior ↳ Collecting microscopes and slides ↳ Home-made microscope adaptations ↳ Identification help ↳ Resources (online, books etc.) ↳ For forum members who want to buy and sell equipment ↳ Miscellaneous Board index All times are UTC Delete cookies Contact us Style developer by forum, Powered by phpBB® Forum Software © phpBB Limited Privacy \| Terms
3808	https://www.youtube.com/watch?v=cCEOefbaLak	Transformations - Composition of a Rotation, Translation, and a Reflection - FishMath.com FishMath 433 subscribers 2 likes Description 636 views Posted: 5 Jul 2019 In this video we look at how to do compositions of transformations on the coordinate grid For endless practice problems with transformations on the coordinate grid visit Transcript: hi everyone we're looking how to do a composition of transformations on the coordinate grid so composition basically is just a combination of more than one transformation so we've got to do is follow the steps here in order so we're gonna first rotate we're gonna then translate and we're going to reflect so we're gonna rotate this shape first 90 degrees counterclockwise about the origin so I'm take a copy of this picture so that I can turn it and we can look at the new version after we do there our rotation so here's the original the rotate it 90 degrees counterclockwise that looks like this where the new location of everything is just kind of turned 90 degrees counterclockwise you'll notice here everything moves one box a and B were in quadrant three now they're in quadrant four C was in quadrant two and now it jumped to quadrant three during 90 degrees counterclockwise that's how it kind of rotates so we're going to graph these new locations that can get our answer over here they'll tell us how to graph it on the other grid on the right our original graph so when you're doing this on paper just simply take your paper turn it according to the directions to look at the new locations and then kind of turn it back for each point so when you turn it you'll see a is now from our center point from the origin a is one to the right and three down then turn it back and want to go from the origin one to the right and three down that'll be our new location at a a prime B from the origin if you turn it again 90 degrees counterclockwise B is at seven to the right and one down from the origin it's always from the center point so seven the right one down would be right over here that's B Prime and then C again turning at 90 degrees counterclockwise we can get the turn version over here C is six to the left of zero and two down so we go back to the original six to the left two down and that's the new location of C over here now we're going to connect these to make our triangle and it should look like what we've got on the left it looks pretty accurate right then okay so step one is done now we're going to do step two step two is the translation of X minus three y minus five so what does this mean well the X's are always left's and rights so if it's X minus three that's going to be left three if it was plus would be right Y minus five is gonna be down five so if it was plus their to go up so we're going to take our new version of the shape here and we're gonna move it according to those directions so I'm gonna kind of show you this a little bit typically I'm gonna take copy of our shape I'm gonna make it a new color I'm gonna see what happens here if I take this original shape the second one I mean I'm gonna take this I want to translate according to our direction so we want to go left 3 so 1 2 3 then we go down 5 there's a 1 2 3 4 5 and then we'll get our new location of our shape after both transformations have now been done we've got our rotation done and our translation done and that works pretty well of course you can't do is on paper the exact same way but it's the same thing just take each coordinate go left 3 down 5 and you'll get the new locations just make sure for each one you label it with a new prime this is a second version after the transformation the second transformation our last step is to take the last newest one and we're going to reflect it over the y axis so the y axis is the vertical axis right here so we're going to use this as our a line of reflection and we're going to flip everything it's on the left of the line to the right anything that's on the right of the line will go to the left it's gonna make kind of a mess here so looking at be double prime that's 4 to the right so it's gonna go 4 to the left and it's not gonna change height to maybe the same height she's going to flip over the line and go left instead of right so there we are that at negative 4 will be B triple prime a a is to the left so it becomes 2 to the right a triple prime and C is 9 to the left so it goes nine to the right again not changing the y-value though it's still negative seven down so C triple prime goes here and then we get this very complicated looking shape where it's overlapping a lot and the y-axis there should be our line of symmetry and you can see it looks a little bit off because my lines are not perfect if I try and move those down a little bit hopefully it looks a little bit better let's see if this looks a little better yeah a little bit better there so again the line of reflection is the y-axis that should be our line of symmetry and it's about right this this tool is not the best on my iPad here so it's a little bit hard to tell but that is the line of reflection and we're done we've followed all three steps we did all three transformations
3809	https://www.ecmm.info/wp-content/uploads/Cornely-TLID-2019-Global-guideline-for-the-diagnosis-and-management-of-mucormycosis-an-initiative-of-the-ECMM-in-cooperation-with-theMSG-ERC_OC.pdf	www.thelancet.com/infection Published online November 4, 2019 1 Review Global guideline for the diagnosis and management of mucormycosis: an initiative of the European Confederation of Medical Mycology in cooperation with the Mycoses Study Group Education and Research Consortium Oliver A Cornely, Ana Alastruey-Izquierdo, Dorothee Arenz, Sharon C A Chen, Eric Dannaoui, Bruno Hochhegger, Martin Hoenigl, Henrik E Jensen, Katrien Lagrou, Russell E Lewis, Sibylle C Mellinghoff, Mervyn Mer, Zoi D Pana, Danila Seidel, Donald C Sheppard, Roger Wahba, Murat Akova, Alexandre Alanio, Abdullah M S Al-Hatmi, Sevtap Arikan-Akdagli, Hamid Badali, Ronen Ben-Ami, Alexandro Bonifaz, Stéphane Bretagne, Elio Castagnola, Methee Chayakulkeeree, Arnaldo L Colombo, Dora E Corzo-León, Lubos Drgona, Andreas H Groll, Jesus Guinea, Claus-Peter Heussel, Ashraf S Ibrahim, Souha S Kanj, Nikolay Klimko, Michaela Lackner, Frederic Lamoth, Fanny Lanternier, Cornelia Lass-Floerl, Dong-Gun Lee, Thomas Lehrnbecher, Badre E Lmimouni, Mihai Mares, Georg Maschmeyer, Jacques F Meis, Joseph Meletiadis, C Orla Morrissey, Marcio Nucci, Rita Oladele, Livio Pagano, Alessandro Pasqualotto, Atul Patel, Zdenek Racil, Malcolm Richardson, Emmanuel Roilides, Markus Ruhnke, Seyedmojtaba Seyedmousavi, Neeraj Sidharthan, Nina Singh, János Sinko, Anna Skiada, Monica Slavin, Rajeev Soman, Brad Spellberg, William Steinbach, Ban Hock Tan, Andrew J Ullmann, Jörg J Vehreschild, Maria J G T Vehreschild, Thomas J Walsh, P Lewis White, Nathan P Wiederhold, Theoklis Zaoutis, Arunaloke Chakrabarti, for the Mucormycosis ECMM MSG Global Guideline Writing Group Mucormycosis is a difficult to diagnose rare disease with high morbidity and mortality. Diagnosis is often delayed, and disease tends to progress rapidly. Urgent surgical and medical intervention is lifesaving. Guidance on the complex multidisciplinary management has potential to improve prognosis, but approaches differ between health-care settings. From January, 2018, authors from 33 countries in all United Nations regions analysed the published evidence on mucormycosis management and provided consensus recommendations addressing differences between the regions of the world as part of the “One World One Guideline” initiative of the European Confederation of Medical Mycology (ECMM). Diagnostic management does not differ greatly between world regions. Upon suspicion of mucormycosis appropriate imaging is strongly recommended to document extent of disease and is followed by strongly recommended surgical intervention. First-line treatment with high-dose liposomal amphotericin B is strongly recommended, while intravenous isavuconazole and intravenous or delayed release tablet posaconazole are recommended with moderate strength. Both triazoles are strongly recommended salvage treatments. Amphotericin B deoxycholate is recommended against, because of substantial toxicity, but may be the only option in resource limited settings. Management of mucormycosis depends on recognising disease patterns and on early diagnosis. Limited availability of contemporary treatments burdens patients in low and middle income settings. Areas of uncertainty were identified and future research directions specified. Introduction Suspected mucormycosis requires urgent intervention, because of the often rapidly progressive and destructive nature of the infection.1,2 Delayed initiation of therapy is associated with increased mortality.1 Maximising survival rates requires rapid diagnostic and therapeutic inter vention, including immediate involvement of a multi disciplinary medical, surgical, radiological, and laboratory-based team.3 Readily available guidance is important to ensure efficient diagnosis and treatment, and to optimise patient prognosis. Optimal management depends on recognising disease patterns and the available diagnostic and therapeutic options, which differ between the regions of the world. Currently available guidelines are limited to specific patient groups in haematology,4 or a specific geographical region,5 or require an update.6–8 Recently, several critical developments have fundamentally changed the manage ment of this condition. These include the development of new and more widely used molecular techniques for the diagnosis of mucormycosis, the licensing of isavuconazole for treatment of mucormycosis, and the availability of new formulations of posaconazole. More over, previous guide lines did not include comprehensive clinical and radio logical imaging, pathological and histological findings, nor did they provide details on surgery as a core element of mucormycosis management. The European Confederation of Medical Mycology (ECMM), together with the Mycoses Study Group Education & Research Consortium (MSG ERC), issues this comprehensive guidance document to facilitate clinical decision-making, and simultaneously provides an overview of the areas of uncertainty in the field.9,10 We aimed to address limitations of previous recom mendations, by engaging physicians and scientists involved in various aspects of mucormycosis manage ment, representing the fields of microbiology, patho logy, radio logy, infectious diseases, surgery, paediatrics, haema tology, intensive care, dermatology, and pharm cology. In addi tion, the guideline group comprises experts from all parts of the world and provides manage ment pathways for different regional environ ments (panel; for further infor mation on guideline development, systematic Lancet Infect Dis 2019 Published Online November 4, 2019 S1473-3099(19)30312-3 Department I of Internal Medicine, University Hospital of Cologne, Cologne, Germany (O A Cornely MD, D Arenz PhD, J J Vehreschild MD, M J G T Vehreschild MD, S C Mellinghoff MD, D Seidel PhD); German Centre for Infection Research (DZIF) partner site Bonn-Cologne, Cologne, Germany (O A Cornely, J J Vehreschild, M J G T Vehreschild); CECAD Cluster of Excellence, University of Cologne, Cologne, Germany (O A Cornely, D Arenz, S C Mellinghoff, D Seidel); Clinical Trials Center Cologne, University Hospital of Cologne, Cologne, Germany (O A Cornely); Mycology Reference Laboratory, National Centre for Microbiology, Instituto de Salud Carlos III, Madrid, Spain (A Alastruey-Izquierdo PhD); Centre for Infectious Diseases and Microbiology Laboratory Services, New South Wales Health Pathology, and the Department of Infectious Diseases, Westmead Hospital, School of Medicine, University of Sydney, Sydney, NSW, Australia (S C-A Chen PhD); Université Paris-Descartes, Faculté de Médecine, APHP, Hôpital Européen Georges Pompidou, Unité de Parasitologie-Mycologie, Service de Microbiologie, Paris, France (E Dannaoui MD); 2 www.thelancet.com/infection Published online November 4, 2019 Review Radiology, Hospital São Lucas da Pontificia Universidade Catolica do Rio Grande do Sul (PUCRS), Escola de Medicina, Porto Alegre, Brazil (B Hochhegger MD); Radiology, Universidade Federal de Ciências da Saúde de Porto Alegre (UFCSPA), Porto Alegre, Brazil (B Hochhegger); Section of Infectious Diseases and Tropical Medicine and Division of Pulmonology, Medical University of Graz, Graz, Austria (M Hoenigl MD); Division of Infectious Diseases and Global Public Health, Department of Medicine, University of California San Diego, San Diego, USA (M Hoenigl); Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark (H E Jensen PhD); Department of Microbiology, Immunology and Transplantation, KU Leuven and Clinical Department of Laboratory Medicine and National Reference Center for Mycosis, University Hospitals Leuven, Leuven, Belgium (K Lagrou PharmD); Infectious Diseases Clinic, Sant’Orsola-Malpighi Hospital, Department of Medical and Surgical Sciences, University of Bologna, Bologna, Italy (R E Lewis PharmD); Divisions of Critical Care and Pulmonology, Department of Medicine, Charlotte Maxeke Johannesburg Academic Hospital and Faculty of Health Sciences University of the Witwatersrand, Johannesburg, South Africa (M Mer MD); Infectious Diseases Unit, 3rd Department of Paediatrics, Faculty of Medicine, Aristotle University School of Health Sciences, Thessaloniki, Greece; Hippokration General Hospital, Thessaloniki, Greece (Z D Pana MD; E Roilides MD); Division of Infectious Diseases, Department of Medicine, Microbiology and Immunology, McGill University, Montreal, Quebec, Canada (D C Sheppard MD); Department of General, Visceral and Cancer Surgery, University Hospital of Cologne, Cologne, Germany (R Wahba MD); Department of Medical Microbiology, Postgraduate Institute of Medical Education & Research, Chandigarh, India approach, authors and contributors, literature search terms and work flow, see appendix pp 1–4). Epidemiology of mucormycosis Patient populations As medical science advances, the patient populations most at risk for mucormycosis expand accordingly. In the mid-20th century, diabetes evolved as a major risk factor for mucormycosis, while in more recent years, underlying malignancy emerged as another important risk factor due to the increasing number of patients undergoing chemotherapy or cancer immuno therapy.11–13 Further more, with more solid organ and haematopoietic stem-cell transplantations (HSCT) being performed, increasing numbers of cases have also been reported in these patient groups.14 At the same time, diabetes continues to represent the predominant risk factor for mucormycosis in settings where health-care access for diabetes management is more limited.13 For further information on patient populations, incidence and prevalence of mucormycosis and incidence rates compared to other mould infections, see appendix pp 4–6. Pathogens causing mucormycosis The term mucormycosis is frequently used inter changeably with zygomycosis. The latter term referred to infections caused by fungi of the former phylum Zygomycota (comprising Mucorales, Entomophthorales, and others), which became obsolete with phylogenetic reanalysis of the kingdom Fungi.15,16 Today, mucor mycosis describes infections caused by fungi of the order Mucorales. The most frequently reported patho gens in mucormycosis are Rhizopus spp, Mucor spp, and Lichtheimia spp (formerly of the genera Absidia and Mycocladus), followed by Rhizomucor spp, Cunninghamella spp, Apophysomyces spp, and Saksenaea spp.11,17,18 Lichtheimia spp were identified as the major cause of mucormycosis in a single hospital in Spain, indicating geographical variation and the need to know local epidemiology.19 Clinical manifestations of mucormycosis For further information on clinical manifestations, see appendix p 6. In immunocompromised patients, the main route of infection seems to be through inhalation of sporan giospores causing pulmonary infection. Pulmonary mucor mycosis typically develops in patients with profound neutropenia11 and graft-versus-host disease,20 whereas diabetic patients typically present with rhino-orbital disease. Prolonged fever is seen in most patients, although some patients might be asymptomatic.21 Radiological findings often vary in configuration, size, number, and distribution of lesions; typical examples are given below.22–25 Pulmonary mucormycosis can spread Panel: How the guideline group worked In December, 2017, experts were identified based on their publication activity in the field of mucormycosis in the previous 5 years, their involvement in patient management, and their distribution over world regions defined by the United Nations. Experts were invited to develop this guideline in January, 2018. This guideline follows the structure and definitions of previous guidelines on invasive fungal infections which are in accordance with the Grading of Recommendations Assessment, Development and Evaluation (GRADE) and Appraisal of Guidelines for Research & Evaluation (AGREE) systems. The PICO (population, intervention, comparison, and outcome) approach is reflected by the tables. Both, diagnostic assays and treatment strategies might alter patient course, and are thus regarded as interventions. First, a population is defined; then the intention or objective is stated, followed by the intervention. For such logical sequence, strength of recommendation (SOR) and quality of evidence (QOE) are provided, followed by the references on which the recommendation is based. SOR and QOE are results of two independent evaluations, thus allowing a strong recommendation even in the absence of the highest quality evidence (table 1). Search strings used were “mucormyc OR zygomyc”, “cavernous sinus syndrome OR orbital apex syndrome AND etiology”, and for the epidemiological section “mucormyc OR zygomyc AND (case[Title/Abstract] OR patient[Title/ Abstract] OR report[Title/Abstract]) AND (“2013/01/01”[PDat] : “2017/12/31”[PDat])”. From January to February, 2018, video conferences on the methodology were held, and a video tutorial added in March, 2018. Assistance and supervision to the group were provided by the coordinators (OAC, AC). Documents were shared among the authors on a password-protected OneDrive (Microsoft Corp, Redmont WA, USA) repository, and were updated several times per day. Updates on PICO tables were written in red font; after spellcheck and formatting font colour was changed to blue for consideration by the group. Contents discussed and agreed on were changed to black font. Once all tables were finalised, a writing group (OAC, AAI, DA, SCAC, ED, BH, MH, HEJ, KL, REL, SCM, MMe, ZP, DS, DCS, RW, AC) contributed the first draft, which was circulated to all participants for approval in October, 2018. Recommendations were consensus-based. If no consensus was found, majority vote was used. In November, 2018, a 4-week public consultation phase ensued. Comments received were evaluated, and either dismissed or used to change the manuscript, resulting in a final author review in December, 2018. 51 scientific societies from 33 countries reviewed and endorsed the guidance document. www.thelancet.com/infection Published online November 4, 2019 3 Review (A Chakrabarti MD); Department of Infectious Diseases, Hacettepe University School of Medicine, Ankara, Turkey (M Akova MD); Institut Pasteur, National Reference Center for Invasive Mycoses and Antifungals, Department of Mycology, CNRS UMR2000, Parasitology-Mycology Laboratory, Lariboisière, Saint-Louis, Fernand Widal Hospitals, Assistance Publique-Hôpitaux de Paris (AP-HP), Université de Paris, Paris, France (A Alanio MD, S Bretagne MD); Westerdijk Fungal Biodiversity Institute, Utrecht, The Netherlands (A M S Al-Hatmi PhD); Centre of Expertise in Mycology RadboudUMC/Canisius Wilhelmina Hospital, Nijmegen, The Netherlands (A M S Al-Hatmi); Ministry of Health, Directorate General of Health Services, Ibri, Oman (A M S Al-Hatmi PhD); Department of Medical Microbiology, Hacettepe University School of Medicine, Sıhhiye Ankara, Turkey (S Arikan-Akdagli MD); Department of Medical Mycology/Invasive Fungi Research Center (IFRC), School of Medicine, Mazandaran University of Medical Sciences, Sari, Iran (H Badali PhD, S Seyedmousavi PhD); Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel (R Ben-Ami MD); Infectious Diseases Unit, Tel Aviv Medical Center, Tel- Aviv, Israel (R Ben-Ami); Dermatology Service & Mycology Department, Hospital General de México “Dr. Eduardo Liceaga”, Mexico City, Mexico (A Bonifaz MD); Infectious Diseases Unit, Istituto Giannina Gaslini Children’s Hospital, Genoa, Italy (E Castagnola MD); Department of Medicine, Faculty of Medicine Siriraj Hospital, Mahidol University, Bangkok, Thailand (M Chayakulkeeree MD); Special Mycology Laboratory, Division of Infectious Diseases, Department of Medicine, Universidade Federal de São Paulo (UNIFESP), São Paulo, Brazil (A L Colombo MD); Department of Epidemiology and Infectious Diseases, Hospital General Dr Manuel Gea González, Mexico City, Mexico (D E Corzo-Leon MD); Medical contiguously into other organs, for example through the diaphragm into the abdomen. Cutaneous and soft-tissue mucormycosis are the most common forms of mucormycosis in immunocompetent patients, primarily after skin disruption due to traumatic injury (eg from natural disasters, motor vehicle accidents, improvised explosive devices in theatres of war, or iatrogenic sources), surgery, or burns.26–31 Abscesses, skin swelling, necrosis, dry ulcers, and eschars are characteristic presentations (figure 1A and G).32–34 For further information on cutaneous and soft-tissue mucormycosis, see appendix p 6. Figure 1: Cutaneous and rhino-orbito-cerebral mucormycosis (A) Extensive primary cutaneous mucormycosis of the left leg due to Apophysomyces variabilis, after a car accident. (B) Erythematous skin, ptosis, palpebral oedema, limited ocular motility, and right maxillary pain, 6 days after symptom onset in uncontrolled diabetes. (C) Proptosis, palpebral erythema, and cavernous sinus syndrome, 7 days after symptom onset in uncontrolled diabetes. (D) Necrotic, purulent palatal ulcer and cavernous sinus syndrome, 8 days after symptom onset in uncontrolled diabetes. (E) Rhinocerebral mucormycosis in a female child, 2 years old with acute lymphoblastic leukaemia and lethal outcome. (F) 52-year-old man with persistent neutropenia post chemotherapy, sinusitis, and skin necrosis. (G) Black eschar as typical skin lesion in mucormycosis; one of several lesions on the right forehead, ear and cheek in a non-diabetic, haematopoietic stem cell transplant recipient with pansinusitis due to Lichtheimia corymbifera. Image A courtesy of Alexandro Bonifaz, images B–D courtesy of Dora E Corzo-León, images E and F courtesy of Valentina Arsic Arsenijevic, Belgrade, Serbia, and image G courtesy of University Hospital Cologne. We obtained written permission from patients or parents respectively to publish images, and from ethics committee as appropriate per local regulation. E A D B C F G Definition Grade A The guideline group strongly supports a recommendation for use Grade B The guideline group moderately supports a recommendation for use Grade C The guideline group marginally supports a recommendation for use Grade D The guideline group supports a recommendation against use Quality of evidence Definition Level I Evidence from at least 1 properly designed randomised, controlled trial (orientated on the primary endpoint of the trial); note: poor quality of planning, inconsistency of results, indirectness of evidence etc would lower the SOR Level II Evidence from at least one well designed clinical trial (including secondary endpoints), without randomisation; from cohort or case-controlled analytic studies (preferably from >1 centre); from multiple time series; or from dramatic results of uncontrolled experiments; note: every level II item of evidence must have at least one added index Level III Evidence from opinions of respected authorities, based on clinical experience, descriptive case studies, or reports of expert committees Added Index Defining the source of level II evidence r Meta-analysis or systematic review of randomised controlled trials t Transferred evidence—ie, results from different patient cohorts, or similar immune-status situation h Comparator group: historical control u Uncontrolled trials a For published abstract presented at an international symposium or meeting SOR=strength of recommendation. Table 1: Definition of strength of recommendation and quality of evidence by population type 4 www.thelancet.com/infection Published online November 4, 2019 Review Rhino-orbito-cerebral mucormycosis typically develops in patients with diabetes, whereas such patients very rarely develop lung infection.11 It has been described in haematology patients, too.35 Rhino-orbital-cerebral infection usually originates from the paranasal sinuses, with bone destruction and subsequent invasion of the orbit, eye, and brain.36–39 Unilateral facial oedema, proptosis, and palatal or palpebral fistula developing into necrosis may be present (figure 1B, F). For further information on rhino-orbito-cerebral mucor mycosis see appendix p 6. Primary gastrointestinal disease is a rare manifestation of mucormycosis that can present with symptoms similar to other common gastrointestinal diseases.40,41 However, gastrointestinal mucormycosis is the most common manifestation of mucormycosis in neonates, where it carries a high mortality.42 Figure 2: Diagnostic pathway for mucormycosis Depending on the geographical location not all recommended tests might have regulatory approval for use in clinical settings. HSCT=haematopoietic stem cell translplantation. SOT=solid organ transplantation. PAS=periodic acid Schiff. GMS=Grocott-Gomori’s methenamine-silver strain. qPCR=quantitative PCR. HRM=high resolution melting. ITS=internal transcribed spacer. rDNA=ribosomal DNA. Neutropenic, HSCT or SOT Persistent fever or respiratory symptoms Suspected and conﬁrmed mucormycosis are emergencies and require rapid action Diabetic Facial pain, sinusitis, proptosis, amaurosis Trauma Persistent fever or respiratory symptoms Trauma Eschar Asia, speciﬁcally China and India No underlying diseases: ﬂank pain, fever, haematuria with sterile urine, and blood cultures Asia, speciﬁcally India Diabetic adult on dialysis or malnourished/premature child with broad-spectrum antibiotic therapy with abdominal mass, distension or bilious vomting, with or without gastrointestinal bleeding Chest CT Typical ﬁndings: Reverse Halo >10 nodules pleural eﬀusion Staging CT sinuses, chest, abdomen, pelvis Cranial CT Typical ﬁnding: Bone destruction Cranial MRI Typical ﬁnding: orbit, brain involvement Sampling Biopsy (eg, endoscopic, or CT-guided) Grinding samples causes non-viability Serology Galactomannan, 1,3-β-D-glucan Positive assays support diﬀerential diagnosis or mixed invasive fungal infection Culture Routine media at 30°C and 37°C Typical ﬁndings: cottony white or greyish black colony Susceptibility testing Direct microscopy using ﬂourescent brightener and histopathology with special stains (eg PAS and GMS) Typical ﬁndings: non-septate/ pauci-septate, ribbon-like hyphae (at least 6–16 μm wide) Vessel occlusion Chest CT If persistent fever or respiratory symptoms Molecular identiﬁcation Preferably semi-nested qPCR, HRM, Multiplex Target: 18s, ITS, 28s or rDNA Immunohistochemical staining with speciﬁc primary reagents Imaging of injured regions as appropriate Abdominal CT or MRI Typical ﬁnding: isolated abdominal mass Ultrasound/abdominal CT Typical ﬁndings: enlarged, non-hydronephrotic kidneys with hypodensities, cortical rim sign Strongly recommended Moderately recommended Mycology and Fungal Immunology/Wellcome Trust Strategic Award Program, Aberdeen Fungal Group, University of Aberdeen, King’s College, Aberdeen, UK (D E Corzo-Leon MD); Oncohematology Clinic, Faculty of Medicine, Comenius University and National Cancer www.thelancet.com/infection Published online November 4, 2019 5 Review For further information on gastrointestinal mucor mycosis, see appendix p 6. Cases of isolated renal mucormycosis in immuno competent hosts are extremely rare, but have been reported from China and India.43–48 For further information on renal and abdominal mucormycosis, see appendix p 7 . Mortality All-cause mortality rates for mucormycosis range from 40% to 80% with varying rates depending on underlying conditions and sites of infection.11,19,49–51 The highest survival rates are reported in patients with a healthy immune status and those without comorbidities. The poorest prognosis is observed in patients with haematological malignancies and HSCT recipients11 and in patients with extensive burns.51 Disseminated disease, especially to the CNS is often associated with mortality rates higher than 80%.11 Conversely, lower mortality is seen with localised sinus or skin infection, where earlier tissue-based diagnosis is often feasible and surgical debridement may result in cure. Mortality is also high in neonates and other Figure 3: Radiographic signs of mucormycosis Four imaging signs can suggest pulmonary mucormycosis in an appropriate clinical setting. (A) Halo sign on CT, a ring of ground glass opacity surrounding a nodular infiltrate, which pathophysiologically represents a region of ischaemia, and which is also typical of invasive pulmonary aspergillosis (arrow). (D & B) Reversed halo sign on CT, also known as inversed halo or atoll sign, an area of ground glass opacity surrounded by a ring of consolidation (arrow). (E) Hypodense sign on MRI, T1 weighted, a central hypodensity in a lung consolidation or nodule, corresponding to a central area of necrosis caused by vascular obstruction with secondary lung infarction and sequestration. Magnetic resonance imaging shows pulmonary nodule with central hypodensity in right upper lobe (arrow), corresponding to a central area of necrosis caused by vascular obstruction with secondary lung infarct and sequestration. (C) Vascular occlusion sign on CT angiography, defined as interrupted vessel at the border of a focal lesion without depiction of the vessel inside the lesion or peripheral to the lesion (arrow). Particularly aggressive forms of mucormycosis are F. Contiguous spread on CT, presence of a mass or consolidation exhibiting invasion of adjacent organs by traversing tissue planes, including the diaphragm, chest wall, pleura, and spleen. (G) Typical rapidly progressive pulmonary mucormycosis on CT, associated with clinical deterioration. Day 8 and Day 15 CT scans showing a reversed halo sign. Images A, C, D, and E courtesy of Bruno Hochhegger, images B, F, and G courtesy of University Hospital Cologne. D A E C B F G Day 1 Day 8 Day 15 Institute, Bratislava, Slovakia (L Drgona MD); InfectiousDisease Research Program, Department of Paediatric Hematology/ Oncology and Center for Bone Marrow Transplantation, University Children’s Hospital Münster, Münster, Germany (A H Groll MD); Clinical Microbiology and Infectious Diseases, Hospital General Universitario Gregorio Marañón, Madrid, Spain (J Guinea PharmD); Instituto de Investigación v Sanitaria Gregorio Marañón, Madrid, Spain (J Guinea); Medicine Department, School of Medicine, Universidad Complutense de Madrid, Madrid, Spain (J Guinea PharmD); Diagnostic and Interventional Radiology, Thoracic Clinic, University Hospital Heidelberg, Heidelberg, Germany (C-P Heussel MD); Division of Infectious Diseases, Los Angeles Biomedical Research Institute at Harbor-University of California at Los Angeles (UCLA) Medical Center, Torrance, CA, USA (A S Ibrahim PhD); Department of Internal Medicine, Division of Infectious Diseases, American University of Beirut Medical Center, Beirut, Lebanon (S S Kanj MD); Department of Clinical Mycology, Allergology and Immunology, North Western State Medical University, St Petersburg, Russia (N Klimko MD); Division of Hygiene and Medical Microbiology, Department of Hygiene, Microbiology and Public Health, Medical University Innsbruck, Innsbruck, Austria (M Lackner MD, C Lass-Floerl MD); Infectious Diseases Service, Department of Medicine and Institute of Microbiology, Lausanne University Hospital, Lausanne, Switzerland (F Lamoth MD); Institute of Microbiology, Department of Laboratories, Lausanne University Hospital, Lausanne, Switzerland (F Lamoth); Institut Pasteur, National Reference Center for Invasive Mycoses and Antifungals, Department of Mycology, Paris Descartes University, Necker-Enfants Malades University Hospital, Department of Infectious Diseases and Tropical Medicine, Centre d’Infectiologie 6 www.thelancet.com/infection Published online November 4, 2019 Review immunocompromised patients with gastrointestinal mucor mycosis, possibly related to delay in diagnosis and polymicrobial sepsis. Generally, improved survival is related to earlier diagnosis and application of early, multidisciplinary treatment approaches involving aggre ssive surgical de bridement.19,52–54 Despite improved under standing of the disease and the availability of more therapeutic options, survival rates in mucormycosis remain poor.19,55,56 Diagnosis The capability of diagnosing mucormycosis depends on the availability of imaging techniques, trained personnel, and mycological and histological investi gations. Patients with suspected mucormycosis should be referred immediately to a facility with the highest care level. In case of any delay, manage ment should be initiated following this guidance document. If all diagnostic options are available, one should follow the management pathway depicted in figure 2. For further information on diagnosing mucormycosis, see appendix p 7 . Imaging Radiographical signs suggestive of pulmonary mucor mycosis are shown in figure 3. For further information on imaging see appendix p 7 . Recommendations In patients with haematological malignancy and sus pected pulmonary mucormycosis, pulmonary CT scan is recommended for the detection of the reversed halo sign, an area of ground glass opacity surrounded by a ring of consolidation on thoracic CT, or vessel occlusion on CT pulmonary angiography. In diabetic patients with facial pain, sinusitis, proptosis, ophthalmoplegia, or newly diagnosed amaurosis, or both, cranial CT or MRI is strongly recommended to determine if sinusitis is present. If sinusitis is diagnosed, endoscopy is strongly recommended to diagnose mucormycosis. If disease of the eye or brain is suspected, MRI should be conducted in lieu of a CT scan due to substantially greater sensitivity. If mucormycosis is a potential diagnosis, biopsy is strongly recommended. Once mucormycosis has been proven in a patient with underlying malignancy, cranial, thoracic and abdominal imaging studies to determine the extent of disease are recommended with moderate strength. In view of the rapid progress of mucormycosis, weekly CT scans are strongly recommended, particularly in unstable patients (appendix p 7). Histopathology in mucormycosis Evidence Mucormycosis is usually suspected based on results of direct microscopy of clinical specimens, preferably stained with fluorescent brighteners calcofluor white (Sigma Aldrich, St Louis, MO, USA) or blankophor (Tanatax Chemicals, Ede, The Netherlands). To confirm an infection, non-pigmented hyphae showing tissue invasion must be shown in tissue sections stained with haematoxylin-eosin (HE), periodic acid-Schiff stain (PAS), or Grocott-Gomori’s methenamine-silver Figure 4: Hyphal morphology in mucormycosis and aspergillosis (A) Typical hyphal morphology in mucormycosis lesions (GMS, × 200). Mucorales hyphae are at least 6–16 µm wide, ribbon-like, pauci-septate, and branch irregularly. (B) Hyphal structure covered with Splendore-Hoeppli phenomenon (HE, × 1000). The eosinophilic material likely represents antigen-antibody complexes. First described by Splendore in 1908, and by Hoeppli in 1932. (C) Typical hyphal morphology in aspergillosis lesions (PAS, x 200). Aspergillus hyphae are 3–5 µm wide, regularly septated, with dichotomous branching. (D–F) Sizes and branching angles for Mucorales and aspergillus stained by calcofluor-white. D and F correspond to Rhizopus arrhizus and E to Aspergillus fumigatus. Measurements correspond to the size of the white lines; hyphal diameter were performed with the Leica software LAS-AF and are expressed in µm. Diagnosis needs to be confirmed by culture, molecular techniques, or both. Images A–C courtesy of Henrik E Jensen and images D–F courtesy of Ana Alastruey-Izquierdo. A D B E C F GMS, x200 HE, x1000 7·721 µm 2·020 µm 24·668 µm 10 μm PAS, x200 10 μm 10 μm Necker-Pasteur, Institut Imagine, AP-HP, Paris, France (Lanternier MD); Division of Infectious Diseases, Department of Internal Medicine, Catholic Hematology Hospital, College of Medicine, The Catholic University of Korea, Seocho-gu, Seoul, Korea (D-G Lee MD); Division of Paediatric Haematology and Oncology, Hospital for Children and Adolescents, Johann Wolfgang Goethe-University, Frankfurt, Germany (T Lehrnbecher MD); School of Medicine and Pharmacy, University Mohammed the fifth, Hay Riad, Rabat, Morocco (B E Lmimouni MD); Laboratory of Antimicrobial Chemotherapy, Ion Ionescu de la Brad University, Iași, Romania (M Mares PhD); Department of Hematology, Oncology and Palliative Care, Klinikum Ernst von Bergmann, Potsdam, Germany (G Maschmeyer MD); Department of Medical Microbiology and Infectious Diseases, Centre of Expertise in Mycology Radboudumc/ Canisius Wilhelmina Hospital, Nijmegen, Netherlands (J F Meis MD); Clinical Microbiology Laboratory, Attikon University Hospital, www.thelancet.com/infection Published online November 4, 2019 7 Review stain (GMS), or both.57,58 Histopathologically, Mucorales hyphae have a variable width of 6–16 µm, but may be up to 25 µm, and are non-septate or pauci-septate. In tissue, the hyphae appear ribbon-like with an irregular pattern of branching (figure 4A–C).57 Hyphae can artefactually seem to have septae because tissue can fold over itself during processing, which can create artificial lines that can be confused with septations. Similarly, the historically described 90° branching angle of Mucorales in tissue, versus 45° branching angle of septate moulds, can be difficult to identify in tissue due to interstitial pressures exerted on the fungi by the tissue and alterations in tissue architecture during processing. Thus the wider and irregular (ribbon-like) nature of the hyphae are more reliable distinguishing characteristics than septations and angle of branching. The lesions of mucormycosis are characteristic but non-specific.59–61 In acute lesions, haemorrhagic infarction, coagulation necrosis, angioinvasion, infiltration by neutrophils (in non-neutropenic hosts), and perineural invasion are characteristic features;62 whereas, in chronic lesions, a pyogranulomatous inflammation with presence of giant cells, and sometimes hyphae covered by the Splendore-Hoeppli phenomenon,63,64 which describes deeply eosinophilic material surrounding the pathogen, are seen (figure A–C).17,62,65–67 Obtaining a diagnosis of mucormycosis on histo morphological basis is challenging, and the most common cause for incorrect morphological diagnosis is the mis identification of Mucorales as Aspergillus spp (figure A–C).58 The application of immunohistochemistry with commer cially available monoclonal antibodies68–70 National and Kapodistrian University of Athens, Athens, Greece (J Meletiadis PhD); Department of Medical Microbiology and Infectious Diseases, Erasmus Medical Center, Rotterdam, The Netherlands (J Meletiadis PhD); Department of Infectious Diseases, Alfred Health & Monash University, Melbourne, Australia (C O Morrissey PhD); Department of Internal Medicine, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil (M Nucci MD); Department of Medical Microbiology & Parasitology, College of Medicine, University (Figure 5 continues on next page) Strongly recommended A Moderately recommended Marginally recommended Recommended against Suspected and conﬁrmed mucormycosis are emergencies and require rapid action If brain involvement If SOT Avoid slow escalation of doses Avoid slow escalation of doses Avoid slow escalation of doses Liposomal amphotericin B 5–10 mg/kg per day from day 1 Response assesment (eg weekly imaging) Liposomal amphotericin B 10 mg/kg per day from day 1 Posaconazole oral suspension 4 × 200 mg per day Isavuconazoayle IV 3 × 200 mg day 1–2 1 × 200 mg per day from day 3 Posaconazole IV 2 × 300 mg day 1 1 × 300 mg per day from day 2 Liposomal amphotericin B or amphotericin B lipid complex 10 mg/kg per day from day 1 If preexisting renal compromise Isavuconazoayle IV 3 × 200 mg day 1–2 1 × 200 mg per day from day 3 Posaconazole IV 2 × 300 mg day 1 1 × 300 mg per day from day 2 Continuation of 1st line treatment or change to oral treatment Isavuconazoayle PO 3 × 200 mg day 1–2 1 × 200 mg per day from day 3 or Posaconazole DR tablets 2 × 300 mg day 1 1 × 300 mg per day from day 2 Liposomal amphotericin B 10 mg /kg per day from day 1 Amphotericin B lipid complex or Liposomal amphotericin B 5 mg /kg per day from day 1 Combination with pozaconazole Isavuconazoayle IV 3 × 200 mg day 1–2 1 × 200 mg/day 2 from day 3 or Posaconazole IV or DR tablets 2 × 300 mg day 1 1 × 300 mg per day from day 2 Posaconazole oral suspension 4 × 200 mg per day Isavuconazoayle IV 3 × 200 mg day 1–2 1 × 200 mg/day 2 from day 3 or Posaconazole IV or DR tablets 2 × 300 mg day 1 1 × 300 mg per day from day 2 Posaconazole oral suspension 4 × 200 mg per day Liposomal amphotericin B <5 mg/kg per day Progressive disease Stable disease or partial response Toxicity amphotericin B lipid complex or liposomal amphotericin B 5 mg /kg per day from day 1 Avoid amphotericin B deoxycholate Any dose Surgical debridement with clean margins for 3 purposes: (1) disease control, (2) histopathology, (3) microbiological diagnostics Plus Immediate treatment initiation 8 www.thelancet.com/infection Published online November 4, 2019 Review or PCR techniques on either fresh or formalin-fixed paraffin-embedded tissue19,71–95 have been shown to be highly specific, although a variation in sensitivity has been reported, in addition, these tests might not be widely available (appendix p 9). Recommendations Hyphae of Mucorales can be distinguished from septate hyaline moulds due to their greater width and irregular pattern of branching. However, there are no data available to describe the accuracy of distinguishing Mucorales from other moulds based on these charac teristics. Therefore, it is strongly recommended to confirm the diagnosis of mucormycosis in tissue by culture or by application of molecular or in-situ identification techniques, at centres where such assays are available (appendix p 9). For further information on antigen biomarkers, see appendix p 10. Culture and microscopy Recommendations Culture of specimens is strongly recommended for genus and species identification, and for antifungal susceptibility testing. Homogenisation of tissue should be avoided before culturing. Incubation at 30°C and 37°C separately is strongly recommended (appendix p 11). Direct microscopy with fluorescent brighteners from clinical specimens is strongly recommended mainly focusing on septation, branching angle, and hyphal width. (Figure 5 continues on next page) Strongly recommended B Moderately recommended Marginally recommended Recommended against Suspected and conﬁrmed mucormycosis are emergencies and require rapid action Surgical debridement with clean margins for 3 purposes: (1) disease control, (2) histopathology, (3) microbiological diagnostics Plus Immediate treatment initiation If pre-existing renal compromise Isavuconazole IV 3 × 200 mg day 1–2, 1 × 200 mg per day from day 3 Posaconazole IV 2 × 300 mg day 1, 1 × 300 mg from day 2 Response assesment (eg weekly imaging) Isavuconazole IV 3 × 200 mg day 1–2, 1 × 200 mg per day from day 3 Posaconazole IV 2 × 300 mg day 1, 1 × 300 mg from day 2 Posaconazole oral suspension 4 × 200 mg per day Continuation of ﬁrst-line treatment or change to oral treatment Isavuconazole PO 3 × 200 mg day 1–2, 1 × 200 mg per day from day 3 or Posaconazole DR tablets 2 × 300 mg day 1, 1 × 300 mg per day from day 2 Isavuconazole IV or PO 3 × 200 mg day 1–2, 1 × 200 mg per day from day 3 or Posaconazole IV or DR tablets 2 × 300 mg day 1 1 × 300 mg per day from day 2 Posaconazole oral suspension 4 × 200 mg per day Isavuconazole IV or PO 3 × 200 mg day 1–2, 1 × 200 mg per day from day 3 or Posaconazole IV or DR tablets 2 × 300 mg day 1 1 × 300 mg per day from day 2 Posaconazole oral suspension 4 × 200 mg per day Progressive disease Stable disease or partial response Toxicity Avoid Amphotericin B deoxycholate Any dose of Lagos, Yaba, Lagos, Nigeria (R Oladele MD); Faculty of Biology, Medicine and Health, The University of Manchester, Manchester, UK (R Oladele MD); Department of Hematology, Fondazione Policlinico Universitario A. Gemelli –IRCCS– Universita Cattolica del Sacro Cuore, Roma, Italy (L Pagano MD); Federal University of Health Sciences of Porto Alegre, Hospital Dom Vicente Scherer, Porto Alegre, Brazil (A Pasqualotto MD); Infectious Diseases Clinic, Vedanta Institute of Medical Sciences, www.thelancet.com/infection Published online November 4, 2019 9 Review For further information on culture and microscopy, see appendix p 10. Susceptibility testing For further information on susceptibility testing, see appendix p 11–12. Recommendations The use of standard methods for antifungal sus ceptibility testing to guide antifungal treatment in Mucorales is marginally supported and may be clinically useful in cases of treatment failure. However, we strongly recommend the use of these methods primarily to establish epidemiological knowledge in the field. Currently, commercial methods such as E-test are recommended for use in mucormycosis with marginal strength only (appendix p 11). Molecular-based methods for direct detection For further information on molecular-based methods, see appendix p 13. Currently, in the absence of a standardised test, the use of molecular methods on both fresh clinical material and paraffin sections for the diagnosis of mucormycosis is moderately supported. Fresh material is preferred over paraffin-embedded tissue because formalin damages DNA. Figure 5: Optimal treatment pathways for mucormycosis in adults Depending on the geographical location not all recommended treatments may have regulatory approval for use in clinical settings. (A) When all treatment modalities and antifungal drugs are available, (B) when amphotericin B lipid formulations are not available, and (C) when isavuconazole and posaconazole IV and delayed release tablets are not available. IV=intravenous. PO=per os (taken orally). SOT=solid organ transplantation. DR=delayed release. Strongly recommended C Moderately recommended Marginally recommended Recommended against Suspected and conﬁrmed mucormycosis are emergencies and require rapid action If brain involvement If SOT Avoid slow escalation of doses Avoid slow escalation of doses Avoid slow escalation of doses Liposomal amphotericin B 5–10 mg/kg per day from day 1 Response assesment (eg weekly imaging) Liposomal amphotericin B 10 mg/kg per day from day 1 Posaconazole oral suspension 4 × 200 mg per day Liposomal amphotericin B or amphotericin B lipid complex 10 mg/kg per day from day 1 Posaconazole oral suspension 4 × 200 mg per day Liposomal amphotericin B 10 mg/kg per day from day 1 Amphotericin B lipid Complex or liposomal amphotericin B 5 mg /kg per day from day 1 Combination with pozaconazole Liposomal Amphotericin B <5 mg/kg per day Posaconazole oral suspension 4 × 200 mg per day Progressive disease Toxicity Amphotericin B lipid complex or liposomal Amphotericin B 5 mg /kg per day from day 1 Avoid Amphotericin B deoxycholate Any dose Surgical debridement with clean margins for 3 purposes: (1) disease control, (2) histopathology, (3) microbiological diagnostics Plus Immediate treatment initiation Navarangpura, Ahmeddabad, India (A Patel MD); Institute of Hematology and Blood Transfusion, Prague, Czech Republic (Z Racil MD); UK NHS Mycology Reference Centre, Manchester University NHS Foundation Trust, Manchester, UK (M Richardson PhD); Hämatologie & Internistische Onkologie, Lukas-Krankenhaus Bünde, Onkologische Ambulanz, Bünde, Germany (M Ruhnke MD); Center of Expertise in Microbiology, 10 www.thelancet.com/infection Published online November 4, 2019 Review Detection of DNA in serum as well as in other body fluids is very promising but because of lack of standardisation supported with moderate strength only (appendix p 13). Genus and species identification Evidence Although some genera, such as Cunninghamella, can be associated with an increased mortality rate in patients11,96 and have been shown to be more virulent in experimental models,97 there is currently sparse evidence that identification of the causative Mucorales to the genus or species level, or both, could guide the choice of the antifungal treatment. By contrast, identification to the species level is of importance for improved epidemiological knowledge of the disease. In particular, the clinical picture can be different depending on the species.11,96,98,99 Moreover, species identification is valuable for investigation of health care-associated mucormycosis and outbreaks.100–103 For further information on genus and species iden tification, see appendix p 14–15. Recommendations Identification to the genus and species level is strongly supported for improved epidemiological understanding of mucormycosis. Guiding treatment by identifi cation to the genus level is supported with marginal strength. Molecular identification is strongly supported and preferred over morphology. Because the best tech nique for molecular identification, internal transcribed spacer (ITS) sequencing is strongly supported. Matrix assisted laser desorption ionisation time of flight (MALDI-TOF) identification is moderately supported because it relies mainly on in-house databases, and many laboratories do not have that capacity (appendix p 15). Treatment approaches to mucormycosis The ability to treat mucormycosis effectively depends on the availability of the surgical techniques and antifungal drugs discussed below. If all treatment options are available one should follow the management pathways detailed in figure 5A and appendix p 25. If local or regional capabilities differ, less comprehensive pathways need to be followed; examples are given in figure 5B, C, and appendix p 26. Surgical treatment for mucormycosis For further information on surgical treatment, see appendix p 16. Recommendations—The guideline group strongly supports an early complete surgical treatment for mucormycosis when ever possible, in addition to systemic antifungal treat ment. Resection or debridement should be repeated as required (appendix p 16). Drug treatment for mucormycosis Prophylaxis For further information on prophylaxis, see appendix p 18. Recommendations —In neutropenic patients or those with graft versus host disease, primary prophylaxis with posaconazole delayed release tablets is recommended with moderate strength, and prophylaxis with oral suspension is recommended with marginal strength to prevent mucormycosis. Secondary prophylaxis For further information on secondary prophylaxis, see appendix p 18. Recommendations —In immunosuppressed patients with previous diagnosis of mucormycosis, surgical resection and continuation or restart of the last drug effective in that patient is strongly recommended. Fever-driven treatment For further information on fever-driven treatment, see appendix p 19. Recommendations—The guideline group recommends against initiation of treatment for mucormycosis when fever of unknown origin is the sole evidence of infection. Diagnosis-driven treatment For further information on fever-driven treatment, see appendix p 19. Recommendations—In any immunocompromised patient with suspected mucormycosis, immediate treatment initiation is strong ly ecommended. Every attempt to attain a diagnosis should be made at the time of initiation of therapy, but should not delay therapy. First-line antifungal monotherapy Evidence—In several case series, the use of liposomal ampho tericin B successfully treated mucormycosis with various organ involvement patterns.17,50,67,104–109 Daily doses ranged from 1 mg/kg per day to 10 mg/kg per day.104,110 Recipients of increased doses tended to have increased response rates.104 Patients receiving 10 mg/kg per day had substantial serum creatinine increases that were mostly reversible.104,106 Doses higher than 10 mg/kg per day did not result in higher blood concentrations.111 In CNS involvement, animal models and the above observations support use of liposomal amphotericin B at 10 mg/kg per day.112 In the absence of CNS involvement, amphotericin B lipid complex 5 mg/kg per day has been used success fully.17,112,113 In kidney transplant recipients, amphotericin B lipid complex 10 mg/kg per day has been given.114 Amphotericin B deoxycholate has been the drug of choice for decades.11,17,66,109 It is effective, but its use is limited by its substantial toxicity, specifically in the doses and treatment durations needed for mucormycosis (table 2).115,116 Use of amphotericin B deoxycholate should be restricted to settings in which there is no other antifungal therapy available. Infection Biology and Antimicrobial Pharmacology, Tehran, Iran (S Seyedmousavi PhD); Molecular Microbiology Section, Laboratory of Clinical Immunology and Microbiology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD, USA (S Seyedmousavi); Department of Hemato Oncology, Amrita Institute of Medical Sciences, Amrita Viswa Vidyapeetham University, Kochi, India (N Sidharthan MD); Division of Infectious Diseases, University of Pittsburgh Medical Center and VA Pittsburgh Healthcare System, Infectious Diseases Section, University of Pittsburgh, Pittsburgh, PA, USA (N Singh MD); Infectious Diseases Unit, Szent Istvan and Szent Laszlo Hospital, Budapest, Hungary (J Sinko MD); Department of Infectious Diseases, Laiko General Hospital, National and Kapodistrian University of Athens, Athens, Greece (A Skiada MD); University of Melbourne, Melbourne, VIC, Australia (M Slavin MD); The National Centre for Infections in Cancer, Peter MacCallum Cancer Centre, Parkville, Melbourne, VIC, Australia (M Slavin); P D Hinduja Hospital & Medical Research Centre, Department of Medicine, Veer Sarvarkar Marg, Mumbai, India (R Soman MD); Los Angeles County and University of Southern California (LAC+USC) Medical Center, Los Angeles, CA, USA (B Spellberg MD); Division of Pediatric Infectious Diseases, Department of Pediatrics, Duke University Medical Center, Durham, NC, USA (W Steinbach MD); Department of Infectious Diseases, Singapore General Hospital, Singapur, Singapore (B H Tan MD); Department for Internal Medicine II, University Hospital Würzburg, Würzburg, Germany (A J Ullmann MD); Department of Internal Medicine, Hematology/ Oncology, Goethe University Frankfurt, Frankfurt, Germany (J J Vehreschild); Department of Internal Medicine, Infectious Diseases, Goethe University Frankfurt, Frankfurt, Germany (M J G T Vehreschild); Departments of Medicine, Pediatrics, Microbiology & Immunology, Weill Cornell www.thelancet.com/infection Published online November 4, 2019 11 Review The efficacy of isavuconazole was similar to an external matched control group treated with amphotericin B formulations. This limited size study enrolled 21 patients with isavuconazole first-line treatment, and compared efficacy results to 33 matched patients from the FungiScope registry.49,117 As a result, isavuconazole has been licenced in the USA for first-line treatment of mucormycosis.118 By contrast with other mould-active azoles, isavuconazole is less hepatotoxic although it can result in shortening the QTc interval.119–121 Posaconazole oral suspension has been used successfully in first-line treatment.17,67 Recently, concerns about its oral bio availability led to the development of a delayed release tablet with improved exposure122,123 and an intravenous infusion formulation (table 2).124,125 Recommendations—First-line treatment with liposomal amphotericin B 5–10 mg/kg per day is strongly supported across all patterns of organ involvement. If substantial renal toxicity develops, the dose can be reduced as necessary, but doses below 5 mg/kg per day are recommended with marginal strength only.104,110 Doses should not be slowly increased over several days; rather, the full daily dose should be given from the first treatment day. Amphotericin B lipid complex 5 mg/kg per day is recommended with moderate strength for patients without CNS involvement. Use of amphotericin B deoxycholate is discouraged whenever alternatives are available. Isavuconazole is recommended with moderate strength for the first-line treatment of mucormycosis. The group marginally supports use of posaconazole oral suspension, and moderately supports posaconazole delayed release tablets and infusion for first-line treatment (table 2). First-line antifungal combination therapy Evidence—In animal models, some antifungal combinations have shown the potential to improve cure and survival rates with no antagonism noted.126,127 Results from some patient series are promising.128–130 However, a historical control study55 and a propensity score analysis failed to show benefits of double and triple antifungal combinations in patients with haematological malignancy.108 In trauma patients, specifically in blast injury, more than one mould species can cause mixed See Online for appendix For the video tutorials see watch?v=ZrSd1_rSr0o Intention Intervention SOR QOE Reference Any To cure and to increase survival rates Amphotericin B, any formulation, escalation to full dose over days D IIu Chamilos1 (N=70, give full daily dose from day 1) Any To cure and to increase survival rates Amphotericin B, liposomal, 5–10 mg/kg per day A IIu Gleissner144 (N=16, haematology); Pagano109 (N=5); Cornely106 (N=4); Pagano105 (N=44); Rüping67 (N=21); Shoham50 (N=28); Skiada17 (N=130); Lanternier104 (N=34, 18 haematology, six diabetic); Kyvernitakis108 (N=41); Stanzani107 (N=97, increased renal toxicity with cyclosporine) CNS involvement To cure Amphotericin B, liposomal, 10 mg/kg per day, initial 28 days A III Ibrahim112 (Animal); Lanternier104 (N=9) SOT adults To cure Amphotericin B, lipid formulation; dose not given A IIh Singh145 (N=25); Sun146 (N=14); Lanternier147 (N=3) SOT adults To cure Amphotericin B, lipid complex; 10 mg/kg per day A III Forrest114 (N=6, 3 of 6 died) Any, without CNS involvement To cure Amphotericin B, lipid complex; 5 mg/kg per day B IIu Larkin113 (N=10); Ibrahim112 (animal); Skiada17 (N=7) Haematological malignancy To cure Amphotericin B, liposomal; 1–<5 mg/kg per day ± surgery C III Nosari110 (N=13, 8 of 13 treated, 5/8 died); Li148 (N=7, 2 of 7 died) Any To cure Isavuconazole PO or IV; 3 × 200 mg day 1–2, 1 × 200 mg/d from day 3 B IIh Marty49 (N=21, 11 haematology, 4 diabetes, overall mortality comparable to amphotericin B formulations) Any To cure Posaconazole DR tablet or intravenously 2 × 300 mg day 1, 1 × 300 mg from day 2 B IItu Duarte;122 Maertens;124 Cornely;123 Cornely125 (higher trough levels than oral suspension, intravenous bridging when oral dosing not feasible) Any To cure Posaconazole oral suspension; 4 × 200 mg/day or 2 × 400 mg/day C IIu Rüping67 (N=8); Skiada17 (N=17); Dannaoui149 (animal, emphasises preference of amphotericin B, liposomal) Any To cure Amphotericin B, deoxycholate, any dose (if alternative therapy available) D IIt Walsh116 (renal toxicity); Pagano109 (N=9); Roden11 (N=532); Ullmann115 (renal toxicity); Chakrabarti66 (N=10); Skiada117 (N=21) Orbital mucormycosis To cure Retrobulbar injection of amphotericin B deoxycholate in addition to systemic therapy D III Hirabayashi50 (N=1, post-injection inflammatory response, risk for acute compartment syndrome) IV=intravenous. PO=per os (taken orally). SOR=strength of recommendation. QOE=quality of evidence. N=number of individuals. SOT=solid organ transplantation. DR=delayed release. Table 2: Recommendations on first-line antifungal monotherapy for mucormycosis by population type Medicine, and New York Presbyterian Hospital, New York City, NY, USA (T J Walsh MD); Public Health Wales Microbiology Cardiff, UHW, Heath Park, Cardiff, UK (P L White PhD); Fungus Testing Laboratory, University of Texas Health Science Center, San Antonio, TX, USA (N P Wiederhold PharmD); and Division of Infectious Diseases, The Children’s Hospital of Philadelphia, Philadelphia, PA, USA (T Zaoutis MD) Corrrespondence to: Oliver A Cornely MD, Department I of Internal Medicine, University Hospital Cologne, 50937 Cologne, Germany oliver.cornely@uk-koeln.de 12 www.thelancet.com/infection Published online November 4, 2019 Review infection warranting empirical combination therapy with liposomal amphotericin B and either posaconazole or voriconazole.29,131 The downsides of combination therapy are unclear aside from potential added toxicity, drug interactions, and cost. Recommendations—There are no definitive data to guide the use of antifungal combination therapy. Limited data support combinations of polyenes and azoles or polyenes plus echinocandins. Combination therapy can be rationally given due to lack of enhanced toxicity with possible but unproven benefit; however, data are too limited to support this beyond a marginal recommendation. For further information on first-line combination therapy, see appendix p 19. Antifungal salvage treatment Evidence—In general, there are two drug-related reasons for treatment failures, refractory mucormycosis or toxicity of first-line regimens—ie, intolerance to a drug. For amphotericin B formulations, particularly renal toxicity can be a limiting factor, while for the azole class hepatic toxicity has the highest prevalence. Toxicity can be caused by previous antifungals, or expected due to pre-existing organ damage. Only two drug classes have proven efficacy in mucor mycosis, thus salvage treatment mostly means switching to the other class. Isavuconazole salvage treatment was successful in both clinical scenarios, refractory disease, and intolerance or toxicity.49,132 In Europe, isavuconazole is licenced for salvage treatment of mucormycosis only. Posaconazole treatment with oral suspension achieved cure in two non-randomised clinical trials133,134 and in case series.17,135 Liposomal amphotericin B was effective as salvage treatment,109 as was amphotericin B lipid complex,113,136 and amphotericin B colloidal dispersion.137 Recommendations—Isavuconazole is strongly supported as salvage treatment. Posaconazole delayed release tablets or infusions are strongly supported for salvage treatment, and when available should be preferred over posaconazole oral suspension, which in turn is marginally supported for salvage treatment. In cases of primary treatment failure with isavuconazole or posaconazole, the guideline group supports recommendations for all three lipid-based amphotericin B formulations with strong to moderate strength. For further information on salvage treatment, see appendix p 20. Treatment duration for mucormycosis Evidence—The duration of therapy necessary to treat mucormycosis is unknown. In general, weeks to months of therapy are given. If immune defect is resolved—eg diabetes is controlled, neutropenia definitively resolved, immuno suppression can be tapered or stopped, therapy can be continued until resolution of signs and symptoms of infection, and substantial radiographical improvement. Median duration of isavuconazole first-line or salvage treatment was 84 days intravenous or oral route or both.49 Across several posaconazole oral suspension studies, treatment duration ranged from 1 week to almost 3 years, mean duration was approximately 6 months.113,133,134,138,139 The wide range reflects the pattern of organs involved, with competing risks from underlying conditions. Late relapse in long-term survivors have been documented (appendix p 21).140 Recommendations—The guideline group strongly supports treatment until permanent reversal of immuno suppression and complete response on imaging, which might be difficult to determine because of scarring and postoperative changes. Treatment duration is a personalised decision. There is moderate support for intravenous treatment until stable disease is achieved. When switching to oral treatment, use of isavuconazole or posaconazole delayed release tablets is strongly supported. Posaconazole oral suspension can be used, but is marginally supported, especially when formu lations with higher exposure are available (appendix p 21). Therapeutic drug monitoring in mucormycosis (appendix p 22), specific considerations on treatment of mucormycosis in children (appendix p 23), adjunctive treatments for mucormycosis (appendix p 27), intensive care and crtically ill patients with mucormycosis (appendix p 29), health economics (appendix p 29), and future directions (appendix p 30) are available in the appendix where indicated. Treatment pathways for mucormycosis The proposed treatment algorithms for adult (appendix p 25; figure 5) and for paediatric patients (appendix p 25) are based on case series, retrospective studies, and expert opinion. Large, randomised controlled trials investigating efficacy of treatment regimens are lacking. Surgical debridement should be performed whenever feasible in parallel to antifungal treatment.11,17,141,142 The drug of choice is liposomal amphotericin B.67,109 In case of renal failure, posaconazole or isavuconazole were shown to be effective. If a patient is intolerant to liposomal amphotericin B, its dose can be reduced, but should stay ≥5 mg/kg bodyweight. In case of extensive disease, rapid pro gression, or poor general condition, the addition of isavuconazole or posaconazole can be considered.133–135 Treatment should be continued until resolution of initially indicative findings on imaging and reconstitution of host immune system. Isavuconazole or posaconazole may be administered as maintenance therapy.143 Contributors OAC and AC coordinated the work of the authors and guided the development of the guideline. OAC, AC, AAI, DA, SCAC, ED, BH, MH, HEJ, KL, REL, SCM, MMe, ZP, DS, DCS, and RW wrote the initial manuscript draft. All authors contributed to the literature review, compilation of data tables and interpretation and assessment of recommendations. All authors participated in review and revisions, www.thelancet.com/infection Published online November 4, 2019 13 Review approved the final manuscript, and are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. Declaration of interests OAC reports research grants from Actelion, Amplyx, Arsanis, Astellas, AstraZeneca, Basilea, Bayer, Cidara, F2G, Gilead, GSK, Leeds University, Matinas, Medicines Company, MedPace, Melinta, Merck/MSD, Miltenyi, Pfizer, Rempex, Roche, Sanofi Pasteur, Scynexis, Seres; is a consultant to Allecra Therapeutics, Amplyx, Actelion, Astellas, Basilea, Cidara, Da Volterra, Entasis Therapeutics, F2G, Gilead, IQVIA, Janssen, Matinas, Menarini, Merck/MSD, Paratek, PSI, Scynexis, Seres, Summit, Tetraphase, Vical, and received lecture honoraria from Astellas, Basilea, Gilead, Merck/MSD and Pfizer ED reports grants from Gilead, MSD; personal fees from Pfizer, Astellas; non-financial support from MSD and Pfizer. AM reports grants from Sanofi and ROCHE. AAI reports grants and personal fees from GILEAD, personal fees from Pfizer, grants from F2G, grants from Scynexis, personal fees from Astellas, personal fees from MSD. SAA reports grants from Pfizer. SCAC reports grants from MSD Australia. MH reports reports personal fees from Basilea, Merck, Practitioner Network; and grants and personal fees from Gilead. KL reports grants, personal fees, and non-financial support from MSD, Gilead, and Pfizer; and personal fees from Abbott. REL reports personal fees from Gilead and grants from Merck. DCS reports grants from Merck and personal fees from Merck, Astellas, and AVIR. AA reports non-financial support from MSD, Gilead, and Pfizer; and personal fees from Gilead sciences and Pathoquest. RB reports grants and personal fees from Merck and Pfizer. SB reports grants from MSD; personal fees from Gilead and other from Pfizer. EC reports personal fees from Astellas and Basilea. MC reports personal fees from Astellas, Pfizer, LF Asia, Meiji, and MSD, and non-financial support from Astellas, Pfizer, and LF Asia. ALC reports grants from Astellas; grants, personal fees, and non-financial support from Pfizer; personal fees and non-financial support from Biotoscana; personal fees and non-financial support from MSD; and personal fees and non-financial support from Gilead. LD reports personal fees and non-financial support from MSD and Pfizer; and non-financial support from Teva. AHG reports grants and personal fees from Gilead, Merck, Sharp & Dohme, and Pfizer; and personal fees from Astellas and Basilea. JG reports grants from Scynexis, CIDARA; and personal fees from Gilead, Pfizer, Astellas, MSD, and United Medical. CPH reports personal fees from Schering-Plough; grants and personal fees from Pfizer, Boehringer Ingelheim, Siemens; personal fees from Basilea, Novartis, Roche, Astellas, Gilead, MSD, Lilly, Intermune, Fresenius, Essex, AstraZeneca, Bracco, MEDA Pharma, Chiesi, Covidien, Pierre Fabre, Grifols, Bayer; and grants from MeVis, German Center for Lung Research. ASI reports grants from Amplyx Pharmaceuticals, grants from Astellas Pharma USA and is founder and shareholder from Vitalex Biosciences. NK reports personal fees from Astellas, Gilead, Merck, and Pfizer. FLan reports personal fees from Gilead, MSD, and Basilea. CLF reports reports grants from Gilead and Astellas; and personal fees from Gilead, Merck Sharp & Dohme, Basilea. DGL reports consultant fees from Astellas, GILEAD, MSD, Pfizer, and Yuhan; has served as a board member for Gilead and Yuhan; and has received research support, travel support and payment for lectures, including service on Speaker’s bureaus, from Astellas, GILEAD, MSD, Pfizer, and Yuhan. TL reports grants from Gilead; personal fees and non-financial support from Gilead, Astellas, and MSD; and personal fees from Basilea. GM reports personal fees from Gilead and Pfizer. JFM reports personal fees from Scynexis, Gilead, Merck, United Medical, and Teva; grants from F2G, Pulmocide, and Amplyx. JM reports grants from Astellas, Gilead, MSD, and Pfizer. COM reporsts grants from Gilead and Merck. MN reports grants from Pfizer; and personal fees from Gilead, Scynexis, Cidara, Teva, United Medical, MSD, and Jansen. LP reports grants from Gilead, MSD, and Pfizer. APas reports grants from Gilead; and personal fees from Gilead, United Medical. ZR reports grants from Astellas and Teva. MRi reports personal fees from Gilead, MSD, and Basilea. ER reports grants from Gilead, Pfizer, Merck, and Sanofi; personal fees and non-financial support from Pfizer, Merck, and Astellas. MRu reports personal fees from Scynexis, Daiichi Sankyo, and Kedplasma GmbH. JS reports personal fees from Pfizer and MSD. MS reports grants and personal fees from Gilead and Merck. BS reports personal fees from Cempra, Bayer, Forge, Shionogi, Alexion, Synthetic Biologics, Paratek, Ovagene, Accuryx, and Bioversys; and is shareholder for Motif, BioAIM, Synthetic Biologics, Mycomed, and ExBaq. WS reports fees from Astellas and Merck. BHT reports grants from Pfizer. AJU reports personal fees from MSD, Basilea, and Aicuris. JJV reports personal fees from Merck/MSD, Gilead, Pfizer, Astellas Pharma, Basilea, Deutsches Zentrum für Infektionsforschung, Uniklinik Freiburg/Kongress und Kommunikation, Akademie für Infektionsmedizin, Universität Manchester, Deutsche Gesellschaft für Infektiologie, Ärztekammer Nordrhein, Uniklinik Aachen, Back Bay Strategies, and Deutsche Gesellschaft für Innere Medizin; and grants from Merck/MSD, Gilead, Pfizer, Astellas Pharma, Basilea, Deutsches Zentrum für Infektionsforschung, Bundesministerium für Bildung und Forschung. MJGTV reports having been on speakers’ bureau for Pfizer, MSD/Merck, Gilead Sciences, Organobalance and Astellas Pharma; received research funding from 3M, Astellas Pharma, DaVolterra and Gilead Sciences; and is a consultant to Berlin Chemie, MSD/Merck and Astellas Pharma. TJW reports grants from Amplyx, Astellas, Merck, Scynexis, Allergan, Medicines Company, Lediant, and Tetraphase; and having served on Advisory Boards of Astellas, Merck, Scynexis, Allergan, Medicines. PLW reports personal fees from Gilead, MSD; and grants from Bruker. NPW reports grants from Astellas, bioMerieux, F2G, and Viamet; and personal fees from Mayne Pharma. All other authors declare no competing interests. Acknowledgments The following authors are fellows of the European Confederation of Medical Mycology (ECMM): OAC, AAI, SCAC, ED, BH, MH, HEJ, KL, DCS, AC, MA, AA, AMSA, SAA, HB, SB, MC, ALC, LD, AHG, ASI, SSK, NK, ML, FL, FL, CLF, MM, GM, JFM, JM, COM, MN, RO, LP, AP, MR, ER, SS, JS, JJV, MJGTV, TJW, PLW, and NPW. The following authors are members of the Mycoses Study Group and Research Consortium (MSG ERC): OAC, SCAC, DCS, ASI, BS, WS, TJW, NPW, TZ. 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3810	https://www3.risc.jku.at/publications/download/risc_6711/MAIN.pdf	Author Philipp Nuspl 01556359 Submission Research Institute for Symbolic Computation Supervisor and First Evaluator Assoc. Prof. Dr. Veronika Pillwein Second Evaluator Prof. Dr. James Worrell May 2023 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Austria www.jku.at DVR 0093696 Algorithms for linear recurrence sequences Doctoral Thesis to obtain the academic degree of Doktor der Technischen Wissenschaften in the Doctoral Program Technische Wissenschaften Abstract In the past few decades, numerous tools for automatically discovering and proving identities involving sequences and special functions were developed. These tools are often based on algorithms which manipulate sequences satisfying linear recurrences. If the recurrences have constant coefficients, these sequences are called C-finite and in the case of polynomial coefficients they are called D-finite. We study sequences satisfying recurrences with coefficients which are C-finite themselves and call them C2-finite. We investigate which properties and algorithms carry over from the classical C-finite and D-finite cases to this new setting. In particular, we show that most so-called closure-properties, which are known for the classical cases, also hold for C2-finite sequences, i.e., they are closed under termwise addition, termwise multiplication, interlacing and taking subsequences at arithmetic progressions. In many cases these operations are effective and we present algorithms for performing them. In general, however, these algorithms are closely related to and limited by certain decision procedures of C-finite sequences. Deciding whether every term of a sequence is positive or nonzero is not known to be decidable in theory. Nevertheless, we show that it is often easy to decide these properties in practice. Restricting the ring of C2-finite sequence to sequences which satisfy a monic (i.e., having constant leading coefficient) linear recurrence with C-finite coefficients, we obtain a sub-ring where all closure properties can be performed effectively. On the other hand, we can allow more general sequences as coefficients. This way we obtain increasingly larger rings where the operations are more difficult to perform. Most of the theoretical results are also implemented in a package for the computer algebra system SageMath. The thesis contains a tutorial for this package. The tutorial shows how the examples given throughout the thesis can be performed automatically on the computer. i Kurzfassung In den letzten Jahrzehnten wurden zahlreiche Werkzeuge für das automatische Entdecken und Beweisen von Identitäten von Folgen und speziellen Funktionen entwickelt. Diese Werkzeuge basieren oft auf Algorithmen, die Folgen, die lineare Rekursionen erfüllen, manipulieren. Falls die Rekursionen konstante Koeffizienten haben, nennt man die Folgen C-finit und im Falle von polynomiellen Koeffizienten nennt man sie D-finit. Wir untersuchen Folgen, die Rekursionen mit Koeffizienten erfüllen, die selbst C-finit sind und nennen sie C2-finit. Wir untersuchen, welche Eigenschaften und Algorithmen wir vom klassischen C-finiten und D-finiten Fall auf diese neue Klasse übertragen kön-nen. Insbesondere zeigen wir, dass die meisten sogenannten closure properties, die für die klassischen Fälle bekannt sind, auch für C2-finite Folgen gelten. D.h., sie sind abge-schlossen bezüglich termweiser Addition, termweiser Multiplikation und Verflechtung. Außerdem ist die Teilfolge von C2-finiten Folgen wieder C2-finit. Diese Operationen sind häufig effektiv und wir stellen Algorithmen vor, die diese Berechnungen durchführen. Im Allgemeinen sind diese Algorithmen jedoch durch bestimmte schwierige Entschei-dungsprobleme für C-finite Folgen eingeschränkt. Es ist zum Beispiel nicht bekannt, ob die Probleme, dass jeder Term einer Folge positiv oder ungleich null ist, entscheidbar sind. Wir zeigen jedoch, dass diese Probleme in der Praxis oft einfach zu entscheiden sind. Schränkt man den Ring der C2-finiten Folgen auf Folgen ein, die eine normierte (d.h. mit konstantem Leitkoeffizienten) lineare Rekursion mit C-finiten Koeffizienten erfüllen, so erhält man einen Unterring, in dem alle closure properties effektiv durchgeführt werden können. Andererseits können wir auch allgemeinere Folgen als Koeffizienten zulassen. Auf diese Weise erhält man größere Ringe, in denen die Operationen schwieriger durch-zuführen sind. Die meisten der theoretischen Ergebnisse sind auch in einem Softwarepaket für das Computeralgebrasystem SageMath implementiert. Die Dissertation enthält ein Tutorial für dieses Paket. Das Tutorial zeigt, wie die in der Dissertation gegebenen Beispiele automatisch auf dem Computer ausgeführt werden können. ii Acknowledgments First and foremost I would like to thank my advisor Veronika Pillwein. She gave me the freedom to investigate topics I had a strong interest in and at the same time guided me to fruitful and exciting problems in these areas. Thanks to her I learned a lot about mathematics and how research is and should be done. Furthermore, I thank James Worrell for agreeing to serve as the second evaluator of this thesis. It is an honor to have him as a leading expert for linear recurrence sequences and their computational problems in my committee. Thanks to Antonio Jimenéz Pastor my start in the PhD program went very smoothly. Our discussions over the past years heavily influenced a lot of the research I have done. Likewise, I am very thankful for numerous discussions with Manuel Kauers who has shaped my understanding of computer algebra since the first lectures in linear algebra almost eight years ago. I am grateful for ample constructive feedback on my work during his algebra seminar. Furthermore, I greatly appreciate the comments I got during the combinatorics seminar organized by Peter Paule and Carsten Schneider as well as the DK seminar organized by Veronika Pillwein. In the past three years I had the pleasure to be part of three different institutes, the DK, RISC and the Institute for Algebra. I enjoyed my time in all three very much and I am very thankful to my colleagues (many of them I now consider good friends) for giving me such a wonderful experience in my PhD program. Finally, I am very grateful for the help and support I have received over the years from Katharina as well as from my family. The research presented in this thesis was funded by the Austrian Science Fund (FWF) under the grants W1214-N15, project DK15, P33530 and P31571. iii Contents 1 Introduction 1 2 Preliminaries 4 2.1 D-finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 C-finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 C2-finite sequences 13 3.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Asymptotics and counterexamples . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Algebraic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 Computations with C2-finite sequences 31 4.1 Setting up the linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Ring computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1.3 Sparse subsequences of C-finite sequences . . . . . . . . . . . . . . 39 4.2 Solving the linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Order bounds for C2-finite closure properties 47 5.1 The exponent lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Order bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2.1 Interlacing and subsequence . . . . . . . . . . . . . . . . . . . . . . 55 5.2.2 Ring operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2.3 Sparse subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 A computable subring: simple C2-finite sequences 63 6.1 Algebraic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Computable ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Extension to Ck-finite and Dk-finite sequences 72 7.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.2 Ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 iv Contents 8 Positivity of C-finite sequences 78 8.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.1.1 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.1.2 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.1.3 D-finite reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.1.4 Classical algorithm for sequences with unique dominant eigenvalue 84 8.1.5 Combination of Algorithm 1 and Algorithm 2 . . . . . . . . . . . . 86 8.1.6 Decomposition into nondegenerate sequences . . . . . . . . . . . . 91 8.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.2.1 Recurrence sequences in the OEIS . . . . . . . . . . . . . . . . . . . 93 8.2.2 Positive sequences in the OEIS . . . . . . . . . . . . . . . . . . . . . 97 9 Implementation 100 9.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.2 C-finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.3 C2-finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 List of Symbols 106 Bibliography 108 Index 122 v 1 Introduction Linear recurrence sequences have been a subject of significant interest in mathematics for a long time and the Fibonacci sequence is undoubtedly the most well-known and meticulously studied sequence of this kind. The sequence which was only later (from 1876 on, coined by Lucas [Kos11]) known as the Fibonacci sequence was apparently already studied several hundred years earlier in Indian poetry [Sin85]. Leonardo of Pisa (also called Fibonacci, around 1170–1240) introduced the sequence in his book Liber Abaci in the context of a combinatorial problem. He described the development of a rabbit population by the famous recurrence [Sig03]. Later, also Kepler (1571–1630) studied the Fibonacci numbers in his book Harmonices Mundi (1619). He observed that the ratio of two consecutive Fibonacci numbers tends to the golden ratio. Furthermore, Kepler found the identity which is known as the Cassini identity today [KADF97]. In the first half of the 18th century, de Moivre (1667–1754) and D. Bernoulli (1700–1782) were among the first ones to investigate similar sequences as the Fibonacci numbers. Both of them also found the closed form of the Fibonacci numbers (later known as Binet’s formula) [Ber28, Kos11]. Moreover, de Moivre studied the generating functions of linear recurrence sequences [Moi22, Moi30]. According to [EPSW15], one of the key steps for developing our modern understanding of linear recurrence sequences and in particular their arithmetic properties are the works of Lucas (1842–1891). Today, we call sequences of this type, i.e., satisfying linear recurrences with constant coefficients, C-finite [Zei90]. The solutions of linear differential equations (with polynomial coefficients) were a topic of many articles in the 19th century. Deriving a linear recurrence (with polynomial coefficients) for the coefficients of the series solutions of such differential equations were routinely done by mathematicians such as Fuchs (1833–1902) and Frobenius (1849– 1917) [Fuc66, Fro75]. In modern language such functions and sequences are called D-finite or holonomic (the precise correspondence between functions and their coefficient sequences is given in Theorem 2.7). Likewise, arithmetic operations using D-finite objects were already performed in the 19th century, for instance by Hurwitz (1859–1919) and Beke 1 1 Introduction (1862–1946) [Jun31, Bek94]. Nowadays, the operations which can be performed in the ring of D-finite sequences/functions are called closure properties (cf. Theorem 2.6). These closure properties of D-finite (or C-finite) sequences are implemented in all major computer algebra systems and are used to automatically prove identities of sequences and special functions. For instance, the identity [Rao53] 2n ∑ k=0 f (k) f (k + 1) = f (2n + 1)2 −1, where f (n) = ⟨0, 1, 1, 2, 3, 5, . . . ⟩denotes the Fibonacci numbers, can be shown routinely. Combining methods for finding closed form solutions of linear recurrences (cf. [Pet92, Hoe99]) together with a method called creative telescoping (cf. [Zei91, Chy14]) identities of the form n ∑ k=0 (n k )2 = (2n n ) can be discovered and proven automatically [PWZ96]. Considering the usefulness of D-finite and C-finite sequences it is natural to consider generalizations of these classes. Notable extensions from the past two decades include admissable sequences [Kau07a], sequences built by nested expressions [Sch07], coefficient sequences of DD-finite functions [JPP19] or poly-recursive sequences [CMP+21]. In our approach we keep the aspect of considering linear recurrences but we allow more general coefficients. In particular, we consider sequences which satisfy a linear recurrence with coefficients which are C-finite themselves. Such sequences are called C2-finite and numer-ous examples can be found in combinatorics and in other areas of research. For instance, the sequence f (n2) where f, again, denotes the Fibonacci sequence is C2-finite satisfying the recurrence f (2n + 3) f (n2) + f (4n + 4) f ((n + 1)2) −f (2n + 1) f ((n + 2)2) = 0. In this thesis we investigate the computational properties of C2-finite sequences. First, in Chapter 2 we give an overview of important results of C-finite and D-finite sequences which will be used throughout the thesis. Then, in Chapter 3, we introduce C2-finite sequences and prove some important facts on their asymptotic behavior, the ring-structure 2 1 Introduction and their generating functions. Chapter 4 discusses how (and which) closure properties of C2-finite sequences can be performed. One of the differences to C-finite and D-finite sequences is that the same order bounds do not seem too hold. Therefore, in Chapter 5, we show a different way to compute with C2-finite sequences. Using this approach we can also derive order bounds. Even though closure properties can often be performed in practice, in theory it seems that we are limited by the so-called Skolem Problem (which is the problem of deciding whether a given C-finite sequence contains a zero term, cf. Section 2.2). In Chapter 6 we show that if we fix the leading coefficient in the recurrence of a C2-finite sequence to be constant, we get a subring where all closure properties can be computed effectively. On the other hand, by allowing other sequences as coefficients in the recurrence we can obtain a chain of increasingly larger difference rings as shown in Chapter 7. The Skolem Problem, which plays an important role in the computations of C2-finite sequences, can be reduced to showing that a C-finite sequence is positive. In Chapter 8 we compare several, mostly well-known, algorithms for automatically proving positivity of certain sequences. To this effect, we use sequences from The On-Line Encyclopedia of Integer Sequences (OEIS, cf. [OEI23]) for testing the implementations. Furthermore, we study how many of these sequences from the OEIS are in fact C-finite or D-finite. Most of the algorithms discussed throughout the thesis are implemented in a software package for the computer algebra system SageMath [Sag23]. Finally, Chapter 9 can be seen as a tutorial and showcase of this software package. 3 2 Preliminaries In this chapter we introduce the notions that we work with throughout the thesis. In particular, we discuss D-finite and C-finite sequences. These are sequences satisfying a linear recurrence with polynomial or constant coefficients, respectively. Furthermore, we introduce important properties of these sequences that we need later. There are numerous expositions on sequences satisfying linear recurrences with constant or polynomial coefficients. Some of these, that were also used to prepare this thesis, include [Sta80, Zei90, Sta99, FS09, KP11, Kau13, EPSW15]. Throughout this thesis, if not further specified, K denotes a field of characteristic zero. Of-ten we think about sequences arising from combinatorics, then K can usually be thought of as the field of rational numbers Q. The natural numbers are denoted by N = {0, 1, 2, . . . , }. The K-algebra of sequences is denoted by KN. For a sequence a = (a(n))n∈N ∈KN, we sometimes simply write a(n). It is always clear from the context if a(n) denotes the sequence (a(n))n∈N or the specific term at index n. 2.1 D-finite sequences The shift-operator σ: KN →KN acts on a sequence a := (a(n))n∈N in the natural way as σ(a) := (a(n + 1))n∈N. Let R ⊆KN be a subring of the ring of sequences. The linear recurrence operators R[σ] over R form an (in general) noncommutative ring under the usual (i.e., polynomial) addition and multiplication obeying the commutation rule σ · a = σ(a) · σ for a ∈R. An element A := ∑r i=0 ciσi ∈R[σ] acts on a sequence a as Aa := r ∑ i=0 ci(n)a(n + i). 4 2 Preliminaries If cr ̸= 0, then r is called the order of the operator A. If Aa = 0, then we call A an annihilator of a and say that A annihilates a. Definition 2.1. A sequence a ∈KN is called D-finite (or P-recursive or holonomic) if there is a nonzero linear recurrence operator with polynomial coefficients A ∈K[n][σ] which annihilates a, i.e., Aa = 0. A D-finite sequence a with annihilator A is also annihilated by BA for any B ∈K[n][σ]. Usually we are interested in an operator with minimal order. The order of such an operator is then called the order of the sequence a and is denoted by ord(a). Let A = ∑r i=0 piσi ∈K[n][σ]. We can define the characteristic polynomial of A as χ(A) = lcn ( r ∑ i=0 pi(n)yi ) ∈K[y]. (2.1) The roots of this polynomial are called the eigenvalues of A. The eigenvalues of a minimal annihilating operator A of the sequence a are also called the eigenvalues of a. If an annihilating operator A of the sequence a is fixed, we denote the characteristic polynomial of the annihilator of a simply by χ(a), i.e., χ(a) := χ(A). Let a be a D-finite sequence annihilated by A = ∑r i=0 pi(n)σi. Equivalently we can say that a satisfies a linear recurrence with polynomial coefficients p0(n)a(n) + p1(n)a(n + 1) + · · · + pr(n)a(n + r) = 0, for all n ∈N. A D-finite sequence can always be described by finite amount of data, namely by the coefficients of the recurrence p0, . . . , pr and finitely many initial values a(0), . . . , a(m). The number of initial values that are needed to uniquely determine the sequence depends on the order r of the recurrence and integer roots of the leading coefficient pr. D-finite sequences often appear in combinatorics counting certain objects like graphs, paths, permutations or tilings. Numerous example can be found in the OEIS, The On-Line Encyclopedia of Integer Sequences [OEI23]. This database, based on books by Neil Sloane and Simon Plouffe [Slo73, SP95], contains about 360 000 integer sequences at the time of writing (spring 2023), many with detailed information and references. 5 2 Preliminaries Example 2.2. Polynomial sequences p(n) with p ∈K[n], geometric sequences αn with α ∈K and the factorial sequence n! are all D-finite. Example 2.3. Let H(n) = ∑n k=1 1 k be the sequence of harmonic numbers. The sequence is D-finite of order 2 and satisfies the recurrence (n + 1)H(n) −(2n + 3)H(n + 1) + (n + 2)H(n + 2) = 0, for all n ∈N. Example 2.4. The Catalan numbers (A000108 in the OEIS [OEI23]) defined as C(n) = 1 n+1(2n n ) are D-finite of order 1 satisfying the recurrence 2(2n + 1)C(n) −(n + 2)C(n + 1) = 0. The Catalan numbers have numerous combinatorial interpretations [Sta15]. For proving computational properties of D-finite sequences the following well-known equivalent characterization via vector spaces is often useful [Sta80, Zei90]: Theorem 2.5. The sequence a ∈KN is D-finite if and only if the K(n)-vector space ⟨σi(a) \| i ∈N⟩K(n) has finite dimension. In fact, the dimension of this vector space corresponds precisely to the order of the recurrence. As an application of Theorem 2.5 the following so-called closure properties of D-finite sequences can be shown [Sta80, Zei90, Mal96]: Theorem 2.6. Let a(n), b(n), a0(n), . . . , am−1(n) be D-finite sequences. Then, 1. σ(a(n)) = a(n + 1) is D-finite of order at most ord(a), 2. a(n) + b(n) is D-finite of order at most ord(a) + ord(b), 3. a(n)b(n) is D-finite of order at most ord(a) ord(b), 6 2 Preliminaries 4. ∑n k=0 a(k) is D-finite of order at most ord(a) + 1, 5. ∑n k=0 a(k)b(n −k) is D-finite, 6. a(kn + ℓ) is D-finite of order at most ord(a) for all k, ℓ∈N and 7. the interlacing e(n) = ar(q) where n = qm + r for 0 ≤r < m is D-finite of order at most m ∑m−1 r=0 ord(ar). In order to give an idea how these properties are usually shown we prove part 2 on the addition of two sequences. Proof. By Theorem 2.5 the vector spaces Va := ⟨σi(a) \| i ∈N⟩K(n), Vb := ⟨σi(b) \| i ∈N⟩K(n) both have finite dimension. In fact, Va has dimension at most ord(a) and Vb has dimension at most ord(b). By linearity of σ we have ⟨σi(a + b) \| i ∈N⟩K(n) ⊆Va + Vb. The vector space Va + Vb has dimension at most ord(a) + ord(b) as does the subspace. Hence, by Theorem 2.5 the sequence a + b is D-finite with order at most ord(a) + ord(b). All the properties from Theorem 2.6 are algorithmic and implemented in various computer algebra systems (e.g., GeneratingFunctions [Mal96] and HolonomicFunctions [Kou10a, Kou10b] for Mathematica, gfun [SZ94] for Maple and ore_algebra [KJJ15] for Sage-Math). D-finite sequences not only often appear in combinatorics but also as the coefficient sequences of functions satisfying linear differential equations: Theorem 2.7 (Theorem 1.5 in [Sta80]). The sequence a(n) is D-finite if and only if the corre-sponding generating function f (x) = ∑n∈N a(n)xn ∈KJxK satisfies a linear differential equation of the form q0(x) f (x) + q1(x) f ′(x) + · · · + qs(x) f (s)(x) = 0 7 2 Preliminaries with q0, . . . , qs ∈K[x] not all zero. Using Theorem 2.7 we can see that the coefficient sequences of many special functions are D-finite. Example 2.8. The trigonometric functions sin(x), cos(x), the exponential function exp(x) and the Bessel functions all satisfy a linear differential equation with polynomial coeffi-cients [DLMF21]. The corresponding coefficient sequences are therefore all D-finite. Due to the recurrence satisfied by a D-finite sequence, the terms of such a sequence cannot grow arbitrarily. In fact, for every such sequence a(n) there is a constant α ∈Q such that \|a(n)\| ≤n!α for all n ≥2 [Ger05, Proposition 1.2.1]. We can find even more precise asymptotics for D-finite sequences. For sequences a(n), b(n) ∈KN we write a(n) ∼b(n) if limn→∞ a(n) b(n) = 1. Using this notion of asymptotic equivalence the following theorem can be proven [WZ85, FS09, MS10, Mel21]: Theorem 2.9 (Theorem 2 in [Kau13]). Let a(n) be D-finite. Then, there are constants c1, . . . , cm, polynomials p1, . . . , pm, natural numbers r1, . . . , rm, constants γ1, . . . , γm, φ1, . . . , φm, α1, . . . , αm and natural numbers β1, . . . , βm such that a(n) ∼ m ∑ k=1 ckepk(n1/rk)nγknφn knαk log(n)βk. These asymptotics can be used to determine that a sequence is not D-finite. Other tech-niques for proving that a sequence is not D-finite include the analysis of the corresponding generating function. Example 2.10. As the terms grow too fast, the sequence 2n2 is not D-finite. Furthermore, the sequences nn, log(n) and P(n) where P(n) denotes the n-th prime number are not D-finite [Ger05, FGS05]. A technique that is commonly used for constructing D-finite recurrences is guessing. The idea is to guess an operator ∑r i=0 pi(n)σi = ∑r i=0 ∑d k=0 pi,knkσi of order r and degree d which annihilates the known terms a(0), . . . , a(N −1) of a sequence. A straightforward approach is to set up a linear system for the (r + 1)(d + 1) many unknown variables pi,k using the N −r many equations ∑r i=0 ∑d k=0 pi,knka(n + i) = 0 for n = 0, . . . , N −r −1. If 8 2 Preliminaries this system is overdetermined, i.e., if N −r ≥(r + 1)(d + 1), then any solution of this system is a reasonable guess for an annihilating operator of the sequence a. More advanced techniques for guessing use, for instance, Hermite-Padé approximation, homomorphic images or methods from lattice theory [Kau13, Yur22, KK22]. 2.2 C-finite sequences A difference ring R is a subring of the ring of sequences KN which is closed under shifts, i.e., σ(a) ∈R for all a ∈R. The closure properties in Theorem 2.6 show that the set of D-finite sequences forms a difference ring under termwise addition and termwise multi-plication (also called the Hadamard product). Unless specified otherwise, these termwise operations are always our ring operations. A particularly interesting and well-studied subring of the ring of D-finite sequences is the ring of C-finite sequences. Definition 2.11. A sequence c ∈KN is called C-finite if there is a nonzero linear recurrence operator with constant coefficients C ∈K[σ] which annihilates c, i.e., Cc = 0. Equivalently, c satisfies a linear recurrence with constant coefficients. The order ord(c) can again be defined as the order of the minimal nonzero operator which annihilates c. We denote the ring of C-finite sequences by RC. Example 2.12. Polynomial sequences p(n) with p ∈K[n] and geometric sequences αn with α ∈K are C-finite. Example 2.13. The Fibonacci sequence f (n) ∈QN (A000045 in the OEIS) satisfying the recurrence f (n) + f (n + 1) −f (n + 2) = 0, for all n ∈N, with initial values f (0) = 0, f (1) = 1 is C-finite of order 2. The C-finite sequence l(n) satisfying the same recurrence but having initial values l(0) = 2, l(1) = 1 is called the Lucas sequence (A000032 in the OEIS). Example 2.14. The Perrin numbers (A001608 in the OEIS) p(n) ∈QN are C-finite of order 3 satisfying the recurrence p(n) + p(n + 1) −p(n + 3) = 0, for all n ∈N, 9 2 Preliminaries with initial values p(0) = 3, p(1) = 0, p(2) = 2. Analogous to Theorem 2.5 for D-finite sequences, a sequence c is C-finite if and only if the K-vector space ⟨σi(c) \| i ∈N⟩K has finite dimension. From this property, the same closure properties (including the same bounds for the orders) as for D-finite sequences, Theorem 2.6, can be derived. Further, a sequence c(n) is C-finite if and only if its generating function f (x) = ∑n∈N c(n)xn ∈KJxK is a rational function [Zei90, KP11]. In contrast to D-finite sequences, C-finite sequences have a nice closed form expression. Namely, they can be written as polynomial-linear combination of geometric sequences. Theorem 2.15 (Theorem 4.1 in [KP11]). Let c be a C-finite sequence over the field K with characteristic polynomial r ∑ i=0 γiyi = yn0 m ∏ i=1 (y −λi)di ∈K[y] where λ1, . . . , λm ∈L ⊇K are the pairwise different nonzero eigenvalues of c and d1, . . . , dm their multiplicities. Then, there are p1, . . . , pm ∈L[n] with deg(pi) = di −1 for i = 1, . . . , m such that c(n + n0) = m ∑ i=1 pi(n)λn i , for all n ∈N. Theorem 2.15 shows in particular that every C-finite sequence c ∈CN can be bounded as \|c(n)\| ≤αn for all n ≥1 for some α ∈Q. In fact, more precise asymptotics can be derived. Namely, the asymptotic behavior is completely governed be the eigenvalues of maximal modulus. Using the notions from Theorem 2.15, let \|λ1\| = · · · = \|λk\| > \|λk+1\| ≥· · · ≥\|λm\| (2.2) 10 2 Preliminaries and d = maxi=1,...,k deg(pi). Then, c(n) ∼nd k ∑ i=1 coeff(pi, d)λn i (2.3) where coeff(pi, d) denotes the coefficient of nd in pi ∈K[n] [KP11]. Example 2.16. Let f be the Fibonacci sequence as in Example 2.13. The closed form of f is given by the well known Binet’s formula (which was already known to de Moivre and D. Bernoulli in the first half of the 18th century before it was rediscovered by Binet in 1843 [Kos11]) f (n) = 1 √ 5 ( 1+ √ 5 2 )n − 1 √ 5 ( 1− √ 5 2 )n , for all n ∈N. Clearly, 1+ √ 5 2 is the unique dominant eigenvalue, so f (n) ∼ 1 √ 5 ( 1+ √ 5 2 )n . An important notation for C-finite sequences is degeneracy. A C-finite sequence with pairwise different eigenvalues λ1, . . . , λm is called degenerate if there is a quotient λk λi with k ̸= i which is a root of unity. Otherwise the sequence is called nondegenerate. Theorem 2.17 ([BM76, EPSW15]). 1. Every C-finite sequence c(n) can be written (effec-tively) as the interlacing of nondegenerate subsequences c(dn), . . . , c(dn + d −1) for some d ∈N. 2. Let c(n) be a nondegenerate C-finite sequence. The set Zc := {n ∈N \| c(n) = 0} is either equal to N (i.e., c is the zero sequence) or finite. One of the most celebrated theorems on C-finite sequences is the Skolem-Mahler-Lech theorem which gives a description of the set of indices whose corresponding terms are zero. It was first proven by Skolem for sequences over the rational numbers [Sko33] and later extended by Mahler to number fields (finite algebraic extensions of the rational numbers) [Mah35] and by Lech to fields of characteristic zero [Lec53]. The theorem can be seen as a consequence of Theorem 2.17. 11 2 Preliminaries Theorem 2.18 (Skolem-Mahler-Lech). Let c be a C-finite sequence. The set Zc := {n ∈N \| c(n) = 0} can be written as the union of a finite set S and a finite number of arithmetic progressions, i.e., Zc = S ∪{n1 + pn \| n ∈N} ∪· · · ∪{nk + pn \| n ∈N} (2.4) for some n1, . . . , nk, p ∈N. Equivalently, the Skolem-Mahler-Lech theorem states that the zeros of a C-finite sequence are cyclic from some term onwards. Several computational problems are closely related to the Skolem-Mahler-Lech theo-rem [EPSW15]. First, since the arithmetic progressions in (2.4) can be computed, it can be decided whether or not a given C-finite sequence has infinitely many zeros [BM76]. More difficult is the problem of determining the finite set S [OW12, HHHK05]. The Skolem Prob-lem asks whether a given C-finite sequence has any zeros (i.e., provided that the sequence only has finitely many zeros S, is this set S empty). Decidability of the Skolem Problem is only known for special cases, most notably for sequences of order at most 4 [MST84, Ver85]. For sequences of higher order it remains open whether the Skolem Problem is decidable. Some decidability results can be achieved by restricting the C-finite input sequences further or by restricting the set where zeros are sought [LLN+22, BLN+22, KLOW20, LOW21]. The situation for D-finite sequences is even more unclear. Even the Skolem-Mahler-Lech theorem is only known for special cases of D-finite sequences and remains open for general D-finite sequences [BBY12, BCH21]. Futhermore, decidability of checking whether a sequence has infinitely many zeros is only known for certain D-finite sequences of order 2 [NOW21]. 12 3 C2-finite sequences D-finite sequences are a natural generalization of C-finite sequences by considering linear recurrences with polynomial instead of constant coefficients. In the same way, we can generalize D-finite sequences by considering linear recurrences with coefficients that are C-finite themselves. These are called C2-finite sequences. To our knowledge they were described for the first time in the context of graph polynomials [KM14]. Their computational properties and their generating functions were also studied in [TZ20]. The approach we present in this thesis is more computational compared to [KM14]. Our goal is always to check how operations on C2-finite sequences can be implemented and used in practice for studying concrete problems, e.g., in combinatorics. Here, we give a basic introduction to C2-finite sequences and prove some basic facts about them. This chapter is mostly based on [JPNP21, JPNP23, NP22b]. 3.1 Definition and examples Let K be some field of characteristic zero and RC the ring of C-finite sequences over K. For a ring R ⊆KN we denote the set of sequences which are invertible (i.e., not zero divisors) as elements in KN by R×. In particular, R× C denotes the multiplicatively closed set of all C-finite sequences which do not contain any zeros. The localization of R w.r.t. R× is denoted by Q(R) and known as the total ring of fractions of R. An element c(n)/d(n) ∈Q(R) can be interpreted naturally as a sequence in KN. Definition 3.1. A sequence a ∈KN is called C2-finite if there is a linear recurrence operator A ∈RC[σ] with lc(A) ∈R× C which annihilates a, i.e., Aa = 0. 13 3 C2-finite sequences Equivalently, the sequence a is C2-finite if there are C-finite sequences c0, . . . , cr such that cr(n) ̸= 0 for all n ∈N and c0(n)a(n) + · · · + cr(n)a(n + r) = 0, for all n ∈N. (3.1) The order is again defined analogously to the C-finite and D-finite cases, namely as the order of the minimal operator which annihilates the sequence. Example 3.2. As polynomials are C-finite, all D-finite sequences (and as such C-finite sequences) are C2-finite. If the leading coefficient pr(n) of the linear recurrence has any roots n ∈N, the recurrence can be shifted such that all these roots disappear. Example 3.3. Let L := K(q) with q transcendental. Then, a ∈LN is called q-holonomic if it satisfies a linear recurrence p0(qn)a(n) + · · · + pr(qn)a(n + r) = 0, for all n ∈N, with coefficients p0, . . . , pr ∈L[x], not all zero [KK09]. As all coefficients pi(qn) are C-finite over L, such a sequence a is also C2-finite over L. Example 3.4. Let a(n) count the number of graphs on n labeled nodes (A006125 in the OEIS). Then, a(n) = 2n(n−1)/2 = 2(n 2) and a is C2-finite as 2na(n) −a(n + 1) = 0, for all n ∈N. Similarly, it can be easily seen that all sequences αn2 for α ∈K are C2-finite. The se-quence 2n2 is not D-finite. Hence, the set of C2-finite sequences is certainly a strict general-ization of D-finite sequences. Example 3.5. Let c(n) be a C-finite sequence. Then, the sequence of partial products a(n) = ∏n k=0 c(k) is C2-finite of order 1 satisfying the recurrence c(n + 1)a(n) −a(n + 1) = 0, for all n ∈N. If c(n) denotes the Fibonacci sequence, the terms a(n) are also known as fibonorials or Fibonacci factorials (A003266 in the OEIS). This sequence and similar sequences have been studied in several works, e.g., in [Bro72, Mar13, BCMS20]. 14 3 C2-finite sequences Example 3.6. Let c(n) be a C-finite sequence with c(n) ̸= 0 for all n ∈N. The se-quence a(n) = 1 c(n) is not C-finite in general (unless c(n) is the interlacing of geometric sequences [LT90]). The sequence a(n) is, however, C2-finite satisfying c(n)a(n) −c(n + 1)a(n + 1) = 0, for all n ∈N. In particular, by the closure properties of C2-finite sequences (Theorem 5.8), the sequence a(n) = f (n) f (n + 1) + n ∑ k=1 (−1)k f (k) f (k + 1) is C2-finite (where f denotes the Fibonacci numbers). In fact it was shown that a(n) = 0 for all n ∈N [Kau05, Example 4.7]. Example 3.7. Let f (n) denote the Fibonacci numbers. It was observed in [KM14] that f (2n + 3)( f (2n + 1) f (2n + 3) −f (2n + 2)2) f (n2) + f (2n + 2)( f (2n + 3) + f (2n + 1)) f ((n + 1)2) −f (2n + 1) f ((n + 2)2) = 0 holds for all n ∈N. Hence, f (n2) is C2-finite (A054783 in the OEIS). In fact, the C-finite coefficients can be simplified and we obtain the recurrence: f (2n + 3) f (n2) + f (4n + 4) f ((n + 1)2) −f (2n + 1) f ((n + 2)2) = 0. Example 3.8. Let f (n) denote the Fibonacci numbers and l(n) the Lucas numbers (as in Example 2.13). Let Fib(n, k) = ∏k i=1 f (n −i + 1)/ f (i) be the fibonomial coefficient. It has been shown [KAO12, Theorem 1] that a(n) = n ∑ k=0 Fib(2n + 1, k) = n ∏ k=1 l(2k), for all n ∈N. (3.2) Hence, the sequence a (one half of the sequence A294349 in the OEIS) is C2-finite and satisfies the recurrence l(2n + 2)a(n) −a(n + 1) = 0, for all n ∈N. Numerous other identities of fibonomial coefficients can be found in [ST05, BR14]. 15 3 C2-finite sequences Identities of the form of Example 3.8 can also be viewed as a multibasic hypergeometric identity. The difficulty comes from the relations among the bases [BP99]. The identity (3.2) can be proven fully automatically using the tools developed in [AS21] or using a form of multibasic creative telescoping. Open Question 3.9. Can one develop a version of creative telescoping for C2-finite se-quences which can prove identities as in Example 3.8 automatically? Such a tool could, for instance, be used to verify the following example. Example 3.10. The sequence a(n) = ∑n k=0 Fib(n, k) seems to be C2-finite of order 4 satisfy-ing the recurrence −(l(2n + 2) + 2)a(n) −l(n + 2)a(n + 2) + a(n + 4) = 0. The recurrence was obtained by guessing. Example 3.11. Let c(n) be a sequence and let a(n) = [c(0), c(1), . . . , c(n)] = c(0) + 1 c(1) + 1 ... + 1 c(n) be the n-th convergent of the continued fraction [c(0), c(1), c(2), c(3), . . . ] = c(0) + 1 c(1) + 1 c(2) + 1 c(3) + . . . The n-th convergents a(n) = p(n) q(n) can be described by the recurrences p(n) + c(n + 2)p(n + 1) −p(n + 2) = 0, p(0) = c(0), p(1) = c(0)c(1) + 1, q(n) + c(n + 2)q(n + 1) −q(n + 2) = 0, q(0) = 1, q(1) = c(1). 16 3 C2-finite sequences In particular, if c is a C-finite sequence, these sequences p, q are C2-finite. For instance, if c (A003417 in the OEIS) satisfies c(n + 1) −2c(n + 4) + c(n + 7) = 0 with initial values a(n) = (2, 1, 2, 1, 1, 4, 1, . . . ), then a(n) →e [Eul44, Coh06]. From recurrence (3.1) it is clear that every C2-finite sequence can be described by finite data: The coefficients of the recurrence can be described by finite amount of data and the initial values uniquely define the sequence. I.e., given the coefficients c0, . . . , cr of the recurrence and the initial values a(0), a(1), . . . , a(r −1), the term a(n) of the sequence can be computed. Substantial amount of research has been done to see how was fast terms c(n) for a C-finite, D-finite or q-holonomic sequence c can be evaluated [BGS07, Bos20, BM21]. Open Question 3.12. Let a be a C2-finite sequence and n ∈N. How fast can we com-pute a(n)? It is in general not known whether we can decide that the leading coefficient cr(n) in the recurrence has no zeros (cf. Skolem Problem on page 12). Alternatively, we could allow the leading coefficient cr(n) to have at most finitely many zeros (as in [KM14] and later in Chapter 5). This alternative definition is equivalent to the definition given here as the recurrence can be shifted in order to attain a recurrence with a leading coefficient without any zeros. The advantage in allowing finitely many zeros is that this property is decidable [BM76]. For computations in practice, we would still have to compute the finite set of zeros to know how many initial values we have to save (where we would be limited by the Skolem Problem again). A sequence is called X-recursive if it satisfies a linear recurrence with C-finite coeffi-cients [TZ20]. I.e., the leading coefficient can have infinitely many zeros in this case. The disadvantage when dealing with X-recursive sequences is that finite amount of data might not suffice to uniquely define a sequence. Also, sequences might grow arbitrarily fast. 17 3 C2-finite sequences Example 3.13. Let c(n) be the interlacing of the constant 0 and the constant 1 sequence. Then, c is C-finite of order 2 satisfying c(n) −c(n + 2) = 0 with c(0) = 0, c(1) = 1. Any sequence a(n) satisfying c(n)a(n) −c(n)a(n + 1) = 0, for all n ∈N, is X-recursive. The given recurrence only forces a(n) = a(n + 1) for odd n ∈N. Hence, there are uncountably many X-recursive sequences over Q whereas there are only count-ably many C2-finite sequences over Q. 3.2 Asymptotics and counterexamples Proving that a sequence satisfies a recurrence of a certain type is usually significantly easier than proving that a recurrence does not satisfy any recurrence of a certain type. Example 3.14. Let a(n) = (−1)⌈log(n+1)⌉+ 1. Then, for every r ∈N the sequence contains a run of at least r consecutive zeros. Hence, if the sequence would be C2-finite of order r, then the sequence would be eventually zero. Therefore, the sequence cannot be C2-finite. Example 3.15. By definition, every term of a C2-finite sequence can be obtained by field operations of finitely many elements from the ground field K (the initial values of the C-finite coefficients, the coefficients of the C-finite recurrences and the initial values of the C2-finite recurrence). Hence, all terms of a C2-finite sequence have to be in a finite extension field of Q (cf. [Ger05, Proposition 1.3.3]). Therefore, the sequence a(n) = √n cannot be C2-finite. Another method for showing that a sequence is not C2-finite is by showing that it grows too fast asymptotically. Lemma 5 in [KM14] states, without a proof, that every C2-finite sequence a(n) with leading coefficient cr(n) ∈Z for all n ∈N can be bounded by \|a(n)\| ≤αn2 for some α ∈Q. In fact, such a bound can be shown for any C2-finite sequence (our proof follows the proof for D-finite sequences [Ger05, Proposition 1.2.1]). Lemma 3.16. Let a ∈CN be C2-finite. Then, there is an α ∈Q such that \|a(n)\| ≤αn2 for all n ≥1. 18 3 C2-finite sequences Proof. Suppose a satisfies the recurrence c0(n)a(n) + · · · + cr−1(n)a(n + r −1) + cr(n)a(n + r) = 0 for all n ∈N with c0, . . . , cr ∈RC and cr(n) ̸= 0 for all n ∈N. Then, for all ci(n) with i = 0, . . . , r −1 there exists an αi ∈Q such that \|ci(n)\| ≤αn i for all n ≥1. Furthermore, there exists an αr ∈Q such that 1 \|cr(n)\| ≤αn r for all n ∈N (cf. [EPSW15, Theorem 2.3]). Let 1 ≤α ∈Q be large enough such that r−1 ∑ i=0 αn i αn r ≤r ( max i=0,...,r−1 αi αr )n ≤αn for n ≥1 and large enough such that \|a(n)\| ≤αn2 holds for n = 1, . . . , r −1. We show \|a(n)\| ≤αn2 by induction on n. Suppose the inequality holds for all a(i) with i ≤n + r −1. In the induction step we have \|a(n + r)\| = ⏐ ⏐ ⏐ ⏐ ⏐ r−1 ∑ i=0 ci(n) cr(n)a(n + i) ⏐ ⏐ ⏐ ⏐ ⏐≤ r−1 ∑ i=0 \|ci(n)\| \|cr(n)\| \|a(n + i)\| ≤ r−1 ∑ i=0 αn i αn r α(n+i)2 ≤α(n+r−1)2 r−1 ∑ i=0 αn i αn r ≤α(n+r−1)2αn ≤α(n+r)2. Example 3.17. Lemma 3.16 shows that the sequences 2n3 and ∏n i=0 i! are not C2-finite. For a given C2-finite sequence a it is, in general, not clear whether such an α with \|a(n)\| ≤ αn2 can be computed. The obstacle is the computation of a number αr ∈Q such that 1 \|cr(n)\| ≤αn r for all n ∈N [EPSW15]. In special cases, however, such an αr can be found, e.g., if cr(n) is a polynomial sequence. The sequence αn2 is C2-finite. Hence, there are examples where the bound in Lemma 3.16 is tight. Determining more precise asymptotics for C2-finite sequences could turn out useful to determine that certain sequences are not C2-finite. For instance, the asymptotics of D-finite sequences (Theorem 2.9) can be used to show that the sequence of prime numbers is not D-finite [Mel21, Example 2.26]. Open Question 3.18. Determine the asymptotics of C2-finite sequences (or a subclass) analogously to the asymptotics of D-finite sequences. 19 3 C2-finite sequences D-finite sequences in general exhibit more complex asymptotics than C-finite sequences. Let a(n) = n ∑ k=0 (n k )2(n + k k )2 denote Apéry’s numbers (A005259 in the OEIS). The sequence is D-finite of order 2 but is not C-finite as can be seen by the asymptotics (cf. [Hir12]) a(n) ∼2−9/4π−3/2( √ 2 + 1)4n+2n−3/2. An analogous example for C2-finite sequences could answer a question from Armin Straub (Open Question 3.18 might also shed some light on this problem): Open Question 3.19. Find a C2-finite integer sequence a ∈ZN which does not grow faster than a D-finite (C-finite) sequence but is not D-finite (C-finite) itself. The sequence nn is another common counterexample for a sequence which is not D-finite [Ger04]. It turns out, this sequence is also not C2-finite. Example 3.20. The sequence nn is neither polynomial nor rational recursive, i.e., it cannot be described by a certain system of polynomial or rational difference equations [CMP+21]. As C2-finite sequences are examples of rational recursive sequences, the sequence nn is not C2-finite. 3.3 Algebraic characterization In this section we derive an equivalent characterization of C2-finite sequences in terms of finitely generated modules which is analogous to the characterization in Theorem 2.5 for D-finite sequences. This characterization can then be used to show that the set of C2-finite sequences forms a difference ring. For C-finite sequences c0, . . . , cr ∈RC we denote the smallest K-difference algebra (i.e., a K-algebra which is additionally closed under shifts) which contains the se-20 3 C2-finite sequences quences c0, . . . , cr by Kσ[c0, . . . , cr]. For a sequence a ∈KN and a subring S ⊆RC, we consider the module of shifts over the ring Q(S), ⟨σi(a) \| i ∈N⟩Q(S) where the scalar multiplication is given by the Hadamard product of sequences in KN. In Theorem 3.23 below, we prove that this module (with S = RC) is finitely generated if and only if the sequence is C2-finite. For this purpose, we show two auxiliary lemmas first. Lemma 3.21. Let a be C2-finite with annihilating operator A = c0 + · · · + crσr and let R := Kσ[c0, . . . , cr]. If S ⊇R is a subring of the ring of sequences KN, then ⟨σi(a) \| i ∈N⟩Q(S) is finitely generated. Proof. By assumption we have lc(A) = cr ∈R× C and Aa = 0. Let i ∈N. Then, σiA = σi(c0)σi + · · · + σi(cr)σi+r and lc(σiA) = σi(cr) ∈R× C . Since (σiA)a = σi(Aa) = 0, we can write σi+r(a) = −σi(c0) σi(cr) σi(a) −· · · −σi(cr−1) σi(cr) σi+r−1(a). Hence, for all i ∈N the sequence σi+r(a) is a Q(R)-linear combination of the sequences σi(a), . . . , σi+r−1(a). By induction, σi+r(a) is a Q(R)- and therefore a Q(S)-linear com-bination of a, σ(a), . . . , σr−1(a). Thus, the module ⟨σi(a) \| i ∈N⟩Q(S) is generated by a, σ(a), . . . , σr−1(a). Lemma 3.22. Let a ∈KN and S ⊆RC a subring. If ⟨σi(a) \| i ∈N⟩Q(S) is finitely generated, then a is C2-finite. Proof. As the module is finitely generated, we can write ⟨σi(a) \| i ∈N⟩Q(S) = ⟨g1, . . . , gm⟩Q(S) for some m and some sequences g1, . . . , gm. There exists an r ∈N such that the ele-ments gj can be written as gj = ∑r−1 i=0 ci,jσi(a) for some ci,j ∈Q(S). Then, σr(a) is a Q(S)-linear combination of g1, . . . , gm, so in particular a linear combination of the sequences 21 3 C2-finite sequences a, σ(a), . . . , σr−1(a). Hence, there exist sequences c0, . . . , cr−1 ∈S and d0, . . . , dr−1 ∈S× with σr(a) = c0 d0 a + c1 d1 σ(a) + · · · + cr−1 dr−1 σr−1(a). Clearing denominators shows that a is C2-finite of order at most r. Theorem 3.23. The following are equivalent: 1. The sequence a is C2-finite. 2. There exists A ∈RC[σ] with lc(A) ∈R× C and a C2-finite sequence b with Aa = b. 3. The module ⟨σi(a) \| i ∈N⟩Q(RC) over the ring Q(RC) is finitely generated. Proof. (1) ⇒(2): We can choose the C2-finite sequence b = 0. (2) ⇒(1): Since b is C2-finite, there exists an operator B ∈RC[σ] with lc(B) ∈R× C and Bb = 0. Then, (BA)a = B(Aa) = Bb = 0. Furthermore, lc(BA) = lc(B)σord(B)(lc(A)) ∈R× C . (1) ⇒(3): Follows from Lemma 3.21 with S = RC. (3) ⇒(1): Follows from Lemma 3.22 with S = RC. In the case of D-finite and C-finite sequences, the base ring of the finitely generated module is a field. As we have seen in Theorem 2.6 the key step for proving that these sets form rings makes use of the fact that submodules of finitely generated modules over fields (i.e., finite dimensional vector spaces) are again finitely generated. This holds more generally for Noetherian rings. However, the rings RC and Q(RC) are not Noetherian. Example 3.24. Let ck ∈RC be defined by ck(n) −ck(n + k) = 0 for every n ∈N, and ck(0) = · · · = ck(k −2) = 1, ck(k −1) = 0 (i.e., ck has a 0 at every k-th term and 1 else). Let Lm = ⟨c2, . . . , c2m⟩be ideals in RC for m ∈N. By construction, c2m / ∈Lm−1 since c(2m−1 −1) ̸= 0 and for every sequence d(n) ∈Lm−1 we have d(2m−1 −1) = 0. Hence, L1 ⊊L2 ⊊L3 ⊊· · · 22 3 C2-finite sequences is an infinitely properly ascending chain of ideals in RC. Therefore, RC is not a Noetherian ring. However, instead of considering the whole ring of C-finite sequences, we can instead limit the base ring to a ring of the form Kσ[c0, . . . , cr]. By the properties of C-finite sequences, these algebras are Noetherian rings. Lemma 3.25. Let c0, . . . , cr ∈RC. Then, Kσ[c0, . . . , cr] is a Noetherian ring. Proof. All the K-vector spaces ⟨σi(cj) \| i ∈N⟩K are finitely generated. Hence, also the difference algebras Kσ[cj] are finitely generated. Therefore, also Kσ[c0, . . . , cr] is finitely generated and a Noetherian ring [AM69, Corollary 7.7]. Theorem 3.26. The set of C2-finite sequences is a difference ring under termwise addition and termwise multiplication. Proof. Let a, b be C2-finite sequences and A = c0 + c1σ + · · · + cr1σr1 and B = d0 + d1σ + · · · + dr2σr2 the corresponding annihilating operators with c0, . . . , cr1, d0, . . . , dr2 ∈RC. Now, let S = Kσ[c0, . . . , cr1, d0, . . . , dr2]. By Lemma 3.25, this ring S is Noetherian. Therefore, also Q(S) is a Noetherian ring [AM69, Proposition 7.3]. By Lemma 3.21, the modules ⟨σi(a) \| i ∈N⟩Q(S) and ⟨σi(b) \| i ∈N⟩Q(S) are both finitely generated Q(S)-modules. Hence, also the modules ⟨σi(a + b) \| i ∈N⟩Q(S) ⊆⟨σi(a) \| i ∈N⟩Q(S) + ⟨σi(b) \| i ∈N⟩Q(S) and ⟨σi(ab) \| i ∈N⟩Q(S) ⊆⟨σi(a)σj(b) \| i, j ∈N⟩Q(S) are finitely generated as they are submodules of finitely generated modules over a Noethe-rian ring. By Lemma 3.22, the sequences a + b and ab are C2-finite. Therefore, the set of C2-finite sequences is a ring. 23 3 C2-finite sequences The operator ˜ A = σ(c0) + σ(c1)σ + · · · + σ(cr1)σr1 ∈RC[σ] annihilates σ(a) as ˜ A(σ(a)) = ( ˜ Aσ)a = (σA)a = σ(Aa) = 0. Furthermore, we have lc( ˜ A) = σ(cr1) ∈R× C . Hence, the ring of C2-finite sequences is also closed under shifts. The statement that C2-finite sequences form a ring can also be found in [KM14, Corol-lary 15]. Our proof is different and closer resembles the proofs for the classical C-finite and D-finite cases. Furthermore, their proof of Lemma 14, on which their result builds, seems to contain a mistake. In the proof they choose a certain sequence sn which is not guaranteed to exist. 3.4 Generating functions Switching between the generating function representation g(x) = ∑n≥0 a(n)xn ∈KJxK of a D-finite (or C-finite) sequence a(n) ∈KN and back often turns out to be a useful technique when proving properties of sequences. In this section we examine the generating functions of C2-finite sequences. For a C2-finite sequence a over the field K with annihilating operator c0 + · · · + crσr, the smallest field L ⊇K which contains all the splitting fields of the characteristic polynomials of c0, . . . , cr is called the splitting field of a. For natural numbers n, k ∈N we write nk = n(n −1) · · · (n −k + 1) 24 3 C2-finite sequences for the falling factorial. Let g(x) = ∑n≥0 a(n)xn ∈LJxK. Then, for λ ∈L we write g(d)(λx) for the d-th derivative of the formal power series g(λx) ∈LJxK, i.e., g(d)(λx) = ∑ n≥d ndλna(n)xn−d. Theorem 3.27. Let a be a C2-finite sequence over K with splitting field L. Let g(x) = ∑n≥0 a(n)xn be its generating function. Then, g(x) satisfies a functional equation of the form m ∑ k=1 pk(x)g(dk)(λkx) = p(x) (3.3) for p, p1, . . . , pm ∈L[x], d1, . . . , dm ∈N and λ1, . . . , λm ∈L. Proof. Consider the defining recurrence of a: c0(n)a(n) + · · · + cr(n)a(n + r) = 0, for all n ∈N. Multiplying by xn and summing over all n ∈N yields ∑ n≥0 c0(n)a(n)xn + · · · + ∑ n≥0 cr(n)a(n + r)xn = 0. (3.4) The coefficients c0, . . . , cr have some closed form for all n ≥n0. Hence, the left-hand side of equation (3.4) is just an L-linear combination of power series of the form ˜ h(x) = ∑n≥n0 njλna(n + i)xn for j ∈N, i ∈{0, . . . , r}, λ ∈L. The first terms n = 0, . . . , n0 −1 in (3.4) just yield some polynomial factors. Furthermore, it is sufficient to consider formal power series of the form h(x) = ∑n≥0 njλna(n + i)xn as h(x) −˜ h(x) is again polynomial. Hence, also these factors h(x) −˜ h(x) contribute to the right-hand side of (3.3). 25 3 C2-finite sequences Let S(k, l) denote the Stirling numbers of the second kind. Then, nk = ∑k l=0 S(k, l)nl. Therefore, h(x) = ∑ n≥i (n −i)jλn−ia(n)xn−i = ∑ n≥i ( j ∑ k=0 (j k ) nk(−i)j−k ) λn−ia(n)xn−i = j ∑ k=0 k ∑ l=0 (j k ) (−i)j−kS(k, l) ∑ n≥i nlλn−ia(n)xn−i = j ∑ k=0 k ∑ l=0 (j k ) (−i)j−kS(k, l) xl−i λi ∑ n≥i nlλna(n)xn−l = j ∑ k=0 k ∑ l=0 (j k ) (−i)j−kS(k, l) xl−i λi ( g(l)(λx) + pl(x) ) where pl(x) ∈L[x] is defined as pl(x) = ⎧ ⎨ ⎩ −∑i−1 n=l nlλna(n)xn−l, if i > l, 0, otherwise. Hence, h(x) = ∑ j l=0 ql(x)g(l)(λx) + q(x) with q0, . . . , qj, q ∈L(x). Using this in equa-tion (3.4) and clearing the denominator xr yields a functional equation of the desired form. This functional equation is nontrivial, i.e., the left-hand side of (3.3) does not simplify to zero: Fix some λ and consider a term njλna(n + i) with j maximal and i minimal among these maximal j. This term yields a nonzero term xj−i+rg(j)(λx) in the functional equation which cannot be canceled because of the choice of j and i. The proof of Theorem 3.27 uses the closed form representation of the C-finite coefficients and generalizes the classical proof for D-finite sequences [Sta80, Mal96, KP11]. A close investigation of the proof shows the following bounds (in the special case of D-finite sequences we get precisely the known bounds): 1. We have deg(pk) ≤r + maxi(ord(ci)) −1. 2. The λk are exactly the eigenvalues of the C-finite coefficients ci. 26 3 C2-finite sequences 3. The derivatives dk are each bounded by the multiplicity of the eigenvalue λk in any ci. In particular, maxk(dk) ≤maxi(ord(ci)) −1. 4. Let n0 be minimal such that all c0, . . . , cr have closed forms from n0 on. We have deg(p) < max(r, n0). If we differentiate the functional equation max(r, n0) times, we get a homogeneous functional equation (i.e., p = 0). The functional equation then satisfies maxk(dk) ≤maxi(ord(ci)) −1 + max(r, n0). Theorem 3.27 also generalizes the result for q-holonomic sequences: Every generating function of a q-holonomic sequence satisfies a q-shift equation [KK09]. In this case we would have λk = qk. Example 3.28. Let a(n) := f (n2) be the sparse subsequence of the Fibonacci sequence f (cf. Example 3.7). The generating function g of a satisfies the functional equation ( ϕ3x2 −ϕ−3) g ( ϕ2x ) − ( ψ3x2 −ψ−3) g ( ψ2x ) + xg ( ϕ4x ) −xg ( ψ4x ) = (ψ −ϕ)x where ϕ := 1+ √ 5 2 denotes the golden ratio and ψ := 1− √ 5 2 its conjugate. Example 3.29. Since 1 n! is C2-finite (as it is D-finite), the coefficient sequence of the expo-nential generating function ∑n≥0 a(n) n! xn of a C2-finite sequence a is again C2-finite. Let b be the coefficient sequence of the exponential generating function of the fibonorial numbers ∏n i=1 f (i), where f denotes the Fibonacci numbers (cf. Example 3.5). Then, b satisfies f (n + 1)b(n) −(n + 1)b(n + 1) = 0, for all n ∈N. Let h(x) = ∑n≥0 b(n)xn be the generating function of b. Then, h satisfies ϕ h(ϕx) −ψ h(ψx) −(ϕ −ψ) h′(x) = 0 where ϕ, ψ are as in Example 3.28. Theorem 3.27 shows that the generating function g(x) of a C2-finite sequence satisfies a linear differential equation with polynomial coefficients which can also contain terms involving the generating function with scaled argument g(λx). On the contrary, we can ask whether the coefficient sequence of any function satisfying an equation of this type is C2-finite. This is not necessarily the case. In general, the coefficient sequence only satisfies 27 3 C2-finite sequences a linear recurrence with C-finite coefficients where the leading coefficient might have infinitely many zeros. I.e., such sequences are always X-recursive. Theorem 3.30. Let g(x) = ∑n≥0 a(n)xn satisfy a functional equation of the form m ∑ k=1 pk(x)g(dk)(λkx) = p(x) for p, p1, . . . , pm ∈L[x], d1, . . . , dm ∈N and λ1, . . . , λm ∈L. Then, the coefficient se-quence a(n) satisfies a linear recurrence with C-finite coefficients over L. Proof. The functional equation is an L-linear combination of functions xjg(d)(λx) = xj ∑ n≥d ndλna(n)xn−d = ∑ n≥j (n + d −j)dλn+d−ja(n + d −j)xn. We can compute this for every factor appearing in the functional equation. Comparing the coefficients yields a linear recurrence with C-finite coefficients. Example 3.31. Let g(x) = ∑n≥0 a(n)xn satisfy the equation xg(2x) + g(x) = 1. Then, a(0) = 1 and 2na(n) + a(n + 1) = 0, for all n ∈N. I.e., a(n) is the sequence from Example 3.4. The equation satisfied by even and odd functions are of the form (3.3). A function g(x) = ∑n≥0 a(n)xn satisfies the equation g(x) = g(−x) (i.e., is even) if and only if the coefficient sequence a(n) satisfies (1 −(−1)n)a(n) = 0 for all n ∈N (i.e., a(n) = 0 for all odd n ∈N). By construction, C2-finite sequences are uniquely defined by finitely many elements α ∈K. This means in particular that there are only countably many C2-finite sequences if the underlying field K is countable. On the other hand, there are uncountably 28 3 C2-finite sequences many even functions. Hence, the coefficient sequences of functions satisfying a functional equation of the form (3.3) are not C2-finite in general. D-finite and C-finite sequences a, b are closed under both termwise multiplication (ab)(n) = a(n)b(n) and the Cauchy product (a ⊙b)(n) := ∑n i=0 a(i)b(n −i). It is not clear whether C2-finite sequences are closed under the Cauchy product. Open Question 3.32. Is the Cauchy product of two C2-finite sequences C2-finite again? The sequences a(n) = 2n2, b(n) = 3n2 are both C2-finite. it is not known whether a ⊙b is C2-finite. In the restricted setting that one of the two C2-finite sequences is in fact C-finite, we know that the Cauchy product is C2-finite again. Lemma 3.33. Let a be C2-finite and b be C-finite over K. Then, the Cauchy product c := a ⊙b is again C2-finite over the splitting field L of the characteristic polynomial of b. Proof. First, let b(n) = ndλn for all n ∈N for some k ∈N, λ ∈L and c = a ⊙b. Furthermore, we denote al(n) := ∑n i=0 a(i)(n −i)lλn−i for l = 0, . . . , d. Then, for all j ∈N, n ∈N we have σj(c(n)) = n+j ∑ i=0 a(i)(n + j −i)dλn+j−i = d ∑ l=0 λj (d l ) jd−l n+j ∑ i=0 a(i)(n −i)lλn−i = d ∑ l=0 λj (d l ) jd−lal(n) + d ∑ l=0 λj (d l ) jd−l j ∑ i=1 a(n + i)(−i)lλ−i. Let A = c0 + c1σ + · · · + crσr be an annihilating operator of a. With Lemma 3.25, the ring R := Lσ[c0, . . . , cr] is Noetherian. The computation above shows ⟨σj(c) \| j ∈N⟩Q(R) ⊆⟨a0, . . . , ad⟩Q(R) + ⟨σj(a) \| j ∈N⟩Q(R). With Lemma 3.21, the module on the right-hand side is finitely generated, hence also the module on the left-hand side is finitely generated. Therefore, with Lemma 3.22, the sequence c is C2-finite. As C2-finite sequences are closed under termwise addition (Theorem 3.26) and every C-finite sequence is just a linear combination of such exponential sequences from some term n0 on, the Cauchy product of a C2-finite sequence with a C-finite sequence is again C2-finite. 29 3 C2-finite sequences Example 3.34. Let a(n) := 2n2, b(n) := 3n, c := a ⊙b. Then, c is again C2-finite and satisfies 4nc(n) −( 1 34n + 1 8)c(n + 1) + 1 24c(n + 2) = 0, for all n ∈N and c(0) = 1, c(1) = 5. C2-finite sequences generalize the sequence definition of a D-finite/C-finite sequence. As these sequences have an equivalent characterization in terms of their generating functions (Theorem 2.7), one could also aim to generalize these. I.e., one could, for instance, study power series which satisfy a linear differential equation with coefficients that are D-finite themselves. Such functions are called DD-finite [JPP19, JPP18]. These functions are, however, not closed under termwise multiplication [BJP20]. Hence, there cannot be a one-to-one correspondence of a generalization of the sequence definition of D-finite sequences and of the function definition of D-finite functions. It might, however, still be interesting to investigate the relationship between these two variants. Open Question 3.35. Study the relationship between C2-finite (or D2-finite sequences) and DD-finite functions. 30 4 Computations with C2-finite sequences The closure properties of D-finite and C-finite sequences are all entirely computable. The situation is more complicated for C2-finite sequences. In this chapter we study how closure properties can, in principle, be computed for C2-finite sequences. The chapter is mostly based on the papers [JPNP21, JPNP23]. Suppose we consider a C2-finite sequence c for which no recurrence is known yet (think about c being the sum of two C2-finite sequences, for instance). We can make an ansatz for a C2-finite recurrence y0c + y1σ(c) + · · · + ys−1σs−1(c) + ysσs(c) = 0 with coefficients y0, . . . , ys ∈RC and ys ∈R× C that we have yet to determine. Dividing by the leading coefficient yields a recurrence of the form x0c + x1σ(c) + · · · + xs−1σs−1(c) + σs(c) = 0 (4.1) with unknown x0, . . . , xs−1 ∈Q(RC). By Theorem 3.23, since c is C2-finite, there are finitely many sequences g1, . . . , gr such that ⟨σi(c) \| i ∈N⟩Q(RC) = ⟨g1, . . . , gr⟩Q(RC). 31 4 Computations with C2-finite sequences In particular, there are sequences wi,k ∈Q(RC) such that σi(c) = ∑r k=1 wi,kgk for all i ∈N. Using these in equation 4.1 yields 0 = x0 r ∑ k=1 w0,kgk + x1 r ∑ k=1 w1,kgk + · · · + xs−1 r ∑ k=1 ws−1,kgk + r ∑ k=1 ws,kgk = r ∑ k=1 gk ( s−1 ∑ i=0 wi,kxi + ws,k ) . Hence, if we find x0, . . . , xs−1 ∈Q(RC) such that s−1 ∑ i=0 wi,kxi = −ws,k, for all k = 1, . . . , r, (4.2) then we have found a solution of (4.1) and therefore a C2-finite recurrence for c. To write the linear system (4.2) more concisely we denote wi = ( wi,1, . . . , wi,r )⊤ ∈Q(RC)r, for i = 0, . . . , s, and x = ( x0, . . . , xs−1 )⊤ . Then, (4.2) reads as an inhomogeneous system of linear equa-tions of size r × s over Q(RC) ( w0, . . . , ws−1 ) x = −ws. (4.3) In Section 4.1 we show how this linear system, in particular the vectors wi, can be computed for different closure properties. The difference compared to the classical D-finite or C-finite case is that the order of the ansatz s is not known a priori. In Section 4.2 we show that for big enough s the linear system (4.3) has a solution, so a C2-finite recurrence of order s can be found. Further, we show a method (which is limited by the Skolem Problem again) for solving such linear systems. 32 4 Computations with C2-finite sequences 4.1 Setting up the linear system Suppose we have a C2-finite sequence c. By Theorem 3.23 there are finitely many sequences G = (g1, . . . , gr) such that ⟨σi(c) \| i ∈N⟩Q(RC) = ⟨g1, . . . , gr⟩Q(RC). In this section we show how sequences wi,k ∈Q(RC) with σi(c) = ∑r k=1 wi,kgk can be computed. First, we study the case where a recurrence of order r for c is given and we want to find these coefficients with respect to the generators Gc := ( c, σ(c), . . . , σr−1(c) ) . Based on this, we consider the case where c = a + b and c = ab where a, b are C2-finite sequences of order r1, r2, respectively. The generating sets in these cases are given by Ga ⊕Gb = ( a, σ(a), . . . , σr1−1(a), b, σ(b), . . . , σr2−1(b) ) in the case of addition and Ga ⊗Gb = ( ab, aσ(b), . . . , aσr2−1(b), . . . , σr1−1(a)b, σr1−1(a)σ(b), . . . , σr1−1(a)σr2−1(b) ) in the case of multiplication. Next, we consider the case a(kn) where a is a C2-finite sequence of order r and k ∈N. The generating set is given by (a(kn), . . . , a(kn + r −1)). Finally, we consider the case c(jn2 + kn + ℓ) for a C-finite sequence c where j, k, ℓ∈N. Here, the generating set is given by (c(jn2), . . . , c(jn2 + r −1)). 4.1.1 Ring computations Let a be C2-finite of order r with annihilating operator c0 + · · · + cr−1σr−1 + σr ∈Q(RC)[σ]. We write the components of a vector ui ∈Q(RC)r as ui,k for k = 0, . . . , r −1. The componentwise shift of a vector ui is simply denoted by σ(ui), i.e., (σ(ui))k = σ(ui,k). The i-th unit vector is denoted by e(r) i ∈Q(RC)r for i = 0, . . . , r −1. Note that, e.g., e(r) 0 = (1, 0, . . . , 0)⊤. 33 4 Computations with C2-finite sequences Lemma 4.1. Initialize ui := e(r) i ∈Q(RC)r with the unit vectors for i = 0, . . . , r −1 and define ui := − r−1 ∑ k=0 σi−r(ck)ui+k−r (4.4) inductively for i ≥r. These ui ∈Q(RC) satisfy σi(a) = r−1 ∑ k=0 ui,kσk(a) (4.5) for all i ∈N or σi(a) = Gaui written concisely. Proof. We show equation (4.5) by induction on i. It clearly holds for i = 0, . . . , r −1 by the definition of the ui. Shifting the defining recurrence of a yields σi(a) = − r−1 ∑ j=0 σi−r(cj)σi+j−r(a), for i ≥r. Let us assume that equation (4.5) holds for a, . . . , σi−1(a). Then, r−1 ∑ k=0 ui,kσk(a) = − r−1 ∑ k=0 r−1 ∑ j=0 σi−r(cj)ui+j−r,kσk(a) = − r−1 ∑ j=0 σi−r(cj)σi+j−r(a) = σi(a). A different way to compute the vectors uj is to use the companion matrix of a sequence. The companion matrix Ma of the sequence a with annihilator c0 + c1σ + · · · + cr−1σr−1 + σr ∈ Q(RC)[σ] is defined as Ma := ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 −c0 1 0 . . . 0 −c1 0 1 . . . 0 −c2 . . . . . . ... . . . . . . 0 0 . . . 1 −cr−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈Q(RC)r×r. 34 4 Computations with C2-finite sequences Lemma 4.2. Let Ma be the companion matrix of a. Let u0 := e(r) 0 = (1, 0, . . . , 0)⊤ and define ui := Maσ(ui−1) inductively for i ≥1. 1. These ui are identical to the vectors from Lemma 4.1. 2. The ui satisfy equation (4.5). Proof. (1): Clearly ui = e(r) i for i = 0, . . . , r −1 by the definition of the companion matrix. For i ≥r we show that equation (4.4) from Lemma 4.1 is satisfied using induction on i. For i = r we have ur = ( −c0, . . . , −cr−1 )⊤ by the definition of the companion matrix. Therefore, − r−1 ∑ k=0 ckuk = − r−1 ∑ k=0 cke(r) k = ur. Now, we assume that equation (4.4) from Lemma 4.1 holds for i −1, i.e., ui−1 = − r−1 ∑ k=0 σi−1−r(ck)ui−1+k−r. (4.6) Using equation (4.6) and the definition of the ui we have ui = Maσ(ui−1) = −Ma r−1 ∑ k=0 σi−r(ck)σ(ui−1+k−r) = − r−1 ∑ k=0 σi−r(ck)ui+k−r. (2): Follows directly from part (1) and Lemma 4.1. 35 4 Computations with C2-finite sequences Consider two C2-finite sequences a, b. To compute the vectors wi in the linear system (4.3) for a + b we can concatenate the vectors we get from Lemma 4.1. Alternatively, we can use a similar approach as in [JPP18]. Suppose a, b have orders r1, r2, respectively. Define M := Ma ⊕Mb = ( Ma 0 0 Mb ) ∈Q(RC)r1+r2×r1+r2 to be the direct sum of the companion matrices of a and b and w0 := e(r1) 0 ⊕e(r2) 0 = ( e(r1) 0 e(r2) 0 ) ∈Q(RC)r1+r2. Let us define wi := Mσ(wi−1) iteratively. If we denote the first r1 components of wi by ui and the last r2 components by vi, these ui, vi clearly satisfy (4.5) by the construction of M and Lemma 4.2 for Ga, Gb, respectively. Therefore, (Ga ⊕Gb)wi = (Ga ⊕Gb)(ui ⊕vi) = Gaui + Gbvi = σi(a) + σi(b) = σi(a + b). Analogously, for the multiplication we can define M as the Kronecker product M := Ma ⊗Mb ∈Q(RC)r1r2×r1r2 of the two companion matrices. Again, defining w0 := e(r1) 0 ⊗e(r2) 0 and wi := Mσ(wi−1), we have wi = ui ⊗vi where ui, vi satisfy Gaui = σi(a) and Gbvi = σi(b). Therefore, (Ga ⊗Gb)wi = (Ga ⊗Gb)(ui ⊗vi) = (Gaui)(Gbvi) = σi(a)σi(b) = σi(ab). Algorithm 1 summarizes the arguments from the introduction of Chapter 4 and this section. The algorithm computes a recurrence for the addition or multiplication of two C2-finite sequences a, b of orders r1, r2 provided that we can solve linear systems of equations over Q(RC). The termination of Algorithm 1 follows from Lemma 4.9 which we show in the next section. 36 4 Computations with C2-finite sequences Input :C2-finite sequences a, b of order r1, r2, respectively output:C2-finite recurrence satisfied by a + b (or ab, respectively) M ←Ma ⊕Mb (or Ma ⊗Mb for the multiplication) A ←empty matrix w ←e(r1) 0 ⊕e(r2) 0 (or e(r1) 0 ⊗e(r2) 0 for the multiplication) for s = 0, 1, 2, . . . do if solution x ∈Q(RC)s of Ax = −w exists then return ∑s−1 i=0 xiσi + σs else A ←(A \| w) w ←Mσ(w) end end Algorithm 1: Computing C2-finite ring operations Example 4.3. Let b(n) := 1 f (n+1) where f denotes the Fibonacci numbers (cf. Example 2.13 and Example 3.6). The sequence b is C2-finite satisfying f (n + 1)b(n) −f (n + 2)b(n + 1) = 0. We want to compute a recurrence for c(n) := f (n)b(n). The companion matrices of f and b are given by Mf = ( 0 1 1 1 ) , Mb = ( f (n+1) f (n+2) ) . Therefore, M = Mf ⊗Mb = ⎛ ⎝ 0 f (n+1) f (n+2) f (n+1) f (n+2) f (n+1) f (n+2) ⎞ ⎠, w = ( 1 0 ) ⊗ ( 1 ) = ( 1 0 ) . Hence, the linear system corresponding to an ansatz of order 2 is ( 1 0 0 f (n+1) f (n+2) ) ( x0(n) x1(n) ) = ⎛ ⎝−f (n+1) f (n+3) −f (n+1) f (n+3) ⎞ ⎠. 37 4 Computations with C2-finite sequences The system clearly has the solution x0(n) = −f (n+1) f (n+3), x1(n) = −f (n+2) f (n+3) which gives rise to the recurrence −f (n + 1)c(n) −f (n + 2)c(n + 1) + f (n + 3)c(n + 2) = 0. Analogously we can find that d(n) := ∑n k=1 (−1)k f (k) f (k+1) satisfies the recurrence −f (n + 2)d(n) −f (n + 3)d(n + 1) + f (n + 4)d(n + 2) = 0. The sequence c(n) + d(n) is then again C2-finite and we can find a recurrence of order 3. As the first 3 initial values are 0, we have c(n) + d(n) = 0 for all n ∈N as shown in [Kau05, Example 4.7]. 4.1.2 Subsequences Let a be C2-finite of order r with annihilating operator c0 + · · · + cr−1σr−1 + σr ∈Q(RC)[σ]. Let b(n) = a(kn) for some k ∈N. Then, σi(b(n)) = a(kn + ki). Let G := ( a(kn), . . . , a(kn + r −1) ) . The next lemma shows that ⟨σi(b) \| i ∈N⟩Q(RC) = ⟨G⟩Q(RC). In particular, we find vectors wi ∈Q(RC)r such that Gwi = σi(b). Lemma 4.4. Define w0 := e(r) 0 and wi(n) := Ma(kn) · · · Ma(kn + k −1)wi−1(n + 1), for all n ∈N, iteratively. Then, Gwi = σi(b) for all i ∈N. 38 4 Computations with C2-finite sequences Proof. We use induction on i. Clearly, Gw0 = Ge(r) 0 = b. Suppose that Gwi−1 = σi−1(b). By the definition of the companion matrix Ma, we have G(n)wi(n) = G(n)Ma(kn) · · · Ma(kn + k −1)wi−1(n + 1) = G(n + 1)wi−1(n + 1) = b(n + i) where we use the shifted induction hypothesis in the last step. Hence, Gwi = σi(b). Lemma 4.4 shows in particular that ⟨σi(b) \| i ∈N⟩Q(RC) is finitely generated. Hence, the sequence b is C2-finite by Theorem 3.23. Theorem 4.5. Let a be C2-finite and k ∈N. Then, the sequence b(n) = a(kn) is C2-finite. For computing a recurrence for a(kn) we can adjust Algorithm 1. Choosing M = Ma(kn) · · · Ma(kn + k −1) ∈Q(RC)r×r and w = e(r) 0 ∈Q(RC)r gives an explicit algorithm for computing a recurrence for the subsequence of a C2-finite sequence. 4.1.3 Sparse subsequences of C-finite sequences Let c be C-finite of order r with annihilating operator c0 + · · · + cr−1σr−1 + σr ∈K[σ]. Let a(n) = c(jn2 + kn + ℓ) for some j, k, ℓ∈N and G := ( c(jn2), . . . , c(jn2 + r −1) ) . The next lemma shows that ⟨σi(a) \| i ∈N⟩Q(RC) = ⟨G⟩Q(RC). In particular, we find vectors wi ∈Q(RC)r such that Gwi = σi(a). The proof is based on ideas from [KM14, Theorem 1]. Since c is C-finite, Mc ∈Kr×r. We use the fact that 39 4 Computations with C2-finite sequences (Mp(n) c )n∈N for a linear polynomial p ∈N[n] can also be viewed as a matrix of C-finite sequences [KM14, Lemma 11]. Lemma 4.6. Let wi(n) = M2jni+ji2+kn+ki+ℓ−r+1 c e(r) r−1. These wi ∈Q(RC)r satisfy Gwi = σi(a) for all i ∈N. Proof. Let Gc(n) = ( c(n), . . . , c(n + r −1) ) . Then, Gc(jn2) = G(n). By the definition of the companion matrix we have Gc(n + 1) = Gc(n)Mc (4.7) for all n ∈N. Using n →jn2 we have Gc(jn2 + 1) = Gc(jn2)Mc. Repeated application of equation (4.7) yields Gc(j(n + i)2 + k(n + i) + ℓ−r + 1) = Gc(jn2)M2jni+ji2+kn+ki+ℓ−r+1 c . Multiplying by e(r) r−1 and using the definition of the Gc(n) yields Gwi = σi(a). Lemma 4.6 shows in particular that ⟨σi(a) \| i ∈N⟩Q(RC) is finitely generated. Hence, the sequence a is C2-finite by Theorem 3.23. Theorem 4.7. Let c be C-finite and j, k, ℓ∈N. Then, the sequence a(n) = c(jn2 + kn + ℓ) is C2-finite. An alternative method for proving a variant of Theorem 4.7 using the closed form of a C-finite sequence is given in [JPNP21, Corollary 3.6] (and a more general version in Theorem 7.11 in this thesis). This alternative proof, however, only guarantees a C2-finite recurrence for a sequence c(jn2 + kn + ℓ) ∈KN over an extension field L ⊇K. Suppose a 40 4 Computations with C2-finite sequences sequence a ∈KN is D-finite over some extension field L ⊇K, then a is actually D-finite over K [Ger05, Lemma 1.3.2]. It would be interesting to see whether the same holds for C2-finite sequences. Open Question 4.8. Suppose a ∈KN is C2-finite with C-finite coefficients over L ⊇K. Can we compute a C2-finite recurrence with C-finite coefficients over K for a? For computing a recurrence for a(n) we can, again, adjust Algorithm 1. By Lemma 4.6 we have w0(n) = Mkn+ℓ−r+1 c e(r) r−1 and wi(n) = Mj(2n+1) c wi−1(n + 1). Hence, choosing M = Mj(2n+1) c and w = w0 in Algorithm 1 gives an explicit algorithm for computing a recurrence for a(n). By [KM14, Lemma 11], w ∈Q(RC)r, M ∈Q(RC)r×r. These recurrences can be computed by using either the Cayley-Hamilton theorem (as suggested by the proof of the lemma) or using guessing. In the very same way we could compute recurrences for a(n) = c(j(n 2) + kn + ℓ) for j, k, ℓ∈N. 4.2 Solving the linear system So far, we have shown that we can set up a linear inhomogeneous system of equations over the ring Q(RC) such that a solution to this system gives rise to a recurrence for the closure properties which we want to compute. First, we show that this algorithm terminates, i.e., if the linear system is big enough, then the system has a solution. Lemma 4.9. The order of the ansatz s in (4.1) can be chosen big enough such that the corresponding linear system (4.3) for the addition and multiplication of two C2-finite sequences has a solution. Proof. Let c0 + · · · + cr1σr1 ∈Q(RC)[σ], d0 + · · · + dr2σr2 ∈Q(RC)[σ] be annihilators of the C2-finite sequences a, b. Writing As = (w0, . . . , ws), equation (4.3) reads as Asx = −ws. Let R := Kσ[c0, . . . , cr1, d0, . . . , dr2]. 41 4 Computations with C2-finite sequences In Section 4.1.1 we have shown that all sequences in the linear system As = −ws are in the ring Q(R). By Lemma 3.25 this ring is Noetherian. Hence, writing Im A for the image of a matrix A, the increasing chain of modules Im A0 ⊆Im A1 ⊆Im A2 ⊆· · · has to stabilize. In particular, there is an s such that Im As = Im As+1. Therefore, ws ∈ Im As+1 = Im As, so Asx = −ws has a solution x ∈Q(R)s. The proof of Lemma 4.9 is not constructive as the properties of the Noetherian ring only gives us the existence of a number s. The same proof, however, can also be used to show that the ansatz for computing a subsequence of a C2-finite sequence or a sparse subsequence of a C-finite sequence can be chosen big enough such that the corresponding linear system has a solution. In the case of D-finite and C-finite sequences, the s in Lemma 4.9 can be chosen as at most ord(a) + ord(b) in the case of addition a + b and ord(a) ord(b) in the case of multipli-cation ab. These results follow directly from a dimension argument of the corresponding vector spaces. For C2-finite sequences these order bounds cannot be used anymore. Example 4.10. Let c ∈RC be the cyclic sequence of order m defined by c(n) −c(n + m) = 0, c(0) = −1, c(1) = · · · = c(m −1) = 1 and let a, b be C2-finite sequence defined by c(n)a(n) −a(n + 1) = 0, a(0) = 1, b(n) −b(n + 1) = 0, b(0) = 1. Suppose we make an ansatz of order s < m + 1 for a + b. With the definition of c, the corresponding linear system at n = m −s + 1 is of the form ( 1 1 · · · 1 1 1 · · · 1 ) ⎛ ⎜ ⎜ ⎝ x0(n) . . . xs−1(n) ⎞ ⎟ ⎟ ⎠= ( 1 −1 ) . Hence, the linear system has no solution. For s = m + 1 we get a solution for every n and therefore a C2-finite recurrence for a + b of order m + 1. 42 4 Computations with C2-finite sequences Example 4.10 indicates that any order bounds for C2-finite sequences should depend not only on the orders of the C2-finite recurrences but also on the orders of the C-finite coefficients. In Chapter 5 we derive such order bounds. Next, we study how the linear system (4.3) can be solved. The main obstacle here is, again, the Skolem Problem. If the zeros of all minors of the linear system can be computed, then we have an algorithm for computing a solution of the linear system. This algorithm is based on methods from [KM14]. Theorem 4.11. Let A ∈Q(RC)r×s and w ∈Q(RC)r. Suppose the linear system Ax = w has a solution x ∈Q(RC)s. Provided that the Skolem Problem is decidable, we can compute a solution x ∈Q(RC)s. Proof. All minors of A are sequences in Q(RC). Consider the set of all these. By the Skolem-Mahler-Lech theorem, the zeros of these minors are cyclic. Let p ∈N be such that all minors have cycle-length p from the term n0 ∈N on. These numbers p, n0 can be computed if the Skolem Problem is decidable. We write A = (w0, . . . , ws−1) for w0, . . . , ws−1 ∈Q(RC)r. Now, for every m ∈{n0, . . . , n0 + p −1} we can compute a subset jm ⊆{0, . . . , s −1} such that the vectors {wj(m) \| j ∈jm} ⊆Kr are maximally linearly independent, i.e., they are linearly independent and generate the same subspace as {w0(m), . . . , ws−1(m)}. By the choice of n0 and p this is also true for all n = m + pk for k ∈N, i.e., the vectors {wj(m + pk) \| j ∈jm} ⊆Kr are maximally linearly independent for every k ∈N. Let us denote by Am ∈Q(RC)r×\|jm\| the submatrix of A where we keep the columns wj with j ∈jm. For every m, we can solve the system Am(m + pk)xm(k) = w(m + pk), for all k ∈N, (4.8) 43 4 Computations with C2-finite sequences using the Moore-Penrose-Inverse [BIG03]: By the choice of m, p, n0, the matrix Am(m + pk) has linear independent columns for every k ∈N. Therefore, the Gramian matrix G(k) = Am(m + pk)⊤Am(m + pk) is regular for every k and (det(G(k)))k∈N ∈R× C . Now, let xm(k) = 1 det(G(k)) cof (G(k)) Am(m + pk)⊤w(m + pk) where cof(·) denotes the transposed cofactor matrix. Then, since equation (4.8) has a termwise solution, (xm(k))k∈N ∈Q(RC)\|jm\| satisfies equation (4.8) by the theory of Moore-Penrose-Inverses. Let x′ m ∈Q(RC)s be the vector where we add 0 ∈Q(RC) at the indices j ∈{0, . . . , s −1} \ {jm}. Now, the solution x for the entire system can be computed as the interlacing of x′ n0, . . . , x′ n0+p−1 from n0 on and the first n0 values can be computed explicitly. Then, x ∈Q(RC)s as Q(RC) is closed under interlacing and specifying finitely many initial values. The proof of Theorem 4.11 in fact also shows that whenever the linear system A(n)x(n) = w(n) with A ∈Q(RC)r×s and w ∈Q(RC)r has a solution x(n) ∈Ks for every n ∈N, then a solution x ∈Q(RC)s exists. Of course, also vice versa, if a sequence solution exists, then we have a termwise solution. In Algorithm 2 we give the algorithm suggested by Theorem 4.11 in pseudocode. For a sequence c ∈Q(RC) we denote the start and the length of the zero-cycle of the sequence c by period_start(c) and period_length(c), respectively, i.e., the numbers n0, p ∈N such that (c(pk + n0), . . . , c(pk + p −1 + n0)) has the same zero-pattern for every k ∈N. Even though Theorem 4.11 heavily relies on the Skolem Problem in theory, in practice the algorithm can, in many cases, be used for solving linear systems over the C-finite sequence ring. Using the techniques from Chapter 8, the zeros of C-finite sequences can often be found. The problem of the algorithm is rather that it is computationally too expensive if the Moore-Penrose inverses are computed explicitly. However, also if a classical row reduction algorithm such as the fraction-free Bareiss algorithm (cf. [Bar68]) is used, the 44 4 Computations with C2-finite sequences Input : A ∈Q(RC)r×s, w ∈Q(RC)r output: x ∈Q(RC)s with Ax = w if it exists and false otherwise Φ ←minors of A p ←lcm(period_length(ϕ) \| ϕ ∈Φ) n0 ←max(period_start(ϕ) \| ϕ ∈Φ) if A(m)x(m) = w(m) is not solvable for an m ∈{0, . . . , n0 −1} then return false end for m = n0, . . . , n0 + p −1 do jm ←indices of columns of A(m) which are maximally linearly independent Am ←matrix built by columns jm of A A′ m ←(Am(m + pk))k∈N w′ ←(w(m + pk))k∈N G ←(A′ m)⊤A′ m xm ← 1 det(G) cof(G)(A′ m)⊤w′ ∈Q(RC)\|jm\| x′ m ←insert 0 in xm at indices j ∈{0, . . . , s −1} \ {jm} end x ←interlacing of sequences x′ n0, . . . , x′ n0+p−1 with prepended terms x(0), . . . , x(n0 −1) if Ax = w then return x else return false end Algorithm 2: Solving linear systems over Q(RC) 45 4 Computations with C2-finite sequences computations are very expensive due to a blowup of the order of the C-finite sequences and their coefficients in the linear system. Furthermore, the solutions x ∈Q(RC)s that are computed are often too big. Cancelling common factors is, however, again a difficult problem [KZ08, KZ18]. 46 5 Order bounds for C2-finite closure properties In the previous chapters a C2-finite sequence a was annihilated by an operator A with lc(A)(n) ̸= 0 for all n ∈N. As discussed on page 17 we can equivalently allow annihilating operators A with lc(A)(n) ̸= 0 for almost all n ∈N. The set of sequences annihilated by these operators are the same, but the order of a sequence might be different. For deriving order bounds for C2-finite sequences, we need the latter definition, i.e., we need to allow finitely many zeros in the leading coefficient of the recurrence. Example 5.1. Let a(n) := 2(n+1 2 ) (A006125 in the OEIS) and b(n) := 4(n 2) (A053763). Both sequences are C2-finite satisfying the recurrences 2n+1 a(n) −a(n + 1) = 0, 4n b(n) −b(n + 1) = 0. The coefficients for a recurrence of c := a + b are given by an element in the kernel of ( 1 2n+1 22n+3 1 22n 24n+2 ) . A recurrence is therefore, for instance, given by 23n+3(2n −1)c(n) −2n+2(22n −2)c(n + 1) + (2n −2)c(n + 2) = 0. The recurrence has order ord(a) + ord(b) = 2 but the leading coefficient has a zero term at n = 1. Shifting the recurrence yields a recurrence of higher order with a leading coefficient which does not have any zero terms anymore. Theorem 2.6 shows that the closure properties of D-finite sequences satisfy very simple order bounds. The same bounds hold for C-finite sequences. In Example 4.10 we have 47 5 Order bounds for C2-finite closure properties seen that these order bounds do not hold in the C2-finite case any longer. However, in this chapter we derive similar bounds which also depend on the C-finite coefficients which appear in the recurrences of the C2-finite sequences. The content of this chapter is based on [KNP23]. 5.1 The exponent lattice For proving the order bounds for C2-finite sequences, we heavily rely on the fact that a C-finite sequence c can be written as interlacing of nondegenerate sequences (cf. [EPSW15, Theorem 1.2]) c(dn), . . . , c(dn + d −1). More generally, if we have a finitely generated difference algebra of C-finite sequences, we determine a number d ∈N (which we call the torsion number) such that every sequence in the algebra can be written as the interlacing of d nondegenerate subsequences. Let c0, . . . , cr ∈RC with eigenvalues λ1, . . . , λm and let Rd := Kσ[c0(dn), . . . , cr(dn)] be the smallest difference algebra which contains the sequences c0(dn), . . . , cr(dn). Suppose c ∈Rd. Then, every eigenvalue λ of c is of the form λ = λe1 1 · · · λem m for some e1, . . . , em ∈N. We want to find a d such that every sequence c ∈Rd is nondegenerate. Equivalently, we want to find a d such that ( λ de1 1 ···λdem m λ d f1 1 ···λd fm m )k = 1 = ⇒λde1 1 · · · λdem m = λd f1 1 · · · λd fm m for all k, e1, . . . , em, f1, . . . , fm ∈N. In order to write this more concisely we define the multiplicative group G := ⟨λ1, . . . , λm⟩≤(C×, ·). Then, this condition reads as ∀k ∈N≥1∀λ ∈G: λkd = 1 = ⇒λd = 1. The next lemma shows that this number d also has a purely group-theoretical and a purely lattice-theoretical description. A lattice is a Z-submodule of Zm. Every lattice L admits a finite basis v1, . . . , vℓ∈Zm, i.e., a set of linearly independent generators of the module L. We call ℓthe rank of the lattice L. 48 5 Order bounds for C2-finite closure properties Lemma 5.2. Let G := ⟨λ1, . . . , λm⟩≤(C×, ·). The following conditions on d ∈N≥1 are equivalent: 1. The number d satisfies ∀k ∈N≥1∀λ ∈G: λkd = 1 = ⇒λd = 1. 2. Let T(G) := {λ ∈G \| ord(λ) < ∞} be the torsion subgroup of G. Then, d satisfies ord(λ) \| d for all λ ∈T(G). 3. Let L := L(λ1, . . . , λm) := {(e1, . . . , em) ∈Zm \| λe1 1 · · · λem m = 1} ⊆Zm be the lattice of integer relations among λ1, . . . , λm. Then, d satisfies ∀k ∈N≥1 ∀v ∈Zm : kdv ∈L = ⇒dv ∈L. (5.1) Proof. 1 = ⇒2: Let λ ∈T(G) and let m ∈N≥1 be minimal with λm = 1. Then, clearly λmd = 1. By assumption, λd = 1. As m was chosen minimal, we have m \| d. 2 = ⇒3: Let k ∈N≥1, v = (e1, . . . , em) ∈Zm and kdv ∈L. Let λ = λe1 1 · · · λem m . By definition of L, λkd = λkde1 1 · · · λkdem m = 1. Hence, λ ∈T(G). Therefore, by assumption, ord(λ) \| d, so λd = 1 and dv ∈L. 3 = ⇒1: Let k ∈N≥1, λ = λe1 1 · · · λem m ∈G and λkd = 1. Defining v := (e1, . . . , em) ∈Zm yields kdv ∈L. By assumption, dv ∈L, i.e., λd = 1. Considering condition 2 of Lemma 5.2, we can see that the smallest d which satisfies the condition is the exponent of the torsion group. 49 5 Order bounds for C2-finite closure properties Definition 5.3. The torsion number d ∈N≥1 of λ1, . . . , λm ∈Q is defined as d := exp(T(G)) := lcm(ord(λ) \| λ ∈T(G)) where G := ⟨λ1, . . . , λm⟩≤(C×, ·). We also call d the torsion number of the lattice L if it is the smallest number satisfying (5.1). A useful tool in studying lattices is the Smith normal form of a matrix. Suppose V ∈ Zm×ℓand r = min(m, ℓ). Then, we can compute unimodular (i.e., invertible) matrices P ∈Zm×m, Q ∈Zℓ×ℓand a diagonal matrix D = diag(d1, . . . , dr) ∈Zm×ℓwith di \| di+1 for all i = 1, . . . , r −1 such that PVQ = D. The unique matrix D is called the Smith normal form of V and the largest diagonal entry dr is called the invariant factor of V. If ei denotes the i-th determinantal divisor of V, i.e., the greatest common divisor of all i-by-i minors of V, then dr = er er−1 [New72, Mid19]. Let v1, . . . , vℓ∈Zm be a basis of the lattice L = ⟨v1, . . . , vℓ⟩⊆Zm and let V := (v1, . . . , vℓ) ∈Zm×ℓwith Smith normal form PVQ = D with nonzero invariant fac-tors d1, . . . , dr. Since Q is unimodular we have L = VZℓ= P−1DQ−1Zℓ= P−1DZℓ. Let p1, . . . , pm ∈Zm be the columns of P−1. Since P is unimodular, these columns form a basis of Zm and d1p1, . . . , drpr form a basis of L [Tao22]. A lattice L is called pure if for all k ∈Z, v ∈Zm the condition kv ∈L implies v ∈L. Equivalently, L is pure if and only if L is a direct summand of Zm [CR66, Chapter III.16A]. The pure closure L of L is the smallest pure lattice which contains L, i.e., the intersection of all pure lattices that contain L. We have (cf. [CFQ15]) L = {v ∈Zm \| ∃k ∈Z \ {0}: kv ∈L}. Using the terminology of pure modules, (5.1) is equivalent to Ld := {v ∈Zm \| dv ∈L} ⊇L being a pure lattice. We show that for suitable d the lattice Ld is precisely the pure closure of L. The property of being pure is closely related to the invariant factors of the matrix built by a basis of a lattice. The following lemma is already given, without a proof, in [CMDS84] on page 80. For the sake of completeness we include a proof here. 50 5 Order bounds for C2-finite closure properties Lemma 5.4. Let v1, . . . , vℓ∈Zm be a basis of the lattice L = ⟨v1, . . . , vℓ⟩⊆Zm. Let V := (v1, . . . , vℓ) ∈Zm×ℓ. Then, L is pure if and only if all invariant factors of V are 1. Proof. Let PVQ = D be the Smith normal form of V with invariant factors d1, . . . , dℓ. Furthermore, let p1, . . . , pm ∈Zm denote the columns of the unimodular matrix P−1. = ⇒: The set {d1p1, . . . , dℓpℓ} forms a basis of L. As L is pure, p1, . . . , pℓalso form a basis of L. Hence, there is a unimodular change-of-basis matrix U with Udℓpℓ= pℓ. In particular (cf. Corollary 158 in [Mid19]), gcd(Udℓpℓ) = dℓgcd(pℓ) = gcd(pℓ). Therefore, dℓ= 1 and by the divisibility property of the invariant factors d1 = · · · = dℓ= 1. ⇐ =: As {p1, . . . , pℓ} form a basis of L and {p1, . . . , pm} form a basis of Zm, L is a direct summand of Zm and therefore pure. Now, we want to show that the torsion number of algebraic numbers λ1, . . . , λm ∈Q can actually be computed. First, there are algorithms which compute a basis v1, . . . , vℓ∈Zm for the lattice L := L(λ1, . . . , λm) [Ge93, Kau05, Fac14, ZX19, Zhe20, Zhe21, KNP23]. Then, the invariant factor of the matrix built by the basis is precisely the torsion number of the lattice: Theorem 5.5. Let v1, . . . , vℓ∈Zm be a basis of the lattice L = ⟨v1, . . . , vℓ⟩⊆Zm. Let V := (v1, . . . , vℓ) ∈Zm×ℓwith invariant factor d. Then, Ld = {v ∈Zm \| dv ∈L} is the pure closure of L. In particular, d is the torsion number of L. Proof. The lattices L and Ld have the same rank ℓ. Let d1, . . . , dℓ−1, dℓ= d 51 5 Order bounds for C2-finite closure properties be the invariant factors of L and d1, . . . , dℓthe invariant factors of Ld. The lattice Ld has a basis of the form d1p1, . . . , dℓpℓ. Let V := (d1p1, . . . , dℓpℓ). Then, VS = V for some matrix S ∈Zℓ×ℓas L ⊆Ld. Therefore, dℓ\| dℓ= d [New72, Lemma II.2]. Hence, dpℓ∈Ld, so pℓ∈Ld. By the same argument as in Lemma 5.4, d = dℓ= 1, so Ld is pure. As Ld is pure, the pure closure L of L is contained in Ld. Let v ∈Ld. Then, dv ∈L, so v ∈L. Therefore, Ld = L. Example 5.6. Let λ1 = 21/2, λ2 = (−2)1/3, λ3 = i, λ4 = −i. The columns of V := ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 −2 0 0 3 1 2 −1 1 −2 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = P−1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 1 0 0 0 4 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ Q−1 are a basis of L(λ1, λ2, λ3, λ4). Hence, d = 4 is the torsion number of λ1, . . . , λ4. Let c0, . . . , cr ∈RC with eigenvalues λ1, . . . , λm. Then, we have seen that we can compute a number d ∈N≥1 (namely the torsion number) such that the algebra R := Kσ[c0(dn), . . . , cr(dn)] only contains sequences which are nondegenerate, i.e., sequences which contain only finitely many zeros (cf. Theorem 2.17). A nondegenerate sequence might be a zero divisor in the ring KN. However, we can still define the localization Q(R) := { c d \| c ∈R, d ∈ R \ {0}}. This localization Q(R) is a field. Note, that an element of Q(R) can be interpreted only as a sequence in KN from some term on (cf. the discussion in Section 8.2 in [PWZ96] or [Sch20]). For instance, the sequence 3n 2n−1 cannot be evaluated at the term n = 0. This is 52 5 Order bounds for C2-finite closure properties not a problem for our applications as we see in Section 5.2. We summarize the discussions of the section in the following theorem: Theorem 5.7. Let c0, . . . , cr ∈RC with eigenvalues λ1, . . . , λm. Then, we can compute a number d ∈N≥1 (namely the torsion number) such that the localization Q(R) of the algebra R := Kσ[c0(dn), . . . , cr(dn)] is a field. The nonzero elements of the field Q(R) can be considered as sequences which are nonzero from some term on. From the closed form of C-finite sequences it is clear that these sequences can be seen as special cases of sums of single nested product expressions. The torsion number can be used to find a certain algebraic independent basis of these sequences [Sch20]. 5.2 Order bounds In this section we derive order bounds for the ring operations and additional closure properties of C2-finite sequences. In Chapter 4 we have seen how computations of closure properties of C2-finite sequences can be reduced to solving linear systems of equations. A C2-finite recurrence (here we use a “homogeneous ansatz” compared to the “inhomogeneous ansatz” in (4.1) which yields an inhomogeneous linear system) x0(n) + x1(n)σ + · · · + xs(n)σs with xi ∈R for some suitable ring of sequences R is obtained by computing an element (x0, . . . , xs) in the kernel of a matrix ( w0, w1, . . . , ws ) ∈Q(R)r×(s+1). (5.2) The wi can be computed iteratively using wi+1 = Mσ(wi) for a suitable matrix M ∈ Q(R)r×r. 53 5 Order bounds for C2-finite closure properties • In the case a recurrence for a + b is computed, we use w0 = e(r1) 0 ⊕e(r2) 0 and M = Ma ⊕Mb where r1 = ord(a), r2 = ord(b). • In the case a recurrence for ab is computed, we use w0 = e(r1) 0 ⊗e(r2) 0 and M = Ma ⊗Mb. • In the case a recurrence for a(kn) with k ∈N is computed, we use w0 = e(r) 0 and M = Ma(kn) · · · Ma(kn + k −1) where r = ord(a). • In the case a C2-finite recurrence for c(jn2 + kn + ℓ) with j, k, ℓ∈N and a C-finite sequence c (which does not have 0 as an eigenvalue) of order r is computed, we use w0 = Mkn+ℓ−r+1 c e(r) r−1 and M = Mj(2n+1) c . (5.3) The underlying ring R is the difference algebra Kσ[c0, . . . , cr] generated by the C-finite sequences c0, . . . , cr appearing in w0 and M. In the next sections we present order bounds for the closure properties that we discussed in Chapter 4. In particular, we prove the following theorem (cf. Theorem 2.6 for the D-finite order bounds): Theorem 5.8. Let a(n), b(n), a0(n), . . . , am−1(n) be C2-finite sequences. Let da be the torsion number of the eigenvalues appearing in the recurrence of a and da,b the torsion number of the eigenvalues appearing in the recurrences of a, b. Then, 1. σ(a(n)) = a(n + 1) is C2-finite of order at most ord(a), 2. a(n) + b(n) is C2-finite of order at most da,b(ord(a) + ord(b)), 3. a(n)b(n) is C2-finite of order at most da,b ord(a) ord(b), 4. ∑n k=0 a(k) is C2-finite of order at most ord(a) + 1, 5. a(kn + ℓ) is C2-finite of order at most da ord(a) for all k, ℓ∈N and 6. the interlacing e(n) = ar(q) where n = qm + r for 0 ≤r < m is C2-finite of order at most m maxr=0,...,m−1 ord(ar). 54 5 Order bounds for C2-finite closure properties Proof. 1: Clear from the proof of Theorem 3.26. 2, 3: Theorem 5.14. 4: Let b(n) := ∑n k=0 a(k). Then, σ(b) −b = σ(a). If A is an annihilator of σ(a), then A · (−1 + σ) is an annihilator of b of order ord(A) + 1. 5: Theorem 5.13. 6: Theorem 5.10. Example 5.9. Let p denote the Perrin sequence (cf. Example 2.14). With Theorem 5.8 and Theorem 5.19, the sequence 5n+1 ∑ k=0 ( p((2k + 1)2) + p(k2) p(3k + 2) ) is C2-finite of order at most 7. 5.2.1 Interlacing and subsequence Theorem 5.10. Let a1(n), . . . , ad(n) be C2-finite sequences of maximal order r. Let b be the interlacing of these sequences. We can compute a C2-finite recurrence of order at most dr for b. Proof. By shifting the recurrences of the as appropriately, we can assume that they all satisfy a C2-finite recurrence of order r of the form cs,0(n)as(n) + · · · + cs,r(n)as(n + r) = 0 for s = 1, . . . , d for C-finite sequences cs,i where the cs,r only have finitely many zeros. Let edi be the interlacing of c1,i, . . . , cd,i for i = 0, . . . , r. These edi are then C-finite and edr only has finitely many zeros. Then, b satisfies the recurrence e0(n)b(n) + ed(n)b(n + d) + · · · + edr(n)b(n + dr) = 0. 55 5 Order bounds for C2-finite closure properties As seen in the proof of Theorem 5.10, computing the interlacing of C2-finite sequences is simpler than in the case of C-finite and D-finite sequences. This is because the coefficients of the recurrence, namely C-finite sequences, are closed under interlacing themselves. Example 5.11. Let c be C-finite satisfying c(n) −c(n + r) = 0, c(0) = 1, c(1) = · · · = c(r −1) = 0. Furthermore, let a be the interlacing of c and d −1 times the zero sequence. Theorem 5.10 shows that a is C2-finite of order at most dr. The sequence a is cyclic and has rd −1 consecutive zeros. Hence, the sequence a also has to have order at least rd as otherwise, the sequence would be constantly zero. The bound in Theorem 5.10 is therefore tight in general. Lemma 5.12. Let a be C2-finite of order r and let d be the torsion number of the eigenvalues appearing in the recurrence of a. Let k ∈N. We can compute a C2-finite recurrence of order at most r which is satisfied by all sequences a(dkn + i) for i = 0, . . . , dk −1. Proof. The sequences a(n + i) for i = 0, . . . , d −1 all satisfy the same recurrence. By the choice of d, all sequences in the ring R generated by the sequences appearing in M = Ma(dkn) · · · Ma(dkn + dk −1) are nondegenerate. By Theorem 5.7, Q(R) is a field. Therefore, if s = r, then the linear system (5.2) is underdetermined and we can compute an element (after clearing denomi-nators) (x0, . . . , xr) ∈Rr+1 in the kernel with xt ̸= 0 and xt+1 = · · · = xr = 0 for some t ≤r. This gives rise to a C2-finite recurrence x0(n) + x1(n)σ + · · · + xt(n)σt as xt only has finitely many zeros by the choice of d. To extend Lemma 5.12 to subsequences at arbitrary arithmetic progressions we write such an arbitrary subsequence as the interlacing of certain subsequences for which Lemma 5.12 can be applied. 56 5 Order bounds for C2-finite closure properties Theorem 5.13. Let a be C2-finite of order r and let d be the torsion number of the eigenvalues appearing in the recurrence of a. Let k ∈N. We can compute a C2-finite recurrence of order at most dr which is satisfied by the sequence a(kn). Proof. By Lemma 5.12 we can compute a recurrence of order at most r satisfied by a(dkn + i) for i = 0, . . . , dk −1. Let b be the interlacing of the d sequences a(dkn), a(dkn + k), . . . , a(dkn + (d −1)k). By Theorem 5.10, b has order at most dr. We show that b(n) = a(kn): Let n = qd + s with 0 ≤s < d. Then, by the definition of b b(n) = b(qd + s) = a(dkq + sk) = a(k(dq + s)) = a(kn). 5.2.2 Ring operations Theorem 5.14. Let a, b be C2-finite of order r1, r2, respectively and let d be the torsion number of the eigenvalues appearing in the recurrences of a, b. Then, 1. the sequence a + b is C2-finite of order at most d(r1 + r2) and 2. the sequence ab is C2-finite of order at most dr1r2. Furthermore, such recurrences can be computed. Proof. We can compute C2-finite recurrences of maximal order r1, r2 for a(dn + i), b(dn + i) by Lemma 5.12. The closure properties a(dn + i) + b(dn + i) and a(dn + i)b(dn + i) can be computed again by solving a linear system of equations over the field Q(R). Then, the same order bounds as in the C-finite and D-finite case apply, so the sequences a(dn + i) + b(dn + i), a(dn + i)b(dn + i) have maximal orders r1 + r2, r1r2, respectively. By Theorem 5.10, we can interlace these sequence and obtain a recurrence of order d(r1 + r2), dr1r2 for a + b and ab, respectively. 57 5 Order bounds for C2-finite closure properties In the special case that both C2-finite sequences are C-finite or D-finite, the torsion number is 1 and the bounds simplify to the known order bounds for these rings. Example 5.15. Let c be C-finite of order 2 satisfying c(n) −c(n + 2) = 0, c(0) = −1, c(1) = 1. Let a, b be C2-finite satisfying a(n) = 1 c(n)b(n) −b(n + 1) = 0, b(0) = 1. The eigenvalues that appear are 1 and −1. The torsion number is therefore d = 2. Let ai(n) = a(2n + i) and bi(n) = b(2n + i) for i = 0, 1. These are even C-finite of order 1 satisfying ai(n) −ai(n + 1) = 0, bi(n) + bi(n + 1) = 0. Let si = ai + bi. These si are C-finite of order 2 satisfying si(n) −si(n + 2) = 0. The interlacing s = a + b of s0, s1 satisfies the C-finite recurrence of order 4 = d(ord(a) + ord(b)) s(n) −s(n + 4) = 0. However, s also satisfies a C2-finite recurrence of order 3, namely c0(n)s(n) + c2(n)s(n + 2) + s(n + 3) = 0 with c0(n) −c0(n + 2) = 0, c0(0) = −1, c0(1) = 0, c2(n) −c2(n + 2) = 0, c2(0) = 0, c2(1) = −1. There cannot be a shorter recurrence for s(n) as it contains 2 consecutive zeros. 58 5 Order bounds for C2-finite closure properties The order bound is not reached in the previous example. In fact, we could not find any example where the bounds from Theorem 5.14 are reached. This, of course, yields the obvious question whether the bounds we have found are, in general, sharp. Open Question 5.16. Let d, r1, r2 ∈N. Can we find C2-finite sequences a, b of orders r1, r2 such that the torsion number of the eigenvalues appearing in the recurrences is d and ord(a + b) = d(ord(a) + ord(b)) (or ord(ab) = d ord(a) ord(b)). Even a single example with d ̸= 1 where the bound is reached would already be interest-ing. Theorem 5.14 does not imply that the ring of C2-finite sequences is computable. We can compute C2-finite recurrences for the sum and the product. These recurrences, however, have leading coefficients which can have finitely many zeros. To uniquely determine the sequences a + b, ab we might need to define additional initial values at these singularities. However, by the Skolem Problem, we do not know whether these singularities can be computed. This is also illustrated by Example 5.1. Hence, the following question is still open. Open Question 5.17. Is the ring of C2-finite sequences computable? I.e., suppose we are given C2-finite sequences a, b with their recurrences and enough initial values. Can we compute a C2-finite recurrence for c = a + b or c = ab together with a number r such that the initial values c(0), . . . , c(r) uniquely determine the sequence c? Closely related is also the following problem: Suppose we have C2-finite sequences a, b. For checking whether these sequences are identical we can compute a recurrence for c = a −b. Of course, a = b if and only if c = 0. If we can compute the zeros of the leading coefficient of the recurrence of c, we can check whether c = 0 by checking sufficiently many initial values. Hence, identity checking of C2-finite sequences can be reduced to the Skolem Problem. We do not know if the converse holds: Open Question 5.18. Can the Skolem Problem be reduced to identity checking of C2-finite sequences? 59 5 Order bounds for C2-finite closure properties 5.2.3 Sparse subsequences Theorem 5.19. Let c be C-finite of order r and λ1, . . . , λm its eigenvalues and λi ̸= 0 for all i = 1, . . . , m. Let d be the torsion number of the eigenvalues. Then, we can compute a C2-finite recurrence of c(jn2 + kn + ℓ) of maximal order dr for all j, k, ℓ∈N. Proof. In a first step, we show how we can find a recurrence of order r for the sequence a(n) = c(d(jn2 + kn) + ℓ). Lemma 11 in [KM14] shows that Mpn+q for p, q ∈Z is a matrix of C-finite sequences. The proof shows that the characteristic polynomials of the sequences is the characteristic polynomial of Mp. Let Mc be the companion matrix of c. Suppose (x −λ1)d1 · · · (x −λm)dm is the characteristic polynomial of c which, by definition of the companion matrix, is also equal to the characteristic polynomial of Mc. Then, by the closed form of C-finite sequences, the characteristic polynomial of c(pn) is given by (x −λp 1)d1 · · · (x −λp m)dm which, in turn, is equal to the characteristic polynomial of Mp c . By (5.3), the sequences that generate the underlying ring R used for computing a recurrence for a(n) all have characteristic polynomial equal to the characteristic polynomials of Mdk c and M2dj c . An element in the kernel of the linear system over the field Q(R) can easily be computed if s = r. This gives rise to a C2-finite recurrence of order r for a. An arbitrary sequence b(n) = c(jn2 + kn + ℓ) 60 5 Order bounds for C2-finite closure properties can be written as interlacing of sequences ar(n) = c(d(djn2 + (2jr + k)n) + jr2 + kr + ℓ) for r = 0, . . . , d −1 as the term at index n = qd + r of the interlacing is precisely given by ar(q) = c(d(djq2 + (2jr + k)q) + jr2 + kr + ℓ) = c(j(d2q2 + 2rq + r2) + k(dq + r) + ℓ) = c(jn2 + kn + ℓ). We can compute C2-finite recurrences of order r for these sequences ar by the first part of the proof (choosing j = dj, k = 2jr + k, ℓ= jr2 + kr + ℓ). By Theorem 5.10 we can therefore compute a C2-finite recurrence of order dr for b. Example 5.20. Let c be the C-finite sequence (A006131 in the OEIS) satisfying 4 c(n) + c(n + 1) −c(n + 2) = 0, c(0) = c(1) = 1. The sequence has eigenvalues 1± √ 17 2 and their torsion number is 1. The sparse subse-quence a(n) = c(n2) is C2-finite of order 2 satisfying c0(n)a(n) −c(4n + 3)a(n + 1) + c(2n)a(n + 2) = 0 where c0 is C-finite of order 2 satisfying 4096 c0(n) −144 c0(n + 1) + c0(n + 2) = 0, c0(0) = −20, c0(1) = −1856. Computing a C2-finite recurrence for c(n2) where c is a C-finite sequence of order 2 is usually possible as the corresponding linear system is small. However, if c has order 3 it can already be difficult. Open Question 5.21. Compute a C2-finite recurrence for p(n2) where p(n) is the sequence of Perrin numbers (cf. Example 2.14). In fact, setting up the linear system and solving it via guessing yields a recurrence for p(n2) of order 3 having coefficients with maximal order 32. Checking the first 1000 terms in-dicates that this recurrence is indeed correct. The recurrence is, however, much more 61 5 Order bounds for C2-finite closure properties complicated than the easy recurrence we have for the sparse Fibonacci numbers (cf. Exam-ple 3.7) and other sequences of order 2. 62 6 A computable subring: simple C2-finite sequences We have seen that many computations with C2-finite sequences are limited by the Skolem Problem. The problem stems from possible zeros in the leading coefficient of the recurrence. This can be avoided by only considering sequences which satisfy a linear recurrence with C-finite coefficients and constant leading coefficient. This chapter is based on the article [NP22b]. 6.1 Algebraic characterization Our notion for simple C2-finite sequences is based on the analogous notion of simple P-recursive sequences for D-finite sequences [Kot12]. Definition 6.1. A sequence a ∈KN is called simple C2-finite if there is a linear recurrence operator A ∈RC[σ] with lc(A) = 1 which annihilates a, i.e., Aa = 0. Equivalently, we could restrict lc(A) ∈K in Definition 6.1. As C-finite sequences are closed under multiplication with a field element, multiplying the operator A by 1 lc(A) yields an annihilating operator with leading coefficient 1. Many of the C2-finite sequences that we have considered earlier are in fact simple C2-finite. Example 6.2. Let f denote the Fibonacci sequence. In Example 3.7 we have seen that a(n) := f (n2) satisfies a C2-finite recurrence of order 2 with coefficients having maximal order 2. The sequence a is even simple C2-finite and satisfies a recurrence of order 3 with coefficients having order at most 4: −f (6n + 11)a(n) −c1(n)a(n + 1) + f (6n + 9)a(n + 2) + a(n + 3) = 0 63 6 A computable subring: simple C2-finite sequences with c1(n) −54c1(n + 1) + 331c1(n + 2) −54c1(n + 3) + c1(n + 4) = 0 and initial values c1(0) = 136, c1(1) = 6710, c1(2) = 317434, c1(3) = 14927768. This recurrence can be found using guessing and fixing the coefficients of the recurrence to only involve C-finite sequences which have certain powers of the golden ratio (and its conjugate) as roots. The recurrence can then be verified using closure properties of C2-finite sequences. Using an algorithm for computing algebraic relations of C-finite sequences due to Kauers and Zimmermann [KZ08], we can write c1 in terms of the Fibonacci sequence as c1(n) = f (4n + 6)(−1 −2f (4n + 4) + 3f (4n + 6)). In fact, for any C-finite sequence c and j, k, ℓ∈N we can find a simple C2-finite recurrence for the sequence c(jn2 + kn + ℓ): In this section we show that simple C2-finite sequences form a ring. Furthermore, this ring clearly contains all C-finite sequences. Hence, the proof of Corollary 3.6 in [JPNP21] can be adjusted to show that the subsequence c(jn2 + kn + ℓ) is simple C2-finite. However, not all C2-finite sequences are simple C2-finite. In fact, not all D-finite sequences are simple C2-finite. Example 6.3. The Catalan numbers (Example 2.4) are D-finite but neither simple P-recursive [Kot12, Section 8.1.5] nor polynomial recursive [CMP+21, Corollary 8]. In particular, the Catalan numbers are not simple C2-finite over Q. By Lemma 3.16, for every simple C2-finite sequence a ∈CN there is an α ∈Q such that \|a(n)\| ≤αn2 for all n ≥1. As discussed after the lemma, such an α can be computed explicitly for simple C2-finite sequences. Analogous to Theorem 3.23, we can find an equivalent characterization for simple C2-finite sequences in terms of finitely generated modules. Theorem 6.4. The following are equivalent: 64 6 A computable subring: simple C2-finite sequences 1. The sequence a is simple C2-finite. 2. There exists A ∈RC[σ] with lc(A) = 1 and a simple C2-finite sequence b with Aa = b. 3. The module ⟨σi(a) \| i ∈N⟩RC over the ring of C-finite sequences RC is finitely generated. Based on this characterization, one can easily prove the following theorem analogous to Theorem 3.26: Theorem 6.5. The set of simple C2-finite sequences is a difference ring under termwise addition and termwise multiplication. In Section 3.4 we have studied the generating functions of C2-finite sequences. In the case of simple C2-finite sequences we can find an equivalent characterization in terms of certain functional equations. Theorem 6.6. The sequence a ∈Q N is simple C2-finite if and only if its generating function g(x) := ∑n≥0 a(n)xn satisfies a functional equation of the form m ∑ k=1 αkxjkg(dk)(λkx) = p(x) (6.1) for 1. α1, . . . , αk, λ1, . . . , λk ∈Q \ {0}, 2. j1, . . . , jm, d1, . . . , dm ∈N, 3. p ∈Q[x] and 4. let s := maxk=1,...,m(dk −jk), then for all k = 1, . . . , m with dk −jk = s we have dk = 0 and λk = 1. Proof. = ⇒: With Theorem 3.27 we can clearly find a functional equation satisfying proper-ties (1)–(3). The contribution from the leading term after clearing the common denomina-tor xr is given by h(x) = ∑n≥0 a(n + r)xn = g(x) −p0,0(x) for some polynomial p0,0(x). In particular, dk = jk = 0 and λk = 1, so dk −jk = 0. The other terms njλna(n + i) with i < r yield contributions with jk = l −i + r, dk = l in (6.1). In particular, dk −jk = i −r < 0. 65 6 A computable subring: simple C2-finite sequences ⇐ =: According to the proof of Theorem 3.30, the leading coefficient in the recurrence is given by terms where dk −jk is maximal. If λk = 1 and dk = 0 in these cases, then this leading coefficient is just 1. 6.2 Computable ring In Chapter 4 we have seen how closure properties for C2-finite sequences can be reduced to solving systems of equations of the form Ax = b (6.2) with A ∈Q(R)m×s, b ∈Q(R)m and R := Kσ[c0, . . . , cr] ⊊RC (6.3) with c0, . . . , cr ∈RC. All the fractional sequences, in fact, originate from the leading coefficient of the C2-finite recurrences. Therefore, if the sequences for which we perform closure properties are simple C2-finite, the linear system (6.2) is of the form A ∈Rm×s and b ∈Rm. We show how such systems can be solved. Our method is based on the closed form of C-finite sequences. Therefore, we assume that the base field is always the field of algebraic numbers Q. Note that every C-finite sequence over Q has again a closed form as Q is algebraically closed itself. First, we consider the special case, where we compute a constant solution of such a system. Lemma 6.7. We can compute all constant solutions x ∈Q s of the linear system Ax = b where A ∈Rm×s C and b ∈Rm C. In particular, we can decide whether such a solution exists. Proof. It is sufficient to consider one equation, i.e., A ∈R1×s C . The set of constant solutions is an affine subspace of Q s. For several equations we can compute the intersection of these affine subspaces to determine all solutions. Using the closed form of the sequences, we can rewrite the equation Ax = b as l ∑ k=1 ( ∑ i∈Sk ϵk,ixi + ϵk )    =:yk (n −n0)dkλn−n0 k = 0, for all n ≥n0 (6.4) 66 6 A computable subring: simple C2-finite sequences with n0 ∈N, and ϵk,i, ϵk, λk ∈Q, dk ∈N and Sk ⊆{1, . . . , s} for all k = 1, . . . , l. Certainly, if yk = 0 for all k = 1, . . . , l we have a solution. On the other hand, evaluating this equation for n = n0, n0 + 1, . . . , yields a linear system for the yk. This linear system is a generalized Vandermonde matrix, in particular it is regular [LT08, Liu68]. Therefore, if equation (6.4) holds, then yk = 0 for all k = 1, . . . , l. This yields a linear system over Q which can be solved. For the initial terms n = 0, 1, . . . , n0 −1 the equation Ax = b can simply be solved over Q. The affine space of all solutions of the single equation is now given as the intersection of the affine subspace arising from solving equation (6.4) and the affine subspaces arising from the initial terms. Suppose the sequence c is C-finite over the algebraic numbers Q with closed form c(n + n0) = m ∑ i=1 pi(n)λn i as in Theorem 2.15, i.e., λ1, . . . , λm ∈Q, p1, . . . , pm ∈Q[n] and deg(p1) = d1, . . . , deg(pm) = dm. Let Bc := {( njλn i ) n∈N \| i ∈{1, . . . , m}, j ∈{1, . . . , di −1} } . (6.5) Then, the sequence c is an L-linear combination of sequences in Bc from n0 on. Suppose n0 is the smallest index such that each ci from (6.3) is a Q-linear combination of sequences in Bci, as defined in (6.5), from n0 on. We write B := Bc0 ∪· · · ∪Bcr ∪{1}. Then, for any sequence c ∈R there is an N ∈N and coefficients xd1,...,dN ∈Q such that c(n) = ∑ d1,...,dN∈B xd1,...,dNd1(n) · · · dN(n), for all n ≥n0. (6.6) Furthermore, if c ∈R is given by a recurrence and initial values such a representation can be computed: We can compute the closed form of c. Now, every term in this closed form has to be a product of the finitely many (not necessarily distinct) sequences from B. Lemma 6.8. Let A ∈Rm×s and b ∈Rm. If Ax = b has a solution x ∈Rs, then we can compute a solution x ∈Rs C. 67 6 A computable subring: simple C2-finite sequences Proof. We can assume that every sequence in the linear system is given in the form (6.6). For N = 1, 2, . . . we write xi = ∑ d1,...,dN∈B xi,d1,...,dNd1 · · · dN for unknown coefficients xi,d1,...,dN ∈Q for i = 1, . . . , s. In particular, for fixed N, we can compute sequences e1, . . . , el ∈RC such that xi = ∑l k=1 xi,kek with xi,k ∈Q unknown for i = 1, . . . , s. Let ˆ x := ( x1,1, . . . , xs,1, . . . , x1,l, . . . , xs,l )⊤ . Then, Ax = b has a solution for the xi,d1,...,dN if and only if the m × ls linear system ( e1A, . . . , elA ) ˆ x = b has a solution for ˆ x. With Lemma 6.7 we can check whether the linear system has a solution for ˆ x. If we have found a solution we can easily compute the corresponding xi. As we know that a solution x of this form exists, this algorithm has to terminate for large enough N. Theorem 6.9. The ring of simple C2-finite sequences over Q is computable. Proof. Suppose a, b are simple C2-finite with annihilating operators ∑r1−1 i=0 ciσi + σr1 and ∑r2−1 i=0 diσi + σr2, respectively. By Lemma 4.9 (which can be proven completely analogously for simple C2-finite sequences), there is a linear system over the computable ring R := Qσ[c0, . . . , cr1−1, d0, . . . , dr2−1] which has a solution and whose solution gives rise to a recurrence for a + b or ab. This linear system can be computed and a solution of the system can be obtained with Lemma 6.8. As we do not know a priori how big this order s from Lemma 4.9 is and how big the N in the proof of Lemma 6.8 has to be chosen, we can simultaneously increase s and N. Eventually, this algorithm terminates and any solution gives rise to a recurrence for a + b or ab. Because we are working with the closed form of C-finite sequences, the recurrences that we compute in Theorem 6.9 might be over a bigger field than we started with. E.g., it might be that the sequences a, b are simple C2-finite over Q, but the coefficients in the 68 6 A computable subring: simple C2-finite sequences recurrence for a + b are C-finite over K ⊋Q. However, we do know by Theorem 6.5 that a recurrence with coefficients over Q exist. Open Question 6.10. Is the ring of simple C2-finite sequences over a field K computable? In fact, a positive answer to Question 4.8 would also give a positive answer to this question by combining Theorem 6.5 and Theorem 6.9. C2-finite sequences are also closed under taking differences, partial sums, subsequences at arithmetic progressions and interlacing. The same proofs carry over to simple C2-finite sequences. Even more, as these operations can be reduced to solving linear systems, these closure properties can be computed effectively. This method for computing closure properties can also yield nicer (i.e., smaller coefficients) recurrences than the method from Chapter 4 as the following example shows. Example 6.11. Consider the sequences 2na(n) + a(n + 1) = 0, b(n) + b(n + 1) = 0. Both are simple C2-finite. We want to compute a recurrence for c := a + b. An ansatz of order 3 yields the linear system ( 1 −2n 2 · 4n 1 −1 1 ) ⎛ ⎜ ⎝ x0 x1 x2 ⎞ ⎟ ⎠= ( 8 · 8n 1 ) . This is the smallest system which has a solution. Using the generalized inverse method from Algorithm 2 to compute the solution we get the recurrence ( −25n+4 + 24n+2 + 23n+3 −22n+1) c(n) + ( 25n+4 −23n+3 −22n+1 + 1 ) c(n + 2) + ( 24n+2 −22n+2 + 1 ) c(n + 3) = 0 (6.7) 69 6 A computable subring: simple C2-finite sequences if we use columns 0 and 2 of the matrix. Using columns 1 and 2 we get ( 23n+4 −3 · 22n+2 + 2n+1) c(n + 1) + ( 23n+4 −22n+3 −2n+1 + 1 ) c(n + 2) + ( 22n+2 −2n+2 + 1 ) c(n + 3) = 0. Both recurrences have coefficients with maximal order 4. By Theorem 3.26, we know that c also has to satisfy a recurrence with leading coefficient 1. We make an ansatz xi = xi,1 + xi,22n and write ˆ x = (x0,1, x1,1, x21, x0,2, x1,2, x22). The linear system for ˆ x ∈Q6 computed in Lemma 6.8 is given by ( 1 −2n 2 · 4n 2n −4n 2 · 8n 1 −1 1 2n −2n 2n ) ˆ x = ( 8 · 8n 1 ) . Comparing the coefficients of 1, 2n, 4n, 8n as in Lemma 6.7 yields the constant system ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 0 0 0 −1 0 1 0 0 0 0 2 0 −1 0 0 0 0 0 0 2 1 −1 1 0 0 0 0 0 0 1 −1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ˆ x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 8 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . This system has the unique solution ˆ x = ( 0, 2, 3, 2, 6, 4 ) which gives rise to the recurrence (2 · 2n) c(n) + (2 + 6 · 2n) c(n + 1) + (3 + 4 · 2n) c(n + 2) + c(n + 3) = 0. (6.8) 70 6 A computable subring: simple C2-finite sequences The ideas of Chapter 5 seem to not carry over easily to the case of simple C2-finite sequences. It would be interesting to see whether similar order bounds can be proven for this case. Open Question 6.12. Can we derive similar order bounds as in Theorem 5.14 for simple C2-finite sequences? Identities of C-finite sequences can often be proven fully automatically by checking a moderate number of initial values. For proving the identity in the introduction of the thesis, we can define the C-finite sequence c(n) := 2n ∑ k=0 f (k) f (k + 1) −f (2n + 1)2 + 1 where f denotes the Fibonacci numbers. By the order bounds of C-finite sequences, this sequence c has order at most 10. Hence, if c(0) = · · · = c(9) = 0, then c is the constant zero sequence which proves the identity. Similar methods for D-finite sequences are more difficult because one has to take possible zeros in the leading coefficient of the recurrence into account. Therefore, for D-finite sequences this method is often not feasible in practice [Yen96, Yen97, GHS08]. However, deriving reasonable order bounds for sequences that are simple (P-recursive or C2-finite), we might be able to extend the method used for C-finite sequences to larger classes. 71 7 Extension to Ck-finite and Dk-finite sequences Instead of considering sequences satisfying linear recurrences with C-finite coefficients we can allow D-finite sequences. Analogous to Dn-finite functions (cf. [JPP19, JPPS20]) the construction can be iterated, i.e., we can, for instance, define C3-finite sequences as sequences satisfying a linear recurrence with C2-finite coefficients. This chapter shows that these Ck-finite and Dk-finite sequences form an increasing chain of difference rings. The results are based on [JPNP23]. 7.1 Definition and examples A sequence is called C0-finite if it is constant and called D0-finite if it is polynomial. Definition 7.1. Let k ≥1. A sequence a ∈KN is called Ck-finite (or Dk-finite) over K if there are Ck−1-finite (or Dk−1-finite) sequences c0, . . . , cr over K with cr(n) ̸= 0 for all n ∈N such that c0(n)a(n) + c1(n)a(n + 1) + · · · + cr(n)a(n + r) = 0 for all n ∈N. Example 7.2. Let a(n) := ∏n k=1 k!. The sequence a is D2-finite satisfying the recurrence (n + 1)! a(n) −a(n + 1) = 0, for all n ∈N. The sequence is called the superfactorial (A000178 in the OEIS). 72 7 Extension to Ck-finite and Dk-finite sequences Example 7.3. Let α ∈K. Every sequence a with a(n) := αn3 is C3-finite satisfying the recurrence c(n)a(n) −a(n + 1) = 0, for all n ∈N, where c(n) = α3n2+3n+1 is C2-finite (cf. Example 3.4). More generally, a(n) = αnk is Ck-finite for every k ∈N. Example 7.4. Using the same argument as in [KM14] one can derive a C3-finite recurrence for f (n3) where f denotes the Fibonacci numbers: c0(n) f (n3) + c1(n) f ((n + 1)3) + c2(n) f ((n + 2)3) = 0, for all n ∈N, with c0(n) = f (3n2 + 9n + 7) f (3n2 + 3n + 3) f (3n2 + 3n + 1) −f (3n2 + 9n + 7) f (3n2 + 3n + 2)2, c1(n) = f (3n2 + 9n + 7) f (3n2 + 3n + 2) + f (3n2 + 9n + 6) f (3n2 + 3n + 1), c2(n) = −f (3n2 + 3n + 1). These coefficients c0, c1, c2 are C2-finite with Theorem 3.26 and Theorem 4.7. Furthermore, clearly c2(n) ̸= 0 for all n. 7.2 Ring structure Adapting Section 3.3 to this more general setting, we show that the sets of Ck-finite and Dk-finite sequences form difference rings. We denote the set of Ck-finite sequences by RCk, the set of D-finite sequences by RD and the set of Dk-finite sequences by RDk. Now, Lemma 3.21, Lemma 3.22 and Theorem 3.23 can be formulated completely analo-gously for Ck-finite and Dk-finite sequences: Lemma 7.5. Let a be Ck-finite (or Dk-finite) with annihilating operator A = c0 + · · · + crσr and let R be the difference ring generated by c0, . . . , cr. If S ⊇R is a subring of the ring of sequences KN, then ⟨σi(a) \| i ∈N⟩Q(S) is finitely generated. 73 7 Extension to Ck-finite and Dk-finite sequences Lemma 7.6. Let a ∈KN and S be a subring of the set of Ck−1-finite (or Dk−1-finite) sequences. If ⟨σi(a) \| i ∈N⟩Q(S) is finitely generated, then a is Ck-finite (or Dk-finite). The proofs of Lemma 7.5 and Lemma 7.6 are analogous to the proofs of the correspond-ing lemmas in Section 3.3. Using Lemma 7.5 and Lemma 7.6 one can again prove a characterization for Ck-finite and Dk-finite sequences. Theorem 7.7. Let a ∈KN. 1. The sequence a is Ck-finite if and only if ⟨σi(a) \| i ∈N⟩Q(RCk−1) is finitely generated. 2. The sequence a is Dk-finite if and only if ⟨σi(a) \| i ∈N⟩Q(RDk−1) is finitely generated. Similarly as in the C2-finite setting, we can use Theorem 7.7 to show that the sets of Ck-finite and Dk-finite sequences form difference rings. Example 3.24 shows that these rings are not Noetherian. Hence, the idea is, again, to restrict the underlying ring to a Noetherian subring. Lemma 7.8. 1. Let A = ∑r i=0 ciσi ∈RCk[σ]. Then, the K-difference-algebra Kσ[c0, . . . , cr] is contained in a Noetherian ring S. 2. Let A = ∑r i=0 ciσi ∈RDk[σ]. Then, the K(n)-difference-algebra K(n)σ[c0, . . . , cr] is contained in a Noetherian ring S. Proof. We use induction on k. For k = 0 we have RC0 = K and RD0 = K[n] which are both Noetherian. Now, let c be a coefficient of A and let C be its annihilator. By induction, the difference-algebra generated by the coefficients of C is contained in a Noetherian ring Sc. Then, also the localization Q(Sc) is Noetherian [AM69, Proposition 7.3]. By Lemma 7.5, the module ⟨σi(c) \| i ∈N⟩Q(Sc) is finitely generated. Hence, also the difference-algebra Ac := Q(Sc)σ[c] is finitely generated and is, in particular, a Noetherian ring containing Kσ[c] (or K(n)σ[c] in the D-finite case). Then, S can be chosen as the smallest ring containing the Noetherian rings Ac0, . . . , Acr. This ring S is again Noetherian [AM69, Corollary 7.7]. Theorem 7.9. The sets of Ck-finite (resp. Dk-finite) sequences are difference rings under termwise addition and termwise multiplication. 74 7 Extension to Ck-finite and Dk-finite sequences Proof. Let a, b be Ck-finite (or Dk-finite) sequences and A = c0 + c1σ + · · · + cr1σr1 and B = d0 + d1σ + · · · + dr2σr2 the corresponding annihilating operators. With Lemma 7.8, there is a Noetherian ring S which contains all difference rings generated by c0, . . . , cr1, d0, . . . , dr2. Hence, with Lemma 7.5, the modules ⟨σi(a + b) \| i ∈N⟩Q(S) ⊆⟨σi(a) \| i ∈N⟩Q(S) + ⟨σi(b) \| i ∈N⟩Q(S) and ⟨σi(ab) \| i ∈N⟩Q(S) ⊆⟨σi(a)σj(b) \| i, j ∈N⟩Q(S) are finitely generated as they are submodules of finitely generated modules over a Noethe-rian ring. By Lemma 7.6, the sequences a + b and ab are Ck-finite (or Dk-finite). The operator ˜ A := σ(c0) + σ(c1)σ + · · · + σ(cr1)σr1 annihilates σ(a). Hence, the ring is also closed under shifts. Using the ansatz method described in Chapter 4 one can reduce the computation of ring operations to solving linear systems. For instance, for D2-finite sequences, we need to solve linear systems over the D-finite sequence ring. The ideas from Theorem 4.5 and Theorem 5.10 can be used to show that Ck-finite and Dk-finite sequences are closed under taking subsequences at arithmetic progressions and interlacings. Example 7.10. We define the D-finite sequences (n2 + 1)c0(n) + c0(n + 1) = 0, c0(0) = 2, (n + 7)c1(n) + (−n −1)c1(n + 1) = 0, c1(0) = 2, (n + 1)d0(n) −d0(n + 1) = 0, d0(0) = 1, (n + 2)d1(n) + (−n2 −3)d1(n + 1) = 0, d1(0) = 4, 75 7 Extension to Ck-finite and Dk-finite sequences and the D2-finite sequences c0(n)a(n) + c1(n)a(n + 1) = 0, a(0) = 3, d0(n)b(n) + d1(n)b(n + 1) = 0, b(0) = 5. By Theorem 7.9, the sequence h := ab is D2-finite. With the methods introduced in Chapter 4 we can compute the recurrence e(n)h(n) + h(n + 1) = 0, h(0) = 15 with (n6 + 2n5 + 5n4 + 8n3 + 7n2 + 6n + 3)e(n) + (n2 + 9n + 14)e(n + 1) = 0 and e(0) = −1 4. By induction, every Ck-finite sequence is Dk-finite and every Dk-finite sequence is Ck+1-finite. Therefore, we get the following chain of rings RC ⊆RD ⊆RC2 ⊆RD2 ⊆RC3 ⊆· · · Example 7.4 is true more generally and the following generalization of Theorem 4.7 can be shown: Corollary 7.11. Let c be a C-finite sequence over the field K and p ∈N[n]. Denote k := deg(p). Then, c(p(n)) is Ck-finite over the splitting field L of the characteristic polynomial of c. Proof. We use induction on k. For k = 1, this is precisely the fact that C-finite sequences are closed under taking subsequences at arithmetic progressions. Let k ≥2. We can write c as an L-linear combination of sequences d(n) = niαn for i ∈N and α ∈L from some term on. Let p(n) = pknk + q(n) with deg(q) ≤k −1. Then, we have d(p(n)) = p(n)iαp(n) = p(n)i (αpk)nk αq(n). The sequence p(n)i is polynomial and therefore Ck-finite. The sequence (αpk)nk is Ck-finite (as seen in Example 7.3). By induction αq(n) is Ck−1-finite, so in particular Ck-finite. 76 7 Extension to Ck-finite and Dk-finite sequences Therefore, with Theorem 7.9, the sequence d(p(n)) is Ck-finite as it is the product of Ck-finite sequences. Since Ck-finite sequences are also closed under L-linear combinations and shifts, c(p(n)) is Ck-finite. The same question from Corollary 7.11 can of course be asked for D-finite sequences. Neither the proof of Corollary 7.11 (as D-finite sequences do not have a nice closed form) nor the proof of Theorem 4.7 (as Lemma 11 in [KM14] does not hold for the D-finite case) carry over to this case. Open Question 7.12. Let a be a D-finite sequence. Is a(n2) a D2-finite sequence? 77 8 Positivity of C-finite sequences In the previous chapters we have seen that the Skolem Problem, i.e., the problem of deciding whether a given C-finite sequence c(n) has a zero, plays an important role when computing with C2-finite sequences. Closely related and similarly difficult is the problem of deciding whether a sequence is positive, i.e., whether c(n) > 0 for all n ∈ N. We call this the Positivity Problem. Methods for showing positivity of a sequence can usually be adjusted to show nonnegativity of a sequence. For the sake of a clear presentation, we focus on positivity here. Decidability for this problem is known for sequences of order at most 5 and for certain other classes of C-finite sequences of order at most 9 [HHH06, LT09, OW14b, OW14a]. In the D-finite case even less is known. Similar methods as in the D-finite case were applied to certain sequences of order 2 [NOW21]. For larger orders, procedures based on quantifier elimination (cf. [GK05, KP10, Pil13]) were successfully applied in practice [Kau07b, Pil08, Pil19]. Other techniques are based on writing a sequence as sum of squares or singularity analysis [Cha14, Hoe21, MM22]. Concerning implementation, only very few software packages are known which support proving inequalities of sequences automatically. Two implementations of the Gerhold-Kauers method for Mathematica and SageMath, respectively, are known [Kau06, Ura20]. These, however, do not implement any special procedures for C-finite sequences. An on-line tool for computing zeros of certain C-finite sequences based on SageMath is presented in [BLN+22]. In the context of the thesis two more software packages were created for dealing with C-finite sequences specifically, the package rec_sequences for SageMath and the package PositiveSequence for Mathematica (the latter is part of the RISCErgoSum collection of packages). The Positivity Problem is not only interesting by itself but also plays an important role because other problems can be reduced to it. A sequence c(n) has no zeros if and only if the sequence c(n)2 is positive. Hence, as C-finite sequences form a computable ring, 78 8 Positivity of C-finite sequences the Skolem Problem can be reduced to the Positivity Problem. Further, an inequality problem of the form c(n) > d(n) for all n ∈N can be reduced to checking whether the sequence c(n) −d(n) is positive. As the Positivity Problem only makes sense over real valued sequences, we assume that the base field K is a real number field. We denote the field of real algebraic numbers by A := Q ∩R ⊋K. Suppose a C-finite sequence c has a unique dominant eigenvalue λ1 ∈A (i.e., we have k = 1 in the setting of equation (2.2)). Then, (2.3) shows that c ∼γndλn 1 for some γ ∈A. The sequence c can only be positive if γ, λ1 > 0. Furthermore, c is positive if and only if c(n)/λn 1 is positive. Therefore, in the case of a unique dominant eigenvalue, it is sufficient to show positivity of a sequence p(n) + s ∑ i=1 ( oi(n)ξn i + oi(n)ξi n) + l ∑ i=1 qi(n)ρn i (8.1) with p ∈A[x], o1, . . . , os ∈Q[x], q1, . . . , ql ∈A[x] and constants ξ1, . . . , ξs ∈Q, ρ1, . . . , ρl ∈ A where the leading coefficient of p is positive [OW14b]. In Section 8.1 we discuss several different algorithms which can be used for proving positivity of certain C-finite sequences. These methods are all well-known or slight variations of algorithms which can be found in the literature. In Section 8.2 we compare these algorithms for C-finite sequences coming from the OEIS. Furthermore, we provide some statistics on how many sequences in the OEIS are C-finite or D-finite based on guessing procedures. This chapter is mostly based on [NP22a]. 8.1 Algorithms In this section we give an overview of some methods which can be used to prove positivity of a C-finite sequence. Algorithms 1 and 2 (as well as their adjusted versions Algorithm 1e and 2e) presented below in Sections 8.1.1 and 8.1.2 can be applied to D-finite sequences. As such they can be used to prove positivity of C-finite sequences. However, sometimes C-finite sequences satisfy a D-finite recurrence of lower order, which is better suited as input for these methods. In Section 8.1.3, we discuss when such a D-finite recurrence exists. A method based on the combination of Algorithms 1 and 2 as well as on the closed form of a C-finite sequence is introduced in Section 8.1.5. The methods described in Sections 8.1.4 79 8 Positivity of C-finite sequences and 8.1.6 also make use of the closed form of C-finite sequences. They are based on known results, but we believe that they had not been implemented so far. 8.1.1 Algorithm 1 In 2005 [GK05] a method based on quantifier elimination (in particular, cylindrical al-gebraic decomposition CAD [Col75, CH91, CJ98, BRPR03]) was introduced which can be used to show positivity of sequences that can be defined recursively along some dis-crete parameter. This procedure, however, is not guaranteed to terminate. For D-finite sequences of small order conditions which guarantee the termination of the algorithm are known [KP10, Pil13]. We give a short description of Algorithm 1 from [KP10]. Suppose c is D-finite of order r. By Theorem 2.5 there are rational functions qρ,0(x), . . . , qρ,r−1(x) ∈K(x) for all ρ ∈N with c(n + ρ) = ∑r−1 i=0 qρ,i(n)c(n + i). The idea of the Gerhold-Kauers method is to check with quantifier elimination whether c(n), . . . , c(n + r −1) > 0 implies c(n + r) > 0 where c(n + r) can be written in terms of the c(n), . . . , c(n + r −1). If this is true, then by induction it would be sufficient to check finitely many initial values to deduce positivity of the entire sequence. If, however, this cannot be shown, then we can add c(n + r) > 0 to the hypothesis and show c(n + r + 1) > 0. This process is iterated. In the iteration step ρ ≥r we try to show positivity of the formula Φ(ρ, c) := ∀y0, . . . , yr−1, x ∈R: ⎛ ⎝x ≥0 ∧ ρ−1 ⋀ j=0 r−1 ∑ i=0 qj,i(x)yi > 0 ⎞ ⎠ = ⇒ r−1 ∑ i=0 qρ,i(x)yi > 0. The formula Φ(ρ, c) is a generalized induction formula over the reals. It is certainly sufficient to prove the induction step and has the advantage of being a valid input for CAD and other quantifier elimination methods. Here, we give a slightly adjusted version which searches for an index n0 such that the sequence σn0(c) is positive, i.e., it checks whether the sequence is eventually positive (hence, we denote the algorithm by Algorithm 1e). If such an n0 can be found by the algorithm, then it is sufficient to check the initial values c(0), . . . , c(n0 −1) of the sequence to prove positivity of c. 80 8 Positivity of C-finite sequences Input : D-finite sequence c of order r output:n0 such that σn0(c) is positive n, n0 ←0 d ←c while n < r or ¬Φ(n, d) do if d(n) > 0 then n ←n + 1 else n0 ←n0 + n + 1 d ←σn+1(d) n ←0 end end return n0 Algorithm 1e: Adjusted version of Algorithm 1 from [KP10] Clearly, Algorithm 1e is not guaranteed to terminate. E.g., if the input sequence c is not eventually positive, then the algorithm never terminates. Suppose the sequence c is eventually positive, i.e., there exists an n0 ∈N such that σn0(c) is positive. As the characteristic polynomials agree, χ(c) = χ(σn0(c)), the same termination conditions for Algorithm 1 in [KP10] now also apply to Algorithm 1e. Example 8.1. The alternating sequence A000034 is C-finite of order 2 satisfying c(n) −c(n + 2) = 0 with initial values c(0) = 1, c(1) = 2. Algorithm 1e terminates for this sequence showing that c is positive. Example 8.2. The sequence A005682 is C-finite of order 6 satisfying c(n) + c(n + 2) −2c(n + 5) + c(n + 6) = 0 with initial values c = ⟨1, 2, 4, 8, 15, 28, . . . ⟩. Algorithm 1e cannot show positivity of c in 60 seconds. 81 8 Positivity of C-finite sequences 8.1.2 Algorithm 2 Algorithm 2 in [KP10] again uses quantifier elimination to prove positivity of a D-finite sequence. The idea is to check whether there is a µ > 0 such that c(n + 1) ≥µc(n) for all n ∈N. By induction, if there is a µ > 0 such that c(n + 1) ≥µc(n) ∧· · · ∧c(n + r −1) ≥µc(n + r −2) = ⇒c(n + r) ≥µc(n + r −1), then it is again sufficient to check finitely many initial values to prove positivity of c. Hence, the important step in the algorithm is to use quantifier elimination to verify whether there exists a µ > 0 such that the formula Ψ(ξ, µ, c) := ∀y0, . . . , yr−1 ∈R ∀x ∈R≥ξ : ( y0 > 0 ∧ r−2 ⋀ i=0 yi+1 ≥µyi ) = ⇒ r−1 ∑ i=0 qi(x)yi ≥µyr−1 is valid where qi ∈K(x) are again such that c(n + r) = ∑r−1 i=0 qi(n)c(n + i) for all n ∈N. Again, we give a slightly adjusted version which searches for an index n0 such that the sequence σn0(c) is positive. If the input sequence c is eventually positive, then the same termination conditions as for Algorithm 2 in [KP10] apply in this adjusted version. Example 8.3. Algorithm 2e can show positivity of the sequence from Example 8.2 (which could not be done with Algorithm 1e). Example 8.4. Algorithm 2e cannot show positivity of the alternating sequence from Example 8.1 (which could be proven with Algorithm 1e). 8.1.3 D-finite reduction Clearly, every C-finite sequence is also D-finite. Sometimes, C-finite sequences satisfy D-finite recurrences of lower order. In these cases it can be helpful to use this shorter D-finite recurrence as the next example shows. Example 8.5. Let c be the sequence defined by c(n) = n2 + 1 for all n ∈N (A002522). If c is considered as a C-finite sequence of order 3, then neither Algorithm 1e nor Algorithm 2e 82 8 Positivity of C-finite sequences Input : D-finite sequence c of order r output:n0 such that σn0(c) is positive n, n0 ←0 d ←c Ψ(ξ, µ) ←quantifier free formula equivalent to Ψ(ξ, µ, d) for n = 0, 1, . . . do if d(n) ≤0 then n0 ←n0 + n + 1 d ←σn+1(d) Ψ(ξ, µ) ←quantifier free formula equivalent to Ψ(ξ, µ, d) n ←0 else if ∃µ > 0: r−2 ⋀ i=0 d(n + i + 1) ≥µd(n + i) ∧Ψ(n, µ) then return n0 end Algorithm 2e: Adjusted version of Algorithm 2 from [KP10] terminate in 60 seconds. If c is, however, considered as a D-finite sequence of order 1 and degree 2, then both algorithms terminate and show that c is indeed positive. The next lemma shows that we can find a shorter D-finite recurrence of a C-finite se-quence c if and only if c has eigenvalues of higher multiplicities or equivalently the characteristic polynomial χ(c) ∈K[y] of c is not squarefree. Lemma 8.6. Let c be a C-finite sequence of order r with y ∤χ(c). Then, c is D-finite of order m < r if and only if χ(c) is not squarefree. Proof. Suppose c is given as in Theorem 2.15. ⇐ =: The sequences pi(n)λn i are D-finite of order 1 and degree di over the algebraic closure K. Hence, by Theorem 2.6, c(n) is D-finite of order at most m over K. Lemma 2 in [Ger05] shows that the sequence is then also D-finite over K with the same order and degree. In particular, if χ(c) is not squarefree, then r = ∑m i=1 di > m. = ⇒: Suppose that c satisfies a D-finite recurrence of order m < r and degree d m ∑ i=0 pi(n)c(n + i) = 0 for all n ∈N (8.2) 83 8 Positivity of C-finite sequences with pi(n) = ∑d k=0 pi,knk where not all pi,k are zero. Furthermore, suppose that c is C-finite of order r with pairwise distinct eigenvalues λ1, . . . , λr ∈K, i.e., c(n) can be written as c(n) = ∑r j=1 γjλn j for some γj ∈K. Using this closed form in (8.2) yields d ∑ k=0 ( m ∑ i=0 r ∑ j=1 pi,kγjλn+i j ) nk = 0. (8.3) Let γk,j := ∑m i=0 pi,kγjλi j, then (8.3) is equivalent to ∑d k=0 ( ∑r j=1 γk,jλn j ) nk = 0. Evaluating n = 0, . . . , r(d + 1) −1 yields a homogeneous linear system for the γk,j. The corresponding matrix is regular (cf. Theorem 2.2.1 in [Li06] or Proposition 2.11 in [HHHK05]), so γk,j = 0 for all k, j. Let k be such that pi,k ̸= 0 for some i. Then, 0 = r ∑ j=1 λn j m ∑ i=0 pi,kγjλi j = m ∑ i=0 r ∑ j=1 pi,kγjλn+i j = m ∑ i=0 pi,kc(n + i). Hence, c satisfies a C-finite recurrence of order m < r, a contradiction to c being C-finite of order r. The proof of Lemma 8.6 shows that precisely the polynomial factors can be reduced in the D-finite recurrence, i.e., the m in the statement of Lemma 8.6 is the number of distinct eigenvalues of the sequence, which is also denoted by m in Theorem 2.15. The degree of the D-finite recurrence can be bounded by (m(m + 1) −m) max i=1,...,m di = m2 max i=1,...,m di ≤r3 using Theorem 2 in [Kau14]. In practice, we can easily check whether χ(c) is squarefree by checking whether χ(c) and its derivative are coprime. The shorter D-finite recurrence can then be either found by guessing or by computing it explicitly from the closed form of c. 8.1.4 Classical algorithm for sequences with unique dominant eigenvalue If a C-finite sequence has a unique dominant eigenvalue, checking positivity of the se-quence is known to be decidable. More details on the concrete time complexity are given 84 8 Positivity of C-finite sequences in [OW14b]. Based on these results, we give a full description of such an algorithm in this section which is readily implemented. A similar description is given in [Kou05]. In the introduction of this chapter we have seen that a C-finite sequence c can be assumed to be in its closed form representation as c(n) = p(n) + r(n) (8.4) where p ∈A[x] with lc(p) > 0 and r(n) = ∑m i=1 pi(n)λn i with pi ∈Q[x], λi ∈Q and 1 > \|λ1\| ≥\|λ2\| ≥· · · ≥\|λm\|. The idea is now to compute an ε ∈(0, 1) and n0, n1 ∈N such that \|r(n)\| < (1 −ε)n for n ≥n0 and p(n) ≥(1 −ε)n for n ≥n1. Then, clearly c(n) is positive from max(n0, n1) on. The initial values can be checked separately again. Input :C-finite sequence c of the form (8.4) output:true if c(n) > 0 for all n ∈N and false otherwise ε ←1−\|λ1\| 2 compute n0 such that \|r(n)\| < (1 −ε)n for all n ≥n0 compute n1 such that p(n) ≥(1 −ε)n for all n ≥n1 if c(n) > 0 for n = 0, . . . , max(n0, n1) then return true else return false end Algorithm C: Positivity for sequences with dominant eigenvalues [OW14b] For a polynomial pi(x) = ∑di j=0 γi,jxj ∈A of degree di we can easily compute a constant ci ∈A such that \|pi(n)\| ≤cindi for all n ≥1. For example, we can choose ci := ∑di j=0 ⏐ ⏐γi,j ⏐ ⏐. Let c := ∑m i=1 ci and d := max(d1, . . . , dm), i.e., the maximal multiplicity of the eigenvalues λ1, . . . , λm. Furthermore, let ε := 1−\|λ1\| 2 . Then, 1 −ε = \|λ1\| + ε. First, we show how n0 can be found such that \|r(n)\| < (1 −ε)n for n ≥n0. Let µ := \|λ1\|+ε \|λ1\| . If d = 0, then \|r(n)\| ≤c \|λ1\|n. Clearly, c \|λ1\|n < (1 −ε)n = (\|λ1\| + ε)n ⇐ ⇒ log(c) log(µ) < n. Hence, we can choose n0 := ⌈ log(c) log(µ) ⌉ in this case. If d > 0, then \|r(n)\| ≤c nd \|λ1\|n. Again, c nd \|λ1\|n < (1 −ε)n ⇐ ⇒log(c1/d) < n d log(µ) −log(n). 85 8 Positivity of C-finite sequences The derivative of the right-hand side of the latter inequality is positive if n > d log(µ), i.e., from ⌈ d log(µ) ⌉ on the sequence on the right-hand side is monotonously increasing. Hence, if the inequality is true for some n0 ≥ ⌈ d log(µ) ⌉ , then it is true for all n ≥n0. Checking these values one by one, we find a suitable n0 eventually. If the polynomial p(x) = p0 is just constant, then p(n) ≥(1 −ε)n if and only if n ≥ log(p0) log(1−ε). Otherwise, we can compute the largest real root x1 of the derivative of p(x). If p(n1) ≥(1 −ε)n1 for any n1 ≥⌈x1⌉, then the inequality holds for all n ≥n1. Note that once we have established that a sequence has a unique dominant eigenvalue, all these computations can be done using arbitrary precision arithmetic. For our implementa-tion in SageMath we make use of the Arb library [Joh17]. Example 8.7. The sequence from Example 8.2 has a unique dominant eigenvalue. Hence, Algorithm C shows positivity of the sequence after checking max(n0, n1) = 12 initial terms. Example 8.8. The sequence from Example 8.1 has dominant eigenvalues ±1. Hence, as the sequence does not have a unique dominant eigenvalue, Algorithm C cannot establish positivity of the sequence. 8.1.5 Combination of Algorithm 1 and Algorithm 2 In the case that the C-finite sequence has a unique dominant eigenvalue, we can combine the closed form representation of the sequence together with Algorithm 1e and Algo-rithm 2e. As we know that the polynomial term p(n) in (8.1) certainly dominates the exponential terms, we can find indices ni using Algorithm 1e and Algorithm 2e from which on the exponential sequences are dominated by the polynomial term. These input sequences have very low order (maximum order 3). Therefore, the termination criteria in [KP10] show that these algorithms terminate in most instances. Before we can prove termination criteria for Algorithm P, we prove some auxiliary results on the characteristic polynomial of D-finite sequences. 86 8 Positivity of C-finite sequences Input :C-finite sequence c of the form (8.1) output:true if c(n) > 0 for all n ∈N and false otherwise for i ←1 to s do ni,Q ←Algorithm 1e applied to p(n) s+l + oi(n)ξn i + oi(n)ξi n end for i ←1 to l do ni,A ←Algorithm 2e applied to p(n) s+l + qi(n)ρn i end n0 ←max(n1,Q, . . . , ns,Q, n1,A, . . . , nl,A) if c(n) > 0 for n = 0, . . . , n0 then return true else return false end Algorithm P: Positivity for sequences with dominant eigenvalues First, we extend the notion of the characteristic polynomial from the ring K[n][σ] to the left Euclidean domain K(n)[σ]. For a rational function p(n) q(n) with coprime p, q ∈K[n] we define the degree as deg(p/q) := deg(p) −deg(q) and call lc(p/q) := coeff (p/q, deg(p/q)) := lc(p)/ lc(q) the leading coefficient of p/q. Now, for an operator A = ∑r i=0 pi(n) qi(n) σi ∈K(n)[σ] with deg(A) := maxi=0,...,r deg(pi/qi) we define the characteristic polynomial as χ(A) := r ∑ i=0 deg(pi/qi)=deg(A) lc(pi/qi)yi ∈K[y]. If A ∈K[n][σ], i.e., if all qi are constants, then this definition is identical to the original definition (2.1). Next, in Lemma 8.9 and Lemma 8.10, we state some basic properties of the characteristic polynomial. Since we could not find references for those, we add the proofs for the sake of completeness. Lemma 8.9. Let A, B ∈K(n)[σ]. Then χ(AB) = χ(A)χ(B). 87 8 Positivity of C-finite sequences Proof. Let A := ∑r i=0 pi(n)σi ∈K(n)[σ] and B := ∑s j=0 qj(n)σj ∈K(n)[σ] and dA := maxi=0,...,r deg(pi), dB := maxj=0,...,s deg(qj) ∈Z their respective degrees. We show that AB has degree dA + dB. By the definition of multiplication in K(n)[σ] and the properties of the degree of a rational function, the degree of AB is certainly bounded by dA + dB. Let i′, j′ be maximal such that deg(pi′) = dA and deg(qj′) = dB. We show that the coefficient of σi′+j′ of AB has degree dA + dB. This coefficient is given by ∑ i′+j′ l=0 pl(n)qi′+j′−l(n + l). Because of the choices of i′, j′ we have deg(pl(n)qi′+j′−l(n)) = deg(pl(n)) + deg(qi′+j′−l(n + l)) < dA + dB for all l ̸= i′. For l = i′, we have deg(pl(n)qi′+j′−l(n)) = dA + dB, so by the properties of the degree we have deg (i′+j′ ∑ l=0 pl(n)qi′+j′−l(n + l) ) = max l=0,...,i′+j′ ( deg (pl(n)) + deg ( qi′+j′−l(n + l) )) = dA + dB. Next, we show that all coefficients of χ(A)χ(B) and χ(AB) agree. Let i ∈{0, . . . , r + s}. Then, coeff (χ(A), i) = coeff (pi(n), dA) , coeff (χ(B), i) = coeff (qi(n), dB) and therefore coeff (χ(A)χ(B), i) = i ∑ j=0 coeff ( pj(n), dA ) coeff ( qi−j(n), dB ) . 88 8 Positivity of C-finite sequences In the first part of the proof we have shown that AB has degree dA + dB. Therefore, coeff (χ(AB), i) = coeff ( i ∑ j=0 pj(n)qi−j(n + j), dA + dB ) = i ∑ j=0 coeff ( pj(n)qi−j(n + j), dA + dB ) = i ∑ j=0 coeff ( pj(n), dA ) coeff ( qi−j(n + j), dB ) = i ∑ j=0 coeff ( pj(n), dA ) coeff ( qi−j(n), dB ) . Suppose A is an annihilator of a and B an annihilator of b. Then, the least common left multiple lclm(A, B) is an annihilator of a + b [Kau15]. Lemma 8.10. Let A, B ∈K[n][σ]. Then χ(A) \| χ(lclm(A, B)) and χ(B) \| χ(lclm(A, B)). In particular, we have lcm(χ(A), χ(B)) \| χ(lclm(A, B)). Proof. Let C ∈K(n)[σ] be such that CA = lclm(A, B). Then, with Lemma 8.9 we have χ(lclm(A, B)) = χ(CA) = χ(C)χ(A). Example 8.11. In Lemma 8.10, divisibility cannot be replaced with equality. Consider A := 1 + σ and B := n + (n + 1)σ. Then, χ(A) = χ(B) = 1 + y, 89 8 Positivity of C-finite sequences but χ(lclm(A, B)) = χ(n + (2n + 2)σ + (n + 2)σ2) = 1 + 2y + y2. An operator A = ∑r i=0 piσi ∈K[n][σ] is called balanced if deg p0 = deg pr = max i=0,...,r deg pi. Equivalently, A is balanced if and only if the degree of χ(A) ∈K[y] equals the order of A and the trailing coefficient of χ(A) is nonzero, i.e., y ∤χ(A). As Algorithm 2e terminates for essentially all sequences of order 2, the real algebraic part of Algorithm P certainly terminates. Theorem 8.12. Algorithm P terminates if s = 0, i.e., if all eigenvalues of c are real algebraic. Proof. Each sequence h(n) := p(n) s+l + qi(n)ρn i is the sum of two balanced D-finite sequences g, f over A satisfying the recurrences −p(n + 1)g(n) + p(n)g(n + 1) = 0, −qi(n + 1)ρi f (n) + qi(n) f (n + 1) = 0 with characteristic polynomials χ(G) = lc(p)(y −1), χ(F) = lc(qi)(y −ρi) where G, F denote the annihilating operators of g, f, respectively. As these characteristic polynomials are coprime, Lemma 8.10 yields χ(H) = χ(G)χ(F) = γ(y −1)(y −ρi) for some constant γ where H denotes the annihilating operator of h. In particular, H is balanced. Furthermore, h ∼p(n) by construction. With [KP10, Theorem 3], Algorithm 2e terminates with input h. It is conjectured that Algorithm 1e terminates for sequences of order 3 if the eigenvalues are complex. This is the case if we apply Algorithm 1e. Hence, if the conjecture is true, Algorithm P terminates for all C-finite sequences with a unique dominant eigenvalue. 90 8 Positivity of C-finite sequences Theorem 8.13. Assume Conjecture 1 from [KP10] is true. Then, Algorithm P terminates. Proof. The proof of Theorem 8.12 already shows that the algorithm terminates for the real algebraic eigenvalues. Analogously, in the complex case, the sequences h(n) := p(n) s+l + oi(n)ξn i + oi(n)ξi n are D-finite of order 3 with a balanced annihilating operator H with characteristic polynomial χ(H) = γ(y −1)(y −ξi)(y −ξi) for some constant γ. With Conjecture 1, Algorithm 1e terminates on this input. Example 8.14. The sequence A002248 is C-finite of order 4 satisfying the recurrence 4c(n) −8c(n + 1) + 7c(n + 2) −4c(n + 3) + c(n + 4) = 0 with initial values c = ⟨2, 8, 14, 16, . . . ⟩. The sequence has the unique dominant eigen-value 2. Neither Algorithm 1e nor Algorithm 2e terminate in 60 seconds. However, both Algorithm C and Algorithm P terminate in negligible time. 8.1.6 Decomposition into nondegenerate sequences Theorem 2.17 states that every C-finite sequence c(n) can be written as the interlacing of nondegenerate sequences c1(n) := c(dn), . . . , cd(n) := c(dn + d −1). Reducing the Positivity Problem for c to the Positivity Problem for the subsequences ck for k = 1, . . . , d often turned out useful [MST84, Ver85, OW14b]. For a given C-finite sequence c we can check whether they are degenerate by computing the ratio of all pairs of eigenvalues and checking whether they are a root of unity [Coh13]. Hence, we can compute the decomposition of c into nondegenerate sequences naively by computing the sequences c1, . . . , cd and checking whether all these are nondegenerate. If they are not, we can increase d. Eventually, for large enough d, all subsequences are nondegenerate. This already works well in practice. A more efficient algorithm is given in [YLN95]. 91 8 Positivity of C-finite sequences If decomposition into subsequences is used together with Algorithm C or Algorithm P, then it is more efficient to check whether every subsequence has a unique dominant root (which can be done numerically with arbitrary-precision arithmetic) instead for checking degeneracy. The main bottleneck is usually the computation of the subsequences. Hence, an efficient implementation should certainly aim to minimize these. Sequences of natural numbers which can be decomposed into subsequences with a unique dominant eigenvalue are N-rational [Kou05, Theorem 2.5.12]. It was already observed that they seem to cover most C-finite sequences appearing in practical examples [Kou05]. Example 8.15. The sequence A000115 is C-finite of order 8 and satisfies the recurrence c(n) −c(n + 1) −c(n + 2)+c(n + 3) −c(n + 5) + c(n + 6) + c(n + 7)−c(n + 8) = 0. with initial values c = ⟨1, 1, 2, 2, 3, 4, 5, 6, . . . ⟩. It has 6 dominant eigenvalues and is degenerate. It can be decomposed into 10 nondegenerate sequences with unique dominant eigenvalues. For these subsequences Algorithm C and Algorithm P both have no problem showing positivity. Example 8.16. The Berstel sequence A007420 is C-finite of order 3 satisfying 4c(n) −4c(n + 1) + 2c(n + 2) −c(n + 3) = 0 with c(0) = c(1) = 0, c(2) = 1. Checking the initial values of the C-finite sequence d(n) := c(n + 53)2 indicates that d(n) is positive and that the only zeros of c are at the in-dices n = 0, 1, 4, 6, 13, 52 (this is, in fact, the maximal number of zeros a nondegenerate sequence of order 3 can have and therefore these already have to be all zeros [Beu91]). The sequence d(n) is nondegenerate and does not have a unique dominant root. In fact, Ge’s algorithm applied to the eigenvalues of c shows that there are no relations among them. Hence, we cannot expect this algorithm to work for the sequence d(n). 8.2 Comparison For comparing the different algorithms discussed in the previous section, we consider C-finite sequences from the OEIS. First, we discuss how we can use guessing to determine 92 8 Positivity of C-finite sequences how many of the around 360 000 sequences are C-finite or D-finite. Next, we use 1000 of these C-finite sequences (where the first terms are positive) as a test set for comparing the various algorithms and testing the implementation. 8.2.1 Recurrence sequences in the OEIS In a talk in 2003 Bruno Salvy estimated that about 25% of the 5488 sequences from the book The Encyclopedia of Integer Sequences (cf. [SP95]) are D-finite [Sal03] (in his master thesis 1991 Simon Plouffe, using guessing, estimated this number to be around 18% for a preliminary draft of the book). These 25% have been cited as an estimate for the ratio of D-finite sequences in the OEIS several times over the past two decades [Sal05, Kau13, Yur22]. To our knowledge, no estimate for the ratio of C-finite sequences in the OEIS is known. At the time of writing (spring 2023) the OEIS contains about 360 000 integer sequences. Due to this large number of sequences, only estimates for these ratios can be found. This can be done, for instance, by either inspecting a smaller subset of these sequences closer by hand or by using guessing routines on the terms saved in the database. We use the latter approach. Guessing routines are limited by the number of terms which are known for a particular sequence. The more terms we have, the bigger recurrences we can guess or the more confident we can be that a guessed recurrence is indeed valid. The number of terms given in the OEIS vary widely. About 6% of the sequences have at most 10 terms, about 50% at most 100 terms and about 13% of the sequences have at least 10 000 terms given (note, however, that these terms are only given in the corresponding “B-files” and not displayed in the database itself). Figure 8.1 gives an overview of the number of terms of sequences which are given in the OEIS. Figure 8.1: Number of sequences for which specific number of terms are given in the OEIS 93 8 Positivity of C-finite sequences For guessing C-finite recurrences, we fix a maximal order of 100, i.e., sequences which might satisfy a higher order recurrence are not considered C-finite. Guessing a D-finite recurrence of order r and degree d yields a linear system of equations with (r + 1)(d + 1) many variables. In order to have some confidence in the guess, we have to make sure that the linear system is overdetermined. The more equations we have, the more confident we can be in our guess. We make sure that corresponding linear systems have ⌊(r + 1)(d + 1)(1 + e 2) ⌋+ e many variables for some e ∈N. I.e., the number of equations needed depends also relatively on the order/degree of the recurrence. The number e can be interpreted as the confidence level of our guess. However, if e is larger it might be that we do not have enough terms given to verify a recurrence that could be guessed with smaller e. This can also be seen in Figure 8.2a which shows the results for C-finite sequences. For instance, going from e = 1 to e = 3 we can see that many recurrences are recognized to be wrong (orange part in the figure). For larger e only a small number of recurrences are recognized as wrong, so we can assume that most of these recurrences are indeed correct. However, the figure also shows that by increasing e we might fail to verify a recurrence simply due to a lack of terms given in the OEIS (light blue parts in the figure). We can estimate that close to 15% of the sequences in the OEIS might be C-finite. (a) (b) Figure 8.2: (8.2a): Number of C-finite sequences in the OEIS according to different confidence levels. The percentages indicate the ratio of C-finite sequences in the OEIS according to the given confidence level (8.2b): Number of C-finite sequences of specific orders in the OEIS 94 8 Positivity of C-finite sequences A sanity check can also be done using the OEIS Wiki page which gives an overview of the sequences which were already recognized to be C-finite.1 For e = 1 we miss about 800 sequences from the Wiki page, about 33 600 sequences are considered C-finite both by the Wiki page as well as by guessing and about 61 600 additional (hypothetical) sequences were found. For e = 5 these numbers are about 5700, 28 800 and 23 600, respectively. We can do a closer investigation of the recurrences that we guessed. For instance (for the confidence level e = 5), around one half of the C-finite sequences have eigenvalues of higher multiplicity. I.e., by Theorem 8.6 these are precisely the sequences which have a shorter (in terms of the order) D-finite recurrence. Almost 20% of the C-finite sequences are just polynomial sequences. The orders of the sequences are shown in Figure 8.2b. As can be expected, most sequences which are guessed have relatively small order (for e = 5 about 70% of the C-finite recurrences have orders at most 10 and only about 6% have order larger than 30). For guessing D-finite recurrences we have to be a bit more careful as zero terms of a sequence might yield wrong guesses. We use the techniques from [KV19] to mitigate this problem. Recurrences for a D-finite sequence can be found on the so-called order-degree curve, i.e., the possible minimal values (r, d) for the order r and degree d of a recurrence lie on a hyperbola [Kau14]. Often, the minimal order recurrence, which is the one we are looking for, has large degree. For our guessing approach we search for the operator with (r + 1)(d + 1) ≤100. The results can be observed in Figure 8.3. According to these, we can estimate that up to 20% of the sequences in the OEIS might be D-finite. For e = 3, about one third of the D-finite recurrences we guessed are hypergeometric, i.e., have order one. Close to 40% have degree zero, i.e., they are C-finite with eigenvalues having only multiplicities one. As we estimated before that around half of the C-finite sequences have eigenvalues of higher multiplicities we can guess that around 80% of the D-finite sequences in the OEIS are in fact C-finite. This agrees with our estimates that around 20% are D-finite and 15% are C-finite. Plotting the ratio of D-finite sequences in the OEIS w.r.t. the OEIS identifier one can see that the ratio dropped slightly over the past twenty years of the database’s existence (cf. Figure 8.3d). The first spike around sequence A042700 is due to a series of sequences related to continued fractions of certain algebraic numbers. The spike around the sequence A170000 is caused by a series of C-finite sequences counting the number of words in 1 95 8 Positivity of C-finite sequences certain Coxeter groups (A168680 to A170731). For many of these only few terms are saved in the OEIS which explains the significant drop for larger confidence levels. Around the sequence A149000, the ratio drops significantly (especially for e = 1). These sequences describe lattice walks and most of them seem to not be D-finite [BK08]. (a) Orders (b) Degrees (c) Order-degrees density for e = 5 (d) Ratio of D-finite sequences in the OEIS Figure 8.3: Number of D-finite sequences of specific orders and degrees in the OEIS in (8.3a) and (8.3b). Combined density plot of orders/degrees for e = 5 in (8.3c) and density of D-finite sequences in the OEIS w.r.t. their index in (8.3d). 96 8 Positivity of C-finite sequences 8.2.2 Positive sequences in the OEIS From the sequences where a C-finite recurrence could be guessed we take the first 1000 where the first 500 terms are strictly positive and are therefore highly likely to be positive altogether. 2 The maximal order of these sequences is 42. The following table shows the number of sequences of each given order: order 1 2 3 4 5 6 7 8 9 10 > 10 73 134 117 139 120 80 87 36 47 27 140 More than half of these sequences, 567, have a unique dominant eigenvalue. There are 102, 40, 70, 32 sequences with 2, 3, 4, 5 distinct dominant eigenvalues, respectively. Hence, there are 139 sequences with more than 6 distinct dominant eigenvalues. About half of the sequences, 513, have a characteristic polynomial which is not squarefree. By Lemma 8.6 these are the sequences which have a shorter D-finite recurrence. We test the positivity methods implemented in the rec_sequences package on these sequences. SageMath provides an interface to QEPCADB which allows CAD computa-tions [Sag23, Bro03]. This is used in the implementations of Algorithm 1 and Algorithm 2. For Algorithm C we rely on fast arbitrary-precision arithmetic using the library Arb which is included in SageMath [Joh17]. To decompose a sequence into subsequences with a unique dominant eigenvalue, we decompose the sequence into k subsequences and check, using arbitrary-precision arithmetic, whether all of these have a unique dominant eigen-value. If they do not have a unique dominant eigenvalue, we increase k by one. The main bottleneck when decomposing is by far the computation of the subsequences. Checking whether a subsequence has a unique dominant eigenvalue or proving positivity of a sequence with a unique dominant eigenvalue using Algorithm C only takes negligible time in our examples. 2A table with these sequences and additional information is given on the website jku.at/people/pnuspl/PositivityCFinite. It also contains the detailed results of the SageMath (using and Mathematica tests. The SageMath results in this thesis are using a more recent version of the package and are therefore slightly different. 97 8 Positivity of C-finite sequences We give a list of the methods that can be used on C-finite sequences to show positivity. Every method has a parameter strict which is True by default and indicates whether strict positivity or nonnegativity should be shown. The additional parameter time can be used to give an upper bound (in seconds) after which the algorithms should be terminated, the default value is −1, indicating that they should not stop prematurely. • is_positive_algo1 implements Algorithm 1 from [KP10]. As an additional pa-rameter bound can be specified which gives an upper bound on the number of iterations. • is_positive_algo2 implements Algorithm 2 from [KP10]. Again, bound can be specified. This method is also implemented for general D-finite sequences and can be called using is_positive on D-finite sequences. • is_positive_dominant_root implements Algorithm C for sequences with a unique dominant eigenvalue. • is_positive_dominant_root_decompose first tries to decompose the sequence into sequences with a unique dominant eigenvalue and zero sequences and calls Algo-rithm C on each of those. Using these methods, all of the 1000 sequences from the test set could be proven to be positive using a time limit of 60 seconds. The following table gives an overview of the number of sequences which could be proven to be positive by each method (a “D” indicates that decomposition of the sequence is used): Algo. 1 D, Algo.1 Algo. 2 D, Algo.2 Algo. C D, Algo. C 384 375 327 556 566 1000 It is clear that decomposing the sequences and using Algorithm C is the most powerful method and it can prove positivity of every single sequence in the test set. The implemen-tation of Algorithm C is very fast and takes at most 0.3 seconds for every example we considered. Similar experiments were done using a Mathematica implementation of some of the meth-ods (more details in [NP22a]). Compared to the SageMath implementation, Algorithm C in the Mathematica implementation is significantly slower as it relies much more on 98 8 Positivity of C-finite sequences exact computations. Due to the powerful quantifier elimination methods in Mathematica, Algorithm P can be tested as well. The results show that for the sequences in discussion the method is similarly powerful as Algorithm C. Clearly, the implemented algorithms are already very powerful and can prove positivity of most C-finite sequences arising in practice. The situation for D-finite sequences is much bleaker. It would certainly be interesting to implement some of the algorithms mentioned in the introduction of this chapter and see how well they work on practical examples. Open Question 8.17. How do other methods for proving positivity of C-finite and D-finite sequences compare to the algorithms presented here? Are they more efficient? Can they prove positivity of a wider range of sequences? 99 9 Implementation As a proof of concept and for practical computations, most of the algorithms presented in the previous chapters are implemented in the computer algebra system SageMath [Sag23] in the package rec_sequences (developed under version SageMath 9.4). For most of the basic operations with C-finite or D-finite sequences the package relies on the ore_algebra package [KJJ15]. For proving inequalities of C-finite and D-finite sequences it relies on Arb (cf. [Joh17]) for efficient arbitrary precision computations and on QEPCADB for quantifier elimination using CAD (cf. [Bro03]). The latter needs to be installed separately if inequality methods based on the Gerhold-Kauers method should be used. The package rec_sequences is also described in the extended abstract [Nus22] on which this chapter is based. 9.1 Installation The package is published under the GPL-3.0 license. The source code and extensive documentation can be found on Github.1 Several different methods can be used for installation. Simplest, if SageMath was built from the sources, the command sage --pip install git+https :// github.com/ PhilippNuspl /rec_sequences .git installs the package together with the ore_algebra package (other methods can be found in the Github readme file). For using the functionality based on CAD, we can install QEPCADB by sage -i qepcad 1 100 9 Implementation 9.2 C-finite sequences After the installation of the package it can be loaded in any SageMath session. A C-finite se-quence ring over a field of characteristic zero K can be created by CFiniteSequenceRing(K). A sequence in this ring C can now be defined by a list of the coefficients of the recurrence and initial values: C([γ0, . . . , γr], [c0, . . . , cr−1]) ↔ ⎧ ⎨ ⎩ γ0c(n) + · · · + γrc(n + r) = 0, c(0) = c0, . . . , c(r −1) = cr−1. Alternatively, a symbolic expression in one variable or a list of initial terms can be used to define a C-finite sequence. In both cases guessing is used to find a recurrence. sage: from rec_sequences. CFiniteSequenceRing import sage: C = CFiniteSequenceRing (QQ) sage: fib = C([1,1,-1], [0,1], name="f") # Fibonacci numbers sage: var("n"); sage: exp2 = C(2^n) sage: alt = C(10[1 , -1]) sage: alt C-finite sequence a(n): (1)a(n) + (1)a(n+1) = 0 and a(0)=1 Terms of a C-finite sequence can be obtained in the same way that elements of lists are obtained in Python. sage: exp2 , fib [:10] (8, [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]) Closure properties of C-finite sequences are computed using the ore_algebra package. These include difference ring operations (using +, and shift), partial sums (using sum), Cauchy product (using cauchy), interlacing (using interlace) and subsequences (using subsequence). Equality of two C-finite sequences can be checked as well. The latter of the following examples proves the identity presented in the introduction of the thesis. sage: fib.sum() == fib.shift (2) -1 True sage: (fibfib.shift ()).sum().subsequence (2) == fib.subsequence (2,1)^2-1 True 101 9 Implementation Furthermore, one can obtain the recurrence coefficients, the initial values and the char-acteristic polynomial of a C-finite sequence (using coefficients, initial_values and charpoly, respectively) or compute the closed form: sage: (fib^2-fib.shift ()fib.shift (-1)).closed_form () # Cassini identity -(-1)^n For proving positivity of a sequence, one can either use the methods presented in Sec-tion 8.2.2 directly or use the operators >, <, >=, <=. If these operators are used the computations abort after a set amount of time. Hence, for proving positivity of more complicated sequences, it can be useful to use the method is_positive explicitly as shown for the sequence A000115 from Example 8.15: sage: fib < exp2 , 10 > fib , alt >= -1 (True , False , True) sage: c = C([1,-1,-1,1,0,-1,1,1,-1], [1, 1, 2, 2, 3, 4, 5, 6]) # A000115 sage: c.is_positive () True By the Skolem-Mahler-Lech theorem, Theorem 2.18, the set of zeros of a C-finite sequence is a finite set together with finitely many arithmetic progressions. In many cases, these zeros can be computed using the package: sage: (alt +1).zeros () Zero pattern with finite set {} and arithmetic progressions : - Arithmetic progression (2n+1)_n Often, the sign pattern of a sequence is eventually cyclic (this is, however, not necessarily the case as Example 2.3 in [AKK+21] shows): sage: alt.sign_pattern () Sign pattern: cycle <+-> More information on any of the methods can be obtained using ?, e.g. fib.interlace?. In many cases, more detailed information on the algorithms which are performed can be viewed via the Python logging module. For instance, using the following command all subsequent methods display also intermediate results: sage: logging.basicConfig(stream=sys.stdout , level=logging.DEBUG) 102 9 Implementation 9.3 C2-finite sequences Analogous to C-finite sequences, C2-finite sequences can again be defined by the coeffi-cients of the recurrence and initial values. sage: from rec_sequences. C2FiniteSequenceRing import sage: C2 = C2FiniteSequenceRing (QQ) sage: fibonorial = C2([fib.shift (), -1], ) sage: fibonorial # A003266 , fibonorial[n]== prod(fib[k] for k in range (1,n +1)) C^2- finite sequence of order 1 and degree 2 with coefficients: > c0 (n) : C-finite sequence c0(n): (1)c0(n) + (1)c0(n+1) + (-1)c0(n+2) = 0 and c0(0)=1 , c0(1)=1 > c1 (n) : C-finite sequence c1(n)=-1 and initial values a(0)=1 sage: fibonorial [:10] [1, 1, 1, 2, 6, 30, 240, 3120 , 65520 , 2227680] If a sequence c(n) is C-finite, then the sparse subsequence c(n2) is C2-finite (cf. Theorem 4.7) and we can compute the C-finite coefficients of the recurrence verifying the recurrence from Example 3.7: sage: sparse_fib = fib. sparse_subsequence (C2) # A054783 sage: sparse_fib [:10] == [fib[n^2] for n in range (10)] True sage: coeffs = [-fib.subsequence (2, 3), -fib.subsequence (4, 4), ....: fib.subsequence (2, 1)] sage: sparse_fib.coefficients () == coeffs True Ring operations of C2-finite sequences can be performed in the same way as for C-finite sequences. The operations are reduced to linear systems of equations over the C-finite sequence ring. Sometimes, these systems can be solved by computing a termwise solution first, using guessing to find a C-finite solution and verifying this solution. When defining a C2-finite sequence ring it can be specified explicitly that this approach should be tried. This is demonstrated on Example 4.3: sage: C2_guess = C2FiniteSequenceRing (QQ , guess=True) sage: b = C2_guess ([fib.shift (), -fib.shift (2)], [1/ fib [0+1]]) sage: a = bfib sage: c = C2_guess ([fib.shift ()fib.shift (2), -fib.shift (2)fib.shift (3)], 103 9 Implementation ....: [1/( fib [0+1] fib [0+2]) ]) sage: d = (calt.shift ()).sum().prepend () sage: c+d == 0 True We can also compute a recurrence for ∑⌊n/3⌋ k=0 f ((2k + 1)2) [JPNP23, Example 5.1]: sage: h = fib. sparse_subsequence (C2).subsequence (2 ,1).sum().multiple (3) sage: h.order (), h.degree (), h. leading_coefficient ().order () (9, 90, 84) sage: 0 not in h. leading_coefficient () True Naive approaches for guessing a C2-finite recurrence from given data yield polynomial systems of equations. However, if we fix the eigenvalues of the C-finite coefficients a potential recurrence can be obtained by solving a linear system. This way, we can, for instance, find and verify the simple C2-finite recurrence from Example 6.2: sage: K. = NumberField(x^2 -5) sage: C2_K = C2FiniteSequenceRing (K) sage: phi , psi = (1+a)/2, (1-a)/2 sage: eigenvalues = set([phi^4, psi^4, phi^6, psi^6, phi^8, psi ^8]) sage: f2Data = [fib[n^2] for n in range (100)] sage: sparse_fib_simp = C2_K.guess(f2Data , eigenvalues , order =3, ....: simple=True) sage: sparse_fib_simp .degree () 4 sage: sparse_fib = C2_guess(sparse_fib) sage: sparse_fib == sparse_fib_simp True Guessing is one of the most important tools that we have for C-finite and D-finite se-quences. It would certainly be interesting to see if similar powerful algorithms can be developed for C2-finite sequences. Open Question 9.1. Can we find an efficient method for guessing a C2-finite recurrence for the given terms of a sequence? 104 105 9 Implementation List of symbols N = {0, 1, 2, . . . } Set of natural numbers N≥1 = {1, 2, . . . } Set of positive natural numbers Z, Q, R, C, Q Sets of integers, rational, real, complex and algebraic numbers K[n], K(n) The ring of polynomials and the field of rational func-tions over K deg(p) The degree of the polynomial p lc(p), lcn(p) The leading coefficient of the polynomial p (w.r.t. n if specified) coeff(p, i) The coefficient of ni of the polynomial p(n) KN The K-algebra of sequences ⟨a1, a2, . . . ⟩R The R-module generated by the elements a1, a2, . . . KJxK The ring of formal power series over the field K σ: KN →KN The shift operator (page 4) ord(A), ord(a) The order of an operator A or a sequence a (page 5) a(n) ∼b(n) Similarity of two sequences, i.e., limn→∞ a(n) b(n) = 1 (page 8) RC Ring of C-finite sequences (page 9) R× Set of sequences of R which are units in KN (page 13) Q(R) The total ring of fractions of R (page 13) Kσ[c0, . . . , cr] The smallest K-difference algebra containing the se-quences c0, . . . , cr (page 21) nk Falling factorial, i.e., nk = n(n −1) · · · (n −k + 1) (page 24) a ⊙b The Cauchy product of the sequences a, b (page 29) Ga Suppose a is a sequence of order r, then Ga = ( a, . . . , σr−1a ) (page 33) e(r) i ∈Rr The i-th unit vector for i ∈{0, . . . , r −1} over the ring R (page 33) Ma Companion matrix of a sequence a (page 34) M ⊕N Direct sum of two matrices M, N (page 36) M ⊗N Kronecker product of two matrices M, N (page 36) 106 9 Implementation RCk, RDk Ring of Ck-finite and Dk-finite sequences, respectively (page 73) A Field of real algebraic numbers (page 79) lclm(A, B) Least common left multiple of the operators A, B (page 89) 107 Bibliography [AKK+21] Shaull Almagor, Toghrul Karimov, Edon Kelmendi, Joël Ouaknine, and James Worrell. 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In Proceedings of the 2019 on In-ternational Symposium on Symbolic and Algebraic Computation, page 371–378, 2019. 121 Index annihilator C-finite sequence, 9 C2-finite sequence, 13 D-finite sequence, 5 ansatz, 31 asymptotics C-finite sequence, 11 C2-finite sequence, 18, 19 D-finite sequence, 8 simple C2-finite sequence, 64 Beke, Emanuel, 2 Bernoulli, Daniel, 1 Berstel sequence, 92 Binet’s formula, 1, 11 C-finite sequence, 9 closed form, 10, 67, 79 implementation, 101 C2-finite sequence definition, 13 generating function, 24 implementation, 103 interlacing, 55 ring, 23, 33, 57 simple, 63 subsequence, 38, 57 CAD, 80 Cassini identity, 1, 102 Catalan numbers, 6, 64 Cauchy product, 6, 29 characteristic polynomial, 5, 87 Ck-finite sequence, 72 closure properties C-finite sequence, 10, 101 C2-finite sequence, 54 D-finite sequence, 6 history, 2 computable, 46, 59, 66, 68 computer algebra system, 2, 7, 78, 100 continued fraction, 16 creative telescoping, 2, 16 D-finite sequence, 5 D2-finite sequence, 72 DD-finite function, 2, 30 decidable, 12, 17, 43, 78 degenerate sequence, 11, 48, 91 Dk-finite sequence, 72 eigenvalue, 5, 48 exponent lattice, 48 falling factorial, 25 Fibonacci factorial, 14, 27, 103 Fibonacci sequence 122 INDEX definition, 9 history, 1 sparse subsequence, 15, 27, 63 fibonomial coefficient, 15 fibonorial, see Fibonacci factorial Frobenius, Ferdinand G., 1 Fuchs, Lazarus Immanuel, 1 generating function D-finite sequence, 8 C2-finite sequence, 24 history, 1 simple C2-finite sequence, 65 Gerhold-Kauers method, 80 guessing C2-finite sequence, 16, 104 D-finite sequence, 8, 93 Hadamard inverse, 15 Hadamard product, 9 Harmonic numbers, 6 holonomic sequence, see D-finite sequence Hurwitz, Adolf, 1 implementation, 100 Kepler, Johannes, 1 lattice, 9, 48 lclm, least common left multiple, 89 linear system, 9, 32, 41, 66, 75 Lucas, Édouard, 1 de Moivre, Abraham, 1 Moore-Penrose-Inverse, 44 Noetherian, 22, 42, 74 OEIS, 5, 93, 97 order C-finite sequence, 9 C2-finite sequence, 14, 47 D-finite sequence, 5 operator, 5 simple C2-finite sequence, 71 P-recursive sequence, see D-finite sequence Perrin numbers, 10, 55, 61 poly-recursive sequence, 2, 20 Positivity Problem, 78, 102 q-holonomic sequence, 14, 27 quantifier elimination, 80 SageMath, 3, 7, 97, 100 shift-operator σ, 4 Skolem Problem, 3, 12, 17, 32, 43, 59, 63, 79 Skolem-Mahler-Lech theorem, 11, 43, 102 Smith normal form, 50 sparse subsequence, 15, 40, 60, 63, 103 splitting field, 24, 76 Stirling number, 26 superfactorial, 19, 72 torsion number, 50 total ring of fractions, 13 X-recursive sequence, 17, 28 123
3811	https://math.stackexchange.com/questions/1183954/intuition-behind-variation-of-parameters-method-for-solving-differential-equatio	Skip to main content Intuition behind variation of parameters method for solving differential equations Ask Question Asked Modified 2 years, 7 months ago Viewed 5k times This question shows research effort; it is useful and clear 31 Save this question. Show activity on this post. I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking about the method for second order ODEs: a2(x)y′′(x)+a1(x)y′(x)+a0(x)y(x)=f(x) Does the motivation for considering a particular solution of the form yp(x)=u1(x)y1(x)+u2(x)y2(x) (where y1,y2 are solutions to the corresponding homogeneous equation) because (yp/y1)≠ constant and likewise (yp/y2)≠ constant, as otherwise we would just obtain the complementary solution again. This suggests that both are instead functions of x, i.e. (yp/y1)=u1(x) and (yp/y2)=u2(x), leading to the form of the ansatz I gave above? Secondly, is the reason why we place a further constraint on the form of yp (other that it be a solution to the original ODE) because, in principal, there will be an infinite number of particular solutions of the form yp(x)=u1(x)y1(x)+u2(x)y2(x) but we only require one particular solution, and so by imposing an additional constraint we have two equations for the two unknowns u1 and u2, thus enabling us to uniquely determine a solution for each of them and subsequently enabling us to find a general solution to the original ODE?! calculus ordinary-differential-equations intuition Share CC BY-SA 3.0 Follow this question to receive notifications edited Mar 11, 2015 at 10:55 Perpetual learner asked Mar 10, 2015 at 15:46 Perpetual learnerPerpetual learner 64977 silver badges1616 bronze badges 7 nice question. i don't know an answer to your question though i have been using useful method a lot. i will look into the history of this. – abel Commented Mar 10, 2015 at 15:59 Thanks :) Likewise, and it's been bugging me that I don't have a deeper understanding of it. Appreciate you looking into it! – Perpetual learner Commented Mar 10, 2015 at 16:01 as with everything in math, it seems to have originated with euler and used by lagrange. i am looking at en.wikipedia.org/wiki/Variation_of_parameters – abel Commented Mar 10, 2015 at 16:09 1 These guys are Math Gods, nothing gets past them! I had a look at the Wiki page, but I can't see any real motivation on there for the parts I'm trying to justify, unfortunately :( – Perpetual learner Commented Mar 10, 2015 at 16:16 For myself I usually explain that the second part is the trick that massively simplifies finding any particular solution of equation and replaces it with something as simple as solving system of linear equations. – Evgeny Commented Mar 11, 2015 at 18:33 \| Show 2 more comments 2 Answers 2 Reset to default This answer is useful 21 Save this answer. Show activity on this post. Here is one way of getting the variation of parameters formulas. Firstly, some motivation mixed with a bit of jargon: We think of a2(x)y′′(x)+a1(x)y′(x)+a0(x)y(x) as a result of applying to y(x) an operation D=a2(x)d2dx2+a1(x)ddx+a0(x)Id. This is a linear operation (aka a linear differential operator), meaning that D(c1y1(x)+c2y2(x))=c1D(y1(x))+c2D((y2(x)). Our equation is then D(y(x))=f(x). Linearity ensures that if ya(x) and yb(x) solve Dya=fa and Dyb=fb then ya+yb solves Dy=fa+fb. So if we can decompose our inhomogeneous term f(x) as a sum f=fa+fb and solve the two "pieces" Dy=fa and Dy=fb we would be done - just add the two solutions and get a solution for f. The key idea now (this is known as Green's function method, or perhaps sometimes Duhamel's principle) is to think of f as being a "sum" of a continuum of delta functions f(x)=∫f(t)δ(x−t)dt. Thus we want to solve Dy(x)=δ(x−t) for each t (these solutions are the Green's functions for our equation), to get solutions yt(x)=y(t,x), and then write our solution to Dy=f as the "sum" over t y(x)=∫f(t)yt(x)dt=∫f(t)y(t,x)dt. Continuing in this way (without paying much attention to technical details or introducing the theory of distributions needed to provide such details and make the above rigorous), we are now tasked with finding yt(x) i.e. solving Dyt(x)=δ(x−t). The solution is of course not unique, but is only unique up to adding yh - an arbitrary solution of the homogeneous equation Dy=0. Note that in fact for x>t and for x<t we have δ(x−t)=0, and our yt will coincide with some yh. But it can not be the same yh, or we won't get δ(x−t), just 0 everywhere. So we need to jump at x=t. Namely, we can start with y=0 for x<t and continue as some yh after. Now, any yh is a combination of fixed homogeneous solutions y1 and y2, i.e. yh(x)=c1y1(x)+c2y2(x). So our Green's function yt=0 for xt. What should c1,c2 be? We want Dyt(x)=a2(x)y′′t(x)+a1(x)y′t(x)+a0(x)yt(x)=δ(x−t). So yt should be continuous, so that y′t has only a step discontinuity at x=t, and y′′t only a delta (and not δ′(x−t)). Continuity imposes c1y1(t)+c2y2(t)=0. We also want to have a unit delta jump and not bigger or smaller. The size of the jump is controlled by a2d2dx2 at x=t and so is a2(t)[c1y′1(t)+c2y′2(t)]. Equating it to 1 imposes c1y′1(t)+c2y′2(t)=1a2(t) To summarize: The Green's function at t is [0 for xt] with c1(t),c2(t) subject to c1(t)y1(t)+c2(t)y2(t)=0 c1y′1(t)+c2y′2(t)=1a2(t) Compare with the variation of parameters: c1=u′1f,c2=u′2f And the solution to Dy=f is y(x)=∫f(t)yt(x)dt=∫f(t)y(t,x)dt Plugging in, we get y(x)=∫x>tf(t)(c1(t)y1(x))+c2(t)y2(x))dt=[∫x−∞f(t)c1(t))dt]y1(x)+[∫x−∞f(t)c2(t))dt]y2(x)=u1(x)y1(x)+u2(x)y2(x). So we recover y as a combination of y1 and y2 with function coefficients, and we know exactly which relations the derivatives of these coefficients should satisfy (they have to combine to give Green's functions times f(x)). Writing out these relations we get the same formulas as for the variation of parameters formulas, which we can then justify independently in an ad hoc manner. Of course this generalizes in a straightforward manner to higher order linear equations - the cis will have more linear `matching of derivatives' constraints to satisfy to give a Green's function, and the rest is the same. Share CC BY-SA 3.0 Follow this answer to receive notifications edited May 11, 2017 at 19:44 answered May 18, 2016 at 9:34 MaxMax 14.5k2323 silver badges4242 bronze badges 5 4 The only decent answer to this question I've found on StackExchange after hours of searching. Amazing how many terrible answers there are from people who don't really understand the method and want to silence anyone's curiosity with dismissive "we do it because it's useful" non-answers, or just misguided and wildly wrong algebraic approaches which again don't explain anything. – Marcel Besixdouze Commented Feb 12, 2020 at 16:57 @MarcelBesixdouze Please see my long answer below. – Michael_1812 Commented Jan 11, 2023 at 2:52 Hi, thanks a lot for the excellent answer! I have a quick question. Why the unit jump is controlled by a2d2dx2 at x=t? – user330928 Commented Nov 20, 2024 at 1:53 What I meant is as follows. We need continuity of yt(x) to get "no worse than a delta" discontinuity of Dyt(x). Then, applying the other (lower order) parts of D to a continuous and piecewise smooth yt can only produce jump discontinuities. The only way we get a delta is from applying the highest order part. And in fact if we applied just d2dx2 we would get the difference in slopes at t as the size the delta (the first derivative has a jump of the slope difference, the second - a delta of that size); applying a2(x)d2dx2 scales that by extra a2(t). – Max Commented Nov 21, 2024 at 2:52 I was having trouble following the logic of solving for a green function until I found this clear exposition: phas.ubc.ca/~berciu/TEACHING/PHYS312/LECTURES/FILES/green.pdf – The Rizzler Commented Feb 12 at 8:44 Add a comment \| This answer is useful 6 Save this answer. Show activity on this post. This is an extremely good and clever question. Amazingly, the raised issue, the freedom in our choice of the constraint, had been waiting for discussion for more than two centuries, and was addressed and exploited in the literature only recently. I begin with an elementary example borrowed from this paper. It was proposed by my coauthor William Newman, so I call it Newman's example. Consider a harmonic oscillator disturbed by a force ΔF(t): x¨+x=ΔF(t),with x(0) and x˙(0) known,(1) and seek its solution by using the ansatz x=C1(t)sint+C2(t)cost.(2) Together, the former and the latter equations will yield: x˙=[C˙1(t)sint+C˙2(t)cost]+C1(t)cost−C2(t)sint.(3) It is common to impose at this point an analogue to the Lagrange constrain, i.e., to set [C˙1(t)sint+C˙2(t)cost] equal to zero. This step is convenient but not obligatory. We instead may agree on C˙1(t)sint+C˙2(t)cost=ϕ(t),(4) ϕ(t) being an arbitrary function of time. This results in x¨=ϕ˙+C˙1(t)cost−C˙2(t)sint−C1(t)sint−C2(t)cost,(5) summation whereof with (2) gives us x¨+x=ϕ˙+C˙1cost−C˙2sint.(6) Plugging this expression into (1) produces, when combined with identity (4), the following system: ϕ˙+C˙1cost−C˙2sint=ΔF(t)(7) C˙1(t)sint+C˙2(t)cost=ϕ(t).(8) This leads to C˙1=ΔFcost−ddt(ϕcost),(9) C˙2=−ΔFsint+ddt(ϕsint),(10) integration of which system entails: C1=∫tΔF(t′)cost′dt′−ϕcost+a1,(11) C2=−∫tΔF(t′)sint′dt′ϕsint+a2,(12) a1 and a2 being constants. Substitution of the two last expressions into (9) results in a complete cancellation of the arbitrary ϕ(t) term: x=C1(t)sint+C2(t)cost= −cost∫tΔF(t′)sint′dt′+sint∫tΔF(t′)cost′dt′+a1sint+a2cost.(13) Naturally, the physical trajectory x(t) comes out invariant under the choice of the gauge function ϕ(t). Does this imply that gauge freedom is unimportant? No way! Sometimes it helps a lot in analytical calculations. On other occasions, a clever choice of the gauge function(s) helps to mitigate the numerical error in computations, see this work. Now, some details and a bit of history. Let us recall the geometric idea underlying the Variation of Parameters (VOP) method in its initial form, as suggested by Euler and Largange. We need to model a perturbed orbit by a sequence of "simple" curves. If the orbit is bound, it is convenient (not obligatory - but convenient) to choose these "simple curves" as ellipses sharing one of their foci. One such ellipse can be parameterised, in the inertial Cartesian frame, as r=f(t,C1,...C6),(14) where Ci are six parameters. A popular choice of these parameters is the so-called Keplerian elements: the major semiaxis a, the eccentricity e, the three Euler angles (the longitude of the node Ω, the inclination i of the orbit on the fiducial plane, argument of the pericentre ω), and the initial condition (the value M0 at the initial time t0). Why does the initial condition enter this set? Because we astronomers are interested not only in the geometric shape of a "simple" orbit, but also in the position of the orbiter on it. So, equation (14) gives not just a "simple" orbit, but a "simple" solution, with the initial position included. In the above equation, f is the implicit function serving as a solution to the unperturbed equation r¨=F(r),(15) where F(r) is the inverse-square force. Euler and Largange were interested in this particular example, and this is why in this case the unperturbed force F is a function of r solely. Now, include a perturbation ΔF. We have to solve a more complex equation of motion r¨=F(r)+ΔF(r,r˙).(16) Watch my hands: while Euler and Lagrange were interested only in position-dependent perturbation, I have set ΔF to depend also on velocities, because the modern celestial mechanics has to deal with such perturbations. Such perturbations emerge, e.g., when we have to consider motions in noninertial frames or when relativistic corrections come into play. This detail, however, will not change the central idea of our story. To solve equation (16) by the VOP method, we endow the "constants" with time-dependencies of their own: r=f(t,C1(t),...C6(t)),(17) while keeping the functional form of f unchanged. In simple words, we model the perturbed orbit with a sequence of "simple solutions" parameterised with t. At the risk of sounding repetitive, I again highlight that a "simple solution" is not just an ellipse with one focus fixed, but an ellipse taken together with an initial condition. And why with one focus fixed? -- because it is natural to assemble a perturbed orbit of "simple solutions" that are elliptic orbits about the fixed point, Sun. Each such simple orbit donates one point to the perturbed trajectory, the actual solution. At this point, Lagrange faced a mathematical difficulty. The first time differentiation rendered r˙=∂r∂t+Φ,Φ≡∑j=16∂f∂CjC˙j,(18) while the second differentiation gave him r¨=∂2r∂t2+∑j=16∂2f∂t∂CjC˙j+Φ˙. The insertion thereof into (16) left Lagrange with ∑j=16∂2f∂t∂CjC˙j+Φ˙=ΔF.(19) This didn't look good, because at his disposal Lagrange had only three equations, the x,y,z projections of (19). These were not enough to obtain solutions for the six functions Cj(t). To overcome this difficulty, he set, by hand, what we now call the Lagrange constraint: Φ=0.(20) The physical meaning of this constraint is simple: it postulates that we model the perturbed trajectory by a sequence of ellipses that are tangent to the trajectory. Each ellipse donates one point to the trajectory -- and is tangent to the trajectory at that point. It does not cross the trajectory. Lagrange termed these tangent ellipses osculating (osculare being Latin for kissing). Mathematically, the constraint is arbitrary. As was emphasised in several recent publications (this, this, and this), it is possible to make Φ and arbitrary function of r. Its presence will alter the mathematical form of the resulting solution, but will not change the actual physical trajectory. Geometrically, the trajectory will be assembled of points provided by nonosculating ellipses, which will now be permitted to cross the trajectory. Hence we are facing an example of gauge freedom. Lagrange surely realised the presence of these three degrees of freedom, the three projections of Φ(r). However, neither he nor Poincare not other great celestial mechanicians of the past bothered to look into this issue. My guess is that they did not expect any calculational gains from using alternative gauges. Gains (big ones) emerged only when it became necessary to solve the problems with velocity-dependent perturbations ΔF(r,r˙). It has turned out that in such situations the imposition of the Lagrange constraint entails very complex equations for the orbital parameters -- while the choice of a non-Lagrange gauge simplifies the equations drastically. I would also mention that the attitude mechanics, i.e., the problem of perturbed rotation of an unsupported body is mathematically analogous to the mechanics of orbits. Just as a perturbed orbit gets assembled of points donated by "simple solutions" (usually, conics) -- so a perturbed rotation of a solid body gets assembled of simple "Eulerian cones", each of which is an unperturbed rotation. Like in orbital mechanics, so in attitude mechanics the "Eulerian cones" can be osculating or nonosculating, see this paper. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Jan 12, 2023 at 1:00 answered Jan 11, 2023 at 1:53 Michael_1812Michael_1812 2,1181313 silver badges1414 bronze badges Add a comment \| You must log in to answer this question. 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3812	https://artofproblemsolving.com/wiki/index.php/Extreme_principle?srsltid=AfmBOop7vlmpdij4rsc58ER-r4H-OgE23GAdGTs6_yGOYSH1wY5n-Itq	Page Toolbox Search Extreme principle The extreme principle (or extremal principle) is a problem-solving technique that involves looking at objects with extreme properties, such as the largest or smallest element. For example, consider the following problem: Example: Imagine an infinite chessboard that contains a positive integer in each square. If the value in each square is equal to the average of its four neighbors to the north, south, west, and east, prove the values in all the squares are equal. Solution: Consider the square containing the minimal value. Then its four neighbors must all have this minimal value. Similarly, their neighbors must also have this minimal value, and so on ad infinitum. Thus, every square in the chessboard has the same value. Problems Something appears to not have loaded correctly. Click to refresh.
3813	https://askfilo.com/user-question-answers-smart-solutions/8-identify-the-missing-number-in-the-given-triangle-a-100-b-3230333037333333	Identify the missing number in the given triangle. a) 100 b) 50 c) 0 900 .. World's only instant tutoring platform Instant TutoringPrivate Courses Tutors Explore TutorsBecome Tutor Login StudentTutor CBSE Smart Solutions Identify the missing number in the given triangle. a) 100 b) 5 Question Question asked by Filo student Identify the missing number in the given triangle. a) 100 b) 50 c) 0 900 90 940 40 990 50 ? Views: 5,786 students Updated on: Jan 16, 2025 Not the question you're searching for? Ask your question Ask your question Or Upload the image of your question Get Solution Text solutionVerified Concepts: Number patterns, Triangular numbers Explanation: To find the missing number in the triangle, we can observe the relationship between the numbers in the triangle. Each number in the upper row seems to be the sum of the two numbers directly below it. Let's analyze the numbers: 900 is the sum of 940 and 40, and 990 is the sum of 940 and 50. Therefore, we can set up the following equation for the missing number (let's call it x): 900 = 940 + 40 and 990 = 940 + x. Solving for x gives us x = 990 - 940 = 50. Step by Step Solution: Step 1 Identify the relationship between the numbers in the triangle. Each number in the upper row is the sum of the two numbers directly below it. Step 2 Set up the equation for the missing number: 990 = 940 + x. Step 3 Solve for x: x = 990 - 940 = 50. Final Answer: 50 Ask your next question Or Upload the image of your question Get Solution Get instant study help from an expert tutor 24/7 Download Filo Found 2 tutors discussing this question James Discussed Identify the missing number in the given triangle. a) 100 b) 50 c) 0 900 90 940 40 990 50 ? 5 mins ago Discuss this question LIVE 5 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Download AppExplore now Trusted by 4 million+ students Students who ask this question also asked Question 1 Views: 5,167 Here's the question image C. Replace the coloured words with an adverb and rewrite: Mrs. Vats keeps her house very shabby. The deer hopped over the rocks with ease. My neighbour spoke to my father in anger. The wrestler held his opponent quite tight. She looked at her daughter receiving the award with pride. The boss left the meeting all of a sudden. D. Tick (✓) the correct adverbs of time given in the brackets: Can you call me (later/sooner)? I am (still/already) in the meeting. We shall go to watch a movie (yesterday/tomorrow). Nandini had been to Switzerland (recently/lately). You should come home (early/late) as a heavy rain is expected. Please come back (soon/late). We will be waiting for you. (Yesterday/Tomorrow) the children bought new clothes. E. Fill in the blanks using adverbs of place given in the brackets: The chairs were moved __ (outside/nearly) My father's friend lives __ (outside/abroad) You can always find his things lying __ (here/around) I kept the bag __, but somebody moved it. (out/here) Kiran lives __, I shall call her immediately. (around/nearby) Come __, I have something for you. (around/here) Topic: Smart Solutions View solution Question 2 Views: 5,922 अनूयीलनी 6.1 1 ব পया आयकु कबि 200 नि कियानों। भूर्ণवर्र मरश्या आष्ट ? (i) 51−N (ii) 99 (iii) 205 (iv) 3400 (v) 1987 (i) 4347 (ii) 24832 (iii) 35493 (iv) 403388 (v) 182000 4. (i) 5 টो সशच्या निशा याद বर्গ যুগ्ना। (iii) 35493 (ii) 5 ঢो সशच्या लिथा याব বर्গ अयूश्न। (i) 1+3+5+7 (ii) 1+3+5+7+9+11+13+15 (iii) 1+3+5+7+9+11+13+15+17+19+21+23+25 সिইँতब डোগফলরোব বর্গ সংখ্যা হ’য়नে পবীक्षा कबा। 6.3 সংথ্যা এটাব বর্গ निর্ণয়া কবাব এটা সহজ পদ্ধতি (An easy method to find the square of a number) এককব घবত 5 थকা সংथ্যাব বর্গ नির্ণয়ব সহজ পদ্ধতি (An easy method to find the square of a number having 5 in the unit place) : 15,25,35,45, তোমারলোকে গতানুগতিক পদ্ধতিবে ইইঁত্ বর্গ কবিढে পাবা যে 1 5 2=225 2 5 2=625 3 5 2=1225 4 5 2=2025 এতিয়া এইকেইঁট তল丁 দিয়া ধবণেরে সজাই চাతঁ 1 5 2=225=(1×2)শउक+25,(1×2)শउক इॅन 2×उब=200 2 5 2=625=(2×3)ॠउक+25,(2×3)उब इंन 6×उब=600 3 5 2=1225=(3×4)सउद+25 4 5 2=2025=(4×5)ॠउক+25 107 Topic: Smart Solutions View solution Question 3 Views: 5,934 (4) ⋅3 x−5=5 x−3 Topic: Smart Solutions View solution Question 4 Views: 5,898 Write the first nine numbers that can be arranged as triangles. Observe the pattern and write the next three rows. 17×9=17×(10−1)=170−17=153 17×99=17×(100−1)=1,700−17=1,683 17×999=17×(1,000−1)=17,000−17=16,983 Observe the pattern and use it to find the products 32×625,32×3,125, and 32×15,625, 32×5=32×2 10=2 320=160 32×25=32×4 100=4 3,200=800 32×125=32×8 1,000=8 32,000=4,000 Topic: Smart Solutions View solution View more Video Player is loading. Play Video Play Skip Backward Mute Current Time 0:00 / Duration-:- Loaded: 0% Stream Type LIVE Seek to live, currently behind live LIVE Remaining Time-0:00 1x Playback Rate 2.5x 2x 1.5x 1x, selected 0.75x Chapters Chapters Descriptions descriptions off, selected Captions captions settings, opens captions settings dialog captions off, selected Audio Track Picture-in-Picture Fullscreen This is a modal window. Beginning of dialog window. Escape will cancel and close the window. Text Color Opacity Text Background Color Opacity Caption Area Background Color Opacity Font Size Text Edge Style Font Family Reset restore all settings to the default values Done Close Modal Dialog End of dialog window. Stuck on the question or explanation? Connect with our 380 tutors online and get step by step solution of this question. Talk to a tutor now 310 students are taking LIVE classes Question Text 8. Identify the missing number in the given triangle. a) 100 b) 50 c) 0 900 90 940 40 990 50 ? Updated On Jan 16, 2025 Topic All topics Subject Smart Solutions Class Class 6 Answer Type Text solution:1 Are you ready to take control of your learning? Download Filo and start learning with your favorite tutors right away! 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3814	https://math.stackexchange.com/questions/280551/primes-congruent-to-1-mod-6	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Primes congruent to 1 mod 6 Ask Question Asked Modified 3 years, 11 months ago Viewed 4k times 16 $\begingroup$ I came across a claim that I found interesting, but can't seem to prove for some reason. I have the feeling it should be easy a prime $p$ can be written in the form $p = a^2 -ab +b^2$ for some $a,b\in\mathbb{Z}$ if and only if $p\equiv 1\bmod{6}$ number-theory elementary-number-theory eisenstein-integers Share edited Nov 3, 2021 at 13:34 qwr 11.4k44 gold badges4949 silver badges8484 bronze badges asked Jan 17, 2013 at 6:15 Math2012pcMath2012pc 34722 silver badges88 bronze badges $\endgroup$ 6 $\begingroup$ The "only if" part is easy; the "if" part, not quite as easy. $\endgroup$ Gerry Myerson – Gerry Myerson 2013-01-17 06:18:36 +00:00 Commented Jan 17, 2013 at 6:18 3 $\begingroup$ Oh, and there's one exception; $a=1$, $b=-1$, $p=3$. $\endgroup$ Gerry Myerson – Gerry Myerson 2013-01-17 06:19:17 +00:00 Commented Jan 17, 2013 at 6:19 1 $\begingroup$ As Gerry points out, the only if part is easy since $a^2 - ab + b^2 \equiv 0,1 \pmod 3$. Hence, primes of the form $3k-1$ equivalently of the form $6k-1$ cannot be written as $a^2 - ab + b^2 \equiv 0,1 \pmod 3$. And all primes, except $2$ and $3$ are $\pm 1 \pmod 6$. $\endgroup$ user17762 – user17762 2013-01-17 06:26:05 +00:00 Commented Jan 17, 2013 at 6:26 $\begingroup$ So what is the trick for the "if" direction? Is it a long proof and is there a reference I can check? $\endgroup$ Math2012pc – Math2012pc 2013-01-17 06:51:36 +00:00 Commented Jan 17, 2013 at 6:51 2 $\begingroup$ @Math2012pc, one possible answer is from the arithmetic of Eisenstein integers. There is probably an answer in the book "Primes of the form $x^2+ny^2$", but it would be better if you say a few words about your background, i.e. do you know what PID is? Do you know anything about splitting of primes? Quadratic reciprocity? $\endgroup$ user27126 – user27126 2013-01-17 07:22:52 +00:00 Commented Jan 17, 2013 at 7:22 \| Show 1 more comment 6 Answers 6 Reset to default 14 $\begingroup$ $a^2 - ab + b^2 = (-a)^2 + (-a)b + b^2$. So it suffices to deal with $a^2 + ab + b^2$. Now, take a prime $p \equiv 1 \pmod{6}$. It is elementary to show their exists an integer $d$ such that $d^2 \equiv -3 \pmod{p}$, now take $z \equiv \frac{-1 + d}{2} \pmod{p}$ (so its a third root of unity modulo $p$). Now define $\mathcal L = {(a,b) \in \mathbb{Z}^2 \| a \equiv zb \pmod{p}}$. It is straightfoward to check $\mathcal L$ is a lattice whose fundamental parallelogram has area $p$. Now by Minkowski's theorem one has $\mathcal L$ contains a nontrivial lattice point inside the ellipse $a^2 + ab + b^2 < 2p$. Call this point $(a,b)$. But then $a^2 + ab + b^2 \equiv 0 \pmod{p}$ based on the definition of the lattice, thus it must be $a^2 + ab + b^2 = p$. The if part follows. For the "only if" part, just check modulo $3$ and note that $a^2 + ab + b^2 \equiv 0,1 \pmod{3}$. Note that the problem statement fails for $p=3$ due to that. Share answered Jan 17, 2013 at 7:31 dinoboydinoboy 2,9242020 silver badges2626 bronze badges $\endgroup$ 3 5 $\begingroup$ Nice, but too much use of disturbing words like "elementary", "straightforward", "just"... $\endgroup$ DonAntonio – DonAntonio 2013-01-17 12:17:41 +00:00 Commented Jan 17, 2013 at 12:17 2 $\begingroup$ Well the proofs for them are quite easy and well-known so there wasn't much reason to put them. $\endgroup$ dinoboy – dinoboy 2013-01-17 15:50:06 +00:00 Commented Jan 17, 2013 at 15:50 $\begingroup$ Indeed $4 > \pi \cdot \tfrac{2}{\sqrt{3}} \approx 3.63$ $\endgroup$ cactus314 – cactus314 2014-05-12 18:00:39 +00:00 Commented May 12, 2014 at 18:00 Add a comment \| 11 $\begingroup$ Here is another solution for the "if" part, using algebraic number theory. Let $p$ be a prime satisfying $p \equiv 1 \pmod 3$ and consider the number field $K = \mathbb Q(\omega)$ where $\omega = (1\pm \sqrt{-3})/2$ is a primitive third root of unity. By quadratic reciprocity, $$\left( \frac{-3}{p} \right) = \left(\frac{-1}{p} \right) \left( \frac{3}{p} \right) = (-1)^\frac{p-1}{2} (-1)^\frac{p-1}{2} \left(\frac{p}{3}\right) = \left(\frac{1}{3}\right) = 1,$$ so $\mathbb Z/p\mathbb Z$ contains a square root of $-3$. Since $-3$ is the discriminant of $X^2+X+1$ (the minimal polynomial of $\omega$), the polynomial splits in $\mathbb F_p[X]$, therefore $p$ splits in $K$: $$(p) = \mathfrak p \overline{\mathfrak p}$$ for a prime $\mathfrak p$ of $K$. Since $K$ has class number one (the Minkowski bound is $<2$), $\mathfrak p$ is principal, say $\mathfrak p = (a+b\omega)$. So we have $$(p) = \mathfrak p \overline{\mathfrak p} = (a+b\omega)(a+b\omega^2) = (a^2+ab+b^2).$$ Since $K$ is imaginary quadratic, the only units in $\mathcal O_K = \mathbb Z[\omega]$ are $\pm 1$, so $$p = \pm (a^2+ab+b^2).$$ Since $a^2+ab+b^2$ is positive, we must in fact have "+": $$p = a^2+ab+b^2 = (-a)^2 - (-a)b + b^2.$$ Share edited Jan 18, 2013 at 14:29 answered Jan 17, 2013 at 15:02 marlumarlu 14.3k11 gold badge4646 silver badges5757 bronze badges $\endgroup$ Add a comment \| 3 $\begingroup$ Here is another, albeit non-elementary, solution for the "if" part due to Ireland and Rosen. If $p\equiv 1\pmod{3}$, there exists a multiplicative character $\chi$ of order $3$. Then, $\chi\in{1,\omega,\omega^2}$, where $\omega=e^{2\pi i/3}=\frac{1}{2}(-1+\sqrt{-3})$. Now, consider the Jacobi sum of two such characters: $$J(\chi, \chi)=\sum_{a+b=1}\chi(a)\chi(b)\in\mathbf{Z}[\omega].$$ Then, we can write $J(\chi, \chi)=a+b\omega$, for some $a,b\in\mathbf{Z}$. We have $$p=N(J(\chi, \chi))^2=N(a+b\omega)=a^2-ab+b^2,$$ as desired. Share answered Dec 24, 2013 at 6:28 tc1729tc1729 3,24711 gold badge2323 silver badges4343 bronze badges $\endgroup$ Add a comment \| 2 $\begingroup$ If $(a,b)=d,d^2\mid (a^2-ab+b^2)$ If $d>1,a^2-ab+b^2$ can not be prime $\implies d=1$ Now, if prime $p=a^2-ab+b^2\implies p\mid (a+b)(a^2-ab+b^2)\implies p\mid (a^3+b^3)$ So, $$a^3\equiv(-b)^3\pmod p\implies \left(-\frac ab\right)^3\equiv 1\pmod p\implies ord_p \left(-\frac ab\right)\mid 3$$ If $ord_p \left(-\frac ab\right)=1,p(=a^2-ab+b^2)\mid (a+b)$ If $(a^2-ab+b^2)\mid(a+b),(a^2-ab+b^2)\mid(a+b)^2$ $\implies (a^2-ab+b^2)\mid 3ab$ $\implies (a^2-ab+b^2)\mid 3$ as $(a^2-ab+b^2,a)=(b^2,a)=1$ as $(a,b)=1$ But, $a^2-ab+b^2>3,$ for $a,b>2$ $$\implies ord_p \left(-\frac ab\right)= 3\implies 3\mid \phi(p)\implies p\equiv1\pmod 3\equiv1,4\pmod 6$$ Hence, $p\equiv1\pmod 6$ as $p\equiv4\pmod 6$ is even and $p>2$ Share edited Jan 18, 2013 at 15:29 answered Jan 18, 2013 at 14:36 lab bhattacharjeelab bhattacharjee 279k2020 gold badges213213 silver badges337337 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ $$(a,b)^2\mid(a^2-ab+b^2)\implies a^2-ab+b^2$$ can not be prime if $(a,b)>1$ Now, $a$ can be of the form $3m,3m+1$ or $ 3m-1$ where $m$ is an integer. Similarly, $b$ can be $3n,3n+1$ or $3n-1$ where $n$ is an integer. $(1)$ If $a=3m,a^2-ab+b^2\equiv b^2\pmod 3$ Now, $b^2\equiv0\pmod 3\iff 3\mid b\implies 3\mid(a,b)$ which is impossible as $(a,b)=1$ So, $b^2\equiv1\pmod 3\implies a^2-ab+b^2\equiv1\pmod 3$ $(2a)$ If $a=3m+1, b=3n-1, a^2-ab+b^2\equiv 1-1(-1)+1\equiv0\pmod 3\implies 3\mid p$ But $a^2-ab+b^2>3$ for $a,b>2$ hence in this case will be composite. $(2b)$ If $a=3m+1, b=3n+1, a^2-ab+b^2\equiv 1-1(1)+1\equiv1\pmod 3$ Clearly, $(2c),a=3m+1,b=3n\implies a^2-ab+b^2\equiv 1\pmod 3$ $(3a),a=3m-1,b=3n\implies a^2-ab+b^2\equiv 1\pmod 3$ $(3b),a=3m-1,b=3n+1\implies a^2-ab+b^2\equiv 0\pmod 3$ $(3c), a=3m-1,b=3n-1\implies a^2-ab+b^2\equiv1\pmod 3$ So, prime $p=a^2-ab+b^2\equiv1\pmod 3$ for $a,b>2$ Now, $p\equiv1\pmod3\implies p\equiv1,4\pmod 6$ Hence, $p\equiv1\pmod 6$ as $p\equiv4\pmod 6$ is even and $p>2$ Share edited Jan 18, 2013 at 15:35 answered Jan 18, 2013 at 15:28 lab bhattacharjeelab bhattacharjee 279k2020 gold badges213213 silver badges337337 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ Here is a relatively elementary solution. Handling $p=2,3$ as special cases, the statement for $p > 3$ is equivalent to $p = a^2 - ab + b^2$ for some integers $a,b$ iff $p \equiv 1 \mod 3$. The forwards part is easy - just consider numbers mod 3. The key idea is that for any Eisenstein integer $a + b\omega$, the norm is $N(a + b \omega) = (a + b\omega)(\overline{a + b\omega}) = a^2 - ab + b^2$. This is similar to the fact that the norm of any Gaussian integer $a + bi$ is $(a+bi)(a-bi) = a^2 + b^2$ and a very similar argument follows for Quick proof to showing that $p\equiv 1\pmod{4}$ implies $p$ is reducible in $\mathbb{Z}[i]$?, Fermat's Christmas theorem on sums of two squares with Gaussian integers. From Does $x^2 + x + 1 \equiv 0 \mod p$ have a solution? there exists integer $x$ such that $p$ divides $x^2 + x + 1 = N(1-\omega) = (1-\omega)(\overline{1-\omega})$. If $p$ were prime in $\mathbb Z[\omega]$, $p \mid 1-\omega$ or $p \mid 1-\overline\omega$, both of which are impossible. So $p$ is not prime, so in the UFD $\mathbb Z[\omega]$, $p$ is not irreducible. Hence $p = (a+b\omega)(c+d\omega)$ for non-units $a+b\omega, c+d\omega$, and by an argument from norm being multiplicative, $N(p) = p^2 = N(a+b\omega)N(c+d\omega)$, and neither norm is $1$ for non-units, so it must be that $p = N(c+d\omega) = N(a+b\omega) = a^2 - ab + b^2$. (By taking $-b$ instead of $b$, considering $a - b\omega$, we also get $p = a^2 + ab + b^2$.) Share answered Nov 3, 2021 at 16:44 qwrqwr 11.4k44 gold badges4949 silver badges8484 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions number-theory elementary-number-theory eisenstein-integers See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Linked $p\equiv 2\pmod 3$ is an odd prime. Prove that there are no integers $x$, $y$ satisfying $p=x^2-xy+y^2$. 1 $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$ 0 $ p = a^{2} + ab +b^{2} \ a, b \in \mathbb{Z} $ Quick proof to showing that $p\equiv 1\pmod{4}$ implies $p$ is reducible in $\mathbb{Z}[i]$? 5 Primes congruent to $1$ modulo $6$ are of the form $3a^2+b^2$ 5 Fermat's Christmas theorem on sums of two squares with Gaussian integers 1 Does $x^2 + x + 1 \equiv 0 \mod p$ have a solution? 2 $x^2-xy+y^2=n$ how many solutions 1 primes of the form $a^2+b^2=x^2-xy+y^2$? 1 Ground plan of Backward direction (<=) - Let $p$ be an odd prime. 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3815	https://www.youtube.com/watch?v=Xlel4XLomn4	Learn how to evaluate using the half angle formula of sine Brian McLogan 1600000 subscribers 942 likes Description 90359 views Posted: 13 Feb 2013 👉 Learn how to evaluate the Sine of an angle using the half-angle formula. The half-angle formula for Sine is helpful when you need to determine the exact value of function given an angle but cannot use a calculator or the angle is not on the unit circle. To evaluate all we need to do is enter the angle into the formula and simplify. Rationalizing the denominator is not required but often is asked to be performed. 👏SUBSCRIBE to my channel here: ❤️Support my channel by becoming a member: 🙋‍♂️Have questions? Ask here: 🎉Follow the Community: Organized Videos: ✅ Half Angle Formulas ✅ Evaluate Half Angle Formulas of and Angle ✅ Evaluate Half Angle Formulas from a triangle ✅ Write the Expression as a single function \| Half Angle Formulas ✅ Solve Equations using Half Angle Formulas 🗂️ Organized playlists by classes here: 🌐 My Website - 🎯Survive Math Class Checklist: Ten Steps to a Better Year: Connect with me: ⚡️Facebook - ⚡️Instagram - ⚡️Twitter - ⚡️Linkedin - 👨‍🏫 Current Courses on Udemy: 👨‍👩‍👧‍👧 About Me: I make short, to-the-point online math tutorials. I struggled with math growing up and have been able to use those experiences to help students improve in math through practical applications and tips. Find more here: analytictrig #brianmlogan 46 comments Transcript: when completing a problem like this ladies and gentlemen we have the the half angle formula which says s of U / 2 is equal to plus or minus the square Ro T of 1us s of U uh 1 - cosine of U / 2 all right and what we're going to Simply do is now I want to find s of 3 pi over 8 now some of you got very close and when getting you know very close to this problem if you don't want to write down this first step you don't have to but I found a a lot of people that didn't even get to the first step and really the first step is one of the easiest parts we can do for this problem so if you really really confused you want to start with this if I say s of U / 2 is equal to this and then I say what is s of 3 pi/ 8 first thing we know is these two are in the same spots right these are equivalent to each other so if you kind of got stuck on this problem the first step you're always going to want to do is set your half angle equal to your other half angle so you can say that U / 2 is equal to 3i 8 okay Rachel all right so now the next step is obviously we have to solve for U because why do we have to solve for youu well if you guys look into the formula that's not asking us to find the cosine of U ID two is it doesn't say uid two it says just tell me what cosine of U is so if we know U over 2 what is U well to do that we multiply by two on both sides those divide out to one this reduces down to over four so therefore I say U equals 3 pi over 4 so now what I can do is I can plug 3 pi over 4 in for my U so therefore I write plus or minus the square < TK of 1us cosine of 3 piun over 4 / two is that right following with that so now we say all right the cosine of angle well we've been practicing finding the cosines of angles for since chapter 4 so what we do is we go to our unit circle and we figure out where is the cosine of 3 piun over 4 well here's 1 pi over 4 2 piun over 4 3 piun over 4 and we know that 3 Pi 4 has a coordinate point of2 over 2 comma 2 over 2 right so therefore you can say the cosine which is dealing with your x coordinate is going to be of 2 over2 so so let's go ahead and plug that in so we have plus or minus the square < TK of 1 - 2 over 2 / 2 is that everybody kind of foll me so far with what I've done all right then the next thing is notice our angle our angle is in the second quadrant in the second quadrant is the sign of an angle positive or negative is the sign of the angle posi netive it'si so instead of doing the plus or minus we're only going to be working with the Positive value all right so we're not going to include the negative value because we're only concerned about the positive value because our angle is in the second quadrant all right so now we look at this we say all right now we need to simplify this well we have a fraction in the numerator and then divided by a number so we need to get rid of this fraction to get rid of the fractions we need to multiply by two cuz two multip by one or two over two the twos are going to divide to one but wherever you multiply in the numerator you have to make sure you multiply in the denominator and then notice you're multiplying not by a number like you are in the denominator but here you're multiply by binomial so you need to make sure you put the whole thing in parentheses and apply distributive property so therefore I now have positive square root of two double negative turns now to a positive now the twos are going to cancel out we got to remember you also multiply that by two so it's going to be two and then you have um plus the of 2 over 2 2 is 4 let's bring this down a little bit F farther all right then we know that or hopefully we know that you can break up the square root into squ < TK of 2 +k two over theare > of 4 right you can break up a square root when using division to the top and to the bottom yes I mtip theator by the other two huh well you got to multiply by two over two if you just multiply one thing if you you when you're multiplying a fraction you have to multiply the same fra you have to multiply the numerator and denominator by the same number right because think of you know two two fours that's equal to 1/2 right so but if you multiply by 2 over2 that gives you 4 8 and those equal to each other yeah so you have to multiply by 2 over two right so you don't have to get the two off the bottom down here yeah no not initially right now um if you did if you no because what's going to happen is we don't have to get it off the bottom we just want to multiply by two I see what you're kind of getting at as getting this two off but remember if you're just going to multiply by one number two then that means you have to multiply by two on the other side of the equation right so what I'm doing is I'm just multiplying by 2 over two right now and what that does that reduces it to here which now I can take the square root of four right the square of 4 is two so therefore now my next point is going to be the square < TK of 2 + > of 2 over two and you could also break that up into 12 the of 2 + of two okay so that's how you work your your problem okay can you ruce any that is going to be your final reduce problem I
3816	https://www.youtube.com/watch?v=2xkHjAMSdSg	Sketch polar curve r = 1 + 2 cos theta, first graph r as function of theta in Cartesian coordinates Ms Shaws Math Class 51100 subscribers 61 likes Description 11213 views Posted: 11 May 2021 1 comments Transcript: hi everyone we're going to sketch the curve with the given polar equation by first sketching the graph r as a function of theta in cartesian coordinates we have r equals 1 plus 2 cosine of theta so for the first part basically this problem is wanting you to compare the graphs so for this one if i just graph this part first this is an amplitude of 2 so i'll go here negative 2 to 2 and cosine starts here so if you graph this one that's how that would be now with a one that means we're shifting it up by one so we're going to go to three and this is going to get shifted up by negative one so you do the same thing but you just shift it up whoops so does our center our center gets shifted to one here all right so let me start again there we go all right and then you just sketch the graph make it a little curvy and there you have it so that's what the graph looks like of r as a function of theta in cartesian coordinates now let's do the polar graph the polar graph this is a limison with a inner loop this number is larger than this number that means it's the numbers are different that tells you it's a limison but when this number is larger it means it has an inner loop it's also in cosine so i know it's going to reflect over the polar axis here so i'm going to just assign my values from 0 to pi which i already did to save time so i filled all this out in for us and we can just plot the points so at 0 we're at 3 2 3. i'll put a big let me do a different color big dot there at pi divided by six we're at two point seven three which is very close to three at pi divided by three we're at two and then we're at one four pi divided by two at two pi divided by three we're at zero at five pi divided by uh six we're at negative uh close to one so you go negative you go this way and at pi we're at negative one so it's here so the upper portion or of the graph is going to look like this all right now that you have that you can use the fact that it's submission symmetric to the polar axis to graph the other part and you just match everything up so basically at um let's see where do we stop at pi at 7 pi divided by 6 let's let's go around this way let's go this way so eleven pi over six that matches up to um pi divided by six so that's going to go to the same value 2.3 2.73 all right 5 pi divided by 3 goes with this one so that's going to go to 1 2 which is here and then we're going to go to 1 for 3 pi divided by 2. so here's here it's coming in at 4 pi divided by 3 we're at 0. so now we go back to 0 and then at 7 pi over 6 that matches up to 5 pi over 6. so now that's where you're going to go negative 1 close to 1. and then at pi we're at negative 1. so it closes up right there and that is your graph we started here and if you went around the circle we should have gone this way but it's easier to look at the other way there's your limousine with the inner loop thank you have a nice day bye bye you
3817	https://math.stackexchange.com/questions/2114358/on-the-number-of-divisors-within-given-bounds	Skip to main content On the number of divisors within given bounds Ask Question Asked Modified 8 years, 6 months ago Viewed 986 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. It is well known, that given an integer n whose prime factorization is n=pa11×pa22×…×pann the number of its positive divisors is given by d(n)=(a1+1)×(a2+1)×…×(ak+1) Now the question is the following: How many of the above divisors of n lie between given bounds a and b? Is there a systematic method for answering this question ? In other words: given two positive integers a<b, and a positive integer n, is there a well defined function d(n,a,b) having as its values the number of positive divisors of n, lying between a and b ? number-theory elementary-number-theory analytic-number-theory Share CC BY-SA 3.0 Follow this question to receive notifications edited Jan 26, 2017 at 2:39 KonKan asked Jan 26, 2017 at 2:07 KonKanKonKan 7,57222 gold badges3131 silver badges5151 bronze badges 8 2 of course it is well defined. – Asinomás Commented Jan 26, 2017 at 2:18 does checking every divisor count? – Asinomás Commented Jan 26, 2017 at 2:19 well, that counts for an algorithm however my question is what would that function be. – KonKan Commented Jan 26, 2017 at 2:20 2 You might as well ask for a function d′(n,b) that counts divisors below b. Then d(n,a,b)=d′(n,b)−d′(n,a) – Ross Millikan Commented Jan 26, 2017 at 3:14 1 @KonKan Sure thing. Fix any a<b. There are only finitely many integers in the range [a,b]. Let L be the lcm of the set {a,a+1,…,b}. Then any integer c with a≤c≤b by definition divides L. In particular, c∣x⟺c∣x+L. It follows that d(n,a,b)=d(n+L,a,b), so d(⋅,a,b) is periodic with period at most L. – Erick Wong Commented Feb 2, 2017 at 7:27 \| Show 3 more comments 2 Answers 2 Reset to default This answer is useful 2 Save this answer. Show activity on this post. Here is an approximation that works well when n is enormous, has few factors, and b is large but small compared with n. We will look for a function d′(n,b) that counts divisors of n that are below b. As an example, we will consider the case where n=2k3m and assume that 2k,3m>b. At each of the lattice points (x,y) in the first quadrant we can associate the log of one potential factor, so at (x,y) we associate xlog2+ylog3. The factors less than b are the lattice points below the line xlog2+ylog3=logb. The area of the triangle is (logb)22log2log3 so we would expect that many factors less than b. The extension to more factors is clear. If there were prime factors 2,3,5 the number of factors less than b would be (logb)33!log2log3log5 because we would have a tetrahedron in a 3D lattice. The limitations are easy to see. If there are not enough factors of 2 or 3 to exceed b our triangular region can become a rectangle with the corner cut off. The line or plane dividing the factors below b from those above will pass between lattice points, so there is uncertainty at the boundary. As the boundary is one dimension lower than the bulk of the lattice points this error reduces as n gets large. A feel for the error comes from the fact that for b=106 the area formula gives 125 factors while there really are 142. For b=109 the area formula gives 282 compared to 306. Share CC BY-SA 3.0 Follow this answer to receive notifications answered Jan 26, 2017 at 3:42 Ross MillikanRoss Millikan 383k2828 gold badges262262 silver badges472472 bronze badges 2 thank you for providing this. Although it does not provide a definite answer to the question, it is certainly a very interesting approximation and nicely presented. I think that the initial problem virtually "breaks" into a system of simultaneous diophantine-like inequalities and what you are doing here is a kind of graphical approximation to the solution of such a problem. +1. Do you know of some further references on similar methods, for this or related problems? – KonKan Commented Feb 1, 2017 at 20:24 On the other hand, I have discussed this problem elsewhere but I have not received significant feedback (not even close). So, i think it is fair -after a few days have passed- to accept your answer. – KonKan Commented Feb 1, 2017 at 23:50 Add a comment \| This answer is useful 2 Save this answer. Show activity on this post. Several mathematicians have done research on this very topic. A good place to start might be this paper of Kevin Ford: "The distribution of integers with divisors in a given interval." Share CC BY-SA 3.0 Follow this answer to receive notifications answered Jan 27, 2017 at 4:48 user397701user397701 70344 silver badges99 bronze badges 1 thank you for the reference. However, after taking a quick look through it, i think that it discusses a similar but different problem. It is interesting in any case. – KonKan Commented Feb 1, 2017 at 20:26 Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions number-theory elementary-number-theory analytic-number-theory See similar questions with these tags. Featured on Meta Upcoming initiatives on Stack Overflow and across the Stack Exchange network... 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3818	https://yufeizhao.com/research/extremal-regular-graphs.pdf	Extremal Regular Graphs: Independent Sets and Graph Homomorphisms Yufei Zhao Abstract. This survey concerns regular graphs that are extremal with respect to the number of independent sets and, more generally, graph homomorphisms. More precisely, in the family of of d-regular graphs, which graph G maximizes/minimizes the quantity i(G)1/v(G), the number of independent sets in G normalized exponentially by the size of G? What if i(G) is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research. 1. INDEPENDENT SETS. An independent set in a graph is a subset of vertices with no two adjacent. Many combinatorial problems can be reformulated in terms of independent sets by setting up a graph where edges represent forbidden relations. A graph is d-regular if all vertices have degree d. In the family of d-regular graphs of the same size, which graph has the largest number of independent sets? This ques-tion was initially raised by Andrew Granville in connection to combinatorial number theory and appeared ﬁrst in print in a paper by Alon , who speculated that, at least when n is divisible by 2d, the maximum is attained by a disjoint union of complete bipartite graphs Kd,d. Some ten years later, Kahn arrived at the same conjecture while studying a problem arising from statistical physics. Using a beautiful entropy argument, Kahn proved the conjecture under the additional assumption that the graph is already bipartite. While attending Joe Gallian’s Research Experience for Under-graduates in Duluth in 2009, the author showed that the bipartite assumption can be dropped . The precise theorem is stated below. We write I (G) to denote the set of independent sets in G and i(G) := \|I (G)\| for its cardinality. See Figure 1. Theorem 1.1 ( for bipartite G; for general G). If G is a bipartite n-vertex d-regular graph, then i(G) ≤i(Kd,d)n/(2d) = (2d+1 −1)n/(2d). Equality occurs when n is divisible by 2d and G is a disjoint union of Kd,d’s. We do not concern ourselves here with what happens when n is not divisible by 2d, as the extremal graphs are likely dependent on number-theoretic conditions, and we do not know a clean set of examples. Alternatively, the problem can phrased as maximizing i(G)1/v(G) over the set of d-regular bipartite graphs G, where v(G) denotes the number of vertices of G. The above theorem says that this maximum is attained at G = Kd,d. Note that i(G)1/v(G) remains unchanged if G is replaced by a disjoint union of copies of G. We provide an exposition of this theorem as well as a discussion of subsequent developments. Notably, Davies, Jenssen, Perkins, and Roberts recently gave a new proof by introducing a powerful new technique, which has already had a number of surprising new consequences [14, 15, 26]. The results have been partially extended to graph homomorphisms, though many intriguing open problems remain. We also MSC: Primary 05C35, Secondary 05C69; 05C60 November 2017] EXTREMAL REGULAR GRAPHS 827 Figure 1. The independent sets of a 4-cycle: i(C4) = 7. graph homomorphism independent set Figure 2. Homomorphisms from G to correspond to independent sets of G. graph homomorphism coloring Figure 3. Homomorphisms from G to Kq correspond to proper colorings of vertices of G with q colors. Figure 4. A conﬁguration for the Widom–Rowlinson model on a grid, corresponding to a homomorphism to , where vertices of the grid that are mapped to the ﬁrst vertex in are marked and those mapped to the third vertex are marked . discuss some recent work on the subject done by Luke Sernau as an undergraduate student at Notre Dame. 2. GRAPH HOMOMORPHISMS. Given two graphs G and H, a graph homomor-phism from G to H is a map of vertex sets φ : V (G) →V (H) that sends every edge of G to an edge of H, i.e., φ(u)φ(v) ∈E(H) whenever uv ∈E(G). Here, V (G) denotes the vertex set of G and E(G) the edge set. Denote the set of graph homomorphisms from G to H by Hom(G, H) := {φ : V (G) →V (H) : φ(u)φ(v) ∈E(H) ∀uv ∈E(G)}. We use lowercase letters for cardinalities: v(G) := \|V (G)\|, e(G) := \|E(G)\|, and hom(G, H) := \| Hom(G, H)\|. We usually use the letter G for the source graph and H for the target graph. It will be useful to allow the target graph H to have loops (but not multiple edges), and we shall refer to such graphs as loop-graphs. The source graph G is usually simple (without loops). By graph, we usually mean a simple graph. Graph homomorphisms generalize the notion of independent sets. They are equiva-lent to labeling the vertices of G subject to certain constraints encoded by H. Example 2.1 (Independent sets). Homomorphisms from G to correspond bijec-tively to independent sets in G. Indeed, a map of vertices from G to is a homo-morphism if and only if the preimage of the nonlooped vertex in forms an inde-pendent set in G. So hom(G, ) = i(G). See Figure 2. In the statistical physics 828 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 literature,1 independent sets correspond to hard-core models. For example, they can be used to represent conﬁgurations of nonoverlapping spheres (“hard” spheres) on a grid. Example 2.2 (Graph colorings). When the target graph is the complete graph Kq on q vertices, a graph homomorphism from G to Kq corresponds to a coloring of the vertices of G with q colors so that no two adjacent vertices of G receive the same color. Such colorings are called proper q-colorings. See Figure 3. Thus, hom(G, Kq) is the number of proper q-colorings of G. For a ﬁxed G, the quantity hom(G, Kq) is a polynomial function in q, and it is called the chromatic polynomial of G, a classic object in graph theory. Example 2.3 (Widom–Rowlinson model). A homomorphism from G to cor-responds to a partial coloring of the vertices of G with red or blue, allowing vertices to be left uncolored, such that no red vertex is adjacent to a blue vertex. Such a coloring is known as a Widom–Rowlinson conﬁguation. See Figure 4. As graph homomorphisms generalize independent sets, one may wonder whether Theorem 1.1 generalizes to graph homomorphisms. It turns out, perhaps surprisingly, that the bipartite case of Theorem 1.1, concerning the number of independent sets in a regular bipartite graph, always extends to graph homomorphisms. Theorem 2.4 (Galvin and Tetali ). Let G be a bipartite d-regular graph and H a loop-graph. Then hom(G, H)1/v(G) ≤hom(Kd,d, H)1/(2d). Can the bipartite hypothesis above be dropped as in Theorem 1.1? The answer is no. Indeed, with H = being two disjoint loops, hom(G, ) = 2c(G) where c(G) is the number of connected components of G. In this case, hom(G, )1/v(G) is maximized when the sizes of the components of G are as small as possible (among d-regular graphs), i.e., when G = Kd+1. The central problem of interest for the rest of this article is stated below. It has been solved for certain targets H, but it is open in general. The analogous minimization problem is also interesting and will be discussed in Section 8. Problem 2.5. Fix a loop-graph H and a positive integer d. Determine the supremum of hom(G, H)1/v(G) taken over all d-regular graphs G. We have already seen two cases where Problem 2.5 has been solved: When H = , the maximum is attained by G = Kd,d (Theorem 1.1), and when H = , the maximum is attained by G = Kd+1. The latter example can be extended to H being a disjoint union of complete loop-graphs. Another easy case is H bipartite, as hom(G, H) = 0 unless G is bipartite, so the maximizer is Kd,d by Theorem 2.4. In his undergraduate senior project, the author extended Theorem 1.1 to solve Prob-lem 2.5 for a certain family of graphs H called bipartite swapping targets. As a spe-cial case, we deﬁne a loop-threshold graph to be a loop-graph whose vertices can be ordered so that its adjacency matrix has the property that whenever an entry is 1, all entries to the left of it and above it are 1 as well. An example of a loop-threshold graph, along with its adjacency matrix, is shown below. 1See for the connection between the combinatorics of graph homomorphisms and Gibbs measures in statistical physics. November 2017] EXTREMAL REGULAR GRAPHS 829 1 2 3 4 5 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. Loop-threshold graphs generalize from Example 2.1. The following result was obtained by extending the proof method of Theorem 1.1. Theorem 2.6 (). Let G be a d-regular graph G and H a loop-threshold graph. Then hom(G, H)1/v(G) ≤hom(Kd,d, H)1/(2d). Sernau recently extended the above results to an even larger family of H (see Section 5). The most interesting open case of Problem 2.5 is H = Kq, concerning the number of proper q-colorings of vertices of G (Example 2.2). Conjecture 2.7. For every d-regular graph G and integer q ≥3, hom(G, Kq)1/v(G) ≤hom(Kd,d, Kq)1/(2d). The conjecture was recently solved for d = 3 by Davies, Jenssen, Perkins, and Roberts using a novel method they developed earlier. We will discuss the method in Section 7. The conjecture remains open for all d ≥4 and q ≥3. The above inequal-ity is known to hold if q is sufﬁciently large as a function of G (the current best bound is q > 2 v(G)d/2 4 ). The ﬁrst nontrivial case of Problem 2.5 where the maximizing G is not Kd,d was obtained recently by Cohen, Perkins, and Tetali . Theorem 2.8 (Cohen, Perkins, and Tetali ). For any d-regular graph G, we have hom(G, )1/v(G) ≤hom(Kd+1, )1/(d+1). Theorem 2.8 was initially proved using the occupancy fraction method, which will be discussed in Section 7. Subsequently, a much shorter proof was given in (also see Sernau ).2 These methods can be used to prove that Kd+1 is the maximizer for a large family of target loop-graphs H (see Section 5). There are weighted generalizations of these problems and results. However, for clarity, we defer discussing the weighted versions until Section 7, where we will see that introducing weights leads to a powerful new differential method for proving the unweighted results. We conclude this section with some open problems. Galvin conjectured that in Problem 2.5, the maximizing G is always either Kd,d or Kd+1, as with all the cases we have seen so far. However, this conjecture was recently disproved by Sernau 3 (see Section 6). As it stands, there does not seem to be a clean conjecture concerning the solution to Problem 2.5 on determining the maximizing G. Sernau suggested the possibility that there is a ﬁnite list of maximizing G for every d. 2Sernau also tackled Theorem 2.8, obtaining an approximate result in a version of that predated and . After the appearance of , Sernau corrected an error (identiﬁed by Cohen) in , and the corrected version turned out to include Theorem 2.8 as a special case. 3A similar counterexample was independently found by Pat Devlin. 830 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 Conjecture 2.9. For every d ≥3, there exists a ﬁnite set Gd of d-regular graphs such that for every loop-graph H and every d-regular graph G one has hom(G, H)1/v(G) ≤max G′∈Gd hom(G′, H)1/v(G′). It has been speculated that the maximizing G perhaps always has between d + 1 and 2d vertices (corresponding to Kd+1 and Kd,d, respectively). Sernau suggested the possibility that for a ﬁxed H, the maximizer is always one of Kd,d and Kd+1 as long as d is large enough. Conjecture 2.10. Let H be a ﬁxed loop-graph. There is some dH such that for all d ≥dH and d-regular graphs G, hom(G, H)1/v(G) ≤max{hom(Kd+1, H)1/(d+1), hom(K2d, H)1/(2d)}. We do not know if the supremum in Problem 2.5 can always be attained. Question 2.11. Fix d ≥3 and a loop-graph H. Is the supremum of hom(G, H)1/v(G) over all d-regular graphs G always attained by some G? It could be the case that the supremum is the limit coming from a sequence of graphs G of increasing size instead of a single graph G on ﬁnitely many vertices. This is indeed the case if we wish to minimize hom(G, )1/v(G) over d-regular graphs G. Csikv´ ari recently showed that the inﬁmum of hom(G, )1/v(G) is given by a limit of d-regular graphs G with increasing girth (i.e., G locally looks like a d-regular tree at every vertex). 3. PROJECTION INEQUALITIES. The original proof of the bipartite case of Theorem 1.1 (as well as Theorem 2.4) uses a beautiful entropy argument, with a key input being Shearer’s entropy inequality . We will not cover the entropy arguments as they would lead us too far astray. See Galvin’s lecture notes for a nice expo-sition of the entropy method for counting problems. The ﬁrst nonentropy proof of these two theorems was given in using a variant of H¨ older’s inequality, which we describe in this section. We begin our discussion with a classical projection inequal-ity. See Friedgut’s MONTHLY article concerning how the projection inequalities relate to entropy. Let Pxy denote the projection operator from R3 onto the xy-plane. Similarly, deﬁne Pxz and Pyz. Let S be a body in R3 such that each of the three projections Pxz(S), Pyz(S), and Pxz(S) has area 1. What is the maximum volume of S? (This is not as obvious as it may ﬁrst appear. Note that we are projecting onto the 2-D coordinate planes as opposed to the 1-D axes.) The answer is 1, attained when S is an axes-parallel cube of side-length 1. Indeed, equivalently (by rescaling), we have vol(S)2 ≤area(Pxy(S)) area(Pxz(S)) area(Pyz(S)). (1) Such results were ﬁrst obtained by Loomis and Whitney . More generally, for any functions f, g, h : R2 →R (assuming integrability conditions) R3 f (x, y)g(x, z)h(y, z) dxdydz 2 November 2017] EXTREMAL REGULAR GRAPHS 831 ≤ R2 f (x, y)2 dxdy R2 g(x, z)2 dxdz R2 h(y, z)2 dydz . (2) To see how (2) implies (1), take f, g, h to be the indicator functions of the pro-jections of S onto the three coordinates planes, and observe that 1S(x, y, z) ≤ f (x, y)g(x, z)h(y, z). Let us prove (2). In fact, x, y, z can vary over any measurable space instead of R. In our application, the domains will be discrete, i.e., the integral will be a sum. It sufﬁces to prove the inequality when f, g, h are nonnegative. The proof is via three simple applications of the Cauchy–Schwarz inequality, to the variables x, y, z, one at a time in that order: f (x, y)g(x, z)h(y, z) dxdydz ≤ f (x, y)2 dx 1/2 g(x, z)2 dx 1/2 h(y, z) dydz ≤ f (x, y)2 dxdy 1/2 g(x, z)2 dx 1/2 h(y, z)2 dy 1/2 dz ≤ f (x, y)2 dxdy 1/2 g(x, z)2 dxdz 1/2 h(y, z)2 dydz 1/2 = ∥f ∥2∥g∥2∥h∥2, where ∥f ∥p := \| f \|p 1/p is the L p norm. This proves (2). This inequality strengthens H¨ older’s inequality since a direct application of H¨ older’s inequality would yield f gh ≤∥f ∥3∥g∥3∥h∥3. (3) What we have shown is that whenever each of the variables x, y, z appears in the argument of exactly two of the three functions f, g, and h, then the L3 norms on the right-hand side of (3) can be sharpened to L2 norms (we always have ∥f ∥2 ≤∥f ∥3 by convexity). The above proof easily generalizes to prove the following more general result (also see [25, Theorem 3.1]). It is also related to the Brascamp–Lieb inequality . Theorem 3.1. Let A1, . . . , Am be subsets of [n] := {1, 2, . . . , n} such that each i ∈[n] appears in exactly d of the sets A j. Let i be a measure space for each i ∈[n]. For each j, let f j : i∈A j i →R be measurable functions. Let Pj denote the projection of Rn onto the coordinates indexed by A j. Then 1×···×n f1(P1(x)) · · · fm(Pm(x)) dx ≤∥f1∥d · · · ∥fm∥d. Using this inequality, we now prove Theorem 2.4. 832 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 Proof of Theorem 2.4. Let V (G) = U ∪W be a bipartition of G. Since G is d-regular, \|U\| = \|W\| = v(G)/2. For any z1, . . . , zd ∈V (H), let f (z1, . . . , zd) := \|{z ∈V (H) : z1z, . . . , zdz ∈E(H)}\| denote the size of the common neighborhood of z1, . . . , zd in H. For any φ : U →V (H), the number of ways to extend φ to a graph homomorphism from G to H can be determined by noting that, for each w ∈W, there are exactly f (φ(u) : u ∈N(w)) choices for its image φ(w), independent of the choices for other vertices in W. Therefore, hom(G, H) = φ : U→V (H) w∈W f (φ(u) : u ∈N(w)). Since G is d-regular, every u ∈U is contained in N(w) for exactly d different w ∈W. Therefore, by applying Theorem 3.1 with the counting measure on V (H), we ﬁnd that hom(G, H) ≤∥f ∥\|W\| d . Note that ∥f ∥d d = z1,...,zd∈V (H) f (z1, . . . , zd)d = hom(Kd,d, H). Therefore, hom(G, H) ≤hom(Kd,d, H)\|W\|/d = hom(Kd,d, H)v(G)/(2d). 4. A BIPARTITE SWAPPING TRICK. In the previous section, we proved Theorem 1.1 for bipartite graphs G. Now we use it to deduce Theorem 1.1 for nonbipartite graphs. The proof follows [30, 31]. The idea is to transform G into a bipartite graph, namely the bipartite double cover G × K2, with vertex set V (G) × {0, 1}. The ver-tices of G × K2 are labeled vi for v ∈V (G) and i ∈{0, 1}. Its edges are u0v1 for all uv ∈E(G). See Figure 5. This construction is a special case of the graph tensor prod-uct, which we deﬁne in the next section. Note that G × K2 is always a bipartite graph. The following key lemma shows that G × K2 always has at least as many independent sets as two disjoint copies of G. Lemma 4.1 (). Let G be any graph (not necessarily regular). Then i(G)2 ≤i(G × K2). Since G × K2 is bipartite and d-regular, the bipartite case of Theorem 1.1 implies i(G)2 ≤i(G × K2) ≤(2d+1 −1)n/d so that Theorem 1.1 follows immediately. See Figure 5 for an illustration of the fol-lowing proof. Proof of Lemma 4.1. Let 2G denote a disjoint union of two copies of G. Label its vertices by vi with v ∈V and i ∈{0, 1} so that its edges are uivi with uv ∈E(G) and i ∈{0, 1}. We will give an injection φ : I (2G) →I (G × K2). Recall that I (G) is the set of independent sets of G. The injection would imply i(G)2 = i(2G) ≤i(G × K2) as desired. November 2017] EXTREMAL REGULAR GRAPHS 833 Fix an arbitrary order on all subsets of V (G). Let S be an independent set of 2G. Let Ebad(S) := {uv ∈E(G) : u0, v1 ∈S}. Note that Ebad(S) is a bipartite subgraph of G since each edge of Ebad has exactly one endpoint in {v ∈V (G) : v0 ∈S} but not both (or else S would not be independent). Let A denote the ﬁrst subset (in the previously ﬁxed ordering) of V (G) such that all edges in Ebad(S) have one vertex in A and the other outside A. Deﬁne φ(S) to be the subset of V (G) × {0, 1} obtained by “swapping” the pairs in A, i.e., for all v ∈A, vi ∈φ(S) if and only if v1−i ∈S for each i ∈{0, 1}, and for all v / ∈A, vi ∈φ(S) if and only if vi ∈S for each i ∈{0, 1}. It is not hard to verify that φ(S) is an independent set in G × K2. The swapping procedure ﬁxes the “bad” edges. It remains to verify that φ is an injection. Let S ∈I (2G) and T = φ(S). We show how to recover S from T . Set E′ bad(T ) = {uv ∈E(G) : ui, vi ∈T for some i ∈{0, 1}} so that Ebad(S) = E′ bad(T ). We ﬁnd A as earlier and then swap the pairs of A back. (Remark: It follows that T ∈I (G × K2) lies in the image of φ if and only if E′ bad(T ) is bipartite.) 2G G× K2 G× K2 Figure 5. The bipartite swapping trick in the proof of Lemma 4.1: swapping the circled pairs of vertices (denoted A in the proof) ﬁxes the bad edges (bolded), transforming an independent set of 2G into an indepen-dent set of G × K2. The above method was used to drop the G bipartite hypothesis in Theorem 2.4 when H belongs to a certain class of loop-graphs called bipartite swapping targets, which includes the threshold graphs in Theorem 2.6 as a special case. See for details. We do not know how to extend the method to H = Kq, corresponding to the number of proper q-colorings. Nonetheless, the analogous strengthening of Conjecture 2.7 is conjectured to hold (the inequality was proved for q sufﬁciently large as a function of G). Conjecture 4.2 (). For every graph G and every q ≥3, hom(G, Kq)2 ≤hom(G × K2, Kq). 5. GRAPH PRODUCTS AND POWERS. We deﬁne several graph operations. • Tensor product G × H: Its vertices are V (G) × V (H), with (u, v) and (u′, v′) ∈ V (G) × V (H) adjacent in G × H if uu′ ∈E(G) and vv′ ∈E(H). This construc-tion is also known as the categorical product. • Exponentiation H G: Its vertices are maps f : V (G) →V (H) (not necessarily homomorphisms), where f and f ′ are adjacent if f (u) f ′(v) ∈E(H) whenever uv ∈E(G). • G◦: Same as G except that every vertex now has a loop. 834 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 • ℓ(H): Subgraph of H induced by its looped vertices, or equivalently, delete all nonlooped vertices from H. Example 5.1. For any loop-graph H, the graph H K2 has vertex set V (H) × V (H), with (u, v) and (u′, v′) ∈V (H) × V (H) adjacent if and only if uv′, u′v ∈E(H). In particular, if Hind = , then H K2 ind = . Here are a few easy yet key facts relating the above operations with graph homo-morphisms. The proofs are left as exercises for the readers: hom(G, H1 × H2) = hom(G, H1) hom(G, H2), (4) hom(G × G′, H) = hom(G, H G′), (5) hom(G◦, H) = hom(G, ℓ(H)). (6) Clique as maximizer. Now we prove Theorem 2.8 concerning the Widom–Rowlinson model. Recall it says that for any d-regular graph G, we have hom G, 1/v(G) ≤hom Kd+1, 1/(d+1) . Proof of Theorem 2.8. We have = ℓ(H K2 ind ) (Example 5.1). For any graph G, hom G, = hom G, ℓ(H K2 ind ) = hom G◦, H K2 ind = hom G◦× K2, . (7) When G is d-regular, G◦× K2 is a (d + 1)-regular bipartite graph, so Theorem 2.4 (or Theorem 1.1) implies that the above quantity is at most hom Kd+1,d+1, v(G)/(d+1) . Since Kd+1,d+1 = K ◦ d+1 × K2, we have by (7), hom(G, )1/v(G) = hom(G◦× K2, )1/v(G) ≤hom(K ◦ d+1 × K2, )1/(d+1) = hom(Kd+1, )1/(d+1). The above proof exploits the connection (7) between the hard-core model (indepen-dent sets) and the Widom–Rowlinson model. This relationship had been previously observed in . More generally, the above proof extends to give the following result. Deﬁnition 5.2. The extended line graph H of a graph H has V ( H) = E(H) and two edges e and f of H are adjacent in H if (1) e = f , or (2) e and f share a common vertex, or (3) e and f are opposite edges of a 4-cycle in G. Theorem 5.3 (). Let H be the extended line graph of a bipartite graph H. For any d-regular graph G, hom(G, H)1/v(G) ≤hom(Kd+1, H)n/(d+1). November 2017] EXTREMAL REGULAR GRAPHS 835 For a simple graph H, let H ◦denote H with a loop added at every vertex. Let Pk denote the path of k vertices and Ck the cycle with k vertices. Example 5.4. One has Pk+1 = P◦ k for all k. Also, Ck = C◦ k for all k ̸= 4. Corollary 5.5 (). Let H = C◦ k with even k ≥6 or H = P◦ k for any k ≥1. For any d-regular graph G, hom(G, H)1/v(G) ≤hom(Kd+1, H)1/v(G). The following related result was established by Sernau using similar techniques. Theorem 5.6 (). Let H = ℓ(AB) where A is any loop-graph and B is a bipartite graph. For any d-regular graph G, hom(G, H)1/v(G) ≤hom(Kd+1, H)1/(d+1). Closure under tensor products. Sernau observed that, for any d, if H = H1 and H = H2 both have the property that G = Kd,d maximizes hom(G, H)1/v(G) over all d-regular graphs G, then H = H1 × H2 has the same property by (4). In other words, the set of H such that G = Kd,d is the maximizer in Problem 2.5 is closed under tensor products. This observation enlarges the set of such H previously obtained in . 6. NEITHER COMPLETE BIPARTITE NOR CLIQUE. In all cases of Prob-lem 2.5 that we have considered, the maximizing G is always either Kd,d or Kd+1. It was conjectured that one of Kd,d and Kd+1 always maximizes hom(G, H)1/v(G) for every H. However, Sernau showed that this is false (a similar construction was independently found by Pat Devlin). Let d ≥4, and let G be a d-regular graph with v(G) < 2d other than Kd+1. Brooks’s theorem tells us that G is d-colorable so that hom(G, Kd) > 0. It follows that for this G, hom(G, kKd)1/v(G) = k1/v(G) hom(G, Kd)1/v(G) > k1/(2d) hom(Kd,d, Kd)1/(2d) = hom(Kd,d, kKd)1/(2d) for sufﬁciently large k (as a function of d) since v(G) < 2d. Also, hom(G, kKd)1/v(G) > 0 = hom(Kd+1, kKd)1/(d+1). Therefore, neither G = Kd,d nor G = Kd+1 maximize hom(G, kKd)1/v(G) over all d-regular graphs G. For d = 3, Csikv´ ari found a counterexample using a similar construction. In general, we do not know which graphs G (other than Kd+1 and Kd,d) can arise as maximizers for hom(G, H)1/v(G) in Problem 2.5. See the end of Section 2 for some open questions and conjectures. 7. OCCUPANCY FRACTION. The original proof of the bipartite case of Theorem 1.1 used the entropy method . The proof in Section 3, following , used a variant of the H¨ older’s inequality and is related to the original entropy method proof. Recently, an elegant new proof of the result was found using a novel method, which was unrelated to previous proofs. We discuss this new technique in this section. It will be necessary to introduce weighted versions of the problems. 836 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 The independence polynomial of a graph G is deﬁned by PG(λ) := I∈I (G) λ\|I\|. Recall that I (G) is the set of independent sets of G. In particular, PG(1) = i(G). Theorem 1.1 extends to this weighted version of the number of independent sets. Theorem 7.1 ( for bipartite G; for general G). If G is a d-regular graph and λ ≥0, then PG(λ)1/v(G) ≤PKd,d(λ)1/(2d). The hard-core model with fugacity λ on G is deﬁned as the probability distribu-tion on independent sets of G where an independent set I is chosen with probability proportional to λ\|I\|, i.e., with probability Pr λ [I] = λ\|I\| PG(λ). The occupancy fraction of I is the fraction of vertices of G occupied by I. The expected occupancy fraction of a random independent set from the hard-core model is αG(λ) := 1 v(G) I∈I (G) \|I\| · Pr λ [I] = I∈I (G) \|I\|λ\|I\| v(G)PG(λ) = λP′ G(λ) v(G)PG(λ). It turns out that Kd,d maximizes the occupancy fraction among all d-regular graphs. Theorem 7.2 (Davies, Jenssen, Perkins, and Roberts ). For all d-regular graphs G and all λ ≥0, we have αG(λ) ≤αKd,d(λ) = λ(1 + λ)d−1 2(1 + λ)d −1. (8) Since the expected occupancy fraction is proportional to the logarithmic derivative of PG(λ)1/v(G), the inequality for the expected occupancy fraction implies the corre-sponding inequality for the independence polynomial. Indeed, Theorem 7.2 implies Theorem 7.1 (and hence Theorem 1.1) since 1 v(G) λ 0 αG(t) t dt = 1 v(G) λ 0 P′ G(t) PG(t) dt = log PG(λ) v(G) . We reproduce here two proofs of Theorem 7.2. They are both based on the following idea, introduced in for this problem. We draw a random independent set I from the hard-core model and look at the neighborhood of a uniform random vertex v ∈ V (G). The expected occupancy fraction is then the probability that v ∈I. We then analyze how the neighborhood of v should look in relation to I. Since the graph is regular, a uniform random neighbor of v is uniformly distributed in V (G). By ﬁnding an appropriate set of constraints on the probabilities of seeing various neighborhood conﬁgurations of v, we can bound the probability that v ∈I. The ﬁrst proof is given below under the additional simplifying assumption that G is triangle-free (which includes all bipartite graphs and much more). See for how to extend this proof to all regular graphs. November 2017] EXTREMAL REGULAR GRAPHS 837 Proof of Theorem 7.2 for triangle-free G. Let I be an independent set of G drawn according to the hard-core model with fugacity λ. For each v ∈V (G), let pv denote the probability that v ∈I. We say v ∈V (G) is uncovered if none of the neighbors of v is in I, i.e., N(v) ∩I = ∅. If v ∈I, then v is necessarily uncovered. Conversely, conditioned on v being uncovered, one has v ∈I with probability λ/(1 + λ). So the probability that v is uncovered is pv(1 + λ)/λ. Let Uv denote the set of uncovered neighbors of v. Since G is triangle-free, Uv is an independent set. Conditioned on Uv being the uncovered neighbors of v, the probability that v is uncovered, which is equivalent to Uv ∩I = ∅, is exactly (1 + λ)−\|Uv\|. Hence, 1 + λ λ pv = E[(1 + λ)−\|Uv\|] ≤1 −E[\|Uv\|] d 1 −(1 + λ)−d , (9) where the inequality follows from 0 ≤\|Uv\| ≤d and the convexity of the function x →(1 + λ)−x so that (1 + λ)−x ≤1 −x d (1 −(1 + λ)−d) for all 0 ≤x ≤d by linear interpolation. If v is chosen from V (G) uniformly at random, then E[pv] = αG(λ) is the expected occupancy fraction. Similarly, E[\|Uv\|]/d is the probability that a ran-dom vertex is uncovered (here we use again that G is d-regular), which equals E[pv] 1+λ λ = αG(λ) 1+λ λ . Setting into (9), we obtain 1 + λ λ αG(λ) ≤1 −αG(λ)1 + λ λ 1 −(1 + λ)−d . Rearranging gives us (8). In , Theorem 7.2 was proved for all d-regular graphs G by considering all graphs on d vertices that could be induced by the neighborhood of a vertex in G and using a linear program to constrain the probability distribution of the neighborhood proﬁle of a random vertex. When G is triangle-free, the neighborhood of a vertex is always an independent set, which signiﬁcantly simpliﬁes the situation. The following conjecture extends Theorem 2.4 to triangle-free graphs. Conjecture 7.3 (). Let G be a triangle-free d-regular graph and H a loop-graph. Then hom(G, H)1/v(G) ≤hom(Kd,d, H)1/(2d). Next, we give an alternative proof of Theorem 7.2 due to Perkins , based on a similar idea. In the following proof, we do not need to assume that G is triangle-free. We introduce an additional constraint, which allows us to obtain the result more quickly. This simpliﬁcation seems to be somewhat speciﬁc to independent sets. Second proof of Theorem 7.2. Let I be an independent set of G drawn according to the hard-core model with fugacity λ, and let v be a uniform random vertex in G. Let Y = \|I ∩N(v)\| denote the number of neighbors of v in I (not including v itself). Let pk = P(Y = k). Since Y ∈{0, 1, . . . , d}, p0 + p1 + · · · + pd = 1. (10) 838 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 However, not all vectors of probabilities (p0, . . . , pd) are feasible. The art of the method is in ﬁnding additional constraints on the probability distribution. As in the previous proof, since v is uncovered if and only if Y = 0, we have αG(λ) = P(v ∈I) = λ 1 + λP(Y = 0) = λ 1 + λ p0. On the other hand, since G is d-regular, a uniform random neighbor of v is also uni-formly distributed in V (G), so we have αG(λ) = 1 d E[Y] = 1 d (p1 + 2p2 + · · · + dpd). Comparing the previous two relations, we obtain λ 1 + λ p0 = 1 d (p1 + 2p2 + · · · + dpd). (11) Now, let us compare the probability that v has k versus k −1 neighbors in I. In an event where exactly k neighbors of v are occupied, we can remove any of the occu-pied neighbors from I and obtain another independent set where v has exactly k −1 neighbors. There are k ways to remove an element, but we overcount by a factor of at most d −k + 1. Also factoring in the weight multiplier, we obtain the inequality (d −k + 1)λpk−1 ≥kpk, for 2 ≤k ≤d. (12) The constraints (10), (11), and (12) form a linear program with variables p0, . . . , pd. Next, we show that these linear constraints together imply p0 ≤ (1+λ)d 2(1+λ)d−1, which gives the desired bound on αG(λ) = λ 1+λ p0. Equality is attained for the probability distribu-tion (p0, . . . , pd) arising from G = Kd,d. To prove this claim, ﬁrst we show that if (p0, . . . , pd) achieves the maximum of value of p0 while satisfying the constraints (10), (11), and (12), then every inequality in (12) must be an equality. Indeed, if we have (d −k + 1)λpk−1 > kpk for some k, then by increasing p0 by ϵ, decreasing pk−1 by ( dλ 1+λ + k)ϵ, increasing pk by ( dλ 1+λ + k −1)ϵ, and leaving all other pi’s ﬁxed, we can maintain all constraints and increase p0, provided ϵ > 0 is sufﬁciently small. Thus, in the maximizing solution, equality occurs in (12) for all 2 ≤k ≤d. It can be checked that the vector (p0, . . . , pd) arising from G = Kd,d satisﬁes all the equality constraints, and it is the unique solution since we have a linear system of equations with full rank. Remark. Conjecture 2.7 about the number of colorings was recently proved for 3-regular graphs using an extension of the above method. 8. ON THE MINIMUM NUMBER OF INDEPENDENT SETS AND HOMO-MORPHISMS. Independent sets. Having explored the maximum number of independent sets in a regular graph, let us turn to the natural opposite question. Which d-regular graph has the minimum number of independent sets? It turns out that the answer is a disjoint union of cliques. Theorem 8.1 (Cutler and Radcliffe ). For a d-regular graph G, i(G)1/v(G) ≥i(Kd+1)1/(d+1) = (d + 2)1/(d+1). November 2017] EXTREMAL REGULAR GRAPHS 839 In fact, a stronger result holds: A disjoint union of Kd+1’s minimizes the number of independent sets of every ﬁxed size. We write aG for a disjoint union a copies of G. Let it(G) denote the number of independent sets of G of size t. Theorem 8.2 (). Let a and d be positive integers. Let G be a d-regular graph with a(d + 1) vertices. Then it(G) ≥it(aKd+1) for every 0 ≤t ≤a(d + 1). Proof. Let us compare the number of sequences of t vertices that form an independent set in G and aKd+1. In aKd+1, we have a(d + 1) choices for the ﬁrst vertex. Once the ﬁrst vertex has been chosen, there are exactly (a −1)(d + 1) choices for the second vertex. More generally, for 1 ≤j ≤a, once the ﬁrst j −1 vertices have been chosen, there are exactly (a + 1 −j)(d + 1) choices for the jth vertex. On the other hand, in G, after the ﬁrst j −1 vertices have been chosen, the union of these j −1 vertices along with their neighborhoods has cardinality at most ( j − 1)(d + 1), so there are at least (a + 1 −j)(d + 1) choices for the jth vertex, at least as many compared to aKd+1. Proof of Theorem 8.1. Theorem 8.2 implies that i(G)1/v(G) ≥i(Kd+1)1/(d+1) whenever v(G) is divisible by d + 1. When v(G) is not divisible by d + 1, we can apply the same inequality to a disjoint union of (d + 1) copies of G to obtain i(G)1/v(G) = i((d + 1)G)1/((d+1)v(G)) ≥i(Kd+1)1/(d+1). The situation changes signiﬁcantly if we require G to be bipartite. In this case, the problem was solved very recently by Csikv´ ari , who showed that the inﬁmum of i(G)1/v(G) over d-regular bipartite graphs G is obtained by taking a sequence of G with increasing girth, i.e., G is locally tree-like. The limit of i(G)1/v(G) for a sequence of bipartite d-regular graphs G of increasing girth was determined by Sly and Sun using sophisticated (rigorous) methods from statistical physics. Colorings. Here is the inﬁmum of hom(G, Kq)1/v(G) over d-regular graphs G due to Csikv´ ari . Theorem 8.3. For a d-regular graph G and any q ≥2, hom(G, Kq)1/v(G) ≥hom(Kd+1, Kq)1/(d+1). Proof. Assume q ≥d + 1 since otherwise the right-hand side is zero. Let σ be a ran-dom permutation of V (G). For each u ∈V (G), let dσ u denote the number of neighbors of u that appears before u in the permutation σ. By coloring the vertices in the order of σ, there are at least q −dσ u choices for the color of vertex u, so hom(G, Kq) ≥ u∈V (G) (q −dσ u ). Taking the logarithm of both sides, we ﬁnd that 1 v(G) log hom(G, Kq) ≥ 1 v(G) u∈V (G) log(q −dσ u ). (13) For each u ∈V (G), the random variable dσ u is uniformly distributed in {0, 1, . . . , d} since the ordering of u ∪N(u) under σ is uniform. Therefore, the expected value of the right-hand side of (13) is 840 c ⃝THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124 1 d + 1(log q + log(q −1) + · · · + log(q −d)) = 1 d + 1 log hom(Kd+1, Kq), which proves the theorem. What is the inﬁmum of hom(G, Kq)1/v(G) over bipartite d-regular graphs G? The following inequality was proved by Csikv´ ari and Lin . For q ≥d + 1, the constant in the inequality is best possible as it is the limit for any sequence of d-regular graphs with increasing girth . Theorem 8.4 (). For any d-regular bipartite graph G and any q ≥2, hom(G, Kq)1/v(G) ≥q(1 −1/q)d/2. Widom–Rowlinson model. In the previous two cases, for independent sets and col-orings, the minimizing G is Kd+1, and if we restrict to bipartite G, the “minimizing” G is locally tree-like. For the Widom–Rowlinson model, we saw in Theorem 2.8 that the quantity hom(G, )1/v(G) is maximized, over d-regular graphs G, by G = Kd+1. Csikv´ ari recently showed that hom(G, )1/v(G) is minimized, over d-regular graphs G, by a sequence of graphs G with increasing girth, even without the bipartite assumption on G. 9. RELATED RESULTS AND FURTHER QUESTIONS. Independent sets of ﬁxed size. We saw in Theorem 1.1 that in the family of d-regular graphs on n vertices, a disjoint union of Kd,d’s maximizes the number of independent sets. It is conjectured that the latter maximizes the number of independent sets of every ﬁxed size. Let it(G) denote the number of independent sets of size t in G. Recall that kG denotes a disjoint union of k copies of G. See [13, Section 8] for the current best bounds on this problem. Conjecture 9.1 (). If G is a d-regular graph with 2ad vertices, then it(G) ≤ it(aKd,d) for every t. Graphs with given degree proﬁle. Kahn made the following conjecture extend-ing Theorem 1.1 to irregular graphs. We write du for the degree of vertex u ∈V (G). Conjecture 9.2 (). For any graph G, i(G) ≤ uv∈E(G) i(Kdu,dv)1/dudv = uv∈E(G) (2du + ddv −1)1/(dudv). By the bipartite reduction in Section 4, it sufﬁces to prove the conjecture for bipar-tite graphs G. Galvin and Zhao proved Conjecture 9.2 for all G with maximum degree at most 5. The following conjecture, due to Galvin ,4 extends Theorem 2.4 and the bipartite case of Conjecture 9.2. Conjecture 9.3. For any bipartite graph G and loop-graph H, hom(G, H) ≤ uv∈E(G) hom(Kdu,dv, H)1/(dudv). 4A bipartite assumption on G is missing in [18, Conjecture 1.5]. November 2017] EXTREMAL REGULAR GRAPHS 841 Graphs with additional local constraints. We saw in Theorem 1.1 and Theorem 8.1 that the maximum and minimum of i(G)1/v(G) among d-regular graphs G are attained by Kd,d and Kd+1, respectively. What if we impose additional “local” constraints to disallow Kd,d and Kd+1? For example, consider the following. • What is the inﬁmum of i(G)1/v(G) among d-regular triangle-free graphs G? • What is the supremum of i(G)1/v(G) among d-regular graphs G that do not contain any cycles of length 4? These two questions were recently answered by Perarnau and Perkins . Theorem 9.4. (a) Among 3-regular triangle-free graphs G, the quantity i(G)1/v(G) is minimized when G is the Petersen graph. (b) Among 3-regular graphs G without cycles of length 4, the quantity i(G)1/v(G) is maximized when G is the Heawood graph. Petersen graph Heawood graph Theorem 9.4 was proved using the occupancy method discussed in Section 7. The following general problem is very much open. Problem 9.5. Let d ≥3 be an integer and F be a ﬁnite list of graphs. Determine the inﬁmum and supremum of i(G)1/v(G) among d-regular graphs G that do not contain any element of F as an induced subgraph. We pose the following (fairly bold) conjecture that the extrema are always attained by ﬁnite graphs. It would be interesting to know which graphs can arise as extremal graphs in this manner. Conjecture 9.6 (Local constraints imply bounded extrema). Let d ≥3 be an inte-ger and F be a ﬁnite list of graphs. Let Gd(F) denote the set of ﬁnite d-regular graphs that do not contain any element of F as an induced subgraph. Then there exist Gmin, Gmax ∈Gd(F) such that for all G ∈Gd(F), i(Gmin)1/v(Gmin) ≤i(G)1/v(G) ≤i(Gmax)1/v(Gmax). ACKNOWLEDGMENTS. The author is grateful to Joe Gallian for the REU opportunity in 2009 where he began working on this problem (resulting in ). 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Sci, Vol. 63, American Mathematical Society, Provi-dence, RI, 2004. 97–104. 22. D. Galvin, Y. Zhao, The number of independent sets in a graph with small maximum degree, Graphs Combin. 27 (2011) 177–186. 23. J. Kahn, An entropy approach to the hard-core model on bipartite graphs, Combin. Probab. Comput. 10 (2001) 219–237. 24. L. H. Loomis, H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc 55 (1949) 961–962. 25. E. Lubetzky, Y. Zhao, On replica symmetry of large deviations in random graphs, Random Structures Algorithms 47 (2015) 109–146. 26. G. Perarnau, W. Perkins, Counting independent sets in cubic graphs of given girth, preprint (2016), arXiv:1610.08496. 27. W. Perkins, personal communication. 28. L. Sernau, Graph operations and upper bounds on graph homomorphism counts, J. Graph Theory, to appear. 29. A. Sly, N. Sun, Counting in two-spin models on d-regular graphs, Ann. Probab. 42 (2014) 2383–2416. 30. Y. Zhao, The number of independent sets in a regular graph, Combin. Probab. Comput. 19 (2010) 315–320. 31. , The bipartite swapping trick on graph homomorphisms, SIAM J. Discrete Math. 25 (2011) 660–680. YUFEI ZHAO received his B.Sc. and Ph.D. from MIT in 2010 and 2015 respectively, and his M.A.St. from Cambridge University in 2011. He is currently an Assistant Professor of Mathematics at MIT, and was previ-ously the Esm´ ee Fairbairn Junior Research Fellow in Mathematics at New College, Oxford. His mathematical research interests include extremal, probabilistic, and additive combinatorics. Department of Mathematics, MIT, Cambridge, MA 02139, USA yufeiz@mit.edu November 2017] EXTREMAL REGULAR GRAPHS 843
3819	https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/resources/lec19/	MIT OpenCourseWare 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: �� Lecture 19 18.01 Fall 2006 Lecture 19: First Fundamental Theorem of Calculus Fundamental Theorem of Calculus (FTC 1) If f (x) is continuous and F �(x) = f (x), then � b f (x)dx = F (b) − F (a) a Notation: F (x) b = F (x) x=b = F (b) − F (a) ax=a � b b3 3 b3 a3 x x2; x2dx =Example 1. F (x) = F �(x) = x = 3 − 33 , 3a a Example 2. Area under one hump of sin x (See Figure 1.) � π 0 sin x dx = − cos x π = − cos π − (− cos 0) = −(−1) − (−1) = 2 0 1 � Figure 1: Graph of f (x) = sin x for 0 ≤ x ≤ π. � 1 16 = 1 1 6 − 0 = 6 5 dx = x Example 3. x 60 0 1�� Lecture 19 18.01 Fall 2006 Intuitive Interpretation of FTC: dx x(t) is a position; v(t) = x�(t) = is the speed or rate of change of x. dt � b v(t)dt = x(b) − x(a) (FTC 1) a R.H.S. is how far x(t) went from time t = a to time t = b (difference between two odometer readings). L.H.S. represents speedometer readings. n i=1 x(b) − x(a) = v(t) cancel each other, whereas an � ( )Δ approximates the sum of distances traveled over times Δ t t tv i th The approximation above is accurate if ( ) is close to ( ) on the interval. The interpretation it tv v i of ( ) as an odometer reading is no longer valid if changes sign. Imagine a round trip so that tx v Then the positive and negative velocities odometer would measure the total distance not the net distance traveled. Example 4. � 2π 0 sin x dx = − cos x 2π = − cos 2 π − (− cos 0) = 0. 0 The integral represents the sum of areas under the curve, above the x-axis minus the areas below the x-axis. (See Figure 2.) + - 12� Figure 2: Graph of f (x) = sin x for 0 ≤ x ≤ 2π. 2Lecture 19 18.01 Fall 2006 Integrals have an important additive property (See Figure 3.) � b � c � c f (x)dx + f (x)dx = f (x)dx aba a b c Figure 3: Illustration of the additive property of integrals New Definition : � a � b f (x)dx = − f (x)dx ba This definition is used so that the fundamental theorem is valid no matter if a < b or b < a . It also makes it so that the additive property works for a, b, c in any order, not just the one pictured in Figure 3. 3� � � � � Lecture 19 18.01 Fall 2006 Estimation: � b � b If f (x) ≤ g(x), then f (x)dx ≤ g(x)dx (only if a < b ) aa x Example 5. Estimation of e x Since 1 ≤ e for x ≥ 0, � 1 � 1 1dx ≤ exdx 00 � 1 � exdx = ex � �1 = e1 − e0 = e − 1 00 Thus 1 ≤ e − 1, or e ≥ 2. x Example 6. We showed earlier that 1 + x ≤ e . It follows that � 1 � 1 (1 + x)dx ≤ exdx = e − 1 00 � 1 � 2 �� 1 x � 3 (1 + x)dx = x + � � = 2 20 0 3 5 Hence, 2 ≤ e − 1,or, e ≥ 2 . Change of Variable: If f (x) = g(u(x)), then we write du = u�(x)dx and g(u)du = g(u(x)) u�(x)dx = f (x)u�(x)dx (indefinite integrals) For definite integrals: x2u2 f (x)u�(x)dx = g(u)du where u1 = u(x1), u 2 = u(x2) x1u1 � 2 � �4 Example 7. x3 + 2 x2dx 1 Let u = x3 + 2 . Then du = 3 x2dx = x2dx = du ;⇒ 3 x1 = 1 , x 2 = 2 = u1 = 1 3 + 2 = 3 , u 2 = 2 3 + 2 = 10 , and ⇒ � 2 �10 � �4 � 10 4 du u5 � � 10 5 − 35 x3 + 2 x2dx = u = = 13 3 15 �3 15 4
3820	https://www.khanacademy.org/math/geometry-fl-best/xba45aeb1cf923a80:hs-geo-transformation-properties-and-proofs/xba45aeb1cf923a80:hs-geo-transformations-definitions/a/gtp--praxis-math--article--counterexamples--lesson	Counterexamples \| Lesson (article) \| Khan Academy Skip to main content If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. 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Course: Geometry (FL B.E.S.T.)>Unit 3 Lesson 3: Properties & definitions of transformations Sequences of transformations Sequences of transformations Defining transformations Precisely defining rotations Counterexamples \| Lesson Defining transformations Identifying type of transformation Math> Geometry (FL B.E.S.T.)> Transformation properties and proofs> Properties & definitions of transformations © 2025 Khan Academy Terms of usePrivacy PolicyCookie NoticeAccessibility Statement Counterexamples \| Lesson Google Classroom Microsoft Teams What is a counterexample? A mathematical statement is a sentence that is either true or false. Examples True statement: All even numbers are divisible by 2‍. False statement: All odd numbers are multiples of 3‍ or 5‍. A mathematical statement has two parts: a condition and a conclusion. Example If an odd number and an even number are added, the sum is odd. Condition: is the sum of an odd number and an even number Conclusion: is odd Showing that a mathematical statement is true requires a formal proof. However, showing that a mathematical statement is false only requires finding one example where the statement isn't true. Such an example is called a counterexample because it's an example that counters, or goes against, the statement's conclusion. What skills are tested? • Identifying a counterexample to a mathematical statement How can we identify counterexamples? When identifying a counterexample, follow these steps: Identify the condition and conclusion of the statement. Eliminate choices that don't satisfy the statement's condition. For the remaining choices, counterexamples are those where the statement's conclusion isn't true. Example Consider the statement, "All even numbers are multiples of 4‍." What are two counterexamples that show that the statement is false? We need to find two numbers that satisfy the condition of the statement, but not its conclusion. Condition: is an even number Conclusion: is a multiple of 4‍ Let's consider some example numbers that satisfy the condition: 2‍, 4‍, 6‍, and 8‍. Next, let's check to see if any aren't multiples of 4‍: 2÷4=0.5‍ 4÷4=1‍ 6÷4=1.5‍ 8÷4=2‍ 2‍ and 6‍ do not divide evenly by 4‍, so these are numbers for which the statement's conclusion isn't true. 2‍ and 6‍ are counterexamples that show the statement is false. Your turn! TRY: IDENTIFYING A COUNTEREXAMPLE The square of an integer is always an even number. Which of the following numbers provides a counterexample showing that the statement above is false? Choose 1 answer: Choose 1 answer: (Choice A) −4‍ A −4‍ (Choice B) −2‍ B −2‍ (Choice C) 1‍ C 1‍ (Choice D) 2‍ D 2‍ (Choice E) 4‍ E 4‍ Check Explain TRY: IDENTIFYING A COUNTEREXAMPLE Which of the following expressions can be used to show that the sum of two numbers is not always greater than both numbers? Choose all answers that apply: Choose all answers that apply: (Choice A) 3 2+1 2‍ - [x] A 3 2+1 2‍ (Choice B) 0+4‍ - [x] B 0+4‍ (Choice C) −2+1‍ - [x] C −2+1‍ Check Explain TRY: IDENTIFYING COUNTEREXAMPLES Bart claims that all numbers that are multiples of 5‍ are also multiples of 10‍. Which of the following numbers can be used to show that Bart's statement is not true? Choose 1 answer: Choose 1 answer: (Choice A) 40‍ A 40‍ (Choice B) 36‍ B 36‍ (Choice C) 21‍ C 21‍ (Choice D) 15‍ D 15‍ (Choice E) 10‍ E 10‍ Check Explain TRY: IDENTIFYING A COUNTEREXAMPLE A student claims that when any two even numbers are multiplied, all of the digits in the product are even. Which of the following shows that the student is wrong? Choose 1 answer: Choose 1 answer: (Choice A) 8×8‍ A 8×8‍ (Choice B) 10×8‍ B 10×8‍ (Choice C) 11×8‍ C 11×8‍ (Choice D) 12×8‍ D 12×8‍ (Choice E) 13×8‍ E 13×8‍ Check Explain Things to remember A counterexample to a mathematical statement is an example that satisfies the statement's condition(s) but does not lead to the statement's conclusion. Identifying counterexamples is a way to show that a mathematical statement is false. When identifying a counterexample, Identify the condition and conclusion of the statement. Eliminate choices that don't satisfy the statement's condition. For the remaining choices, counterexamples are those where the statement's conclusion isn't true. Skip to end of discussions Questions Tips & Thanks Want to join the conversation? Log in Sort by: Top Voted Ku, Evan 4 years ago Posted 4 years ago. Direct link to Ku, Evan's post “What grade math is this, ...” more What grade math is this, because i'm going to 5 grade and I think this is for 9th graders. Answer Button navigates to signup page •6 comments Comment on Ku, Evan's post “What grade math is this, ...” (4 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer IMDABIGGESTBIRD a month ago Posted a month ago. Direct link to IMDABIGGESTBIRD's post “You are cooked.” more You are cooked. Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... alarcon 3 years ago Posted 3 years ago. Direct link to alarcon's post “what is a counter exmple ...” more what is a counter exmple using 100 chart and adding one more or one less Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Camden Schuetz 21 days ago Posted 21 days ago. Direct link to Camden Schuetz's post “NOW they teach us about c...” more NOW they teach us about counterexamples. its not like they had an entire exercise about them (Defining Transformations) this article should be much earlier in the course Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer PAULINE TRAN a month ago Posted a month ago. Direct link to PAULINE TRAN's post “Can even numbers and odd ...” more Can even numbers and odd numbers can be different ? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer IMDABIGGESTBIRD a month ago Posted a month ago. Direct link to IMDABIGGESTBIRD's post “I do not get what you sai...” more I do not get what you said 1 comment Comment on IMDABIGGESTBIRD's post “I do not get what you sai...” (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more anthonia u 5 days ago Posted 5 days ago. Direct link to anthonia u's post “on the last problem it di...” more on the last problem it did not mention anything about even intergers do I just assume that problems like that are always talking about even intergers Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer kereremena114 a year ago Posted a year ago. Direct link to kereremena114's post “If multiplication is to m...” more If multiplication is to make more of, why does it not apply to money when we square 1 pound? But applies to other digits. For example 100 pennies = 1 pound but when you square both sides of the equation, it becomes counterexample of the statemant of 100 pennies = 1 pound because now, 10,000 pennies = 1 pounds Answer Button navigates to signup page •Comment Button navigates to signup page (0 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Warrior Oak a month ago Posted a month ago. Direct link to Warrior Oak's post “it equals 1 poundS” more it equals 1 poundS Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Musaed Alawi Musaed 2 years ago Posted 2 years ago. Direct link to Musaed Alawi Musaed's post “1+1=?” more 1+1=? Answer Button navigates to signup page •Comment Button navigates to signup page (0 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Jurnee Mitchell 3 months ago Posted 3 months ago. Direct link to Jurnee Mitchell's post “maybe 2 idk tho” more maybe 2 idk tho Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Up next: exercise Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. 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3821	https://webstersdictionary1828.com/Dictionary/duck	Websters 1828 - Webster's Dictionary 1828 - Duck Menu Home Menu Home Preface History Quotations Constitution The Bible Literature Grammar Education Science Mathematics Medical American Dictionary of the English Language Dictionary Search Home Preface History Quotations Noah Webster Topics Bible Constitution Literature Grammar Education Science Mathematics Medical Duck DUCK, noun [G, Latin , to weave.] A species of coarse cloth or canvas, used for sails, sacking of beds, etc. DUCK, noun [from the verb, to duck ] 1. A water fowl, so called from its plunging. There are many species or varieties of the duck some wild, others tame. 2. An inclination of the head, resembling the motion of a duck in water. 3. A stone thrown obliquely on the water so as to rebound; as in duck and drake. DUCK, noun A word of endearment or fondness. DUCK, verb transitive [G.] 1. To dip or plunge in water and suddenly withdraw; as, to duck a seamen. It differs from dive, which signifies to plunge ones self, without immediately emerging. 2. To plunge the head in water and immediately withdraw it; as, duck the boy. 3. To bow, stoop or nod. DUCK, verb intransitive 1. To plunge into water and immediately withdraw; to dip; to plunge the head in water or other liquid. In Tiber ducking thrice by break of day. 2. To drop the head suddenly; to bow; to cringe. DUCK with French nods. Websters Dictionary 1828 SITEMAP Home Preface History Quotations INFORMATION About Us Statement of Accessibility Terms of Use Privacy Policy Copyright Notice OUR WEBSITES The Kings Bible King James Bible 1611 King James Bible Dictionary Websters Dictionary 1828 Textus Receptus Bibles Treasury of Scripture Knowledge ©Copyright 2025 MasonSoft Technology Ltd v4 (7.10.2024) X Cookies & Privacy We use cookies to ensure you get the best experience on our website. By using our site you acknowledge that you have read and understand our Privacy Policy. Accept Reject
3822	https://www.ck12.org/flexi/cbse-math/comparing-and-ordering-fractions/how-do-you-use-cross-multiplication-when-comparing-fractions/	Flexi answers - How do you use cross multiplication when comparing fractions? \| CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects CBSE Math Comparing and Ordering Fractions Question How do you use cross multiplication when comparing fractions? Flexi Says: In the cross multiplication method, we cross multiply the numerator of one fraction with the other fraction's denominator. For example, compare 2 3 and 5 8. By cross-multiplication, we have 2 3↖↗5 8 2×8=16 5×3=15 Because 16 is greater than 15, we know that 2 3 is greater than 5 8. Remember that to find out which of the two fractions is larger, cross-multiply and place the two products, in order, under the two fractions. The larger product is always under the larger fraction. Note: When cross-multiplying to compare fractions and find out which is greater, start with the first fraction's numerator. To learn more click here! Analogy / Example Try Asking: What Fraction of 3/5 needs to be added to itself to become 2 1/4?How do you arrange fractions in ascending order with different denominators?How do you order fractions and decimals? How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
3823	https://www.math-only-math.com/general-form-into-intercept-form.html	Math Only Math Learn math step-by-step. Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips. General Form into Intercept Form We will learn the transformation of general form into intercept form. To reduce the general equation ax + by + c = 0 into intercept form ((\frac{x}{a}) + (\frac{y}{b}) = 1): We have the general equation ax + by + c = 0. If a ≠ 0, b ≠ 0, c ≠ 0 then from the given equation we get, ax + by = - c (Subtracting c from both sides) ⇒ (\frac{ax}{-c}) + (\frac{by}{-c}) = (\frac{-c}{-c}), (Dividing both sides by -c) ⇒ (\frac{ax}{-c}) + (\frac{by}{-c}) = 1 ⇒ (\frac{x}{-\frac{c}{a}}) + (\frac{y}{-\frac{c}{b}}) = 1, which is the required intercept form ((\frac{x}{a}) + (\frac{y}{b}) = 1) of the general form of line ax + by + c = 0. Thus, for the straight line ax + by + c = 0, Intercept on x-axis = -((\frac{c}{a})) = - (\frac{\textrm{Constant term}}{\textrm{Coefficient of x}}) Intercept on y-axis = -((\frac{c}{b})) = - (\frac{\textrm{Constant term}}{\textrm{Coefficient of y}}) Note: From the above discussion we conclude that the intercepts made by a straight line with the co-ordinate axes can be determined by transforming its equation to intercept form. To determine the intercepts on the co-ordinate axes we can also use the following method: To find the intercept on x-axis (i.e., x-intercept), put y = 0 in the given equation of the straight line line and find the value of x. Similarly To find the intercept on y-axis (i.e., y-intercept), put x = 0 in the given equation of the straight line and find the value of y. Solved examples on transformation of general equation into intercept form: 1. Transform the equation of the straight line 3x + 2y - 18 = 0 to intercept form and find its x-intercept and y-intercept. Solution: The given equation of the straight line 3x + 2y - 18 = 0 First add 18 on both sides. ⇒ 3x + 2y =18 Now divide both sides by 18 ⇒ (\frac{3x}{18}) + (\frac{2y}{18}) = (\frac{18}{18}) ⇒ (\frac{x}{6}) + (\frac{y}{9}) = 1, which is the required intercept form of the given straight line 3x + 2y - 18 = 0. Therefore, x-intercept = 6 and y-intercept = 9. 2. Reduce the equation -5x + 4y = 8 into intercept form and find its intercepts. Solution: The given equation of the straight line -7x + 4y = -8. First divide both sides by -8 ⇒ (\frac{-7x}{-8}) + (\frac{4y}{-8}) = (\frac{-8x}{-8}) ⇒ (\frac{7x}{8}) + (\frac{y}{-2}) = 1 ⇒ (\frac{x}{\frac{8}{7}}) + (\frac{y}{-2}) = 1, which is the required intercept form of the given straight line -5x + 4y = 8. Therefore, x-intercept = (\frac{8}{7}) and y-intercept = -2. ●The Straight Line Straight Line Slope of a Straight Line Slope of a Line through Two Given Points Collinearity of Three Points Equation of a Line Parallel to x-axis Equation of a Line Parallel to y-axis Slope-intercept Form Point-slope Form Straight line in Two-point Form Straight Line in Intercept Form Straight Line in Normal Form General Form into Slope-intercept Form General Form into Intercept Form General Form into Normal Form Point of Intersection of Two Lines Concurrency of Three Lines Angle between Two Straight Lines Condition of Parallelism of Lines Equation of a Line Parallel to a Line Condition of Perpendicularity of Two Lines Equation of a Line Perpendicular to a Line Identical Straight Lines Position of a Point Relative to a Line Distance of a Point from a Straight Line Equations of the Bisectors of the Angles between Two Straight Lines Bisector of the Angle which Contains the Origin Straight Line Formulae Problems on Straight Lines Word Problems on Straight Lines Problems on Slope and Intercept 11 and 12 Grade Math From General Form into Intercept Form to HOME PAGE Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. New! Comments Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. Share this page: What’s this? FacebookXPinterestWhatsAppReddit FacebookXPinterestWhatsAppReddit Home Math Blog Preschool Activities Kindergarten Math 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math 9th Grade Math 10th Grade Math 11 & 12 Grade Math Algebra 1 Concepts of Sets Matrix Probability Statistics Logarithms Boolean Algebra Math Coloring Pages Multiplication Table Cool Maths Games Math Flash Cards Online Math Quiz Math Puzzles Binary System Math Dictionary Conversion Chart Homework Sheets Math Problem Ans Free Math Answers Printable Math Sheet Funny Math Answers Employment Test Math Patterns Link Partners About Us Contact Us Site Map Privacy Policy [?]Subscribe To This Site © and ™ math-only-math.com. All Rights Reserved. 2010 - 2025. This site uses cookies, some of which are required for its operation. Privacy policy.
3824	https://www.quora.com/What-is-meant-by-holoenzyme-and-give-few-examples	What is meant by holoenzyme and give few examples? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Biochemistry Molecular Biology Apoenzyme Enzyme Substrates Proteins (biochemistry) Biomedical Sciences Enzymes Molecular and Cellular Bi... Coenzymes 5 What is meant by holoenzyme and give few examples? All related (34) Sort Recommended Shivani Sri MSc biochemistry from Rashtrasant Tukadoji Maharaj Nagpur University (Graduated 2017) ·7y Holoenzyme is a complete and catalytically active form of the enzymes. Most of the enzymes require some additional factors for its proper functioning. These factors are called cofactors. Though mostly inorganic ions are referred to as cofactors but coenzymes, prosthetic groups also come under this heading. So any enzyme which require cofactors for it's activity but is not associated with it is termed as apoenzyme. When the enzyme is bound to the cofactor and is completely active, it is called as holoenzyme. A very good example of holoenzyme is RNA Polymerase which is missing specific cofactors (a Continue Reading Holoenzyme is a complete and catalytically active form of the enzymes. Most of the enzymes require some additional factors for its proper functioning. These factors are called cofactors. Though mostly inorganic ions are referred to as cofactors but coenzymes, prosthetic groups also come under this heading. So any enzyme which require cofactors for it's activity but is not associated with it is termed as apoenzyme. When the enzyme is bound to the cofactor and is completely active, it is called as holoenzyme. A very good example of holoenzyme is RNA Polymerase which is missing specific cofactors (apoenzyme form) under inactive condition. Binding of prosthetic groups (heme) and metal ions(magnesium) complete the enzyme and makes it catalytically active (holoenzyme form). Dissociable protein i.e., sigma factor also plays an important role in enzyme activation. Some other examples of holoenzymes are DNA Polymerase, carbonic anhydrase, pyruvate kinase, pyruvate carboxylase etc. Upvote · 9 9 Sponsored by Google Cloud Keahlian Gen AI yang membuka peluang dan karier. Raih sertifikasi, dan berkembang bersama program yang dirancang untuk para talenta siap masa depan. Learn More 99 10 Arnab Karmakar Student (2001–present) ·7y Holoenzyme is a biochemically active compound formed by the combination of an enzyme with a coenzyme. Common examples of holoenzymes are: RNA polymerase, DNA polymerase, Carbonic anhydrase, Haemoglobin, Cytochrome oxidase, Pyruvate carboxylase . Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 2 Barry Gehm Former Asst Prof. Of Chemistry/Biochemistry at Lyon College (2003–2024) · Author has 13.6K answers and 16.6M answer views ·6y Originally Answered: What is a "holoenzyme"? · Some enzymes require a non-peptide component to carry out their catalytic function. These “prosthetic groups” as they are called, are sometimes metal ions, sometimes small organic molecules (FAD, for instance), sometimes both (heme in cytochrome P450s, for instance). The protein-only part of the molecule, without the prosthetic group, is called an apoenzyme. The entire, functional enzyme including the prosthetic group is a holoenzyme. Upvote · 9 2 Related questions More answers below What is a "holoenzyme"? What does holoenzyme consist of? What is apoenzyme, coenzyme and holoenzyme? How do holoenzyme and apoenzyme differ? What is a holoenzyme? How does a holoenzyme differ with a simple enzyme and an apoenzyme? Assistant Bot · 1y A holoenzyme is a complex of an enzyme and one or more additional components that are necessary for its biological activity. These additional components can include cofactors (which may be metal ions or organic molecules), coenzymes (which are often derived from vitamins), or prosthetic groups (which are tightly bound to the enzyme). The holoenzyme is the active form of the enzyme, as it contains all the necessary parts to catalyze a biochemical reaction. Examples of Holoenzymes DNA Polymerase: Components: DNA polymerase enzyme along with magnesium ions (cofactors) and various other protein Continue Reading A holoenzyme is a complex of an enzyme and one or more additional components that are necessary for its biological activity. These additional components can include cofactors (which may be metal ions or organic molecules), coenzymes (which are often derived from vitamins), or prosthetic groups (which are tightly bound to the enzyme). The holoenzyme is the active form of the enzyme, as it contains all the necessary parts to catalyze a biochemical reaction. Examples of Holoenzymes DNA Polymerase: Components: DNA polymerase enzyme along with magnesium ions (cofactors) and various other proteins. Function: Catalyzes the synthesis of DNA molecules from nucleotides during DNA replication. RNA Polymerase: Components: Consists of the core enzyme (which includes several subunits) and a sigma factor (a type of protein). Function: Synthesizes RNA from a DNA template during transcription. Pyruvate Dehydrogenase: Components: A multi-enzyme complex that includes several enzymes and cofactors such as thiamine (vitamin B1), lipoic acid, and FAD. Function: Converts pyruvate into acetyl-CoA, linking glycolysis to the citric acid cycle. Chymotrypsin: Components: The enzyme itself, along with a serine residue that acts as a catalytic site and requires calcium ions for stability. Function: A digestive enzyme that breaks down proteins in the small intestine. Aconitase: Components: Contains iron-sulfur clusters as prosthetic groups. Function: Catalyzes the conversion of citrate to isocitrate in the citric acid cycle. These examples illustrate how holoenzymes are essential for various biochemical processes, relying on their complete structure to function effectively. Upvote · Shivani Sri MSc biochemistry from Rashtrasant Tukadoji Maharaj Nagpur University (Graduated 2017) ·7y Related What is apoenzyme, coenzyme and holoenzyme? Enzymes work efficiently in association with various factors which enhances it's activity. These factors may be; Cofactors which are small non-protein inorganic molecule that carries out chemical reactions that cannot be performed by the standard 20 amino acids. Examples of cofactors include metal ions like iron and zinc. Coenzymes which are organic molecules that are nonproteins and mostly derivatives of vitamins soluble in water by phosphorylation. Example of coenzyme include thiamine pyrophosphate (TPP), flavin adenine dinucleotide (FAD), biotin Apoenzyme is an inactive form of enzyme lacking Continue Reading Enzymes work efficiently in association with various factors which enhances it's activity. These factors may be; Cofactors which are small non-protein inorganic molecule that carries out chemical reactions that cannot be performed by the standard 20 amino acids. Examples of cofactors include metal ions like iron and zinc. Coenzymes which are organic molecules that are nonproteins and mostly derivatives of vitamins soluble in water by phosphorylation. Example of coenzyme include thiamine pyrophosphate (TPP), flavin adenine dinucleotide (FAD), biotin Apoenzyme is an inactive form of enzyme lacking the association of coenzyme and/or cofactors. Activation of the enzyme occurs upon binding of an organic or inorganic cofactor. Holoenzyme is a complete and catalytically active form of enzyme. An apoenzyme together with its cofactor is holoenzyme. Most cofactors are not covalently bound but instead are tightly bound. However, organic prosthetic groups such as an iron ion or a vitamin can be covalently bound. Examples of holoenzymes include DNA polymerase and RNA polymerase which contain multiple protein subunits The figure shows the various forms of enzymes. Upvote · 99 77 9 6 9 2 Sponsored by JetBrains Become More Productive in Jakarta EE. Try IntelliJ IDEA, a JetBrains IDE, and enjoy productive Java Enterprise development! Download 999 649 Ludeman Eng Former Chair, Department of Basic Science at Virginia Tech Carilion School of Medicine and Research Institute (2011–2015) · Author has 1.2K answers and 765.4K answer views ·7y Related What is the difference between a coenzyme and a holoenzyme? Enzymes are (almost always) proteins that act as catalysts for (bio)chemical reactions. So they speed up the rate of these reactions, typically by binding to and bringing reactants into close proximity, or participating in the reaction sequence. As catalysts, they are considered to be unchanged (or restored to original condition) at the end of the reaction, and thus can participate in many repetitions of the reaction as long as reactants are in sufficient supply. Some enzyme reactions involve other chemical nonprotein substances that participate in the reaction but cannot by themselves catalyze Continue Reading Enzymes are (almost always) proteins that act as catalysts for (bio)chemical reactions. So they speed up the rate of these reactions, typically by binding to and bringing reactants into close proximity, or participating in the reaction sequence. As catalysts, they are considered to be unchanged (or restored to original condition) at the end of the reaction, and thus can participate in many repetitions of the reaction as long as reactants are in sufficient supply. Some enzyme reactions involve other chemical nonprotein substances that participate in the reaction but cannot by themselves catalyze the reaction. If these are organic (contain carbon) they are called coenzymes and are often vitamins or vitamin derivatives. Another important type of coenzyme consists of nucleotides or their derivatives, such as ATP, NAD, FAD, etc. If non-organic, they are called cofactors and are typically metal ions such as copper, zinc, cobolt, etc. An enzyme complete with its coenzyme or cofactor is termed a holoenzyme. Thus a coenzyme is the nonprotein part of a holoenzyme. Upvote · 9 2 Related questions More answers below What are apoenzymes? Can we say pepsinogen an apoenzyme and pepsin a holoenzyme? Explain it please. What is the difference between a coenzyme and a holoenzyme? What are examples of an enzyme that requires a cofactor and an enzyme that requires a coenzyme? What are metalloenzymes? Khajawaheed Subhani Former Professor of Chemistry at Academic Consulting · Author has 2K answers and 1.8M answer views ·5y Related What is the difference between a coenzyme and a holoenzyme? Coenzymes are organic molecules that are nonproteins and mostly derivatives of vitamins soluble in water by phosphorylation; they bind apoenzyme protein molecule to produce active holoenzyme. A holoenzyme is complete and catalytically active. Most cofactors are not covalently bound but instead are tightly bound. Ref : Google Continue Reading Coenzymes are organic molecules that are nonproteins and mostly derivatives of vitamins soluble in water by phosphorylation; they bind apoenzyme protein molecule to produce active holoenzyme. A holoenzyme is complete and catalytically active. Most cofactors are not covalently bound but instead are tightly bound. Ref : Google Upvote · Sponsored by BestBrokerz.com Which are the largest trading platforms in the world? See the top five largest brokers in the world by daily transaction volume. Learn More 1.4K 1.4K David Thaler Lives in Basel, Switzerland · Author has 703 answers and 562.3K answer views ·7y Related What is the difference between coenzyme and haloenzyme? I get it, there’s a typo! It is not “haloenzyme’ but HOLOENZYME Now you can look it up on google! Upvote · 9 1 Ritwik Sunny Former Customer Support Executive at Ashok Leyland · Author has 39.8K answers and 9.7M answer views ·10mo Related What is the difference between a coenzyme and a holoenzyme? coenzyme is a non-protein molecule that assists an enzyme, while a holoenzyme is a fully functional enzyme that is made up of an enzyme and a coenzyme Upvote · Balasubramanian Kailasam Chief Inspector at Southern Railway Zone, India (SR) (1977–present) · Author has 5K answers and 16.4M answer views ·5y Related What is the difference between a coenzyme and a holoenzyme? Cofactors, mostly metal ions or coenzyme, are inorganic and organic chemicals that function in reactions of enzymes. Holoenzyme- An apoenzyme together with its cofactor. A holoenzyme is complete and catalytically active. Most cofactors are not covalently bound but instead are tightly bound. Upvote · Steve Geo Ph.D. in Chemistry&Organic Chemistry, University of Iowa (Graduated 1964) · Author has 2.9K answers and 612.6K answer views ·2y Related What is a coenzyme? What are some examples of a coenzyme that has an oxidized and reduced form? I think you are confusing coenzymes and cofactors. An example of a coenzyme is coenzyme A (CoA). The structure is complex. You can look it up. The business end of the molecule is a sulfhydryl group (-SH). It forms thiolesters with, for example, acetic acid. The effect is to muzzle the acidic hydrogen and activate the hydrogen on the methyl group. CH3COOH + HS-CoA ===> CH3CO-SCoA CH3CO-SCoA + B ===> BH+ + (-)CH2CO-SCoA An example of a cofactor is nicotinamide adenine dinucleotide (NADH). If an enzyme has to reduce a substrate, then the cofactor gets oxidized and hydrogenates the substrate. NADH ===> Continue Reading I think you are confusing coenzymes and cofactors. An example of a coenzyme is coenzyme A (CoA). The structure is complex. You can look it up. The business end of the molecule is a sulfhydryl group (-SH). It forms thiolesters with, for example, acetic acid. The effect is to muzzle the acidic hydrogen and activate the hydrogen on the methyl group. CH3COOH + HS-CoA ===> CH3CO-SCoA CH3CO-SCoA + B ===> BH+ + (-)CH2CO-SCoA An example of a cofactor is nicotinamide adenine dinucleotide (NADH). If an enzyme has to reduce a substrate, then the cofactor gets oxidized and hydrogenates the substrate. NADH ===> NAD+ + H- In enzymology, coenzymes are like ignition keys; cofactors are like batteries. Upvote · 9 1 Sam Daher Works at The William Paterson University of New Jersey ·8y Related How do coenzymes work? Coenzymes are essential organic compounds that attach to enzymes to help them catalyze reactions. Coenzymes play a role in the functions of cells.Coenzymes, in turn, support the functions of enzymes. They loosely bind to enzymes to help them complete their activities. Coenzymes are nonprotein, organic molecules that facilitate the catalysis, or reaction, of its enzyme. Coenzymes work by binding to the active side of the enzymes, the side that works in the reaction. Since enzymes and coenzymes are nonmetal organic molecules, they bind together by forming covalent bonds. The coenzymes share electr Continue Reading Coenzymes are essential organic compounds that attach to enzymes to help them catalyze reactions. Coenzymes play a role in the functions of cells.Coenzymes, in turn, support the functions of enzymes. They loosely bind to enzymes to help them complete their activities. Coenzymes are nonprotein, organic molecules that facilitate the catalysis, or reaction, of its enzyme. Coenzymes work by binding to the active side of the enzymes, the side that works in the reaction. Since enzymes and coenzymes are nonmetal organic molecules, they bind together by forming covalent bonds. The coenzymes share electrons with the enzymes, rather than lose or gain electrons. When they form this bond, they only help the reaction to occur by carrying and transferring electrons through the reaction. Coenzymes do not become integral parts of the enzymatic reaction. Instead, the covalent bonds are broken at the end of the reaction, and the coenzyme returns back to free circulation within the cell until it is used again. Upvote · 9 4 Ss Kk MS in Biology (college major), University of North Carolina at Chapel Hill (Graduated 1997) ·7y Related Does coenzyme decide the specificity of an enzyme or not? Simple answer,no. Coenzymes enable inherent protein functions. They may change the 3D shape of the protein, or allow a change once all required ligands or cofactors or coenzymes are present. But a coenzyme does not change the amino acid sequence or composition. Specificity usually refers to the particular ligands(s) to which a protein binds. If you want to use it to also refer to the coenzymes with which it interacts, and you want "coenzymes" to refer to concentration of the enzyme, and you want specificity to refer to activity, then maybe the answer is yes. Upvote · 999 139 Related questions What is a "holoenzyme"? What does holoenzyme consist of? What is apoenzyme, coenzyme and holoenzyme? How do holoenzyme and apoenzyme differ? What is a holoenzyme? How does a holoenzyme differ with a simple enzyme and an apoenzyme? What are apoenzymes? Can we say pepsinogen an apoenzyme and pepsin a holoenzyme? Explain it please. What is the difference between a coenzyme and a holoenzyme? What are examples of an enzyme that requires a cofactor and an enzyme that requires a coenzyme? What are metalloenzymes? What is the difference between coenzyme and haloenzyme? What is the difference between a monomeric enzyme and a holoenzyme? What does a lipid profile test represent? What does an apoenzyme require to become a holoenzyme? What is a holoenzyme in a transcription? Related questions What is a "holoenzyme"? What does holoenzyme consist of? What is apoenzyme, coenzyme and holoenzyme? How do holoenzyme and apoenzyme differ? What is a holoenzyme? How does a holoenzyme differ with a simple enzyme and an apoenzyme? What are apoenzymes? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
3825	https://www.youtube.com/shorts/Wv8raojxx1A	3D Myelination in PNS by Schwann cells Animation - Visual Lecture #shorts #biology - YouTube Search Watch later Share Copy link Info Shopping Tap to unmute 2x If playback doesn't begin shortly, try restarting your device. • Share - [x] Include playlist An error occurred while retrieving sharing information. Please try again later. Back Search
3826	https://www.scribd.com/presentation/633081288/Properties-of-Proportion	Properties of Proportion \| PDF Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 100%(1)100% found this document useful (1 vote) 950 views 5 pages Properties of Proportion The document discusses properties of proportions including: 1) The cross-multiplication property which states that if a/b = c/d, then ad = bc. 2) The alternation property which states that … Full description Uploaded by Marilyn Castro AI-enhanced description Go to previous items Go to next items Download Save Save Properties of Proportion For Later Share 100%100% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Properties of Proportion For Later You are on page 1/ 5 Search Fullscreen Fundamental Rule of Proportion If provided that x adDownload to read ad-free Properties of Proportion • Cross-Multiplication Property If , then x • Alternatio n Property If , thenx • Inverse property If , then x adDownload to read ad-free Properties of Proportion • Addition Property If , thenx • Subtraction Property If , thenx • Sum Property If , then adDownload to read ad-free Rewrite the given proportion according to the property indicated in the table and find out if the ratios in t he rewritten proportions are still equal. Property of Proportion Or ig in al Pr op or ti on Us e th e cr os s-mu lt ip li ca ti on property to verify the ratios are equal. Simpli fy if necessary. Alternation Property Inverse Property Addition Property Subtraction Property Sum Property  = 3 4 4  = 3  3  = 4  3  = 4  → 4  = 3   + 3 3 =  + 4 4  − 3 3 =  − 4 4  3 +  4 =  +  7 For 7y = 3y + 3a 7y = 3(y + a) For 7a = 4(y + a)7a = 4y + 4a 3a = 4y 4y = 3a adDownload to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Similar Triangles PPT and Examples No ratings yet Similar Triangles PPT and Examples 38 pages Fundamentals of Proportion No ratings yet Fundamentals of Proportion 8 pages Precise Properties of Proportion No ratings yet Precise Properties of Proportion 16 pages Lesson Plan For Math Grade 9 No ratings yet Lesson Plan For Math Grade 9 4 pages Grade 9 Math Lesson on Proportion No ratings yet Grade 9 Math Lesson on Proportion 3 pages Q3 - Zero Negative Integral Exponents No ratings yet Q3 - Zero Negative Integral Exponents 13 pages Geometry: Proportions & Similarity 100% (1) Geometry: Proportions & Similarity 38 pages Solving Problems Involving Special Parallelogram: Quarter 3 No ratings yet Solving Problems Involving Special Parallelogram: Quarter 3 10 pages Grade 9 Math: Joint & Combined Variations No ratings yet Grade 9 Math: Joint & Combined Variations 30 pages DLL Right Triangle Similarity Theorem 0% (1) DLL Right Triangle Similarity Theorem 6 pages Trigonometric Ratios of Special Angles: Group 1 No ratings yet Trigonometric Ratios of Special Angles: Group 1 14 pages HW SSS SAS and AA Similarity No ratings yet HW SSS SAS and AA Similarity 5 pages Perfect Match Similarity: Grade 9 100% (1) Perfect Match Similarity: Grade 9 9 pages Summative Similarities 2018 No ratings yet Summative Similarities 2018 3 pages DLL in M9 - (M9GE-IIIC-1) - D2 Special Parallelogram (Rectangle) 100% (1) DLL in M9 - (M9GE-IIIC-1) - D2 Special Parallelogram (Rectangle) 2 pages Grade 9 - Characteristics of A Parallelogram 100% (1) Grade 9 - Characteristics of A Parallelogram 2 pages Median Theorem No ratings yet Median Theorem 35 pages 45°-45°-90° Triangle Lesson Plan 100% (1) 45°-45°-90° Triangle Lesson Plan 7 pages Unit 7 - Special Right Triangles PDF No ratings yet Unit 7 - Special Right Triangles PDF 16 pages Grade 9 Trigonometry Lesson Plan No ratings yet Grade 9 Trigonometry Lesson Plan 4 pages Trigonometric Ratios of Special Angles No ratings yet Trigonometric Ratios of Special Angles 33 pages 4th - Similarity. 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Home Questions AI Assist Labs Tags Challenges Chat Articles Users Jobs Companies Collectives Communities for your favorite technologies. Explore all Collectives Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Collectives™ on Stack Overflow Find centralized, trusted content and collaborate around the technologies you use most. Learn more about Collectives Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more How to calculate permutations in linear time, with a twist Ask Question Asked 16 years, 8 months ago Modified12 years, 5 months ago Viewed 4k times This question shows research effort; it is useful and clear 7 Save this question. Show activity on this post. I have a resource scheduling issue in Java where things need to be sequenced, but there are restrictions on what resources can be next to each other. A good analogy is a string of "digits", where only certain digits can be next to each other. My solution was recursive, and works fine for small strings, but run time is O(X^N), where X is the number of possible digits (the base), and N is the length of the string. It quickly becomes unmanageable. Using the compatibility matrix below, here are a few examples of allowed strings Length of 1: 0, 1, 2, 3, 4 Length of 2: 02, 03, 14, 20, 30, 41 Length of 3: 020, 030, 141, 202, 203, 302, 303, 414 0 1 2 3 4 0\| 0 0 1 1 0 1\| 0 0 0 0 1 2\| 1 0 0 0 0 3\| 1 0 0 0 0 4\| 0 1 0 0 0 My solution for counting all strings of length N was start with an empty string, permute the first digit, and make a recursive call for all strings of length N-1. The recursive calls check the last digit that was added and try all permutations that can be next to that digit. There are some optimizations made so that I don't try and permute 00, 01, 04 every time, for example - only 02, 03, but performance is still poor as it scales from base 5 (the example) to base 4000. Any thoughts on a better way to count the permutations other than trying to enumerate all of them? algorithm permutation Share Share a link to this question Copy linkCC BY-SA 2.5 Improve this question Follow Follow this question to receive notifications edited Jan 27, 2009 at 22:20 cmmp cmmp asked Jan 27, 2009 at 15:38 cmmp cmmp 4 what defines whether elements are acceptable? is the compatibility matrix an input to the algorithm?Jason S –Jason S 2009-01-27 16:09:21 +00:00 Commented Jan 27, 2009 at 16:09 To clarify: are you only interested in counting the total number of permutations?Zach Scrivena –Zach Scrivena 2009-01-27 16:09:36 +00:00 Commented Jan 27, 2009 at 16:09 The matrix is the result of a previous function. The one shown above with lengths of 1, 2, and 3 is what I have been using to test other algos. F(1) = 5, F(2) = 6, F(3) = 8 And yes - only counting, not enumerating.cmmp –cmmp 2009-01-27 16:42:32 +00:00 Commented Jan 27, 2009 at 16:42 Dude, you have no understanding of O(.) notation. The algorithm you're describing is exponential O(5^N), not O(N^2). See bellow for a good very fast solution:D user51568 –user51568 2009-01-27 17:15:55 +00:00 Commented Jan 27, 2009 at 17:15 Add a comment\| 5 Answers 5 Sorted by: Reset to default This answer is useful 19 Save this answer. Show activity on this post. If you just want the number of strings of a certain length, you could just multiply the compatibility matrix with itself a few times, and sum it's values. n = length of string A = compatibility matrix number of possible strings = sum of A n-1 A few examples: ``` n = 1 \| 1 0 0 0 0 \| \| 0 1 0 0 0 \| \| 0 0 1 0 0 \| \| 0 0 0 1 0 \| \| 0 0 0 0 1 \| sum: 5 n = 3 \| 2 0 0 0 0 \| \| 0 1 0 0 0 \| \| 0 0 1 1 0 \| \| 0 0 1 1 0 \| \| 0 0 0 0 1 \| sum: 8 n = 8 \| 0 0 8 8 0 \| \| 0 0 0 0 1 \| \| 8 0 0 0 0 \| \| 8 0 0 0 0 \| \| 0 1 0 0 0 \| sum: 34 ``` The original matrix (row i, column j) could be thought of as the number of strings that start with symbol i, and whose next symbol is symbol j. Alternatively, you could see it as number of strings of length 2, which start with symbol i and ends with symbol j. Matrix multiplication preserves this invariant, so after exponentiation, A n-1 would contain the number of strings that start with symbol i, has length n, and ends in symbol j. See Wikipedia: Exponentiation by squaring for an algorithm for faster calculation of matrix powers. (Thanks stefan.ciobaca) This specific case reduces to the formula: number of possible strings = f(n) = 4 + Σ k=1..n 2⌊k-1⁄2⌋ = f(n-1) + 2⌊n-1⁄2⌋ n f(n) ---- ---- 1 5 2 6 3 8 4 10 5 14 6 18 7 26 8 34 9 50 10 66 Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications edited Apr 29, 2013 at 6:44 answered Jan 27, 2009 at 16:35 Markus JarderotMarkus Jarderot 89.5k 23 23 gold badges 141 141 silver badges 142 142 bronze badges 5 Comments Add a comment user49117 user49117Over a year ago Using power of compatible matrix is a very clean idea. But one should note that the formula is only correct for this case. 2009-01-27T17:04:36.043Z+00:00 0 Reply Copy link user51568 user51568Over a year ago Very nice solution. For extra fastness and credit, be sure to use the en.wikipedia.org/wiki/Exponentiation_by_squaring double & add algorithm. 2009-01-27T17:29:06.977Z+00:00 0 Reply Copy link user51568 user51568Over a year ago Also, the answer should mention why this is correct. The correct explanation is that A^i[x][y] is the number of "compatible" strings of length i that start with x and end with y. You can clearly see why multiplication preserves this invariant. 2009-01-27T17:30:47.643Z+00:00 0 Reply Copy link Darius Bacon Darius BaconOver a year ago This is cool, but wasn't the question about the size of a restricted set of permutations? This doesn't enforce the constraint that a digit may be used at most once, which is what I take 'permutation' to mean. Oh, wait, '202' is an example. Just a confusing expression of the question, then. 2009-01-27T18:27:19.137Z+00:00 4 Reply Copy link cmmp cmmpOver a year ago Very good solution - thanks. I have it working as suggested, and now trying to optimize it using the exponentiation by squaring method mentioned by stefan. 2009-01-27T22:15:51.967Z+00:00 1 Reply Copy link Add a comment This answer is useful 3 Save this answer. Show activity on this post. Do you just want to know how many strings of a given length you can build with the rules in the given matrix? If so, that an approach like this should work: ``` n = 5 maxlen = 100 combine = [ [0, 0, 1, 1, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0] ] counts of strings starting with 0,1,...,4, initially for strings of length one: counts = [1, 1, 1, 1, 1] for size in range(2, maxlen+1): # calculate counts for size from count for (size-1) newcount = [] for next in range(n): total = 0 for head in range(n): if combine[next][head]: # \|next\| can be before \|head\|, so add the counts for \|head\| total += counts[head] # append, so that newcount[next] == total newcount.append(total) counts = newcount print "length %i: %i items" % (size, sum(counts)) ``` Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications answered Jan 27, 2009 at 16:05 sthsth 231k 56 56 gold badges 288 288 silver badges 370 370 bronze badges 4 Comments Add a comment cmmp cmmpOver a year ago Thanks. I think this would work, but I believe this has the same issue I have... nested loops/recursions over N elements. Please correct me if I'm missing something. 2009-01-27T16:39:18.497Z+00:00 0 Reply Copy link user51568 user51568Over a year ago You're missing something. It's the same dynamic programming algorithms as suggested by unknown (yahoo). 2009-01-27T17:25:37.45Z+00:00 0 Reply Copy link sth sthOver a year ago It's O(n^2 maxlen), and that's not great. From your description I thought your algorithm might be something like O(n^2 n^(maxlen-1)), but I'm not really sure how you do it. I think the matrix solution above could be implemented in O(n^2log(maxlen)), but that also doesn't really help either... 2009-01-27T17:26:09.197Z+00:00 0 Reply Copy link sth sthOver a year ago But then again: If you have a nn matrix with the compatibility information and need to look at all the elements in it, you need O(n^2) time to do that. Is there maybe a better way to describe the compatibilities? 2009-01-27T17:30:17.537Z+00:00 0 Reply Copy link Add a comment This answer is useful 2 Save this answer. Show activity on this post. Your algorithm seems to be optimal. How are you using these permutations? Are you accumulating them in one list, or using it one by one? Since there are a huge number of such permutations, so the poor performance maybe due to large memory usage (if you are collecting all of them) or it just takes so much time. You just can't do billions of loops in trivial time. Reply to comment: If you just want to count them, then you can using dynamic programming: Let count[n][m] be an array, where count[l][j] is the number of such permutations whose length is l and end with j, then count[l][i] = count[l-1][i1]+count[l-1][i2]+..., where i1, i2, ... are the digits that can precede i (this can be saved in an pre-calculated array). Every cell of count can be filled by summing K numbers (K depends on the compatible matrix), so the complexity is O(KMN), M is the length of the permutation, and N is the total number of digits. Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications edited Jan 27, 2009 at 16:37 answered Jan 27, 2009 at 16:06 user49117user49117 786 3 3 silver badges 9 9 bronze badges 1 Comment Add a comment cmmp cmmpOver a year ago Just counting them for now - not using them. As far as an iterative/recursive solution goes, I think mine is pretty good, but as you say, you can't do 4000^10 of those in reasonable time, so I'm trying to find an alternate. 2009-01-27T16:21:38.5Z+00:00 0 Reply Copy link This answer is useful 1 Save this answer. Show activity on this post. Maybe I don't understand this, but wouldn't this be served by having a table of lists that for each digit has a list of valid digits that could follow it. Then your routine to generate will take an accumulated result, the digit number, and the current digit. Something like: // not really Java - and you probably don't want chars, but you'll fix it void GenerateDigits(char[] result, int currIndex, char currDigit) { if (currIndex == kMaxIndex) { NotifyComplete(result); return; } char[] validFollows = GetValidFollows(currDigit); // table lookup foreach (char c in validFollows) { result[currIndex] = c; GenerateDigits(result, currIndex+1, c); } } The complexity increases as a function of the number of digits to generate, but that function depends on the total number of valid follows for any one digit. If the total number of follows is the same for every digit, let's say, k, then the time to generate all possible permutations is going to be O(k^n) where n is the number of digits. Sorry, I can't change math. The time to generate n digits in base 10 is 10^n. Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications answered Jan 27, 2009 at 15:53 plinthplinth 49.4k 11 11 gold badges 84 84 silver badges 123 123 bronze badges 2 Comments Add a comment cmmp cmmpOver a year ago Thanks. Your suggestion has already been implemented as the optimization that I mentioned where I only look at the digits that can follow the current digit. That improved runtime from O(n^2), but the exponential scale still falls apart when we get to thousands of digits. 2009-01-27T16:08:09.213Z+00:00 0 Reply Copy link user51568 user51568Over a year ago I don't think he made a suggestion. He was trying to understand your algorithm. You use too much the words "linear", "exponential", "O(N^2)", without understanding what they mean. 2009-01-27T17:19:41.893Z+00:00 0 Reply Copy link This answer is useful 1 Save this answer. Show activity on this post. I'm not exactly sure what you're asking, but since there are potentially n! permutations of a string of n digits, you're not going to be able to list them faster than n!. I'm not exactly sure how you think you got a runtime of O(n^2). Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications answered Jan 27, 2009 at 16:18 Brian PostowBrian Postow 12.3k 21 21 gold badges 86 86 silver badges 134 134 bronze badges 1 Comment Add a comment user51568 user51568Over a year ago O(K^N), where K is the number of distinct digits and N is the length of the "permutation". 2009-01-27T17:24:18.513Z+00:00 0 Reply Copy link Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. 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3828	https://www.scivisionpub.com/pdfs/adenomyosis-current-knowledge-recent-advances-and-future-perspective-2768.pdf	Gynecology & Reproductive Health ISSN 2639-9342 Volume 7 \| Issue 3 \| 1 of 8 Gynecol Reprod Health, 2023 Adenomyosis: Current knowledge, Recent Advances and Future Perspective 1Consultant Gynaecologist, Queens Hospital, Romford, London, UK. 2ST2 Trainee in Obstetrics and Gynaecology, Queens Hospital, Romford, London, UK. 3Consultant Gynaecologist, Basildon Hospital, Basildon, UK. 4ST6 Trainee in Obstetrics and Gynaecology, Queens Hospital, Romford, London, UK. 5Lecturer of Obstetrics and Gynecology, Ain-Shams University, Cairo, Egypt. Kunal Rathod1, Michael Magro1, Sarah Shehzad2, Yatin Thakur3 and Sherif Daoud4,5 Citation: Kunal Rathod, Michael Magro, Sarah Shehzad, et al. Adenomyosis: Current knowledge, Recent Advances and Future Perspective. Gynecol Reprod Health. 2023; 7(3): 1-8. Correspondence: Dr. Sarah Shehzad, ST2 Trainee in Obstetrics and Gynaecology, Queens Hospital, Queens Hospital, Romford, London, UK. Received: 28 Mar 2023; Accepted: 30 Apr 2023; Published: 04 May 2023 Case Report ABSTRACT Aim: Adenomyosis is an abnormal overgrowth of the endometrial tissues within the myometrium causing enlargement of the uterus. This present review will focus on clinical symptoms, diagnostic approach, image findings, complications, and management of Adenomyosis. The goal is also to highlight the recent advances in the topic. Methodology: A total of 15 articles published in various journals have been included to write the current review. PubMed, Research Gate, Scopus, Springer are some of the databases used for the literature search. Results: After reviewing the literature Adenomyosis has been discussed under the following topics 1) epidemiology (known and emerging risk factors) 2) Pathogenetic Theories (recent advances such as sequencing analysis of epithelial cells in Adenomyosis) 3) Clinical Manifestations and impact on women's fertility and pregnancy outcome 4)Diagnostic Approach, Current imaging techniques and classifications 5) Medical Management 6) Surgical Interventions (with recent advances such as UAE) 7) Future Perspective. Conclusion: The prevalence of Adenomyosis is still unknown owing to the lack of a validated standard diagnostic approach. Historically, the standard treatment of adenomyosis has been hysterectomy, but this is not always the best option, especially for women who want to preserve their fertility or for those who are poor surgical candidates. Keywords Adenomyosis Introduction The term "Adenomyosis" refers to an abnormal growth of the endometrial tissues into the myometrium causing the uterus to enlarge . A German pathologist by name Carl von Rokitsansky was the first to describe adenomyosis in the year 1860; and he initially referred it as "adenomyomas". The exact etiology of the condition is unknown. However, recent theories about the pathophysiology of endometriosis can alter our understanding of adenomyosis as well . In some patients, adenomyosis coexists with other gynecological pathologies such as endometriosis and uterine fibroids. Endometriosis is defined as the presence of endometrial glands and stroma-like lesions in the outer surface of the uterus . Over the last decade, adenomyosis is mainly identified among young fertile-age women (especially post pregnancy). The most typical clinical manifestations of this disease include pelvic pain, menorrhagia, dysmenorrhea, infertility. However, the frequency and severity of the occurrence of these symptoms and the proportion of adenomyosis sufferers who are completely asymptomatic are unclear . With the emergence of MRI, followed by transvaginal ultrasound (TVUS), the diagnostic approach of adenomyosis has seen an exceptional breakthrough Volume 7 \| Issue 3 \| 2 of 8 Gynecol Reprod Health, 2023 . Medications such as non-steroidal anti-inflammatory drugs and/or hormonal therapy (oral contraceptives, high dose progestins, danazol, gonadotropin-releasing hormone agonists, a levonorgestrel-releasing intrauterine device) are often used to manage the symptoms . Adenomyosis has never been fully characterized until now since hysterectomy has been the primary therapeutic option worldwide . Minimally invasive surgical techniques (uterine artery ligation, endometrial ablation/resection, myometrial excision/reduction, myometrial electrocoagulation) have limited success in the treatment of adenomyosis, however, latest approaches namely uterine artery embolization (UAE), magnetic resonance imaging guided focused ultrasound (MRgFUS) show promising results in the treatment . There are no standard international guidelines to follow for surgical or medical treatment of adenomyosis. This holds a notable importance in the future, as the disease requires a lifelong management process, fertility preservation and pregnancy outcome . In this paper, current and future trends in the diagnostic approach and management plan would be reviewed. Epidemiology In recent years, the diagnosis of adenomyosis was made solely based on histological analysis. Estimates of adenomyosis prevalence vary from 5-70% with the mean incidence of hysterectomy being approximately 20-30% . Gynecological University Clinic in Tubingen conducted a series of consecutive laparoscopic hysterectomies where adenomyosis was histologically diagnosed in 8% of cases (149 out of 1955 women), and associated adenomyosis and leiomyomas was diagnosed in 20% of the women (398 out of 1955 women), 70% of women were premenopausal . There is a wide modification in incidence of adenomyosis between racial and ethnic groups and different geographic regions , however one third of the patients with Adenomyosis are asymptomatic. Additionally with an increase in hysterectomy cases performed as laparoscopic supracervical involvement, resulting in uterine specimens, the dimensional arrangement of the tissue is altered, which leads to different reference to the surface and histologically diagnosing adenomyosis becomes difficult. The likelihood of presence of adenomyosis is directly corresponds to the number of tissue samples accumulated with the diagnosis rate ranging from 31-62% in the same uterus [7,10]. In women, undergoing Assisted Reproductive Techniques (ARTs) adenomyosis prevalence is 20-25%, whereas in women with a history of endometriosis the prevalence varies between 20-80% . Risk Factors Age 70-80% of women who have undergone hysterectomy for adenomyosis are in their 3rd and the 4th decade of life . Currently new reports using MRI imaging for diagnosis suggest that disease may cause symptoms like chronic pelvic pain and dysmenorrhea (menstrual cramps) in adolescents and women of younger fertile age [7,13]. Prior Uterus Surgery There is no evidential proof to indicate a significant increased risk of prior uterine surgeries in women with adenomyosis . However, according to Yilmaz et al., women with a history of pregnancy termination via dilation and curettage depicted higher rates of adenomyosis as compared to women without any pregnancy termination . Non-pregnant women with a history of dilation and curettage also showed higher incidence rate [7,11]. Women post caesarean section did not demonstrate any marked increase in the rates of adenomyosis . Multiparity Multiparous women exhibited a marked increase in adenomyosis rates . Occurrence of adenomyosis during pregnancy is generally due to adenomyotic foci which become included in the myometrium. These foci have a higher ratio of estrogen receptors, which pave way for the formation of islands of ectopic endometrium . Ectopic Pregnancy If the implantation occurs within the focus of adenomyosis, pregnancy can develop within the myometrium [7,14] therefore adenomyosis could also be a risk factor for the development of intramural ectopic pregnancy . Increased likelihood of ectopic pregnancy in association with adenomyosis requires additional evidence. Depression and Antidepressant use There is a significant increase in the incidence of adenomyosis with increased risk of depression in women and use of antidepressants. This may be due to irregularities in prolactin dynamics [7,19]. Prolactin is produced by uterine tissues including endometrium, myometrium, and leiomyomas. A functional prolactin receptor is present in the uterus, and it can act as a smooth muscle cell mitogen . Depression may have a similar pathogenic factor with adenomyosis (inflammation). Tamoxifen Treatment An increased incidence of adenomyosis has been noticed in women treated with Tamoxifen for breast cancer. It is an antagonist of the estrogen receptor in the breast tissue . The rate of adenomyosis depicted among post-menopausal breast cancer patients treated with Tamoxifen is 3-4 times more than the rate reported for pre- and post-menopausal women . Pathogenetic Theories- insight into Next Generation Sequencing (NGS) Until now, the exact pathogenesis of adenomyosis was largely unknown. However, currently two new theories are prevailing: "invagination" and "metaplasia" . According to Garcia-Solaris et al. and Layendecker et al. Tissue Injury and Repair Mechanism (TIAR) is the most common theory wherein adenomyosis is an outcome of invagination of the endometrial basalis into the myometrium. The metaplasia theory states that the adenomyotic lesions may originate from the metaplasia of Volume 7 \| Issue 3 \| 3 of 8 Gynecol Reprod Health, 2023 dislodged embryonic pluripotent Mullerian Rests . High estrogen concentrations, smooth muscle cell hyperplasia and hypertrophy reflect reactive changes secondary to ectopic endometrial proliferation . A latest theory named EMID (endometrial-myometrial interface disruption) has been put forth which revises tissue injury and healing theory. EMID caused by uterine surgeries may lead to iatrogenic adenomyosis later in life . Next-generation sequencing (NGS) studies show that a particular cancer associated gene (KRAS) undergoes mutation causing insufficient PR expression. The KRAS mutation triggers a particular pathway to enhance cell survival and proliferation and is thus linked to progesterone resistance in adenomyosis . NGS technology forms the backbone of all genomic contemporary approaches . Clinical Manifestations of Adenomyosis Adenomyosis is associated with chronic pelvic pain, dysmenorrhoea (menstrual cramps), menorrhagia (heavy and abnormal uterine bleeding) and dyspareunia (painful sexual intercourse) [7,18,29]. Li et al. investigated 710 premenopausal women with adenomyosis and reported that 4.5% of them did not have any symptoms. The most common and frequent complaint was dysmenorrhea (81.7%) , however, the clinical symptoms may vary depending on the patient's age [3,4,7,18,29]. The symptoms of adenomyosis are non- specific and can also be related to conditions such as endometriosis, leiomyomas, dysfunctional uterine bleeding among others [7,18]. Impact on Fertility Presently, infertility is considered as the most strikingly prominent clinical presentation of adenomyosis, and various theories have been out forth to explain the mechanism behind this [1,7,29]. A probable mechanism behind this could be an abnormal utero-tubal transport leading to infertility. This is due to the anatomical disruption of the uterine cavity but also disturbed uterine peristalsis and sperm transport . A meta-analysis published by Vercellini et al. in 2014 reported miscarriage rates of 31% in women with adenomyosis, and 14.1% in non-infected women. In a multicenter study, Mavrelos et al. depicted that the estimated probability of pregnancy decreased from 42.7% in women with no adenomyosis to 22.9% in those with four ultrasound features and 13% in those with all features of adenomyosis . A recent systematic review and meta-analysis on IVF treatment outcomes in adenomyosis [1,34] included 519 patients with TVUS or MRI diagnosis for Adenomyosis and confirmed the damaging effect of the uterine disease on reproductive outcome. The implantation rate, clinical pregnancy per cycle/per embryo transfer, present pregnancy, and live-birth rate among women with adenomyosis were significantly reduced, wherein miscarriage rate was increased [34,35]. Impact on Pregnancy Outcome Adenomyosis is considered a reproductive disorder wherein, not only fertility is affected but it has a prominent negative effect on the pregnancy outcome as well. There is an elevated risk of preterm birth (PTB) with an adjusted odds ratio (aOR) being 1.84,95% CI 1.39-3.15 in adenomyotic patients . The results were confirmed in a small cohort study conducted by Mochimaru et al. where women diagnosed sonographically or by MRI before pregnancy, showed an increased risk of cesarean delivery (OR 4.5,96% CI 2.1-9.7), small gestational age (SGA) fetuses (OR 4.3,95% CI 1.8-10.3), postpartum hemorrhage (OR 6.5,95% CI 2.2-19.0) and fetal malpresentation (OR 4.2,95% CI 1.6-10.8). The type of adenomyosis has an impact on pregnancy outcome with high incidence of pregnancy-induced hypertension . The factors, which are responsible for the obstetric complications in adenomyosis could be (i) role of inflammation (ii) elevated myometrial prostaglandin production (iii) altered uterine contractility (iv) intrauterine pressure. All of these increase the risk of preterm birth in women with adenomyosis . Diagnostic Approach Clinical Examination The classic feature on physical examination is the "boggy" type enlargement of the uterus. This is due to the increased vascularization from the ectopic endometrial tissue and proliferation of the smooth muscles. The uterus appears to be more tender than usual on examination . Histology Histological examination of the specimen obtained from hysterectomy is the gold standard for diagnosing adenomyosis . Histologically there is a presence of endometrial stromal and glandular tissue deep within the smooth muscle of the myometrium . This is associated with smooth muscle hyperplasia. There are no universal criteria for the depth of invasion and number of foci to be included in the diagnosis. However, in some cases invasion depth of more than one third is taken into account and in some, greater than 4mm . The architecture of adenomyosis is distinctively different from that of functional endometrium in which the glands are solitary, non-branching and longitudinally arranged . Imaging The development of imaging techniques (such as TVUS and MRI) has permitted clinicians to make non-invasive diagnosis of adenomyosis in women who are receiving conservative treatment, identifying different phenotypes of the disease . Sonographically adenomyosis is categorized as cystic and non-cystic , furthermore cystic adenomyosis is classified as: intrinsic, extrinsic, intramural, and interdeterminate . Transvaginal Ultrasound Sonography (TVUS) It is the first diagnostic technique since it is widely available, cost-effective with increased accuracy if performed by an expert sonologist. The sensitivity ranges from 65-81% and specificity from 65-100% . The transvaginal scans are 2-dimensional and 3-dimensional, wherein 3D-TVUS is considered more superior to 2D-TVUS. The most common TVUS findings are: heterogeneous Volume 7 \| Issue 3 \| 4 of 8 myometrium, myometrial cysts, a globular and/or asymmetric uterus, abnormal myometrial echo texture, poorly-defined margins between the endometrium and myometrium, echogenic linear striations and focal adenomyomas . Currently, a standardized reporting system of ultrasound findings of adenomyosis has been introduced using Morphological Uterus Sonographic Assessment (MUSA) criteria. According to this, the typical ultrasound features to diagnose adenomyosis are: asymmetrical thickening of uterine walls, intra-myometrial cysts hypoechoic islands or both, fan-shaped shadowing of myometrium, myometrial echogenic sub endometrial lines and buds, translational vascularity [1,44]. Linear striations and parallel shadowing give the appearance of a "Venetian blind" or "rain shower" due to the prior mentioned combination of symptomatic features: heterogeneous, coarsened echotexture of the myometrium, and acoustic shadowing where a hyperplastic reaction is caused by the endometrial tissues. Adenomyosis is also sometimes referred to as "cirrhosis of the uterus"; due to the presence of heterogeneity and sub endothelial echogenic nodular and linear striations. This gives an appearance of chronic liver parenchymal disease . A new reporting system was suggested by Van den Bosch et al. which includes description of the disease location (anterior, posterior, left lateral, right lateral and fundal), categorization of the lesion as focal or diffuse, presence or absence of intraregional cysts, myometrial layer involvement, disease extent (<25%, 25-50% and >50% of the uterine volume affected by Adenomyosis) and lesion size . Magnetic Resonance Imaging (MRI) Pelvic MRI is the preferred choice of diagnosis modality, which provides a detailed classification of adenomyosis. Small field of view T2-weighted images are the most useful, especially sagittal and axial . There is prominent thickening of the junctional zone (JP) which is a clear indicative of an MRI diagnosis for Adenomyosis, however, there may be direct or indirect signs of presence of endometrial glands within the myometrium and the smooth muscle cell hypertrophy . The MRI sequence findings include: (i) T1- foci of high T1 signal are frequently observed, which indicates menstrual hemorrhage into the ectopic endometrial-myometrial tissues (ii) T2- classically a region of adenomyosis presents as an ill-defined ovoid/diffuse region of thickening. Frequently small high T2 signal regions represent small areas of cystic change. The region may also have striated appearance. (iii) T1C+(GD)- MRI evaluation need not be contrast-enhanced for assessment of adenomyosis. However, if performed; it shows enhancement of the ectopic endometrial-myometrial glands . A sensitivity of 46.1%, specificity of 99.2% and a positive predictive value (PPV) is noted in MRI for the diagnosis of adenomyosis. This was described by Stamatopoulos and team in a cohort observational study . The classic MRI findings are the focal or diffuse thickening of the JZ, which is seen as an area of low-signal-intensity in the myometrium and high-signal-intensity spots in the T2-weighted resonance. Bazot et al. reported a sensitivity of 77.5%, specificity of 92.5%, and a PPV of 83.8% in a prospective study with 120 patients. Junctional zone thickness (max) >12mm, a JZ (max) to myometrial thickness ratio > 40% and the presence of high-signal-intensity myometrial spot were the most specific factors, while JZ(max) was the most sensitive value . Normal junctional zone thickness is <8mm . Swiss cheese appearance: It is a type of diffuse adenomyosis that has a "Swiss cheese appearance" with exuberant myometrial cysts and nodules on contrast enhanced and T2 sequences. It is secondary to cross sectional imaging of dilated endometrial glands within the myometrium. There is also a widening and poor definition of JZ and linear striations . Hysteroscopy This surgical approach favors the direct visualization of the uterine cavity, which gives the option of collecting the histological biopsy samples under visual control . Hysteroscopy shows the following typical findings: irregular endometrium with small openings on the endometrial surface, prominent hypervascularization, and a classic endometrial “strawberry pattern, fibrous and hemorrhagic cystic lesions . Hysterosalpingography (HSG) It is a type of X-ray procedure, which views the inside of the uterus and fallopian tubes. It depicts the presence or absence of blockage in the tubes and evaluates the normal size and shape of the uterus. In 1949, the first report was published about using hysterosalpingography (HSG) to diagnose adenomyosis, however, due to its low sensitivity, HSG was never utilized as a prime diagnostic tool . Adenomyosis is seen as irregular branching outpouching radiating from the uterine cavity; representing extension of the endometrial glands of the myometrium . Management Medical treatment Medical treatment is the first line of treatment option for adenomyosis. This helps relieve symptoms and maintain fertility with significantly less side effects. The mechanism involves the disruption of the pathways, which lead to inflammation, neuroangiogenesis, and impaired apoptosis. Presently, various hormonal and non-hormonal options, namely gonadotropin-releasing hormone (GnRH) analogues, selective estrogen receptor modulators (SERMs), aromatase inhibitors (AIs), selective progesterone receptor modulators (SPRMs), combined oral contraceptive, and non-steroidal anti-inflammatory drugs are being used for the symptomatic treatment of adenomyosis . Newer drugs, such as aromatase inhibitors, have also been investigated by Badawy et al. and Rosti et al, while other therapies such as selective progesterone receptor modulators, GnRH antagonist, valproic acid, and antiplatelet therapies are still under investigation . Among progestins, norethisterone acetate (NETA), vaginal danazol, and dienogest (DNG) may be considered . The levonorgestrel-releasing intrauterine system (LNG-IUS) is an effective, reversible, and long-term treatment, which is successfully used to treat adenomyosis. Results show reduced Gynecol Reprod Health, 2023 Volume 7 \| Issue 3 \| 5 of 8 Gynecol Reprod Health, 2023 menstrual bleeding, pain and uterine volume and have an overall satisfaction of 72% . Minimally Invasive Techniques This is a second line of treatment modality after a failed medical treatment, which allows the patient to retain their uterus. Excisional adenomyomectomy involves the complete extraction of the focal lesions (adenomyomas), while myomectrectomy is the surgical debulking of diffuse adenomyosis. Non-excisional treatments aim to induce necrosis of focal or diffuse adenomyosis through selective vascular occlusion or focused ultrasound/thermal energy without direct tissue dissection. In some cases, a combination of surgical and non-excisional methods, i.e., hysteroscopy resection/ ablation, are used to achieve maximum cytoreduction and reduce myometrial tissue damage . Conservative Surgical Treatment The first report of a conservative surgical treatment for adenomyosis in young women was reported in the year 1952. The partial excision of an adenomyosis, as a cytoreduction surgery, became common in which the uterine wall is excised in a V-shape . Several laparotomic techniques have been described, which includes the following: Wedge Resection technique - In this approach, parts of serosa and uterine adenomyomas are excised via wedge resection after identifying the parts of the neuromuscular layer where the adenomyoma is located. The uterine wall wound is sutured together with the remaining muscular layer and serosa . Modified Resection technique- first reported in 1991, this involves the cutting of the adenomatous tissue into thin slices using a microscopic technique in conjunction with laparotomy surgery . Transverse H Incision of uterine wall -in this procedure the transverse incision is made on the uterine fundus, using an electro-surgical scalpel, separating the uterine serosa from the uterine myometrium. The adenomyoma tissue is removed using the electro-surgical scissors or scalpel. Wedge-shaped Uterine wall removal- here the adenomyoma is resected with a thin margin (wedge shaped removal) after a sagittal incision in the uterine body. The radical resection involves the laminate layers on both the endometrial and serosal sides. The suturing technique used is the “baseball” or “continuous Lembert stitch method” . Triple flap method- This method involves reconstructing the uterine wall defect using normal uterine muscle. It has three characteristics: complete extraction of the uterine adenomyosis by performing adenomyomectomy; reconstruction of a uterine cavity, which can sustain future pregnancy (here an endometrial uterine muscle flap is prepared by metroplasty through opening the uterine cavity and removing the uterine adenomyosis under palpation); and reconstruction of a uterine wall resistant to rupture during a subsequent pregnancy . Laparoscopic techniques have also been described in more focal pathology . Laparoscopic adenomyomectomy with hysteroplasty- transverse incision is made in the adenomyotic tissue down to the endometrium. The diseased tissue is surgically removed using a monopolar needle. The normal muscle layer on the serosal membrane side is left as an upper and lower serosal flap. The flaps are overlapped and sutured to counteract the lost muscle layer to reconstruct the uterus . The main issue with conservative surgical methods is the high risk for complications, i.e., uterine rupture and complicated pregnancy (especially in diffuse lesions and on long-term follow-up), making this option safer in focal adenomyomas . Hysteroscopy resection/ablation This is a combined method, which involves the dissection or coagulation of cystic adenomyotic lesions and crypts. It can be performed using yttrium aluminum garnet (YAG) laser, cryoablation, circulated hot fluid ablation, microwave ablation, roller ball resection, thermal balloon resection, electrocoagulation, bipolar radiofrequency ablation . High-Intensity Focused Ultrasound (HIFU) This was first introduced in the year 1942 by Lynn et al. Intense ultrasound energy is used directly which targets the abnormal tissues and their vascularity through heating and cavitation. The normal surrounding tissues are however spared. The method is guided and monitored through MRI or ultrasound . HIFU has been used in the treatment of adenomyosis since 2008 . Recent studies have investigated the use of ultrasound contrast agents (microbubbles) and hormonal (GnRH) and non-hormonal (Metformin) treatments, which enhance the HIFU efficiency. Microbubbles improve the ablative effects of HIFU by changing acoustic characteristics while GnRH and Metformin inhibit cellular proliferation and induce apoptosis . This technique is not commonly used compared to other minimally invasive methods due to limited availability, overall cost, unknown fertility outcome and strict biological indicators . Uterine Artery Embolization (UAE) This is a method, which uses Trans arterial catheters which aims to induce more than 34% necrosis within the adenomyotic tissues . In 1995, the first paper was published by Ravin et al., Which reported a woman being treated by the UAE for symptomatic uterine leiomyomas. The procedure is performed under local anesthesia using the right femoral artery puncture approach. Selective digital subtraction angiography is performed to evaluate the hypogastric and uterine arteries. Embolization is performed through the microcatheter using non-spherical PVA particles. The end point of embolization is complete stasis of blood flow in the uterine artery . A study conducted by Popovic et al, Volume 7 \| Issue 3 \| 6 of 8 Gynecol Reprod Health, 2023 demonstrated long term improvement in patient symptoms (in over 60% of patients) and a short-term decrease in uterine volumes (in over 20% of patients) especially in vascular lesions . Results of an ongoing randomized controlled (QUESTA) trial will soon prove the validity of UAE as a treatment option for adenomyosis . Future Perspective Over the next 5-10 years, ultra-long regulation protocols will be preferred during IVF/ICSI cycles for women with suggestive symptoms or signs of adenomyosis. The diagnosis for adenomyosis should be standardized according to an internationally approved criterion. Significant consequences of Adenomyosis such as preterm birth, spontaneous miscarriage, intrauterine growth retardation, preeclampsia, eclampsia, obstetrical hemorrhages, placental bed, and adherence abnormalities should be evaluated with well-assessed studies . Conclusion There has been a significant improvement in the understanding and management of adenomyosis in recent years. It has become a clinical entity rather than a histological diagnosis, which can be identified through various non-invasive imaging techniques. An increasing amount of evidence suggests the improvement in the treatment modalities owing to minimally invasive techniques, however, there is still a need for a uniform diagnostic criteria profile and reporting system, in order to identify all the clinical and imaging phenotypes of adenomyosis. Clinicians are required to conduct prospective studies on adenomyosis prevalence, effective medical or surgical treatments and impact on fertility and pregnancy outcomes. References 1. Vannuccini S, Petraglia F. Recent advances in understanding and managing adenomyosis. F1000Res. 2019; 8: F1000 Faculty Rev-283. 2. Sharara FI, Kheil MH, Feki A, et al. Current and Prospective Treatment of Adenomyosis. J Clin Med. 2021; 10: 3410. 3. Parasar P, Ozcan P, Terry KL. Endometriosis: Epidemiology, Diagnosis and Clinical Management. Curr Obstet Gynecol Rep. 2017; 6: 34-41. 4. Chen Q, Li YW, Wang S, et al. Clinical Manifestations Of Adenomyosis Patients With Or Without Pain Symptoms. J Pain Res. 2019; 12: 3127-3133. 5. Ryan K Cunningham, Mindy M Horrow, Ryan J Smith, et al. Adenomyosis: A Sonographic Diagnosis. RadioGraphics. 2018; 38: 5. 6. Kuan-Hao Tsui, Wen-Ling Lee, Chih-Yao Chen, et al. Medical treatment for adenomyosis and/or adenomyoma. 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Treatment Mechanism Type Non-Surgical Gonadotropin-Releasing hormone (GnRH) Analogues (53) Dessouky 2017 Endometrial atrophy, Hypoestrogenism Selective estrogen receptor modulators (SERMs) (53) Dessouky 2017 Differential ER expression in target tissue Aromatase inhibitors (AIs) (53) Dessouky 2017 Inhibition of estradiol synthesis (suicidal/ competitive) Selective progesterone receptor modulators (SPRMs) (53) Dessouky 2017 Differential PR expression in target tissue Combined oral contraceptive (53) Dessouky 2017 Atrophy of endometrial tissue causing decreased menstrual bleeding Non-steroidal anti-inflammatory drugs (NSAIDs) Decreased pain and abnormal bleeding due to decreased prostaglandins Norethisterone acetate (NETA) Decidualization and atrophy of endometrial tissue Vaginal danazol (1) Vannucini 2019 Hyperandrogenism Dienogest (DNG) (1) Vannucini 2019 Suppression of estradiol Levonorgestrel-releasing intrauterine system (LNG-IUS) (55) Sheng 2009 Mediated through slow-release progestin Surgical Conservative Wedge Resection (57) Nishida 2007 Parts of serosa and uterine adenomyomas are excised Modified Resection technique (58) Osada H 2018 Involves the cutting of the adenomatous tissue into thin slices Transverse H Incision of uterine wall (58) Osada H 2018 Transverse incision was made on the uterine fundus Wedge-shaped Uterine wall removal (58) Osada H 2018 Adenomyoma is resected with a thin margin Triple flap method Involves reconstructing the uterine wall defect using normal uterine muscle Laparoscopic adenomyomectomy with hysteroplasty (59) Takeuchi H 2006 Transverse incision is made in the adenomyotic tissue down to the endometrium. 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3829	https://www.math.toronto.edu/ssaraf/acute.pdf	ACUTE AND NONOBTUSE TRIANGULATIONS OF POLYHEDRAL SURFACES SHUBHANGI SARAF CSAIL MIT Abstract. In this paper, we prove the existence of acute triangulations for general polyhedral surfaces. We also show how to obtain nonobtuse subtrian-gulations of triangulated polyhedral surfaces. 1. Introduction A triangulation of a 2-dimensional surface is a subdivision of the surface into nonoverlapping triangles in such a way that the intersection of any two distinct triangles is either empty or consists of a vertex or an edge. We are interested in constructing triangulations of a surface with acute or nonobtuse triangles. The construction arises in connection with discretization of partial diﬀerential equations, ﬁnite element mesh generation, and other related areas of computer graphics and simulations. The construction of nonobtuse triangulations of polygons has been investigated in numerous articles, such as those of Baker, Grosse and Raﬀerty , and those of Bern, Mitchell and Ruppert . Later, Maehara and Yuan obtained results on acute triangulations of polygons using a linear number of triangles. Zamﬁrescu’s survey gives a good introduction to the topic. As early as 1960, Burago and Zalgaller proved the existence of acute trian-gulations of general 2-dimensional polyhedral surfaces. The present paper reproves the same result using completely diﬀerent methods and by a very elementary con-struction. In contrast to the original proof, it provides a signiﬁcantly simpler proof of the result. Our result is complementary to the results obtained by Maehara and Yuan, and is motivated by many of the techniques used by them. As a corollary of our construction, we also obtain a nonobtuse subtriangulation of polygons with holes and surfaces of polyhedra that have been subdivided into polygonal regions. In the special case of triangulated polygons with no Steiner points1 in the interior, a construction was given by Bern and Eppstein , who proposed this more general result as a conjecture. In this paper, we have not addressed the issue of the complexity of the number of acute triangles needed to triangulate a polyhedral surface. It is known that a polygon can be triangulated using a linear number of triangles, and it would be interesting to see if such a bound can be achieved in the case of polyhedral surfaces as well. 32 Vassar St, Cambridge, MA 02139, email: shibs@mit.edu. 1By Steiner points, we just mean additional points that are not given as part of the input and serve as vertices in the triangulation. No other special properties are assumed for these points. 1 2 SHUBHANGI SARAF Another interesting generalization of this problem is to consider its analog in higher dimensions. Very little is known about acute tilings of three-dimensional Euclidean space. Very recently it was discovered that it is possible to tile three-dimensional space using tetrahedra having acute dihedral angles, see . However, it is still not even known if a cube can be subdivided into acute tetrahedra. In Section 2, we summarize the main results proved in this paper. In Section 3, we present the preliminary constructions of Steiner points on the edges and the circles. These points are useful in the construction of the ﬁnal triangulation. In Section 4, we present the ﬁnal construction of the nonobtuse triangulation using a grid imposition. 2. Main Results By a (general) polyhedral surface, we mean a 2-dimensional simplicial complex obtained as a ﬁnite union of triangles given with their edge lengths, and a map identifying pairs of edges. There is an intrinsic metric on the surface. Clearly, when two edges are identiﬁed, they have the same edge length. An acute (nonobtuse) triangulation of a polygon or polyhedral surface is a sub-division of the surface into nonoverlapping acute (nonobtuse) triangles such that any two distinct triangles are either disjoint, or they share a common vertex or a common edge. In Figure 1, we see a few examples illustrating the notion of a triangulation. Figure 1. An invalid triangulation, a nonobtuse triangulation of a square, and an acute triangulation of a triangle. The main result of this paper is a new proof of a special case of the Burago-Zalgaller theorem. Theorem 2.1. Every 2-dimensional polyhedral surface can be triangulated into nonobtuse triangles. Maehara and Yuan show how to obtain an acute triangulation of a polygon given a nonobtuse triangulation. In their construction, the number of triangles increases by a constant factor. Combined with Theorem 2.1, these results give a new and elementary proof of the Burago–Zalgaller theorem. Theorem 2.2. Every 2-dimensional polyhedral surface can be triangulated into acute triangles. It is clear that every polygon can be triangulated if there are no restrictions on the angles of the triangles. Also, every obtuse triangle can be subdivided into two nonobtuse triangles by drawing the altitude from the vertex forming the obtuse angle to the opposite side. The diﬃculty arises when we try to ﬁt these triangles together. To obtain a nonobtuse triangulation of a triangle, we may have to add ACUTE TRIANGULATIONS OF SURFACES 3 vertices in the interior of its edges. These vertices need to match those for any other triangle sharing the same edge. There are several elementary constructions of a nonobtuse triangulation of every (not necessarily convex) polygon. An interesting related question is the following. Given a polygon that has been subdivided into polygonal regions, can we obtain a nonobtuse triangulation of all the regions in such a way that the regions ﬁt together to give a nonobtuse triangulation of the original polygon? In this paper, we prove we can. We employ the “divide-and-conquer” technique to obtain a nonobtuse triangu-lation of a general polyhedral surface, by subtriangulating each triangle separately with matching Steiner points on the common edges. As we mentioned earlier, this construction resolves the Bern–Eppstein conjecture, but is also of independent interest. We deﬁne a subdivided polyhedral surface to be a 2-dimensional polyhedral surface that has been subdivided into polygonal regions. Theorem 2.3. Every subdivided polyhedral surface can be subtriangulated into non-obtuse triangles respecting the boundaries of the polygonal regions. Theorem 2.3 follows from the proof of Theorem 2.2. Unfortunately, our methods do not allow us to obtain acute subtriangulations of subdivided polyhedral surfaces. 3. Preliminary Constructions Let P be a polyhedral surface. Let V = {v1, . . . , vn} be the set of its vertices, E = {e1, . . . , em} the set of its edges, and F = {F1, . . . , Fk} the set of its faces. We need to obtain a triangulation of P satisfying the required conditions. In order to do so, in addition to the vertices V of P, we might have to add to the interiors of the faces and edges of P additional vertices that, together with V, comprise the vertices of the triangles in our triangulation. We call these additional points the Steiner points. We ﬁrst give a procedure for choosing vertices in the interiors of all the edges in E. In our ﬁnal triangulation, these vertices are the Steiner points on E. We then show how to triangulate any polygonal face F ∈F using nonobtuse triangles. The procedure ensures that the Steiner points on the boundary of F are precisely those vertices that are chosen initially in the interiors of the edges in E that bound F. For any two faces F1 and F2 with a common edge, the Steiner points along that edge are the same for both those faces. Hence, once we obtain such a triangulation, all the triangulated faces ﬁt together to give a triangulation of the entire polyhedron. We show how to add Steiner points to the face boundaries. Since P is ﬁnite, without loss of generality we may assume that the polygon is scaled such that the distance between any two vertices is at least 10. (A smaller number would have suﬃced.) Let θ be such that, for each face F ∈P, every angle determined by two adjacent edges in F is at least θ. Pick t such that 0 < t < sin(θ/2). Let xy be any edge of P. We pick a ﬁnite number of points, say s1, . . . , sr in that order, in the interior of xy such that ℓ(xs1) = ℓ(sry) = 1 and t < ℓ(sisi+1) < t √ 2 for 1 ≤i ≤r −1. We pick such points in the interior of each edge in P. Figure 2 illustrates this picking of points. This choice of points will be useful later on in the proof of Proposition 4.2. 4 SHUBHANGI SARAF X 1 1 y S1 S2 S3 . . . Sr Figure 2. Picking points in the interior of an edge. Let F be any given face in P. We show how to obtain a nonobtuse triangulation of F where the vertices on the boundary are precisely the original vertices of P in F along with the points added in the interior of each boundary edge. All the rest of the Steiner points are in the interior of F. We construct circles padding the boundary of F in the following manner. We may assume that F is a convex polygon. Let VF be the set of its original vertices, and SF the set of its Steiner points added in the interiors or its boundary edges. For each v ∈VF , we construct a circle of radius 1 with v as center, and for each s ∈SF , we construct a circle of radius t with s as center. By choice of SF and t, these circles cover the entire boundary of F. Moreover, any two adjacent circles intersect at an angle greater than π 2 . Let K be a point of intersection of two adjacent circles. The angle between the two circles is deﬁned as the smallest nonnegative angle through which one of the circles has to be rotated about K so that it becomes externally tangent to the other circle at the point K. We now show how to add nodes to the circle boundaries that will be Steiner points in our triangulation of the face F. Consider the inner boundary B of the region covered by the circles. By this we mean: look at the complement (in F) of the disks that are bounded by the circles we constructed. Then B is the boundary of the resulting region. We identify all the points of intersection of adjacent circles on B as nodes. If the arc of a circle between any two consecutive nodes subtends an angle greater than π 2 at the center of the circle, add a node to the midpoint of the arc which divides the arc into smaller arcs, each subtending a nonobtuse angle at the center. We set up a coordinate system and identify the X and Y directions. For each circle, if the lines through the center in the X and Y directions intersect the inner boundary of the region covered by circles, then we add these points of intersections as nodes as well. We do so to ensure that later on, in Proposition 4.3, certain boundary cases do not arise, thus simplifying the analysis. 4. Nonobtuse Triangulation In this section, we show how to impose a grid, and then use it to obtain the desired nonobtuse triangulation. Given a set H of lines in the horizontal direction and a set V of lines in the vertical direction, we say they form a rectangular grid. The lines are called the lattice lines, and the points of intersection of the lines are called the lattice points. The rectangles deﬁned by two consecutive horizontal lines and two consecutive vertical lines are called the lattice cells. The aspect ratio of the grid is deﬁned to be the supremum taken over all lattice cells, of the ratio of the length of the cell to its breadth. Lemma 4.1. Given a set of n points {(xi, yi) \| 1 ≤i ≤n} in R2, and given ǫ > 0, we can impose a rectangular grid of aspect ratio at most 1+ǫ, such that the n points are lattice points of the grid. ACUTE TRIANGULATIONS OF SURFACES 5 Proof. We’ll prove the lemma by giving a construction of such a grid. First draw lines in the x- and y- directions through each of the n points to form some rect-angular grid G. Our ﬁnal grid will be a reﬁnement of G. Let s be the smallest distance between any two consecutive parallel lines of G. Let m be natural number such that 1/m < ǫ. Choose δ > 0 such that δ < s/(m + 1). Claim: We can get a reﬁnement G′ of G, by adding additional lines in the X and Y directions, such that if d is the distance between some two consecutive parallel lines, then δ < d < δ(1 + 1 m). Proof of Claim: Consider any two consecutive parallel lines with distance s′ ≥s between them. Then s′ δ − s′ δ(1+ 1 m ) = s′ δ (1− m m+1) > s δ ( 1 m+1) > 1. Hence there exists an integer k that lies strictly between s′ δ and s′ δ(1+ 1 m ). Therefore, if k −1 equally spaced lines are added between the two originally given lines, then the distance between any two consecutive ones is between δ and δ(1 + 1 m), as was needed. We repeat this procedure for every pair of consecutive parallel lines in the grid G. This proves the claim. By the choice of d, this new grid G′ clearly has aspect ratio less than 1 + ǫ, and by construction it passes through the given n points. □ Applying Lemma 4.1 to the set of current nodes of F, we can impose a rectangular grid G of aspect ratio at most 1.2 such that each node is a lattice point. We add each point of intersection of the grid lines with the inner boundary B as a node. Let Q be the polygon whose boundary edges are the straight line segments joining any two adjacent nodes. Let R be the region of the face excluding the interior of Q. We next show how to get a nonobtuse triangulation of R and of Q. Proposition 4.2. The region R can be triangulated with nonobtuse triangles with-out adding any Steiner points in addition to those constructed originally on the edges of the face boundary and those constructed on the boundary B of the circles. Proof. By our choice of θ and the points si that are picked in the interiors of the edges of P (in Section 3), we ensure that adjacent circles intersect at an angle greater than π 2 . Hence the angle formed between segments joining centers of adjacent circles to their common point of intersection is less than π 2 . Also, for any two adjacent nodes, the arc between them subtends a nonobtuse angle at the center of the corresponding circle. Hence, when we add segments between the centers of the circles and the nodes on the circumferences of the respective circles, we get a triangulation of R by nonobtuse isosceles triangles without adding any additional Steiner points; see Figure 3. □ We say that Q′ is a reﬁnement of Q if in addition to the nodes of Q, we pick additional nodes on B, and consider the polygon formed by connecting any two adjacent nodes. Let R′ be the region of the the face F excluding the interior of Q′. Then just like R, also R′ can be triangulated by nonobtuse triangles. Therefore, it remains to triangulate Q or a reﬁnement of Q. Proposition 4.3. By adding points to the boundary B the region Q can be reﬁned to a region Q′, such that Q′ can be triangulated with nonobtuse triangles. The vertices of this triangulation are the lattice points of the imposed grid G and points on the boundary B that were added to obtain the reﬁnement. 6 SHUBHANGI SARAF Q Figure 3. The shaded region in the interior is Q. The ﬁgure illustrates the covering of the face boundary with circles. Proof. We introduce all the lattice points lying in the interior of Q as Steiner points. Every grid cell lying wholly in the interior of Q is a rectangle. The rectangles can be divided into two right angled triangles by a diagonal. At the boundary of Q, the grid cells intersect the boundary of Q to form right triangles, trapezoids, and pentagons. Figure 4 shows all the conﬁgurations that the arc of a circle can intersect a grid cell. The grid can be chosen ﬁne enough so that no two nonintersecting circles intersect the same grid cell. By construction, any two intersecting circles that make up the boundary B intersect at an angle greater than π 2 . Also, there is a grid line passing through all the points of intersections of the circles that lie on B. Together, these conditions imply that no grid cell is intersected by the arc of more than one circle. The points of intersection of the diameters in the X and Y directions with the boundaries of the circles were added as Steiner points; hence, clearly Cases 1, 3 and 6 of Figure 4 cannot arise. For Cases 2, 4, 5, 7 and 8, the natural nonobtuse triangulations of the intersection of the grid cell with the boundary of Q is shown in Figure 4. The only case that needs to be resolved is 9, in which the intersection of the grid cell with the boundary results in a pentagon; see Figure 5. This case is handled in Lemma 4.4. □ ACUTE TRIANGULATIONS OF SURFACES 7 1 9 8 7 6 5 4 3 2 Figure 4. Boundary cases. Lemma 4.4. Suppose a grid cell intersects the boundary of Q to form a pentagon as in Figure 4, Case 9. Then it is possible either to triangulate the pentagon with nonobtuse triangles or to reﬁne the pentagonal region by adding one more Steiner point to the arc of B contained in the pentagon and then triangulate the resulting region with nonobtuse triangles. Proof. Let ABCD be the grid cell, and EF be the part of the boundary of Q that is intersected by ABCD. As in Figure 5 below, we have arc EF intersecting the grid cell ABCD. We need to obtain a nonobtuse triangulation of the pentagonal region ABEFD or a region bounded after taking a reﬁnement of the arc EF by adding points in its interior. However, we cannot add points in the interiors of the segments BE, AB, DA, FD. The triangles ∆ABE and ∆ADF are right and hence nonobtuse. If ∆AEF is nonobtuse, then we are done. Consider the case when ∆AEF is obtuse. Without loss of generality, let ∠AEF be greater than π 2 . At F, we construct a ray perpen-dicular to DC. Since the points of intersection of the diameters in the X and Y directions with the boundaries of the circles were added as Steiner points, we note that the constructed ray at F lies in the exterior of the arc EF. Let L be a (moving) point on the arc EF, and let Q be the intersection of the line through L parallel to DC and the line through F parallel to DA. Let T be the intersection of the lines AL and QE. We assert that when L moves along the arc FE from F to E, then, at some stage, the angle ∠AT E becomes the right angle. If L = F, then Q = T = F and ∠AT E = ∠AFE < π 2 . When L →E, then AL →AE, and QL becomes parallel to DC. In this case, the angle α between AL and QE is clearly obtuse. Hence, at some intermediate point, α must equal π 2 . Fix L, Q, T at this stage. 8 SHUBHANGI SARAF A B C D E F T L Q Figure 5. Triangulating the pentagon. We introduce the vertex L on the arc to get a reﬁnement of the arc. We now show how to obtain a nonobtuse triangulation of the hexagonal region ABELFD without introducing any Steiner points on its boundary. It is clear that ∆ABE, ∆AT E, ∆ET L, ∆AT Q, ∆T QL, ∆QLF, and ∆DQF are all right triangles, and hence nonobtuse. Hence it remains to show that ∆AQD is nonobtuse. By assumption, ∠AEF > π 2 . So, ∠BAE > ∠FEC. Hence, \|BE\| \|AB\| > \|F C\| \|EC\|. Hence, \|FC\| < \|BE\| · \|EC\| \|AB\| ≤( 1 2(\|BE\| + \|EC\|))2 \|AB\| = \|BC\|2 4\|AB\|. Now, by Lemma 4.1, we may assume that the aspect ration of the rectangular grid is 1.2. Then, 1 1.2 ≤\|BC\| \|AB\| ≤1.2, and we can conclude that \|FC\| < 1.2\|BC\| 4 = 0.3\|BC\| = 0.3\|AD\|. Since 1 1.2 < \|DF \|+\|F C\| \|AD\| < \|F D\| \|AD\| + 0.3, we have \|FD\| > 0.533\|AD\|. Thus the entire ray FQ lies outside the circle with diameter AD, and hence so does Q. Therefore ∠AQD < π/2. But ∠ADQ and ∠DAQ are less than π/2. Therefore ∆AQD is nonobtuse. The proof of Lemma 4.4 is now complete. □ Together with triangulations of the other boundary cases as shown in Figure 4, Lemma 4.4 completes the proof of the existence of a nonobtuse triangulation of the faces of P with common Steiner points on the edges separating two adjacent faces. Hence, we get a nonobtuse triangulation of P, and thus have proved Theorem 2.1. As an immediate corollary of our proof of Theorem 2.1, we obtain the following result, which doesn’t seem to have been proved previously. Theorem 4.5. Every polyhedral surface that has been divided into polygonal regions can be subtriangulated into nonobtuse triangles respecting the boundaries of the polygonal regions. ACUTE TRIANGULATIONS OF SURFACES 9 Proof. We can view the polygonal regions as the “faces” of the polyhedron, and then apply Theorem 2.1. Thus we obtain a nonobtuse triangulation of the in-dividual faces with the same set of Steiner points on the common edge between any two adjacent faces. Hence the faces can be put together to give a nonobtuse triangulation of the entire surface. □ Once we obtain a nonobtuse triangulation of a polyhedral surface, we can also employ the techniques of Maehara to extend it to give an acute triangulation of the surface and obtain Theorem 2.2. We skip the details here; they are found in Maehara . Once we have a nonobtuse triangulation of the polyhedral surface, Maehara’s method of proof very naturally extends to the case of of a polyhedral surface. However, in the construction of an acute triangulation, the triangles might lie across edges separating two faces, and hence, might get folded and no longer be planar. Therefore, we only get an acute triangulation using geodesic acute triangles. It is not yet known whether one can get an acute subtriangulation of a polygon that has been subdivided into polygonal regions. 5. Acknowledgements Thanks to Prof. Igor Pak for posing the problem and for many valuable discus-sions. Thanks to the UROP department at MIT for funding the summer research. Thanks also to the anonymous referees for many helpful suggestions on improving the presentation of the paper. References B. S. Baker, E. Grosse and C. S. Raﬀerty, Nonobtuse triangulations of polygons, Discrete Comput. Geom., 3 (1998), 147–168. M. Bern, S. Mitchell and J. Ruppert, Linear-size nonobtuse triangulations of polygons, Dis-crete Comput. Geom., 14 (1995), 411–428. Y. D. Burago, V. A. Zallgaller, Polyhedral embedding of a net (Russian), Vestnik Leningrad. Univ., 15 (1960), 66–80. M. Bern, D. Eppstein, Polynomial-size nonobtuse triangualtion of polygons, International Journal of Computational Geometry and Applications, 2(3) (1992), 241–255. D. Eppstein, J. M. Sullivan, A. Ungor, Tiling space and slabs with acute tetrahedra, Com-putational Geometry: Theory and Applications, 27(3) (2004), 237–255. H. Maehara, Acute triangulations of polygons, European J. Combin., 23 (2002), 45–55, doi:10.1006/eujc.2001.0531 L. Yuan, Acute triangulations of polygons, Discrete Comput. Geom., 34 (2005), 697–706. T. Zamﬁrescu, Acute triangulations: a short survey, The sixth national conference of S.S.M.R., (2002),Sibiu, Romania, 9–17.
3830	https://physics.stackexchange.com/questions/620994/doubts-related-to-speed-of-sound-in-different-mediums	temperature - Doubts related to speed of sound in different mediums - Physics Stack Exchange Join Physics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Show activity on this post. since childhood we have been taught that sound travels faster in solid (most dense) as compared to liquids (less dense). and speed of sound is greater in liquids as compared to gases (least dense), sound can't travel in vacuum (0 density). and we were knowing the fact that speed of sound in moist air is more than dry air. and the reason we built was that more dense more speed. but recently i came to know that dry air is more dense than moist air .. why is this breaking the logic? pls help temperature acoustics density speed humidity Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Mar 14, 2021 at 9:27 Qmechanic♦ 222k 52 52 gold badges 636 636 silver badges 2.6k 2.6k bronze badges asked Mar 14, 2021 at 8:41 mr.anonymousmr.anonymous 79 1 1 silver badge 7 7 bronze badges 0 Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 5 Save this answer. Show activity on this post. It's to do with the fact that the dryness (humidity) of the air depends on temperature and the speed of sound in air also depends on temperature, see which has this equation v=331+0.6 T v=331+0.6 T Dry air is typically at a lower temperature than humid air. So even though it may be more dense, the overall effect is that the speed of sound can be lower in dry air. To imagine why warm air can hold more moisture than cold air - imagine a tray of water on a table in a cold room, not much water evaporates and is held by the air. In a warm room the water would quickly evaporate and be held by the air. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Mar 14, 2021 at 9:22 answered Mar 14, 2021 at 9:08 John HunterJohn Hunter 13.9k 2 2 gold badges 26 26 silver badges 58 58 bronze badges 4 The speed of sound in an ideal gas depends on its molecular weight. The MWs of oxygen (32) and nitrogen (28) are both significantly higher than water vapour (18) so the "MW of air" depends on the moisture content.alephzero –alephzero 2021-03-14 11:50:29 +00:00 Commented Mar 14, 2021 at 11:50 thats right but i am taking temp. to be same .. as @alephzero said this is the reason given in my textbooks . so according to this more dense less speed and vice versa(indirectly related) ,.. so why is this not applicable in solids. solids are significantly denser than gas but velocity in solids travel faster (directly related since more density more speed of sound) i am confused mr.anonymous –mr.anonymous 2021-03-14 11:57:19 +00:00 Commented Mar 14, 2021 at 11:57 i am studying speed of sound formula which was given by newton and corrected by laplace.. and i take temo to be same mr.anonymous –mr.anonymous 2021-03-14 11:58:53 +00:00 Commented Mar 14, 2021 at 11:58 imgur.com/a/tM7sIoI (<--- see this) why is this fact in contrary ti the fact that speed of sound is most in solids(even though density is very high)mr.anonymous –mr.anonymous 2021-03-14 12:06:46 +00:00 Commented Mar 14, 2021 at 12:06 Add a comment\| This answer is useful 5 Save this answer. Show activity on this post. The speed of sound in an ideal gas is c=γ k T m−−−−√c=γ k T m where k k is Boltzmann's constant, T T is the absolute temperature, m m is the mass of one molecule of the gas, and γ γ is the "adiabatic index" which depends on the number of internal modes of vibration of the molecule. Dry air is mostly a mixture of oxygen (m=32 m=32) and nitrogen (m=28 m=28) using atomic weight units for m m. Oxygen and nitrogen are both diatomic molecules, and γ=1.4 γ=1.4. Water vapor has m=18 m=18 and is triatomic, with the three atoms in the molecule not arranged in a straight line, and γ=1.333 γ=1.333. At the same temperature, the speed of sound in each gas is therefore proportional to \| Gas \| γ/m−−−−√γ/m \| --- \| \| Oxygen \| 0.21 \| \| Nitrogen \| 0.22 \| \| Water vapor \| 0.27 \| so the speed of sound in a mixture of the three gases (i.e. "air") is higher as the water vapor content increases. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Mar 14, 2021 at 12:12 alephzeroalephzero 10.3k 2 2 gold badges 26 26 silver badges 31 31 bronze badges Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. The speed of sound is given by : B ρ−−√B ρ. ρ ρ is the density and B is the bulk modulus of elasticity. The speed of sound is higher for moist air since its density is lower. This is because the speed does depend inversely on the square root of the density. The speed of sound for liquids in general is higher than it is for gases because the elasticity of liquids (Bulk Modulus) is higher. A liquid has a higher elasticity since it is difficult to compress compared to a gas. To get a comparison of the numbers involved, The bulk modulus of air ranges from about ∼∼ 1-1.5 ×10 5×10 5 Pascal, and the bulk modulus of water is ∼∼ 2 ×10 9×10 9 Pascal so the ratio of the bulk modulus of water to that of air B w a t e r B a i r B w a t e r B a i r is atleast 10000. The density of water is ∼∼ 1000 kg/m^3 and the density of air is ∼∼ 1 k g/m 3 k g/m 3 and so the density of air (dry/moist) is about 1000 times smaller than the density of water. i.e ρ w a t e r ρ a i r ρ w a t e r ρ a i r∼∼ 1000. Since the ratio B w a t e r B a i r>ρ w a t e r ρ a i r B w a t e r B a i r>ρ w a t e r ρ a i r, the speed of sound in water is greater than that in air. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Mar 14, 2021 at 12:55 AnyaCAnyaC 31 4 4 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. It's actually a common myth that density is responsible for the speed of sound in a medium. The belief is that the speed of sound is higher in more dense mediums, and slower in less dense mediums. If this is the case, why is the speed of sound faster in less dense helium gas than it is in air? Popular science YouTuber Cody'sLab explained how the speed of sound works in his video Thunderground 2: Artificial Thunder (explanation starts at 4:56). To summarize his points, he explained that the speed of sound actually depends on the strength of the intermolecular bonds of the material. For a sound wave to propagate, the atoms must be displaced, and then return to their original position. If they don't return, the energy is absorbed and the wave stops. The faster and more efficiently they return (via the pull of stronger intermolecular bonds), the faster the propagation of energy and therefore speed of sound is. It's generally true that solids have stronger intermolecular bonds than liquids which have stronger ones than gases, which is why the speed of sound approximately correlates with density. It's not a direct causation though, as the helium example demonstrates. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Mar 14, 2021 at 19:34 Drake PDrake P 159 5 5 bronze badges Add a comment\| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. 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3831	https://www.quora.com/If-a-function-is-monotonically-increasing-is-its-derivative-always-positive	Something went wrong. Wait a moment and try again. Meaning of Derivatives Function Growth Continuous Functions Calculus 1 Increasing/Decreasing Fun... Derivatives and Different... Calculus (Mathematics) 5 If a function is monotonically increasing, is its derivative always positive? B. S. Thomson Lived in Vancouver, BC · Author has 1.2K answers and 2.9M answer views · 2y A natural question. (Naive, of course, but it comes from a bot, so …) We first learn that a function with a positive derivative on an interval must be [strictly] increasing. So is the converse true? Well no! We learn pretty soon that the function is increasing on the whole real line but the derivative is zero at the origin. With minor effort then you can construct a continuous increasing function with plenty of points where the derivative is zero. How many points are possible? Any finite set? Some infinite set even? The real answer is surprising. There exist continuous increasing functions A natural question. (Naive, of course, but it comes from a bot, so …) We first learn that a function with a positive derivative on an interval must be [strictly] increasing. So is the converse true? Well no! We learn pretty soon that the function is increasing on the whole real line but the derivative is zero at the origin. With minor effort then you can construct a continuous increasing function with plenty of points where the derivative is zero. How many points are possible? Any finite set? Some infinite set even? The real answer is surprising. There exist continuous increasing functions with the property that at MOST points. The actual property is this: Definition. A function is said to be singular if the set of points in at which is false is a set of measure zero (i.e., almost everywhere in Lot's to learn on this topic, all of it interesting. Probably the first to use this terminology: H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928). Wiki reference: [Singular function - Wikipedia The graph of the winding number of the circle map is an example of a singular function. In mathematics , a real-valued function f on the interval [ a , b ] is said to be singular if it has the following properties: f is continuous on [ a , b ]. () there exists a set N of measure 0 such that for all x outside of N, the derivative f ′ ( x ) exists and is zero; that is, the derivative of f vanishes almost everywhere . f is non-constant on [ a , b ]. A standard example of a singular function is the Cantor function , which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the circle map . If f ( x ) = 0 for all x ≤ a and f ( x ) = 1 for all x ≥ b , then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists). Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets , described in a prototypical fashion by the Frenkel–Kontorova model and by the ANNNI model , as well as in some dynamical systems . Most famously, perhaps, they lie at the center of the fractional quantum Hall effect . When referring to functions with a singularity [ edit ] When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations , it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, " 1/x becomes singular at the origin, so 1/x is a singular function." Advanced techniques for working with functions that contain singularities have been developed in the subject called distributional or generalized function analysis. A weak derivative is defined that allows singular functions to be used in partial differential equations , etc. () This condition depends on the references [ 1 ] Lebesgue, H. (1955–1961), Theory of functions of a real variable , F. Ungar Halmos, P.R. (1950), Measure theory , v. Nostrand Royden, H.L (1988), Real Analysis , Prentice-Hall, Englewood Cliffs, New Jersey Lebesgue, H. (1928), Leçons sur l'intégration et la récherche des fonctions primitives , Gauthier-Villars]( "en.wikipedia.org") [3 Freilich, Gerald, Increasing continuous singular functions. Amer. Math. Monthly 80 (1973), 918–919. The author gives a simple construction of a strictly increasing continuous function whose derivative is zero almost everywhere. Related questions Is there a function where the derivative at a point is zero, but it is monotonically increasing at that point? What is (if existent; if not, why?) a monotonically increasing function whose monotonically increasing derivative intersects it infinitely many times? If a function is monotonically increasing (function is a cubic function), why should the discriminant of its derivative be zero? Is a function always strictly increasing if its derivative is positive for all values in its domain? What is the second derivative of a function? Andy Bruckner Studied Mathematics · Author has 763 answers and 759.2K answer views · 2y In his discussion of this question, Brian Thomson mentions an example of a continuous increasing function f such that f’(x) = 0 almost everywhere (a.e.). In particular, f’≥0 a.e. It is not differentiable everywhere. This suggests the question: Can one provide an example of a differentiable function g for which g’ ≥ 0 a.e., but fails to be at least non decreasing? The answer is “NO”. This is so because a derivative, g’ in this case, has a strong version of the IVP. If a set {x: c < g’(x) <d) is not empty, then this set has positive measure. So, if g’ were negative at a single point that would imp In his discussion of this question, Brian Thomson mentions an example of a continuous increasing function f such that f’(x) = 0 almost everywhere (a.e.). In particular, f’≥0 a.e. It is not differentiable everywhere. This suggests the question: Can one provide an example of a differentiable function g for which g’ ≥ 0 a.e., but fails to be at least non decreasing? The answer is “NO”. This is so because a derivative, g’ in this case, has a strong version of the IVP. If a set {x: c < g’(x) <d) is not empty, then this set has positive measure. So, if g’ were negative at a single point that would imply g’ < 0 on a set of positive measure and wouldn’t be ≥0 a.e. (In particular, there can be no differentiable example of the type Brian provides that fails to be non decreasing.) So, on how large a set S can a differentiable function h satisfy h’(x)≥0 and fail to be at least nondecreasing? Can S be dense? IF one restricts attention to functions considered in standard elementary calculus courses, the answer is “no”. In such courses a derivative h’ is continuous, so h’ ≥0 everywhere and h is nondecreasing. But what if we don’t assume h’ continuous? A light comes on! One of Quora’s go-to functions comes to mind. The Cantor function k has k’ = 0 off the Cantor set . Can we make something of that? By considering (- k) we get a non increasing function with 0 derivative on a dense open set. . If we consider m(x) =x/2 - k(x), we get a function that has m’ = 1/2 off the Cantor set yet has m(0) = 0 and m(1) = -1/2. But wait, k is not differentiable on the Cantor set! Can we fix that? Well, there are other nowhere dense perfect sets, some of which have positive measure in every open interval intersecting the set. We can construct a differentiable Cantor-like function relative to such sets. . Now do as we did for k above. The set on which the resulting function has a negative derivative has positive measure. It’s derivative is 1/2 on a dense open set. So this differentiable function is strictly increasing on each component of a dense open set, but is not even non decreasing. On differentiable functions having an everywhere dense set of intervals of constancy, Canad. Math, Bull. 8 (1965), 73–76. Maló Hautus Studied Mathematics at Eindhoven University of Technology (Graduated 1970) · Author has 274 answers and 105.6K answer views · Updated 1y First, you have to assume that the function has a derivative. If this is so, it follows easily that the derivative is nonnegative: The derivative at a point a is If we restrict ourselves to , it follows that , because is increasing. Therefore . (Remember that the limit of a positive function can be zero.) Note, however, that it is possible that . An example is . David Joyce Professor Emeritus of Mathematics at Clark University · Upvoted by Yair Livne , Master's Mathematics, Hebrew University of Jerusalem (2007) and Robby Goetschalckx , Computer scientist for 11+ years and passionate about math since childhood. · Author has 9.9K answers and 68.4M answer views · Updated 9y Related Is there a function where the derivative at a point is zero, but it is monotonically increasing at that point? The function has a zero derivative at but is monotonically increasing everywhere. Addendum. The definition of an increasing function depends only on its values. A function is said to be increasing (sometimes called strictly increasing) on an interval if for every pair of numbers and in that interval with it is the case that . The function satisfies that on the entire real line. Example. The derivative of the function is . Its derivative at 0 is 0, but its derivative everywhere else is positive. The function is increasi The function has a zero derivative at but is monotonically increasing everywhere. Addendum. The definition of an increasing function depends only on its values. A function is said to be increasing (sometimes called strictly increasing) on an interval if for every pair of numbers and in that interval with it is the case that . The function satisfies that on the entire real line. Example. The derivative of the function is . Its derivative at 0 is 0, but its derivative everywhere else is positive. The function is increasing everywhere. The graph has an inflection point at . Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder · Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? 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If f is a monotonically increasing function of g, is g a monotonically increasing function of f? How do you calculate the derivative of a function? Luís Sequeira PhD in Mathematics, University of Lisbon (Graduated 2001) · Author has 1.9K answers and 1.8M answer views · 1y Related Is a function always strictly increasing if its derivative is positive for all values in its domain? Assuming that the domain is an INTERVAL, and the function is differentiable everywhere, with positive derivative at every point, yes. This follows directly from Lagrange’s Theorem. For let [math]f \colon I \to \R[/math] is differentiable with [math]f'(x)>0[/math] for all [math]x \in I[/math]. Suppose, by way of contradiction, that [math]f[/math] is not strictly increasing, and pick [math]a, b \in I[/math] such that [math]a<b[/math] and [math]f(a) \ge f(b)[/math]. Then we have [math]\frac{f(b)-f(a)}{b-a} \le 0[/math]. By Lagrange’s Theorem, there exists some [math]c \in ]a,b[ \subseteq I[/math] with [math]f'(c) = \frac{f(b)-f(a)}{b-a} \le 0[/math], contradicting the assumption that [math]f'[/math] is always positive. Now if the domain is not Assuming that the domain is an INTERVAL, and the function is differentiable everywhere, with positive derivative at every point, yes. This follows directly from Lagrange’s Theorem. For let [math]f \colon I \to \R[/math] is differentiable with [math]f'(x)>0[/math] for all [math]x \in I[/math]. Suppose, by way of contradiction, that [math]f[/math] is not strictly increasing, and pick [math]a, b \in I[/math] such that [math]a<b[/math] and [math]f(a) \ge f(b)[/math]. Then we have [math]\frac{f(b)-f(a)}{b-a} \le 0[/math]. By Lagrange’s Theorem, there exists some [math]c \in ]a,b[ \subseteq I[/math] with [math]f'(c) = \frac{f(b)-f(a)}{b-a} \le 0[/math], contradicting the assumption that [math]f'[/math] is always positive. Now if the domain is not an interval, we can easily show a function that is differentiable, with positive derivative, but NOT strictly increasing (in fact, not increasing, even). For example, let [math]f \colon ]0,1[ \cup ]1,2[ \to \R[/math], defined by [math]f(x) = x - \lfloor x \rfloor[/math]. We have [math]f'(x)=1[/math] for every [math]x \in ]0,1[\cup]1,2[[/math], and yet, for example, [math]f(0.9)=0.9>f(1.1)=0.1[/math]. Romain Mondon-Cancel knows a bit of maths. · Author has 607 answers and 1.3M answer views · 8y Related What is (if existent; if not, why?) a monotonically increasing function whose monotonically increasing derivative intersects it infinitely many times? Okay, so I tried to find such a function. An idea is to find the general solution of the differential equation [math]f' - f = \sin x[/math], and try to find one such solution that match your increasing constraints. Let [math]f_a(x) = -\frac{\sin x + \cos x}{2}[/math]. Then [math]f_a'(x) = -\frac{\cos x - \sin x}{2}[/math] and [math]f_a'(x) - f_a(x) = \sin x[/math]. We hence have a specific solution. The general term of the solution of [math]f' - f = 0[/math] is [math]f_0(x) = ce^x[/math]. Hence the general form of the solution, [math]f(x) = ce^x - \frac{\sin x + \cos x}{2}[/math]. However, because [math]e^x \rightarrow 0[/math] when [math]x \rightarrow -\infty[/math], such an [math]f'[/math] cannot be positive for all [math]x \in [/math] Okay, so I tried to find such a function. An idea is to find the general solution of the differential equation [math]f' - f = \sin x[/math], and try to find one such solution that match your increasing constraints. Let [math]f_a(x) = -\frac{\sin x + \cos x}{2}[/math]. Then [math]f_a'(x) = -\frac{\cos x - \sin x}{2}[/math] and [math]f_a'(x) - f_a(x) = \sin x[/math]. We hence have a specific solution. The general term of the solution of [math]f' - f = 0[/math] is [math]f_0(x) = ce^x[/math]. Hence the general form of the solution, [math]f(x) = ce^x - \frac{\sin x + \cos x}{2}[/math]. However, because [math]e^x \rightarrow 0[/math] when [math]x \rightarrow -\infty[/math], such an [math]f'[/math] cannot be positive for all [math]x \in \mathbb{R}[/math], hence [math]f[/math] cannot be increasing. So we have to find a way around that: the problem being for high negative values of [math]x[/math], let us build a function piecewise, one part for [math]x \geq 0[/math] and one for [math]x < 0[/math]. First, let us choose [math]c[/math] so that the positive part always satisfies our identity: [math]f'(x) = ce^x - \frac{\cos x - \sin x}{2}[/math] [math]f''(x) = ce^x + \frac{\cos x + \sin x}{2}[/math] As [math]\forall x \in \mathbb{R}, \frac{\cos x + \sin x}{2} > -1, - \frac{\cos x - \sin x}{2} > -1[/math], if [math]c = 1[/math] then [math]\forall x \geq 0, f'(x) > 0, f''(x) > 0[/math]. So let us take [math]c = 1[/math]. [math]f[/math] and [math]f'[/math] are therefore both increasing. Now for the function at [math]0[/math], to ensure that the function is continue and differentiable, we should compute [math]f(0)[/math] and [math]f'(0)[/math]: [math]f(0) = e^0 - \frac{\sin0 + \cos0}{2} = 1 - \frac12 = \frac12[/math] [math]f'(0) = e^0 - \frac{\cos0 - \sin0}{2} = 1 - \frac12 = \frac12[/math] So if we define [math]f(x) = \frac{1 + x}{2}[/math] for [math]x < 0[/math], which is increasing, and its derivative being [math]f'(x) = \frac12[/math] also, then the function will be continue on [math]0[/math]. Hence the final form of the function: [math]f(x) = \begin{cases} e^x - \frac{\sin x + \cos x}{2} & x \geq 0 \ \frac{1 + x}{2} & x < 0 \end{cases}[/math] Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. Matthew Bond Ph.D., harmonic analysis, Michigan State, 2011 · Upvoted by Aksh , M.Sc. Mathematics, National Institute of Technology, Calicut (2020) and Pepito Moropo , B.S. / M.S. Mathematics · Author has 359 answers and 1.2M answer views · 7y Related What is happening to a function when the derivative is not defined? It’s a little hard to classify something by the absence of a structure, but differentiability is a very common property of our most popular and easy-to-use functions. “Nice” functions “look straight when you zoom in”, and “bad” functions don’t. And there’s more than one way to be “bad”. I’ll start with some common examples that are closer to basic functions before giving a rough idea of what more advanced cases look like, as these are not usually mentioned in a Calc 1 class. Any place where a function is not continuous is the simplest case. This can be at a jump, or a “compressed spring” shape s It’s a little hard to classify something by the absence of a structure, but differentiability is a very common property of our most popular and easy-to-use functions. “Nice” functions “look straight when you zoom in”, and “bad” functions don’t. And there’s more than one way to be “bad”. I’ll start with some common examples that are closer to basic functions before giving a rough idea of what more advanced cases look like, as these are not usually mentioned in a Calc 1 class. Any place where a function is not continuous is the simplest case. This can be at a jump, or a “compressed spring” shape such as [math]\sin(1/x)[/math], or the function can be all over the place and nowhere continuous, like the function f(rational)=1, f(irrational)=0. (Above: [math]f(x)=\sin(1/x), f(0)=0)[/math]. Limit at 0 DNE.) Continuous-but-not-differentiable-at-a-point is somewhat more interesting, because then we actually get to say something about the derivative. The simplest case is a bounce, such as [math]\|x\|[/math] at the origin - the slope wants to be two different things. Other common cases are cusps (a bounce that approaches pure vertical, such as [math]\sqrt{\|x\|}[/math] or [math]x^{2/3}[/math] at the origin), or vertical tangents (such as [math]x^{1/3}[/math]). There are actually many more cases, and some functions are known to be continuous everywhere, but nowhere-differentiable. They were not discovered until the late 1800s, and can be hard to draw or imagine. These are functions that are “jittery” but still continuous, like a stock market graph. Brownian motion (let’s do the one-variable case, a particle moving up and down over time) is a famous example that moves randomly but continuously, preferring neither up or down. It changes direction too much to stay close to any tangent line, no matter how small you slice it. It always has secant lines sloping both upward and downward. The first example of a function proven to be everywhere continuous but nowhere differentiable is credited to Weierstrass: Weierstrass function. This construction is not random, so you can get a specific example and plug numbers into the function. The idea is to add together a lot of functions that get smaller, but rapidly more “jittery”, so that you throw the derivative into chaos without the function itself blowing up. For example, something like: [math]f(x)= \cos(x)+\frac1{2}\cos(10x)+\frac1{4}\cos(100x)+\frac1{8}\cos(1000x)+...+\frac1{2^n}\cos(10^nx)+...[/math] (The following graph is a Weierstrass function, but not necessarily same one given above) It used to be thought that continuous functions were differentiable-with-maybe-a-few-exceptions, but now we know better. Ivan Pastine Mom made me take math. Now they pay me for it. Thanks mom. · 8y Related What is (if existent; if not, why?) a monotonically increasing function whose monotonically increasing derivative intersects it infinitely many times? [math]e^x[/math] It's derivative is also [math]e^x[/math] I now notice that you are looking for a function who’s derivative intersects at discrete points, not everywhere. You can get something like that by adding an oscillating part to the above function. So for example take: [math]e^x+sin(x)[/math] It’s derivative is [math]e^x+cos(x)[/math] These need to be monotonically increasing (that was your requirement) and both sin and cos have a minimum of -1, so if we define our function only over x>0 (so that [math]e^x>1[/math]) we get this. sin and cos are equal at 45, 45+180, 45+360, etc. An infinite number of points. But adding a different small, smooth, oscillating [math]e^x[/math] It's derivative is also [math]e^x[/math] I now notice that you are looking for a function who’s derivative intersects at discrete points, not everywhere. You can get something like that by adding an oscillating part to the above function. So for example take: [math]e^x+sin(x)[/math] It’s derivative is [math]e^x+cos(x)[/math] These need to be monotonically increasing (that was your requirement) and both sin and cos have a minimum of -1, so if we define our function only over x>0 (so that [math]e^x>1[/math]) we get this. sin and cos are equal at 45, 45+180, 45+360, etc. An infinite number of points. But adding a different small, smooth, oscillating part to [math]e^x[/math] (small enough so that the function and its derivative are still monotonically increasing) would also do the trick. There is nothing magic about sin(x) here. Sponsored by CDW Corporation Want document workflows to be more productive? The new Acrobat Studio turns documents into dynamic workspaces. Adobe and CDW deliver AI for business. Bernd Leps Former scientific official; retired · Author has 5.8K answers and 1.3M answer views · 1y Related Is a function always strictly increasing if its derivative is positive for all values in its domain? The problem lies in the “in its domain”. If the domain are in parts separated by values or ranges not in the domain, the function may be strictly increasing within that parts of the domain, if the derivative is positive within that part of the domain, but that will nothing say about the relative values in different parts of the domain. That is “strictly increasing” must be defined with respect to non-domain regions. If you clear that possibility from the definition, the answer would be “yes”: A continuous function with derivative > 0 is strictly raising there. In the other answer you have an exam The problem lies in the “in its domain”. If the domain are in parts separated by values or ranges not in the domain, the function may be strictly increasing within that parts of the domain, if the derivative is positive within that part of the domain, but that will nothing say about the relative values in different parts of the domain. That is “strictly increasing” must be defined with respect to non-domain regions. If you clear that possibility from the definition, the answer would be “yes”: A continuous function with derivative > 0 is strictly raising there. In the other answer you have an example of that Philip Lloyd Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views · 1y Related Can a function have a negative derivative but always have positive values? The first thing I thought of was the opposite! The first thing I thought of was the opposite! Knows Polish · Author has 4K answers and 1.3M answer views · 1y Related Is a function always strictly increasing if its derivative is positive for all values in its domain? Not always f(x) = -1/x has positive derivative for all x within the domain (1/x^2) and yet is not strictly increasing as for x1 < 0 and x2 > 0, we have x1 < x2, but f(x1) > f(x2) David Joyce Professor Emeritus of Mathematics at Clark University · Upvoted by Justin Rising , PhD in statistics and Jay Wacker , theoretical physicist · Author has 9.9K answers and 68.4M answer views · 9y Related Mathematics: Is the derivative of a positive function also positive? The derivative is the rate of change. When it's positive the function is increasing, when the derivative is negative, the function is decreasing. Different functions will have the same derivative when they differ by a constant. For example [math]f(x)=x^2+100[/math] (graphed in red below) and [math]g(x)=x^2-100[/math] (graphed in blue) both have the derivative [math]2x[/math] (graphed in green). But [math]f(x)[/math] is positive on the interval math[/math] while [math]g(x)[/math] is negative there. Knowing only the derivative a function can't tell you whether the function is positive or negative. And knowing only whether the function is positive or negative can't The derivative is the rate of change. When it's positive the function is increasing, when the derivative is negative, the function is decreasing. Different functions will have the same derivative when they differ by a constant. For example [math]f(x)=x^2+100[/math] (graphed in red below) and [math]g(x)=x^2-100[/math] (graphed in blue) both have the derivative [math]2x[/math] (graphed in green). But [math]f(x)[/math] is positive on the interval math[/math] while [math]g(x)[/math] is negative there. Knowing only the derivative a function can't tell you whether the function is positive or negative. And knowing only whether the function is positive or negative can't tell whether the derivative will be positive or negative. Jack Huizenga Math Professor, Penn State · Author has 613 answers and 4.4M answer views · 11y Related What is a function that is real-valued, positive everywhere, differentiable everywhere, and monotonically increasing? The function whose graph is this (assuming you forgive the limitations of my MSPaint skill): Or maybe this: (This one has the x-axis as a horizontal asymptote too.) Despite the fact that every function in every calculus book ever is something like [math]f(x) =(x^2+1)e^x[/math] (which is another example, by the way), a function doesn't have to have a nice algebraic description. When looking for examples, restricting yourself to things that have nice algebraic names is quite unnecessarily restrictive. If the question is simply "is there a real-valued, positive, differentiable, monotonically increasing functi The function whose graph is this (assuming you forgive the limitations of my MSPaint skill): Or maybe this: (This one has the x-axis as a horizontal asymptote too.) Despite the fact that every function in every calculus book ever is something like [math]f(x) =(x^2+1)e^x[/math] (which is another example, by the way), a function doesn't have to have a nice algebraic description. When looking for examples, restricting yourself to things that have nice algebraic names is quite unnecessarily restrictive. If the question is simply "is there a real-valued, positive, differentiable, monotonically increasing function" the answer is clearly yes. Of course, it might be easier to prove the function works if you have an explicit formula for it. But when proving general things about functions (as in a real analysis course) your functions are these kinds of general things with no "formula" anyways. Related questions Is there a function where the derivative at a point is zero, but it is monotonically increasing at that point? What is (if existent; if not, why?) a monotonically increasing function whose monotonically increasing derivative intersects it infinitely many times? If a function is monotonically increasing (function is a cubic function), why should the discriminant of its derivative be zero? Is a function always strictly increasing if its derivative is positive for all values in its domain? What is the second derivative of a function? How do you find the derivative of a function with respect to X? What is a function that is real-valued, positive everywhere, differentiable everywhere, and monotonically increasing? Is the derivative of a function always smaller than the original function? If f is a monotonically increasing function of g, is g a monotonically increasing function of f? How do you calculate the derivative of a function? What does it mean if a function has an identical derivative? What is a monotonically increasing function? Can a function have more than one derivative? Is the derivative of one polynomial function always another polynomial function? Is the derivative of an exponential function an exponential function? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
3832	https://www.amazon.com/Developing-Human-Clinically-Oriented-Embryology/dp/1437720021	The Developing Human: Clinically Oriented Embryology with Student Consult Online Access, 9th Edition: 9781437720020: Medicine & Health Science Books @ Amazon.com Skip to Main content About this item About this item About this item Buying options Compare with similar items Videos Reviews Keyboard shortcuts Search opt+/ Cart shift+opt+C Home shift+opt+H Orders shift+opt+O Add to cart shift+opt+K Show/Hide shortcuts shift+opt+Z To move between items, use your keyboard's up or down arrows. .us Delivering to North Cha... 29415 Update location Books Select the department you want to search in Search Amazon EN Hello, sign in Account & Lists Returns& Orders0 Cart Sign in New customer? Start here. 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Other sellers on Amazon New & Used (36) from$7.60$7.60&FREE Shipping Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required. Read instantly on your browser with Kindle for Web. Using your mobile phone camera - scan the code below and download the Kindle app. Image Unavailable Image not available for Color: To view this video download Flash Player VIDEOS 360° VIEW IMAGES Read sample Follow the authors Keith L. Moore Keith L. Moore Follow T.V.N.(Vid)Persaud, MD,PhD T.V.N.(Vid)Persaud,… Follow Something went wrong. Please try your request again later. OK The Developing Human: Clinically Oriented Embryology with Student Consult Online Access, 9th Edition 9th Edition by Keith L. Moore(Author), T. V. N. Persaud(Author), Mark G. Torchia(Author)&1 more 4.2 4.2 out of 5 stars58 ratings Sorry, there was a problem loading this page. 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Keith L. Moore, T.V.N. Persaud, and Mark G. Torchia, delivers the world’s most complete, visually rich, and clinically oriented coverage of this complex subject. Written by some of the world’s most famous anatomists, it presents week-by-week and stage-by-stage views of how fetal organs and systems develop, why and when birth defects occur, and what roles the placenta and fetal membranes play in development. You can also access the complete contents online at www.studentconsult.com, along with 17 remarkable animations, downloadable illustrations, additional review questions and answers, and more. Access the full contents of the book online at www.studentconsult.com - as well as 17 remarkable animations that bring normal and abnormal embryological development to life, and hundreds of additional review questions and answers to test your mastery of the material. Acquire a detailed grasp of human embryology with the world’s most comprehensive, richly illustrated, and clinically oriented coverage from a cadre of leading world authorities. Effectively prepare for exams with review questions and answers at the end of each chapter. Understand all of the latest advances in embryology, including normal and abnormal embryogenesis, causes of birth defects, and the role of genes in human development. See how discoveries in molecular biology have affected clinical practice, including the development of sophisticated new techniques such as recumbent DNA technology and stem cell manipulation. Prepare for the USMLE Step 1 with clinical case presentations, highlighted in special boxes, that demonstrate how embryology concepts relate to clinical practice. Read more Report an issue with this product or seller Previous slide of product details ISBN-10 1437720021 ISBN-13 978-1437720020 Edition 9th Publisher Saunders Publication date December 19, 2011 Language English Dimensions 8.75 x 0.75 x 11 inches Print length 560 pages Next slide of product details See all details There is a newer edition of this item: The Developing Human: Clinically Oriented Embryology $60.18 (3) Only 14 left in stock - order soon. Frequently bought together [x] This item: The Developing Human: Clinically Oriented Embryology with Student Consult Online Access, 9th Edition $63.48$63.48 Get it as soon as Friday, Oct 3 Only 1 left in stock - order soon. 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Product details Publisher ‏ : ‎ Saunders Publication date ‏ : ‎ December 19, 2011 Edition ‏ : ‎ 9th Language ‏ : ‎ English Print length ‏ : ‎ 560 pages ISBN-10 ‏ : ‎ 1437720021 ISBN-13 ‏ : ‎ 978-1437720020 Item Weight ‏ : ‎ 2.75 pounds Dimensions ‏ : ‎ 8.75 x 0.75 x 11 inches Best Sellers Rank: #2,849,946 in Books (See Top 100 in Books) 46 in Embryology (Books) Customer Reviews: 4.2 4.2 out of 5 stars58 ratings Brief content visible, double tap to read full content. Full content visible, double tap to read brief content. Videos Help others learn more about this product by uploading a video! Upload your video About the authors Follow authors to get new release updates, plus improved recommendations. Previous page Follow Keith L. Moore -------------- Brief content visible, double tap to read full content. Full content visible, double tap to read brief content. Keith L. Moore, BA,MSc,PhD,DSc(Hon),FAAA,FRSM,FIAC Dr. Moore, Professor Emeritus in the Division of Anatomy, Department of Surgery,Faculty of Medicine,University of Toronto, Toronto ON, Canada. Prior to his retirement in 1991, Dr. Moore was Chair of the Department of Anatomy (1976-1985)and Associate Dean of Basic Medical Sciences(1985-1990). Before moving to the University of Toronto, Dr. Moore was Professor and Head of the Department of Anatomy (1956-1975) in the Faculty of Medicine at the University of Manitoba in Winnipeg,MB. A graduate of Western University in London ON,Canada, Dr. Moore's research was mainly concerned with the causes and prevention of congenital anomalies (birth defects). His work led to innovative methods for the detection of numerical chromosomal anomalies. In fact, he was the first scientist to recognize that males with Klinefelter Syndrome had an extra X chromosome. In 1965, Dr. Moore developed a buccal smear sex chromatin test that is still used for the detection of numerical chromosomal abnormalities. Dr. Moore is an internationally recognized leader in the teaching of human anatomy and embryology. An inspirational teacher, his writings and textbooks are wildly regarded as among the best in the field, and are used in medical colleges around the world. His valuable contributions are a benefit to education, science and society as a whole. Dr. Moore has published numerous scientific papers dealing with prenatal development and birth defects. He is the author of 13 medical textbooks in embryology, anatomy and neuroanatomy. Five of these books are still in print. The 8th edition of The Developing Human has been translated into 12 other languages. The 6th edition of his book, Clinically Oriented Anatomy, has been translated into seven other languages. It is used worldwide and by 80% of medical and dental students in North America. The other books, Essential Clinical Anatomy/4th ed., Before We Are Born/8th ed., and Color Atlas of Clinical Anatomy/2nd ed., are student's digests. Dr. Moore has been invited by universities in many other countries to lecture and to participate in various professional meetings. He is a charter member of the Canadian Association of Anatomists since 1954 (elected President in 1965); a Fellow of the Royal Society of Medicine since 1985 (FRSM); a Fellow of the International Academy of Cytology since 1968 (FIAC); a member of the American Association of Anatomy since 1955 (AAA); a member of the American Association of Clinical Anatomists since 1983 (AACA),Honored Member of the American Association of Clinical Anatomists (1994), Member of the Federative International Committee of Terminology from 1989-2019 (FICAT), and Fellow of the American Association of Anatomists in 2008(FAAA). The rank of Fellow honors distiguished members who have demonstrated excellce in science and their overall contributions to the medical sciences. In 2009, he was appointed Honorary Member of the Italian Society of Anatomy and Histology (SIAI) on the basis of his scientific and academic curriculum. Dr. Moore, one of the best known anatomists in the world today, is the recipient of the J.C.B. Grant Award of the Canadian Association of Anatomists; Honored Member of the American Association of Clinical Anatomists;in 2007 he was the recipient of the Henry Gray/Elsevier Distinguished Educator Award in recognition of sustained excellence & leadership in human anatomy education; Honorary Doctor of Science (DSc) from The Ohio State University in 2012 in recognition of his valuable contibutions to science; the R. Benton Adkins Jr. Distinguished Service Award from the American Association of Clinical Anatomists in 2012, and was presented by Command of Her Majesty The Queen, the Queen Elizabeth 11 Diamond Jubilee Medal in commemoration of the sixtieth anniversary of Her Majesty's accession to the Throne, and in recognition of his contributions to Canada. See more on the author's page 2. Follow T.V.N.(Vid)Persaud, MD,PhD -------------------------- Brief content visible, double tap to read full content. Full content visible, double tap to read brief content. Dr. T. V.N. (Vid) Persaud, Professor Emeritus, Dept. of Human Anatomy and Cell Science, University of Manitoba - received the 2010 Henry Gray/Elsevier Distinguished Educator Award, American Association of Anatomists highest award for human anatomy education. Dr. Persaud was formerly Professor and Head of the Department of Anatomy at the University of Manitoba. He is also recipient the J.C.B. Grant Award of the Canadian Association of Anatomists, and the Honored Member Award of the American Association of Clinical Anatomists. Dr. Persaud is now a Visiting Professor at St. George's University School of Medicine in Grenada where he teaches gross anatomy and embryology See more on the author's page Next page TopAbout this itemSimilarReviews Related books Page 1 of 1Start Over Sponsored Previous page Shop the Store on Amazon › Gerontological Nurse Exam Study Cards: Gerontological Nurse Practitioner Certification Review and Practice Test Questions [Full Color Cards] 4.9 4.9 out of 5 stars 14 $34.99$34.99 List:$39.99$39.99 Shop the Store on Amazon › Gerontological Nursing Certification Review: 3 Full-Length Practice Tests & 450 Questions - Complete Exam Prep for GERO-BC 5.0 5.0 out of 5 stars 7 $23.99$23.99 Shop the Store on Amazon › Nursing Pharmacology Drug Workbook: 800+ Interactive Exercises to Master Nursing Meds – With Clinical Scenarios, Mnemonics, and Patient Safety Strategies 4.7 4.7 out of 5 stars 37 $25.87$25.87 Shop the Store on Amazon › The Complete EKG \| ECG Interpretation for Nurses: Identifying & Treating Arrhythmias with Confidence! 4.1 4.1 out of 5 stars 106 $17.75$17.75 Next page Customer reviews 4.2 out of 5 stars 4.2 out of 5 58 global ratings 5 star 4 star 3 star 2 star 1 star 5 star 65%18%5%4%8%65% 5 star 4 star 3 star 2 star 1 star 4 star 65%18%5%4%8%18% 5 star 4 star 3 star 2 star 1 star 3 star 65%18%5%4%8%5% 5 star 4 star 3 star 2 star 1 star 2 star 65%18%5%4%8%4% 5 star 4 star 3 star 2 star 1 star 1 star 65%18%5%4%8%8% How customer reviews and ratings work Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them. To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness. Learn more how customers reviews work on Amazon Customers say Customers find the book helpful for medical education, with one noting its clinically oriented approach and another mentioning it's required for undergraduate embryology classes. Moreover, the content is well-explained, with one customer highlighting the useful diagrams for understanding timelines. Additionally, the book is perfectly readable and arrives in great condition. Generated from the text of customer reviews Select to learn more Educational valueDetailReadabilityPacing 10 customers mention "Educational value"10 positive 0 negative Customers find the book educational and helpful, with one mentioning it works well for medical school and another noting its clinically oriented approach. "This book was required for my undergrad embryology class. An I have to say that I really like it...." Read more "this book is what I use for my Embryology class. It really is helpful and explains everything there is to know!" Read more "...This books completes perfectly this type of education" Read more "Good resource" Read more 7 customers mention "Detail"6 positive 1 negative Customers appreciate the book's comprehensive coverage of embryology, with one customer noting that the diagrams are especially useful for understanding timelines. "The explanations are fantastic - not too hard, but not simple either...." Read more "...It really is helpful and explains everything there is to know!" Read more "This book is very helpful to the medical student. The diagrams are clear and the book is well written. No regrets with this purchase." Read more "...Strongly recommend! It gives me a clear understanding about embryology." Read more 4 customers mention "Readability"4 positive 0 negative Customers find the book perfectly readable, with one mentioning it is understandable for those with a medical background. "...used (underlined, frayed edges, slight stains) but useable &perfectly readable." Read more "...provided as not to bog you down into details but still be able to understand the book if you get stuck somewhere...." Read more "...Material in the book well presented and able to read by one with a medical background." Read more "Excellent book, clear, concise and easy to read. Want to know everything about embryology in an easy to read manner, this is the book." Read more 3 customers mention "Pacing"3 positive 0 negative Customers are satisfied with the pacing of the book. "Great condition! I thought it was going to be secondhand so I was very pleased to find out it was brand new...." Read more "Intact& Readable..." Read more "Great condition..." Read more View Image Gallery Amazon Customer 5.0 out of 5 stars Images in this review Reviews with images See all photos Previous page Next page All photos Amazon Customer 5 out of 5 stars Intact & Readable Got it as expected, used (underlined, frayed edges, slight stains) but useable & perfectly readable. MoreHide Thank you for your feedback Close Sorry, there was an error Close Sorry we couldn't load the review Try again Top reviews from the United States There was a problem filtering reviews. Please reload the page. JP ##### 5.0 out of 5 stars Works Well for Med School Reviewed in the United States on March 27, 2013 Format: PaperbackVerified Purchase This was a "required" book for school (though nobody actually buys them) but I'm glad I purchased it. There's a ton of info out there on the interwebs, but I've learned that sometimes it's nice to have a concise reference... especially when the professor is using it as a guide to teach. Read more 2 people found this helpful Helpful Report Janine B. ##### 5.0 out of 5 stars Intact & Readable Reviewed in the United States on January 13, 2019 Format: PaperbackVerified Purchase Got it as expected, used (underlined, frayed edges, slight stains) but useable & perfectly readable. Read more Janine B. 5.0 out of 5 stars Intact & Readable Reviewed in the United States on January 13, 2019 Got it as expected, used (underlined, frayed edges, slight stains) but useable & perfectly readable. Images in this review Helpful Report A. Cardona Villanueva ##### 4.0 out of 5 stars An I have to say that I really like it. The student consult online access have some ... Reviewed in the United States on October 13, 2015 Format: PaperbackVerified Purchase This book was required for my undergrad embryology class. An I have to say that I really like it. The student consult online access have some audiovisuals that were really helpful. Also learning from it was facilitated by the clinically oriented approach. The only complain is that the end of chapter questions are out of chapter for some chapters in the online version of the book. Read more Helpful Report Alex Martin ##### 5.0 out of 5 stars Great condition Reviewed in the United States on August 18, 2020 Format: PaperbackVerified Purchase Just what I needed Read more Helpful Report Laureen ##### 1.0 out of 5 stars One Star Reviewed in the United States on June 28, 2014Verified Purchase Just get the textbook - paper back version Read more Helpful Report Danielle Kraack ##### 5.0 out of 5 stars The Developing Human - Great Reviewed in the United States on June 10, 2013 Format: PaperbackVerified Purchase The explanations are fantastic - not too hard, but not simple either. The diagrams are especially useful in understanding timelines and difficult concepts. I highly recommend this text for all college levels. Read more 2 people found this helpful Helpful Report yaigc ##### 5.0 out of 5 stars this book is what I use for my Embryology class ... Reviewed in the United States on October 6, 2016 Format: PaperbackVerified Purchase this book is what I use for my Embryology class. It really is helpful and explains everything there is to know! Read more Helpful Report Jettro77 ##### 3.0 out of 5 stars Why no page numbers? Reviewed in the United States on January 27, 2013Verified Purchase I don't like the fact that all kindle books don't use page numbers. It uses location numbers. This is a text book and my professor assigns reading pages based on page number not location numbers. So I have to find someone with a hard copy to find what I am looking for. I don't recommend using kindle for text books for this exact reason. Read more 4 people found this helpful Helpful Report See more reviews Top reviews from other countries Translate all reviews to English Vishnudevan T S ##### 5.0 out of 5 stars Five Stars Reviewed in India on November 27, 2017Verified Purchase Excellent Read more Report Tracey ##### 5.0 out of 5 stars Very useful Reviewed in the United Kingdom on October 6, 2015 Format: PaperbackVerified Purchase Useful in understanding the physiology of the developing fetus. Explains things well and great illustrations. Great reference tool/go-to book. Read more Report Chris Wright ##### 5.0 out of 5 stars... I didn't need the textbooks but it still looks like it has a lot of good information Reviewed in Canada on August 1, 2015 Format: PaperbackVerified Purchase Found out I didn't need the textbooks but it still looks like it has a lot of good information. It got here quickly and I'm excited to read it Read more Report giovanni ##### 5.0 out of 5 stars gut zum Studium Reviewed in Germany on August 27, 2012 Format: PaperbackVerified Purchase Sehr klarer Text, weiter erklährt mit schönen und gut begreifbaren Zeichnungen, die ihrerseits mit klaren aus der Klinik kommenden Photo begleitet sind. Read more ReportTranslate review to English See more reviews TopAbout this itemSimilarReviews Your recently viewed items and featured recommendations › View or edit your browsing history After viewing product detail pages, look here to find an easy way to navigate back to pages you are interested in. 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3833	https://www.worldcat.org/title/fundamentals-of-electric-circuits/oclc/1050614624?referer=&ht=edition	Fundamentals of electric circuits \| WorldCat.org Skip to main contentSkip to footer Items Create accountSign in Pages Home Libraries Topics Lists About For Librarians North Charleston, South Carolina, US Fundamentals of electric circuits 23 reviews Authors: Charles K. Alexander (Author), Matthew N. O. Sadiku (Author) Summary: "Alexander and Sadiku's sixth edition of Fundamentals of Electric Circuits continues in the spirit of its successful previous editions, with the objective of presenting circuit analysis in a manner that is clearer, more interesting, and easier to understand than other, more traditional texts. Students are introduced to the sound, six-step problem solving methodology in chapter one, and are consistently made to apply and practice these steps in practice problems and homework problems throughout the text."--Publisher's website Show more eBook, English, 2017 Edition: Sixth edition View all formats and editions Publisher: McGraw-hill Education, New York, NY, 2017 Physical Description: 1 online resource (xxii, 903 pages, 63 variously numbered pages .) ISBN: 9781259663918, 1259663914 OCLC Number / Unique Identifier: 1050614624 Subjects: Circuits électriques Electric circuits circuits Additional Physical Form Entry: Print version: Fundamentals of electric circuits. Alexander, Charles K. 920967054 Contents: PART 1 : DC CIRCUITSChapter 1 - Basic ConceptsChapter 2 - Basic LawsChapter 3 - Methods of AnalysisChapter 4 - Circuit TheoremsChapter 5 - Operational AmplifiersChapter 6 - Capacitors and InductorsChapter 7 - First-Order CircuitsChapter 8 - Second-Order CircuitsPART 2 : AC CIRCUITSChapter 9 - Sinusoids and PhasorsChapter 10 - Sinusoidal Steady-State AnalysisChapter 11 - AC Power AnalysisChapter 12 - Three-Phase CircuitsChapter 13 - Magnetically Coupled CircuitsChapter 14 - Frequency ResponsePART 3 : ADVANCED CIRCUIT ANALYSISChapter 15 - Introduction to the Laplace TransformChapter 16 - Applications of the Laplace TransformChapter 17 - The Fourier SeriesChapter 18 - Fourier TransformChapter 19 - Two-Port Networks Notes: Includes index Credits: Produced by the publisher. Held by CAPER-BC, Langara College. More Information: archive.org Free eBook from the Internet Archive openlibrary.org Additional information and access via Open Library Show more information Find a Copy at a Library 55 editions in 681 libraries Find a Copy at a Library 55 editions in 681 libraries Featured libraries All libraries Showing libraries near Clicking this button will open a modal to update the location to find libraries near it North Charleston, South Carolina Regional or National United States Air Force, CharlestonAFB Library FL4418 5.1 miles from your current location. 437 Svs/Svmg, 106 W McCaw St, Charleston Afb, SC, 29404-4700, United States Borrow Favorite Library Academic Jacksonville UniversityCarl S. Swisher Library 198 miles from your current location. Carl S Swisher Library, 2800 University Blvd N., Jacksonville, FL, 32211, United States Borrow Favorite Library Academic University of North Florida, Carpenter LibraryThomas G. 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Sign in Ratings & Reviews 4.05 483 ratings • 23 reviews 12 years ago Nawar I used this book for Electric Circuit 1 and I am going to use it again in Electric Circuit 2. My professor is Charles K. Alexander, one of the authors of this book. He said that they have written the book for different type of courses and it has bee… View More 4 readers found this review helpful 14 years ago Wayland The professor I had chose this book because he saw it had good reviews on Amazon. I was actually a little skeptical in the beginning, but when I bought the book and read it, it was really easy to follow through and understand. I would recommend thi… View More 3 readers found this review helpful 2 years ago Kristhan I read everything except chapter 12, 13, 17-19. This textbook is well written and I'd say it achieves the purpose of teaching students the fundamentals of circuits really well. It was enjoyable to read and rarely felt sluggish. 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3834	https://oshareview.com/2019/08/what-is-red-bag-or-biohazardous-waste-in-a-dental-office-ca-regulations/	What is “Red-Bag” or “Biohazardous” Waste in a Dental Office? – CA Regulations \| OSHA Review Skip to content Open mobile menu Close mobile menu Home Company About Osha Review Testimonials Terms & Conditions Products Spore Check System – Sterilizer Monitoring SUV Ultra 5 Disinfectant & Cleaner OSHA Review Compliance Program Dosimetry Monitoring Service Store Blog Contact Search Blog What is “Red-Bag” or “Biohazardous” Waste in a Dental Office? – CA Regulations August 7, 2019 OSHA Review California Dental Regulation, Dental Board of California, Dental Bytes Most dental offices do not generate “red-bag” biohazardous waste, also known as regulated medical waste.Cal/OSHA’s Bloodborne Pathogens (BBP) Standard (Title 8 CCR Section 5193) and theCalifornia Department of Public Health’s (CDPH’s) Medical Waste Management Act (MWMA) each have a unique, yet similar, definition for regulated“red-bag” “biohazardous” medical waste. Cal/OSHA In the BBP Standard,Cal/OSHA defines regulated “red-bag” medical waste as “liquid or semi-liquid blood or OPIM (other potentially infectious material); contaminated items that either contain liquid or semi-liquid blood or are caked with dried blood or OPIM, and are capable of releasing these materials when handled or compressed; contaminated sharps; and pathological and microbiological wastes containing blood or OPIM.” What regulated medical waste is NOT according to the BBP Standard…inCal/OSHA’s “Frequently Asked Questions about the Bloodborne Pathogens Standard”,Cal/OSHAstates that “in 5193(b), the definition of “regulated waste” makes it clear that some contaminated items may become contaminated with blood or OPIM during the course of their use, but are not within the scope of regulated waste and the disposal provisions of 5193. These include minimally contaminated absorbent items, such as dental drapes, gauze, band-aids, and sanitary napkins, that will dry out and be free of dried blood in quantitiesthat could be considered “caked”.” CDPH The MWMA defines regulated “biohazardous” medical waste as “waste that, at the point of transport from the generator’s site or at the point of disposal contains recognizable fluid human blood, fluid human blood products, containers, or equipment containing human blood that is fluid, or blood from animals suspected by the attending veterinarian of being contaminated with infectious agents known to be contagious to humans.” What regulated medical waste is NOT according to the MWMA… “waste which is not biohazardous, such as paper towels, paper products, articles containing nonfluid blood, and other medical solid waste products commonly found in the facilities of medical waste generators.” In Dentistry… Examples of regulated medical waste found in dental offices that should be disposed of or contained in red biohazard bags include biohazardous waste items that are soaked or saturated with blood or saliva (i.e. gauze saturated with blood following oral surgery), extracted teeth without amalgam if not to be given back to the patient, and surgically removed hard and soft tissues.If an item appears capable of releasing blood or OPIM, then it must be disposed of as regulated medical waste. According to the MWMA, regulated medical waste must be collected and disposed of in biohazard bags, which are disposable red bag that are impervious to moisture and are strong enough to avoid ripping, tearing, or bursting under normal use. The red bags must be labeled with the words “Biohazardous Waste”, or with “Biohazard” and the international biohazard symbol. The labeled red bags must be placed in rigid, labeled containers. If less than 20 lbs of biohazardous “red-bag” waste are generated per month, it may be stored for 30 days if it is above 0°C and for 90 days if below 0°C. If more than 20 lbs per month are generated, then the waste must be stored for no more than 7 days, regardless of the temperature. Regulated medical waste must be transported offsite by a CDPH-approved medical waste hauler. Minimally-contaminated regular trash is not considered medical waste, but solid waste and can be disposed of in regular solid waste trash containers. For our OSHA Review subscribers…more information on medical waste management can be found in Sections IV and IX of your OSHA Review binder. Additionally, online resources such as a list of approved medical waste transporters and a list of sharps waste mail-back systems is available for our subscribers onOSHA Review’s websitein the subscribers-only section. Since 1992, OSHA Review, Inc. has provided dental professionals with comprehensive programs to support regulatory compliance and infection control. We are a registered continuing education provider in the state of California, specializing in Dental Practice Act, infection control, and OSHA training. Share This Twitter Facebook Pinterest LinkedIn OSHA Review Related Posts CA Dental Requirements: Remove Protective Attire Prior to Leaving Patient Treatment Areas Both the Dental Board of California and Cal/OSHA require personal protective equipment (PPE), including protective… Infection Control in Dentistry: Understanding the Use of Surface Barriers Infection control is at the heart of safe, effective dental care. While cleaning and disinfecting… California Requirements for Contaminated Sharps Disposal All contaminated disposable sharps generated in a California dental office must be placed into a… Search Archives Archives Categories Categories Follow Us Twitter (deprecated) Facebook Instagram RSS Recent Posts COVID-19 FAQ for Dental Practitioners March 12, 2020 SUV Ultra 5 Disinfectant & Cleaner (EPA # 6836-366) is Effective for Use in COVID-19 Pandemic March 16, 2020 SUV Ultra Disinfecting Wipes (EPA # 6836-372) is Effective for Use in COVID-19 Pandemic March 16, 2020 FDA Consumer Updates Provide Guidance on COVID-19 March 30, 2020 Recent Projects Tags Aerosol Transmissible DiseaseAmalgam SeparatorsAmerican Dental AssociationBiofilm in Dental Unit Waterlinesbloodborne pathogensCal/OSHACalifornia Dental RegulationCalifornia Division of Occupational Safety and HealthCDCCDPHCenters for Disease Control and PreventionCovid-19CUREScybersecurityDental Board of Californiadental ce creditsDental Continuing EducationDental Disinfectant ProductsDental Infection Controldental surface disinfectantEPAFDAFluHand HygieneHazard Communication StandardHazcom RequirementsHIPAAHIPAA security risk analysisInfection controlInfection Control in Dental Health–Care SettingsNational Prescription Drug Take-Back DayOpioid abuseOSHA ComplianceOSHA ReviewOSHA Review Newsletterprescription drugRadiologic Health BranchSpore Check SystemSpore TestingSterilizer monitoringSUV DisinfectantSUV Disinfectant and CleanerSUV Disinfecting WipesUS DEAvaccination Lorem ipsum dolor sit amet, consectetur adipiscing elit. 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3835	https://stats.stackexchange.com/questions/584319/theory-of-runs-and-probability	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. current community your communities more stack exchange communities Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Theory of 'Runs' and probability I want to know how did author compute the probability of eleven runs if all arrangements are equally probable. My attempt to answer my own question: This example indicates wide applicability of the probability model of placing randomly r balls into n cells. Here n cells = 16 seats and r balls = 5 persons. Such an event is completely described by its occupancy numbers $ r_1,r_2,..., r_n$ where $r_k$ stands for the number of persons in the kth seat. Every n-tuple of integers satisfying $r_1 + r_2 + ...+ r_n= r, r_k \geq 0 $ desribes the possible configuration of occupancy numbers. With indistinguishable persons, two distributions are distinguishable only if the corresponding n-tuples $(r_1, r_2, ..., r_n)$ are not identical. The number of distinguishable distributions is $$Arrangement_{r,n} =\binom{n + r -1}{r} =\binom{n + r -1}{n-1}$$ So, in this case, the total sample space is $\binom{16 + 5 -1}{5} = \binom{20}{5} = \binom{16 + 5 -1 = 20}{ 16 - 1 =15}= 15504.$ Now, we can choose 5 seats to occupy from 11 runs in $\binom{11}{5}=462$ ways. We can choose 5 seats from 10 runs in$ \binom{10}{5}= 252$ ways. We can choose from 5 seats from 9 runs in $ \binom{9}{5}=126$ ways. And lastly we can choose 5 seats from 8 runs in $ \binom{8}{5}= 56$ ways. So, the total number of ways of selecting 11 runs are 462 + 252 + 126 + 56 = 896. The total sample space is 15504. Hence the probability of eleven runs if all arrangements are equally probable is $\frac{896}{15504} = 0.0578 $ which matches with author's answer as well. Is this answer computed by me and its method of computation correct? Note: This paragraph is taken from the book " An Introduction to Probability Theory and its Application Volume 1" written by William Feller.Third edition Page number 59 1 Answer 1 Is this answer computed by me and its method of computation correct? No, because it doesn't have a sample space of equally-probable outcomes. The stated problem is to find the probability that there will be $11$ runs in a random arrangement of $5$ 'O's and $11$ 'E's, these arrangements being equally probable. Thus, the sample space is the set of such arrangements, of which there are $\binom{16}{5}=8736$. The correct answer is then $${\binom{5-1}{5-1}\binom{11-1}{5}\over \binom{16}{5}}={\binom{10}{5}\over \binom{16}{5}}={504\over 8736}={3\over 52}=0.05769...$$ Obviously (since 'O'- and 'E'-runs must alternate), generally the number of 'O'-runs and the number of 'E'-runs must either be equal or they must differ by exactly $1$; thus, here $11$ runs can occur only with $5$ 'O'runs and $6$ 'E' runs (and not vice versa, because there are only $5$ 'O's altogether). Now, the number of arrangements of $a$ 'O's and $b$ 'E's, such that there are $n_1$ 'O'-runs and $n_1+1$ 'E'-runs is (by a "stars-and-bars" argument), $${{a-1}\choose{n_1-1}}{{b-1}\choose{n_1}}$$ which gives the stated result. This is consistent with Feller's Theorem 11.18, which he states as a problem at the end of the section; so the answer "0.0578..." given in the book is not quite correct. (For what it's worth, I also confirmed this result by computing it via Python by brute force, inspecting all the length-$16$ binary sequences having $5$ 'O's and $11$ 'E's, tallying those having exactly $11$ runs.) EDIT: I just discovered an old posting at math.s.e. that draws the same conclusions as above. Your Answer Thanks for contributing an answer to Cross Validated! But avoid … Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Sign up or log in Post as a guest Required, but never shown Post as a guest Required, but never shown By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Explore related questions See similar questions with these tags. Related Hot Network Questions Subscribe to RSS To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Cross Validated Company Stack Exchange Network Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA . rev 2025.9.26.34547
3836	https://physicspages.com/pdf/Thermal%20physics/Paramagnets%20and%20coin%20flips%20-%20peak%20and%20width%20of%20multiplicity%20function.pdf	PARAMAGNETS AND COIN FLIPS - PEAK AND WIDTH OF MULTIPLICITY FUNCTION Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog and include the title or URL of this post in your comment. Post date: 12 July 2021. For a couple of examples of Stirling’s approximation we’ll return to the two-state paramagnet. First, suppose we have a paramagnet with N = 1023 dipoles and that exactly half are in the spin-up state, so that N↑= 1 2 ×1023. The number of microstates in this system is Ω= N! (N/2)!(N/2)! (1) ≈ √ NNNe−N √ 2π(N/2)(N/2)N e−N (2) = 2N+1 √ 2πN (3) ≈2N (4) where the last line is true for N values around 1023 since this makes 2N a very large number, so that dividing by the square root won’t change things much. [Note that in this approximation, Ωfor the peak is actually equal to the total number of microstates summed over all macrostates, so the result says that an equal distribution is virtually certain.] If the system was in a different microstate a billion times per second over a period of ten billion years, it would explore 109 ×1010 ×365.25×24×3600 = 3.16×1026 (5) microstates. The number of available microstates is 21023 ≈103×1022 (6) so the actual number of microstates visited in 10 billion years is a vanish-ingly small proportion of those available. In theory, if we wait long enough, I suppose it’s possible to visit all the “accessible” microstates, but we’d have to wait an almost inﬁnite time (at least on a time scale of the current age of the universe) for this to happen. 1 PARAMAGNETS AND COIN FLIPS - PEAK AND WIDTH OF MULTIPLICITY FUNCTION 2 If we ﬂip N coins, the multiplicity function peaks sharply at a number Nh of heads equal to N/2. The actual multiplicity function in this case has the value given in 1, so this is the height of the peak. To get an estimate of the width of the peak, we can deﬁne x ≡Nh −N 2 (7) to be the deviation of the observed number of heads from the most probable value of N 2 . The multiplicity function in terms of x is therefore Ω(x) = N! N 2 −x ! N 2 +x ! (8) ≈ √ NNN √ 2π q N 2 −x q N 2 +x N 2 −x N 2 −x N 2 +x N 2 +x (9) = √ NNN √ 2π qN 2 2 −x2 N 2 2 −x2 N 2 N 2 −x −x N 2 +x x (10) We can now work with logarithms and consider the case of large N, so we can neglect the square root terms. Thus the approximation becomes Ω(x) ≈ NN N 2 2 −x2 N 2 N 2 −x −x N 2 +x x (11) lnΩ= N lnN −N 2 ln N 2 2 −x2 ! −xln N 2 +x N 2 −x (12) = N lnN −N 2 ln N2 4 1−4x2 N2 −xln 1+ 2x N 1−2x N (13) = N lnN −N 2 ln N 2 2 −N 2 ln 1−4x2 N2 −x ln 1+ 2x N −ln 1−2x N (14) = ln2N −N 2 ln 1−4x2 N2 −x ln 1+ 2x N −ln 1−2x N (15) We can now use the approximation ln(1+z) ≈z for \|z\| ≪1 and get PARAMAGNETS AND COIN FLIPS - PEAK AND WIDTH OF MULTIPLICITY FUNCTION 3 lnΩ≈ln2N + N 2 4x2 N2 −x 2x N − −2x N (16) = ln2N −2x2 N (17) Thus the multiplicity function near the peak is approximately Ω(x) ≈2Ne−2x2/N (18) Note that for x = 0, we get the peak height in 4. [If we’d retained the square root factors in 10, we’d get the √ 2πN factor back.] Thus the peak falls to 1/e of its peak value when x = r N 2 (19) so the width of the peak is twice this, or w = √ 2N (20) If we ﬂipped 106 coins, a result of 501,000 heads is just outside the peak, since x = q 106 2 ≈700, so this wouldn’t be a particularly surprising result. However, a result of 510,000 heads is well outside the peak and would be a surprising result. PINGBACKS Pingback: Random walks and diffusion Pingback: Multiplicity of a 2-dim ideal gas Pingback: Entropy Pingback: Two-state paramagnet - numerical solution Pingback: Two-state paramagnet - analytic solution
3837	https://stats.stackexchange.com/questions/645014/optimal-strategy-win-probability-for-sequential-uniform-random-variable-comparis	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Optimal strategy/win probability for sequential uniform random variable comparisons Ask Question Asked Modified 1 year, 5 months ago Viewed 529 times 6 $\begingroup$ Suppose you are sequentially presented with $n$ random variables, $x_1, x_2, \dots, x_n$, which are all drawn from U(0,1). After being presented with each variable, $x_i$, you must declare where the number would sit on an ordered list of all $x_1$ to $x_n$ (ie. largest, second largest, smallest, etc.). What is your optimal strategy for declaring the relative positions and what is the probability of success (no incorrect answers across all $x_i$). For $n=3$, I think the algorithm should be as follows $x_1$ max if $x_1 > 2/3$ min if $x_1 < 1/3$ mid if $1/3 < x_1 < 2/3$ $x_2$ if $x_1$ was max if $x_2 > x_1$, $x_2$ is max (and we lose game) if $x_2 < 0.5$, $x_2$ is min if $0.5 < x_2 < x_1$, $x_2$ is mid if $x_1$ was min, we operate similarly to if $x_1$ was max if $x_1$ was mid if $x_1 < x_2 < 0.5$ or $0.5 < x_2 < x_1$, $x_2$ is mid (and we lose) if $x_2 > x_1 \land x_2 > 0.5$, $x_2$ is max if $x_2 < x_1 \land x_2 < 0.5$, $x_2$ is min $x_3$ trivially compare against $x_1$ and $x_2$ to see position and declare as min/mid/max accordingly probability Share Improve this question edited Apr 18, 2024 at 21:29 olivarbolivarb asked Apr 14, 2024 at 22:21 olivarbolivarb 17355 bronze badges $\endgroup$ 16 $\begingroup$ A friend gave me this problem and I tried for $n=3$. My strategy was to declare $x_1$ as max if $x_1>2/3$, min if $x_1<1/3$, or middle if $1/3x_1$ or middle if $x_2x_1$; if $x_1$ was middle, declare $x_2$ as max if $x_2>x_1$ or min if $x_2 $\endgroup$ olivarb – olivarb 2024-04-14 22:27:36 +00:00 Commented Apr 14, 2024 at 22:27 3 $\begingroup$ "Optimal" could mean a few distinct things. For instance, would you like to maximize the expected number of correct answers? Or perhaps you would like to maximize the chance of getting at least $k$ correct answers, where you choose $k$ beforehand? The strategies are not necessarily the same... . $\endgroup$ whuber – whuber ♦ 2024-04-15 03:18:10 +00:00 Commented Apr 15, 2024 at 3:18 1 $\begingroup$ @whuber - I interpreted the question to mean we want to maximize the expected number of correct answers $\endgroup$ olivarb – olivarb 2024-04-15 17:10:31 +00:00 Commented Apr 15, 2024 at 17:10 1 $\begingroup$ You have confused me, because earlier you stated you wanted to optimize the expected number of correct answers, but now you assert "we lose [the] game" as soon as just one of the $n$ answers is incorrect. Which is it?? $\endgroup$ whuber – whuber ♦ 2024-04-18 20:21:50 +00:00 Commented Apr 18, 2024 at 20:21 1 $\begingroup$ @whuber - updated $\endgroup$ olivarb – olivarb 2024-04-18 21:29:36 +00:00 Commented Apr 18, 2024 at 21:29 \| Show 11 more comments 3 Answers 3 Reset to default 5 $\begingroup$ The following outlines a further improvement of the strategy derived by @whuber. It turns out that divisions into subinterval of equal length is not optimal. Consider the situation at the beginning of the game with $n$ numbers and let $a_0,a_1,a_2,a_n$ with $a_0=0$ and $a_n=1$ denote the yet to be determined locations of the subinterval divisions. The probability of winning is then given by \begin{aligned} W(n) &= \sum_{j=0}^{n-1} \binom{n-1}{j} W(j)W(n-1-j) \int_{a_j}^{a_{j+1}} x^j(1-x)^{n-1-j}\,\mathrm dx. \end{aligned} Differentiating with respect to each $a_j$ we obtain \begin{aligned} \frac{\partial W(n)}{\partial a_j} &={n-1 \choose j-1}W(j-1)W(n-j)a_j^{j-1}(1-a_j)^{n-j}\&-{n-1 \choose j}W(j)W(n-j-1)a_j^{j}(1-a_j)^{n-j-1} \&=\frac{(n-1)!a_j^{j-1}(1-a_j)^{n-j-1}}{(j-1)!(n-j-1)!}\left(\frac{W(j-1)W(n-j)(1-a_j)}{n-j}-\frac{W(j)W(n-j-1)a_j}{j}\right) \end{aligned} Equating to zero and solving we obtain an optimal value of $$ a_j=\frac{jW(j-1)W(n-j)}{(n-j)W(j)W(n-j-1)+jW(j-1)W(n-j)}. $$ For $n=3$, this gives $$ a_1=\frac3{11}, \quad a_2=\frac{8}{11} $$ for which $W(3)=377/726\approx 0.5192837$, slightly larger than the winning probability based on equal subdivisions. For $n$ up to 15, optimal divisions and winning probability (compared to same quantities based on equal subdivisions in grey) are as follows: library(index0) strategy <- function(nmax=10, equal=FALSE) { p <- function(a1, a2, alpha, beta) { (pbeta(a2, alpha, beta)-pbeta(a1, alpha, beta))beta(alpha, beta) } w <- index_from_0(c(1,1,.75)) aa <- list() for (n in 3:nmax) { j <- 1:(n-1) if (equal) a <- 1:(n-1)/n else a <- jw[j-1]w[n-j]/((n-j)w[j]w[n-j-1] + jw[j-1]w[n-j]) a <- index_from_0(c(0,a,1)) j <- index_from_0(0:(n-1)) w[n] <- sum(choose(n - 1,j)w[j]w[n - 1 - j]p(a[j], a[j + 1], j + 1, n - 1 - j + 1)) aa <- a } list(w=w, a=aa) } nmax <- 15 s <- strategy(nmax) se <- strategy(nmax, TRUE) par(mfrow=c(1,2)) plot(0:nmax, se$w, col="grey",pch=16, log="y", xlab="n", ylab="W(n)") points(0:nmax, s$w, pch=16) plot(NA, xlim=c(0,1), ylim=c(1,nmax), xlab="Subdivision limits", ylab="n") for (n in 3:nmax) { k <- length(s$a) points(se$a, rep(n, k), pch=16,col="grey") points(s$a, rep(n, k), pch=16) } Created on 2024-04-25 with reprex v2.1.0 Share Improve this answer edited Apr 26, 2024 at 10:17 answered Apr 25, 2024 at 10:50 Jarle TuftoJarle Tufto 12.9k22 gold badges4040 silver badges5959 bronze badges $\endgroup$ 1 3 $\begingroup$ +1 Many thanks for finding my error and pursuing it to such an effective result! $\endgroup$ whuber – whuber ♦ 2024-04-25 11:56:18 +00:00 Commented Apr 25, 2024 at 11:56 Add a comment \| 7 $\begingroup$ Let's attempt a recursive solution. Notation Because this can get complicated, suitable notation will help. To that end, For $k=0, 1, \ldots, n,$ let $\mathbf x_{(k)}$ be the sequence of the first $k$ values of the $x_i$ along with the endpoints $0$ and $1,$ in ascending order. Thus this vector has $k+2$ components which I will write $$\mathbf x_{(k)} = (0=x_{k,0},\ x_{k,1},\ \ldots,\ x_{k,k},\ x_{k,k+1}=1).$$ Let $\mathbf r_{(k)}$ represent an ordered sequence of integers $$\mathbf r_{(k)}: 0 = r_{k,0} \lt r_{k,1} \lt r_{k,2}\lt \cdots \lt r_{k,1} \lt r_{k,k+1} = n+1.$$ This represents the ranks we have assigned to $\mathbf x_{(k)},$ where $0$ is the rank of $0$ and $n+1$ is the rank of $1.$ When we observe one of the input numbers $x,$ we will assign a likely final rank (between $1$ and $n,$ with $1$ the smallest and $n$ the largest) to it. Call this $\operatorname{rank}(x).$ The decision tree for this problem contains nodes of two types, "event" and "decision," that alternate down the tree. All paths from the root through the tree correspond to sequences $$(x_1, \operatorname{rank}(x_1);\ x_2, \operatorname{rank}(x_2);\ \ldots,\ x_n, \operatorname{rank}(x_n)).$$ The value of each such path indicates whether it is a winning path. That is, when all the $n$ ranks are the correct ones, the value is $1$ and otherwise the value is $0.$ As usual, let's use capital letters $X_k$ to denote the (uniform, independent) random variables and $x_k$ to denote their realizations. After observing $k$ realizations, then, we have knowledge of $x_1, \ldots, x_k$ and the ranks we have assigned to them; but we know only the distributions of the unobserved random variables $X_{k+1}, \ldots, X_n.$ Analysis In these terms, the challenge for any $k= 0, 1, \ldots, n-1$ is to choose $\operatorname{rank}(x_{k+1})$ to maximize the expected value of the path (which equals the chance of winning), given the vectors $\mathbf x_{(k)}$ and $\mathbf r_{(k)}.$ The expectation is taken over the future observations $(X_{k+1}, \ldots, X_n).$ Write $$f(\mathbf x_{(k)}, \mathbf r_{(k)}, n)$$ for this expected value of winning. The question asks for a strategy to maximize the winning probability $$W(n) = f(\mathbf x_{(0)}, \mathbf r_{(0)}, n) = f((0,1),\ (0,n+1),\ n).$$ The first $k$ values we observe, along with $0$ and $1,$ partition the interval $[0,1)$ into $k+1$ smaller intervals $\mathcal I_{k,j} = [x_{k,j}, x_{k,j+1}),$ $j=0, 1, \ldots, k.$ Explicitly, $$[0,1] = [0, x_{k,1}) \cup [x_{k,1}, x_{k,2}) \cup \cdots \cup [x_{k,k-1}, x_{k,k}) \cup [x_{k,k}, 1).$$ Let $$\delta_{k,j} = x_{k,j+1} - x_{k,j}$$ be the widths of these intervals. The ranks we have previously chosen create gaps. The gap between $r_{k,j}$ and $r_{k,j+1}$ is the number of unassigned ranks between these two ranks, equal to $$s_{k,j} = r_{k,j+1} - r_{k,j} - 1.$$ For each gap indexed by $j,$ in addition to guessing the ranks correctly, we need exactly $\delta_{k,j}$ of the remaining $n-k$ random variables $X_{k+1}, \ldots, X_n$ to lie in the interval $\mathcal I_{k,j}$ in order to win. If this does not occur, then no strategy can possibly win. We can compute that chance. In terms of multinomial coefficients it equals $$\binom{n-k}{s_{k,0},\ s_{k,1},\ \ldots,\ s_{k,k}} \prod_{j=0}^k \delta_{k,j}^{s_{k,j}}.\tag{}$$ The crux of the matter is that, conditional on this event occurring, the problem is reduced (recursively) to finding an optimal strategy to rank the $s_{k,j}$ future (independent uniform) values that will fall into its corresponding gap. But, up to a change of scale, this is the very same problem, but with only $s_{k,j} \lt n$ variables. The last step of this analysis is to establish the strategy for the first number observed within any interval. As is already clear in the question statement, that strategy is to partition that interval into $n$ equal subintervals and assign the rank according to the subinterval. For instance, with an interval $[x, x+\delta)$ of width $\delta$ starting at $x$ that requires $s=3$ future values between ranks $r+1$ and $r+3,$ we will guess $x_j$ has rank $r+1$ when $0 \le x_j-x\lt\delta/3,$ that it has rank $r+2$ when $\delta/3\le x_j-x\lt 2\delta/3,$ and it has rank $r+3$ when $2\delta/3\le x_j-x\lt \delta.$ It's easy to prove this is optimal (exploit the recursion), so I leave it as an exercise. Computations What are the winning chances? It turns out that as $n$ grows, the chance increases astronomically compared to pure guessing (which has a chance of just $1/n!$). Let's work some out explicitly to illustrate the analysis. Note at the outset that $W(0)=1$ vacuously. (This will be useful for the generalization below.) $n=1.$ We assign rank $1$ to $x_1$ and we're done, winning all the time. $W(1) = 1.$ $n=2.$ We assign rank $1$ to $x_1$ when $x_1\lt 1/2$ and otherwise guess it will have rank $2.$ We win in the first case when $x_2 \gt x_1$ and in the second case when $x_2 \lt x_1.$ The full expectation is $$W(2) = \int_0^{1/2} \int_{x_1}^1 \mathrm dx_2\,\mathrm dx_1 + \int_{1/2}^1 \int_0^{x_1} \mathrm dx_2\,\mathrm dx_1 = \frac{3}{8} + \frac{3}{8} = \frac{3}{4}.$$ $n=3.$ Now it gets a little interesting. There are three cases for the first step. $x_1 \le 1/3.$ The strategy assigns rank $1$ to $x_1.$ The remaining two numbers must lie in the interval $[x_1,1)$ of length $\delta_{1,1} = 1 - x_1$ and that occurs with probability $(1 - x_1)^2$ because those two values are independent. We have already determined that our chances, conditional on this event, are $3/4.$ $1/3\lt x_1\le 2/3.$ The strategy assigns rank $2$ to $x_1.$ There are two gaps, each of which must be filled with one number. This occurs with probability $\binom{2}{1,\ 1} (x_1-0)(1-x_1) = 2x_1(1-x_1).$ If this event occurs, we are sure to win (by step 1 above). $2/3\le x_1 \lt 1).$ This is analyzed as in the first case $x_1\le 1/3$ and has the same value. Consequently, the chance of winning when $n=3$ is the sum of three integrals corresponding to these events: $$\begin{aligned} W(3) &= \binom{2}{0,2}W(2)\int_0^{1/3} (1-x_1)^2\,\mathrm dx_1 \&\quad + \binom{2}{1,2}W(1)W(1)\int_{1/3}^{2/3} x_1(1-x_1)\,\mathrm dx_1 \&\quad + \binom{2}{2,0}W(2)\int_{2/3}^1x_1^2\,\mathrm dx_1 \ &= (1)\left(\frac{3}{4} \right)\left(\frac{19}{81}\right) + (2)\left(1\right)^2\frac{13}{162} + (1)\left(\frac{3}{4} \right)\left(\frac{19}{81}\right) \ &= \frac{83}{162} \approx 0.5123457. \end{aligned}$$ General solution The pattern becomes clear. Here is the expected chance of winning, without any further elaboration: $$\begin{aligned} W(n) &= \sum_{j=0}^{n-1} \binom{n-1}{j} W(j)W(n-1-j) \int_{j/n}^{(j+1)/n} x^j(1-x)^{n-1-j}\,\mathrm dx \end{aligned}$$ This comes from taking expectations in formula $()$ after observing $x_1$ and guessing its rank according to the optimal strategy. It employs all known values $W(0), W(1), \ldots, W(n-1)$ recursively. The integrals are differences of incomplete Beta functions, for which there are efficient accurate algorithms. This yields a quadratic ($O(n^2)$) algorithm to compute all values $W(0), W(1), \ldots, W(n),$ shown here in an R implementation. p <- function(a, b) { n <- a + b + 1 (pbeta((a+1)/n, a+1, b+1) - pbeta(a/n, a+1, b+1)) beta(a+1, b+1) } nmax <- 39 w <- c(1, rep(NA, nmax)) for (n in 1:nmax) { w[n+1] <- sum(choose(n - 1, 0:(n-1)) w[1:n] w[n:1] p(0:(n-1), (n-1):0)) } plot(0:nmax, w, log = "y", xlab = "n") Approximations The red line (in these log-linear coordinates) graphs an approximate asymptotic function $W(n) \approx \exp(3 - 0.55 n).$ It gets more accurate than shown here as $n$ grows. Including additive correction terms (for $\log W(n)$) proportional to $1/n,$ $1/n^2,$ and $\log n$ gives an approximation good to better than three decimal places across the full range of values that can be computed in double-precision arithmetic (roughly $n$ from $3$ through $1000$). That is, $$\log W(n) \approx 0.3676 - 0.552564 n + 0.2336/n - 0.0489/n^2 + 0.4998\log n,\quad n = 3, 4, \ldots, 1000.$$ The (fully accurate) values continue $W(4) = 55537/165888,$ $W(5) = 11049709/51840000,$ etc., suggesting no simple closed formula is available. Share Improve this answer answered Apr 20, 2024 at 19:54 whuber♦whuber 342k6666 gold badges821821 silver badges1.4k1.4k bronze badges $\endgroup$ 5 $\begingroup$ So it seems like my amended strategy was correct for $n=3$? This seems like a mean problem to give verbally. I'm not sure how I was supposed to pin down the EV in 10 minutes... $\endgroup$ olivarb – olivarb 2024-04-22 08:31:42 +00:00 Commented Apr 22, 2024 at 8:31 2 $\begingroup$ @olivarb The strategy is different from what you propose. For $n=3$, if we observe $x_1=0.7$ and thus assign a rank of 3 to $x_1$, the remaining numbers $x_2$ and $x_3$ are iid uniform on $[0,0.7)$ conditional on $x_2$ and $x_3$ both being smaller than $x_1$ such that we have a chance to win the game. Thus we are faced with a problem of the same type as the initial problem (up to a change of scale) and we should assign a rank of 2 (from the remaining ranks 1 and 2) to $x_2$ if $x_2>0.35$, otherwise a rank of 1. $\endgroup$ Jarle Tufto – Jarle Tufto 2024-04-22 10:35:39 +00:00 Commented Apr 22, 2024 at 10:35 1 $\begingroup$ @whuber +1 But for $n=3$, partitioning the interval $[0,1)$ at $a$ and $1-a$ instead of $1/3$ and $2/3$ and maximising the resulting $W(3)$ with respect to $a$ I get a slightly larger $W(3)=377/726\approx 0.5192837$ for $a=3/11 \approx 0.2727$. So equal subintervals turns out not be optimal! $\endgroup$ Jarle Tufto – Jarle Tufto 2024-04-23 08:12:36 +00:00 Commented Apr 23, 2024 at 8:12 $\begingroup$ @Jarle That's interesting: I might have been mistaken in my assertion about equal divisions--and, in retrospect, that makes sense. This requires more analysis... . $\endgroup$ whuber – whuber ♦ 2024-04-23 12:40:47 +00:00 Commented Apr 23, 2024 at 12:40 1 $\begingroup$ @whuber I found a nice recursion for the optimal divisions and added a second answer. $\endgroup$ Jarle Tufto – Jarle Tufto 2024-04-25 10:56:06 +00:00 Commented Apr 25, 2024 at 10:56 Add a comment \| 3 $\begingroup$ This is not an answer but only considers a general strategy coinciding with that of the OP that turns out to be suboptimal. After having observed the first $k$ random variables out of all $n$ variables $x_1,x_2,\dots,x_n$, let $r_{k,k}$ denote the current rank of $x_k$ among $x_1,x_2,\dots,x_k$. Conditional on $x_1,x_2,\dots,x_k$, the final rank $r_k$ of $x_k$ among $x_1,x_2,\dots,x_n$ is clearly distributed as $$ r_k \sim r_{k,k} + \operatorname{bin}(n-k,x_k) \tag{1} $$ since each of the remaining $n-k$ variables $x_{k+1},x_{k+2},\dots, x_n$ will be smaller than $x_k$ independently and with probability $x_k$ and each of these events increases the final rank of $x_k$ by one. A possible strategy is to assign as rank $a_k$ to $x_k$ the mode of the conditional probability mass function of the final rank $r_k$ given by $$ a_k=r_{k,k}+\lfloor (n-k + 1)x_k\rfloor. $$ For $n=3$ this coincides with the strategy of the OP. While this choice of $a_k$ makes no regard to the later choices for $a_{k+1},\dots a_n$, it follows from linearity of expectation, that it is optimal in terms of maximising the expected number of correct rankings. However, as the aim of the game is to maximise the probability that all assigned ranks are correct ($a_k=r_k$ for $k=1,2,\dots,n$), it is not, as shown by the following counter example. Suppose $n=3$ and $x_1=0.6$. The above strategy then leads to $a_1=2$. If then observe $x_2=0.55$, then $r_{2,2}=1$ and the most likely final rank of $x_2$ is $r_2=2$ since $P(x_3 < x_2)=0.55$. So by the above strategy we should choose $a_2=2$. But then we have $a_1=a_2=2$ and we loose the game with probability one. On the other hand, if we instead choose $a_2=1$ we still have a chance of get all ranks correct with probability $P(x_3>x_1)=1-x_1=0.4$. Share Improve this answer edited Apr 23, 2024 at 8:39 answered Apr 19, 2024 at 11:57 Jarle TuftoJarle Tufto 12.9k22 gold badges4040 silver badges5959 bronze badges $\endgroup$ Add a comment \| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions probability See similar questions with these tags. 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3838	https://ocw.mit.edu/courses/1-010-uncertainty-in-engineering-fall-2008/f20b1a4fa12a99d17cbb35cb3cf28a43_app_15.pdf	MIT OpenCourseWare 1.010 Uncertainty in Engineering Fall 2008 For information about citing these materials or our Terms of Use, visit: Example Application 15 (Conditional second-moment analysis) UNCERTAINTY UPDATING USING NOISY OBSERVATIONS One of the uses of conditional distributions is in updating uncertainty on a variable of interest X based on observation of one or more other variables. For example, one may want to update uncertainty on rainfall tomorrow based on observation of rainfall today, the strength of beam 1 based on observation of the strength of beam 2, or soil compressibility at location A given soil compressibility at some other location B. In certain cases, the observed variable is itself a measurement of X. For example, one may measure the strength of a concrete column by some nondestructive test, measure topographic elevation at a point using a satellite instrument with limited accuracy, or sample the water of a stream with an imprecise device to determine its degree of contamination. In all these cases, the measurement is not exact. We want to see how, based on such “noisy data”, one can update uncertainty on the quantity of interest X. The method described below is exact if the random variables involved are normally distributed, but is often used as an approximation for variables with any distribution. Conditional Distributions of Variables with Joint Normal Distribution Let X1 and X2 be jointly normal variables with mean values m1 and m2, variances σ1 2 and σ2 2, and correlation coefficient ρ. 0ne can show that the conditional distribution of (X1\|X2=x2) is also normal, with mean value m1\|2 and variance σ1\|2 2 given by σ1 m1\|2 = m1 +ρ σ2 (x2 −m2 ) (1) σ1 2 \|2 = σ1 2(1 −ρ2) 1 Notice that the conditional mean depends on the observed value x2 of X2, whereas the conditional variance does not. Moreover, the conditional variance differs from the unconditional variance by the factor (1 - ρ2), which is smaller than 1 whenever X1 and X2 are dependent. Application to Noisy Observations Next we show how Eq. 1 can be used to update uncertainty on a quantity of interest X (e.g., X = load bearing capacity of the soil or concentration of a pollutant at a given location) after making a measurement of it. The quantity of interest, X, is initially uncertain with mean value m and variance σ2. To reduce this uncertainty (and for example determine whether X is below a critical level x with probability at least P), a measurement Z of X is made. If the measurement had no error, then X could be recovered exactly from Z, but in practice measurements are affected by errors (they are “noisy”). A simple model with noise is the so-called linear model, according to which Z is related to X as Z = a + bX + ε (2) where a and b are given deterministic constants and ε is an error term independent of X, with mean value zero and variance σε 2. The problem is to update uncertainty on X based on the observed value of Z, say z. To use the conditional moment results in Eq. 1, we need to find the mean value and variance of Z and the correlation coefficient between X and Z. After this is done, we may rename X → X1 and Z → X2 and use that equation. Second-moment propagation of uncertainty through linear functions gives mZ = a + bm σZ 2 = b2σ2 + σε 2 (3) Cov[X,Z] = bσ2 2 Using these results and the relationships σ1 ρσ1σ2 Cov[X1,X2] ρσ2 = σ2 2 = Var[X2] σ1 2(1−ρ2) = Var[X1] −{ Cov[X1,X2]} 2 (4) Var[X2] one obtains from Eq. 1: E X Z = z]= m + h   z − b a −m   (5) [ Var XZ = z]= σ2(1−h) [  σ2  -1  2 2 . Like Eq. 1, Eq. 5 holds exactly if both X and ε have normal where h= 1+ ε   b σ  distribution and in approximation for other distributions. A key role in Eq. 5 is played by the quantity h, for which some special cases may be noted: 1. suppose that σε 2 = 0, or more in general that σε 2 << b2σ2. This means that observations are without error or the contribution from X to the variance of Z far exceeds the contribution from ε (high “signal-to-noise ratio”). In this case h = 1 and Eq. 5 gives E[X\|Z = z] = (z - a)/b and Var[X\|Z = z] = 0. This is of course the solution to the deterministic problem; 2. At the other extreme is the case of very noisy measurements, when σε 2 >> b2σ2. In this case h is close to zero and Eq. 5 gives E[X\|Z = z] = m and Var[X\|Z = z] = σ2, i.e. no change in the state of uncertainty on X as a result of observing Z. 3 Problem 15.1 (a) Cases of practical interest are intermediate between the above two limiting cases. To understand the role of different factors in the informativeness of a linear experiment, set b = 1 and plot the posterior-to-prior variance ratio γ = Var[X\|Z = z]/σ2 against σε 2/b2σ2. Notice that γ is a measure of the information value of the experiment and that the ratio σε 2/b2σ2 can be decreased by either reducing the variance of the measurement error σε 2 or increasing the “gain” b. (b) Think of an application of the observation model presented above to a context of interest to you. Postulate a plausible prior uncertainty state and realistic observation model parameters. Derive the uncertainty updating equation and the posterior variance using Eq. 5. Problem 15.2 (a) Extend the previous analysis to the vector case, i.e. consider X to be a vector with n components and Z to be a vector with r components. Assume a linear relation between X and Z of the type Z = a + BX + ε, where a is a given vector, B is a given matrix, and ε is a random measurement error vector. Assume that X has joint normal distribution, ε has joint normal distribution, and X and ε are independent. (b) Extend the results for Part (a) to include dependence between X and ε. Best Linear Unbiased Estimation (BLUE) Theory Equation 1 is often used also when X1 and X2 do not have joint normal distribution. In that case Eq. 1 may be regarded as an approximation or may be used with a different interpretation. Specifically, we show that, irrespective of the type of distribution, the expression for the conditional mean in Eq. 1 has the meaning of best linear unbiased estimator of X1 from X2 and the conditional variance in Eq. 1 has the meaning of associated estimation error variance. 4 Suppose that X1 and X2 have mean values, variances, and correlation coefficient as above, but are not necessarily normally distributed. Based on the observation of X2, we form a linear estimator of X1, X ˆ 1 = a + bX2, and look for coefficients a and b such that 1. The estimator is (unconditionally) unbiased, i.e. E[ X ˆ 1] = E[X1] = m1. This gives a + bE[X2] = a + bm2 = m1. Therefore, a = m1 - bm2. 2. Among all linear unbiased estimators, X ˆ 1 has minimum error variance. The error is e = X ˆ 1 - X1 and its variance is σe 2 = var[bX2 − X1] = σ1 2 + b2σ2 2 − 2bρσ1σ2. Taking the derivative with respect to b and setting it to zero gives 2bσ2 2 − 2ρσ1σ2 = 0. Hence b = ρ σ1/σ2 and a = m1 - ρ m2 (σ1/σ2). We conclude that the BLUE estimator of X1 from X2 is X ˆ 1 = m1 + ρ(σ1/σ2)(x2 - m2) (6) The associated error variance is obtained by substituting b = ρ σ1/σ2 into the expression for σe 2. This gives σe 2 = σ1 2(1 - ρ2) (7) Comparison of Eqs. 6 and 7 with Eq. 1 shows that the BLUE estimator for any joint distribution of X1 and X2 is identical to the conditional mean m1\|2 for jointly normal variables and that the conditional variance in Eq. 1 is also the error variance of the BLUE estimator. This correspondence significantly broadens the applicability of the normal distribution results. 5
3839	https://www.geeksforgeeks.org/engineering-mathematics/moment-generating-functions/	Moment Generating Functions - GeeksforGeeks Skip to content Tutorials Python Java DSA ML & Data Science Interview Corner Programming Languages Web Development CS Subjects DevOps Software and Tools School Learning Practice Coding Problems Courses DSA / Placements ML & Data Science Development Cloud / DevOps Programming Languages All Courses Tracks Languages Python C C++ Java Advanced Java SQL JavaScript Interview Preparation GfG 160 GfG 360 System Design Core Subjects Interview Questions Interview Puzzles Aptitude and Reasoning Data Science Python Data Analytics Complete Data Science Dev Skills Full-Stack Web Dev DevOps Software Testing CyberSecurity Tools Computer Fundamentals AI Tools MS Excel & Google Sheets MS Word & Google Docs Maths Maths For Computer Science Engineering Mathematics Switch to Dark Mode Sign In Aptitude Engineering Mathematics Discrete Mathematics Operating System DBMS Computer Networks Digital Logic and Design C Programming Data Structures Algorithms Theory of Computation Compiler Design Computer Org and Architecture Sign In ▲ Open In App Moment Generating Functions Last Updated : 26 Jul, 2025 Comments Improve Suggest changes 1 Like Like Report Moment Generating Functions (MGFs) are a powerful tool in probability theory used to analyze random variables. They transform a random variable into a function that simplifies the calculation of important characteristics, such as the mean, variance, skewness, and kurtosis. What are Moments? Moments are quantitative measures that describe the shape and characteristics of a probability distribution: First moment: Mean Second moment: Variance Third moment: Skewness Fourth moment: Kurtosis The moment generating function M X(t) of a random variable X is defined as: M X(t)=E[e t X]M_X(t) = \mathbb{E}[e^{tX}]M X(t)=E[e tX] Where: M X(t) is the moment generating function of X. E denotes the expected value. t is a real number. The domain D x of M x is defined as: D X = {t ∈ R ∣ M X (t) < ∞} MGFs exist for all values of t in some open interval around 0 where the expectation is finite. MGFs for Discrete and Continuous Random Variables If X is a discrete random variable with PMF pX, then, M X(s) = ∑x e s x p X(x)\sum_x e^{sx}pX(x)∑xe s x pX(x) If X is a continuous random variable, with PDF fX, then M X(s) = ∫e s x f X(x)d x\int e^{sx} fX(x)dx∫e s x f X(x)d x Examples of Moment Generating Functions of Common Distributions Moment generating functions for some of the most common distributions are listed in the following table: \| Distribution \| Moment Generating Function \| --- \| \| Bernoulli Distribution \| M X(t)=p e t+(1−p)M_X(t) = p e^t + (1 - p)M X(t)=p e t+(1−p) \| \| Binomial Distribution \| M X(t)=(p e t+(1−p))n M_X(t) = (p e^t + (1 - p))^n M X(t)=(p e t+(1−p))n \| \| Geometric Distribution \| M X(t)=p e t 1−(1−p)e t,t<−ln⁡(1−p)M_X(t) = \frac{p e^t}{1 - (1 - p) e^t}, \quad t < -\ln(1 - p)M X(t)=1−(1−p)e t p e t,t<−ln(1−p) \| \| Poisson Distribution \| M X(t)=exp⁡(λ(e t−1))M_X(t) = \exp(\lambda (e^t - 1))M X(t)=exp(λ(e t−1)) \| \| Uniform Distribution \| M X(t)=e t b−e t a t(b−a),t≠0 M_X(t) = \frac{e^{tb} - e^{ta}}{t(b - a)}, \quad t \neq 0 M X(t)=t(b−a)e t b−e t a,t=0 \| \| Exponential Distribution \| M X(t)=λ λ−t,t<λ M_X(t) = \frac{\lambda}{\lambda - t}, \quad t < \lambda M X(t)=λ−t λ,t<λ \| \| Normal Distribution \| M X(t)=exp⁡(μ t+σ 2 t 2 2)M_X(t) = \exp(\mu t + \frac{\sigma^2 t^2}{2})M X(t)=exp(μ t+2 σ 2 t 2) \| \| Gamma Distribution \| M X(t)=(1−t θ)−k,t<θ M_X(t) = (1 - \frac{t}{\theta})^{-k}, \quad t < \theta M X(t)=(1−θ t)−k,t<θ \| \| Beta Distribution \| M X(t)=∑n=0∞t n n!Γ(α+n)Γ(α+β)Γ(α)Γ(α+β+n),∣t∣<1 M_X(t) = \sum_{n=0}^\infty \frac{t^n}{n!} \frac{\Gamma(\alpha + n) \Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\alpha + \beta + n)}, \quad \|t\| < 1 M X(t)=∑n=0∞n!t nΓ(α)Γ(α+β+n)Γ(α+n)Γ(α+β),∣t∣<1 \| \| Chi-Square Distribution \| M X(t)=(1−2 t)−k/2,t<1 2 M_X(t) = (1 - 2t)^{-k/2}, \quad t < \frac{1}{2}M X(t)=(1−2 t)−k/2,t<2 1 \| Moments from Moment Generating Functions Moments of a random variable can be derived from its Moment Generating Function (MGF). The n th moment of a random variable X is given by taking the n th derivative of the MGF with respect to t and evaluating it at t = 0: E[X n]=M X(n)(0)\mathbb{E}[X^n] = M_X^{(n)}(0)E[X n]=M X(n)(0) Deriving Moments from the Moment Generating Function (MGF) The moment generating function (MGF) of a random variable X is defined as: M X(t)=E[e t X]M_X(t) = \mathbb{E}[e^{tX}]M X(t)=E[e tX] The n-th moment of X, E[X n] \mathbb{E}[X^n]E[X n] , can be obtained by taking the n-th derivative of M_X(t) with respect to t and evaluating it at t = 0: E[X n]=M X(n)(0)\mathbb{E}[X^n] = M_X^{(n)}(0)E[X n]=M X(n)(0) Example: Exponential Distribution Let X∼Exp(λ)X \sim \text{Exp}(\lambda) X∼Exp(λ), where its MGF is: M X(t)=λ λ−t,t<λ M_X(t) = \frac{\lambda}{\lambda - t}, \quad t < \lambda M X(t)=λ−t λ,t<λ Step 1: Find the First Moment (Mean, (E[X]\mathbb{E}[X] E[X])) Compute the first derivative of M_X(t): M X′(t)=d d t(λ λ−t)=λ(λ−t)2 M'_X(t) = \frac{d}{dt} \left( \frac{\lambda}{\lambda - t} \right) = \frac{\lambda}{(\lambda - t)^2}M X′(t)=d t d(λ−t λ)=(λ−t)2 λ At t = 0: E[X]=M X′(0)=λ(λ−0)2=1 λ\mathbb{E}[X] = M'_X(0) = \frac{\lambda}{(\lambda - 0)^2} = \frac{1}{\lambda}E[X]=M X′(0)=(λ−0)2 λ=λ 1 This matches the known mean of an exponential distribution. Step 2: Find the Second Moment( E[X 2]\mathbb{E}[X^2]E[X 2] ) Compute the second derivative of M X(t)M_X(t)M X(t): M X′′(t)=d d t(λ(λ−t)2)=2 λ(λ−t)3 M''_X(t) = \frac{d}{dt} \left( \frac{\lambda}{(\lambda - t)^2} \right) = \frac{2\lambda}{(\lambda - t)^3}M X′′(t)=d t d((λ−t)2 λ)=(λ−t)3 2 λ At t = 0: E[X 2]=M X′′(0)=2 λ(λ−0)3=2 λ 2\mathbb{E}[X^2] = M''_X(0) = \frac{2\lambda}{(\lambda - 0)^3} = \frac{2}{\lambda^2}E[X 2]=M X′′(0)=(λ−0)3 2 λ=λ 2 2 This matches the known second moment of an exponential distribution. Step 3: Find the Variance (Var(X)) Using the moments: Var(X)=E[X 2]−(E[X])2=2 λ 2−(1 λ)2=1 λ 2\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = \frac{2}{\lambda^2} - \left( \frac{1}{\lambda} \right)^2 = \frac{1}{\lambda^2}Var(X)=E[X 2]−(E[X])2=λ 2 2−(λ 1)2=λ 2 1 This matches the known variance of an exponential distribution. First moment (Mean): M X′(0)=1 λ M'_X(0) = \frac{1}{\lambda} M X′(0)=λ 1 Second moment: M X′′(0)=2 λ 2 M''_X(0) = \frac{2}{\lambda^2}M X′′(0)=λ 2 2 Variance: Var(X) =1 λ 2= \frac{1}{\lambda^2}=λ 2 1 Properties of Moment Generating Functions Some of the common properties of moment generating functions are: If Y = aX + b, then MY(s) = esbMX(as) If X and Y are independent, then MX + Y(s) = MX(s) MY(s) Let X and Y be the independent random variables. Let Z be equal to X, with probability p, and equal to Y, with probability 1-p. Then, MZ(s) = pMX(s) + (1 - p) MY(s) Solved Examples of Moment Generating Functions Example 1: Let X be a random variable with MGF M X(s)=1 1−s MX(s)=\frac{1}{1-s}MX(s)=1−s 1(exponential with λ=1). Define Y=2X+3. Then, the MGF of Y is: M Y(s)=e 3 s M X(2 s)=e 3 s⋅1 1−2 s,s<1 2 M_Y(s) = e^{3s}M_X(2s) = e^{3s} ⋅\frac{1}{1-2s}, s<\frac{1}{2}M Y(s)=e 3 s M X(2 s)=e 3 s⋅1−2 s 1,s<2 1 Example 2: Let X∼Poisson(λ 1) with M X(s)=e λ 1(e s−1)M_X(s)=e^{λ_1(e^s−1)}M X(s)=e λ 1(e s−1) Y∼Poisson(λ 2)(independent of X) with Then, the MGF of X+Y is: M X+Y(s)=M X(s)M Y(s)=e λ 1(e s−1)⋅e λ 2(e s−1)=e(λ 1+λ 2)(e s−1)M_{X+Y}(s) = M_X(s)M_Y(s) = e^{λ_1(e^s−1)}⋅e^{λ_2(e^s−1)}=e^{(λ1+λ2)(e^s−1)}M X+Y(s)=M X(s)M Y(s)=e λ 1(e s−1)⋅e λ 2(e s−1)=e(λ 1+λ 2)(e s−1) This shows X+Y∼Poisson(λ 1+λ 2) Example 3: Z =_X_(prob p) or Z = _Y_(prob 1−p) ⇒M z(s) = pM X(s) + (1 − p) M Y(s) Let X∼N(0,1)(standard normal) with M X(s)=e s 2/2 M_X(s)=e^{s^2/2}M X(s)=e s 2/2 and Y∼N(1,1)(normal with mean 1) with M Y(s)=e s+s 2/2 M_Y(s)=e^{s+s^2/2}M Y(s)=e s+s 2/2 Suppose Z equals X with probability 0.6 and Y with probability 0.4. Then, the MGF of Z is: M Z(s)=0.6⋅e s 2/2+0.4⋅e s+s 2/2 M_Z(s)= 0.6⋅e^{s^2/2}+0.4⋅e^{s+s^2/2}M Z(s)=0.6⋅e s 2/2+0.4⋅e s+s 2/2 Inverse Theorem for Moment Generating Function Suppose that M X(s) is finite for all s in an interval of the form [-a, a], where a is a positive number. Then, M X determines uniquely the CDF of the random variable X. In particular, if M X(s) = M Y(s) < ∞, for all s ϵ [-a, a], where a is a positive number, then the random variables X and Y have the same CDF. For example, let us take an example of two Bernoulli functions, X and Y, both with the same parameter p = 0.4. MGF of bernoulli random variabe is: M X(s) = E[e sX] = (1 − p) e 0 + p⋅e s = (1 − p) + pe s For both X and Y, with p = 0.4: M X(s) = M Y(s) = 0.6 + 0.4e s This MGF is finite for all real s, so in particular, it's finite for all s ϵ [-a,a] for any a>0. M X(s)=M Y(s) on an interval around 0, ⇒ X and Y must have the same distribution, ⇒ Therefore, FX(t)=FY(t) for all t, i.e., they have the same CDF. Moment Cumulant and Probability Generating Function Moment Cumulant and Probability Generating Functions are three different mathematical tools used in probability theory and statistics to characterize and analyze the properties of random variables and their distributions. Each function provides unique insights into the underlying distribution, helping to simplify the computation and understanding of moments, cumulants, and probabilities. 1. Moment Generating Function (MGF): It is used to calculate the moments of a distribution, which are useful in understanding the central tendency (mean), spread (variance), and higher-order properties of the distribution. M X(t)=E[e t X]M_X(t) = \mathbb{E}[e^{tX}]M X(t)=E[e tX] For normal distribution: M X(t)=exp⁡(t μ+t 2 σ 2 2)M_X(t) = \exp\left( t\mu + \frac{t^2 \sigma^2}{2} \right)M X(t)=exp(t μ+2 t 2 σ 2) 2. Cumulant Generating Function (CGF): This provides a way to calculate cumulants, which describe the shape of a distribution and are particularly useful for sums of random variables. It is the logarithm of the MGF and helps in simplifying complex distributional properties. The CGF, denoted as K X(t), is the natural logarithm of the MGF: K X(t)=ln⁡(M X(t))K_X(t) = \ln(M_X(t))K X(t)=ln(M X(t)) The n th cumulant of a random variable X is given by the n th derivative of the CGF evaluated at t = 0: κ n=d n K X(t)d t n∣t=0\kappa_n = \frac{d^n K_X(t)}{dt^n} \Bigg\|_{t=0}κ n=d t n d n K X(t)∣∣t=0 Cumulants are used to describe the shape of the probability distribution, similar to moments, but they have properties that often make them more convenient for certain types of statistical analysis, especially when dealing with sums of random variables. 3. Probability Generating Function (PGF): Probability Generating Function (PGF) is another useful tool in probability theory, particularly for discrete random variables. The PGF of a discrete random variable X is defined as: G X(s)=E[s X]=∑k=0∞P(X=k)s k G_X(s) = \mathbb{E}[s^X] = \sum_{k=0}^{\infty} P(X = k) s^k G X(s)=E[s X]=∑k=0∞P(X=k)s k Applications of Moment Generation Function in CS Randomized Algorithms Analyzing expectations, variances, and tail bounds In Quicksort, MGFs help analyze the expected number of comparisons. Probabilistic Data Structures Estimating error and behavior of sketches/filters MGFs can help assess the probability of error in approximating cardinalities or set memberships. Performance Modeling Analyzing queues, latency, and throughput. MGFs are used to derive the distribution of waiting times and queue lengths. Machine Learning Moment matching, feature extraction, and convergence analysis Used in moment-matching techniques and feature engineering. Cryptography Randomness testing, distribution analysis Used in analyzing pseudo-random number generators (PRNGs) and detecting bias in output. 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3840	https://www.coursesidekick.com/statistics/196473	Understanding Stratified Sampling in Statistics School Drexel UniversityWe aren't endorsed by this school Course STAT 642 Subject Statistics Date May 14, 2023 Pages 15 Uploaded by GrandResolveWren32 Home/ Statistics Students also studied Week 2 Assignment central tendency calculations.docx 4 pages Answers should be in red font color. The doctor claims that the average man in his practice is overweight. Complete this week's reading assignments prior to completing this assignment. Statistics is a very powerful topic that is used on a daily basis in 6/12/2023Analysis Workbook and Checklist 2022.docx 6 pages DATA ANALYSIS CHECKLIST #1 Data checking & loading Download the data Check .xls data using descriptive statistics (Week 2) Load cleaned data in SPSS #2 Factor analysis (Variable Questions) Run descriptive statistics (Week 2) Run reliability analysis ( 6/12/20232_-_8.2_powerpoint.pptx 17 pages 8.2 ESTIMATING A POPULATION PROPORTION HW: P. 496 (27, 31-37 ODD, 41, 43, 47, 49-52) OVERVIEW... In the last section, you learned the basic ideas behind confidence intervals. In the next two sections, you will learn how to construct and interpret confide 6/12/2023Week 2.docx 8 pages Week 2 Week 2 PRINT Exploring Statistical Software and Descriptive Statistics Creating a Picture of Your Data Imagine you're leading a project for the Office of Learner Affairs at an online university that is interested in learner success. You want to sta 6/12/2023STA13 HW4 solution.pdf 5 pages STA13 Homework 4 December 1, 2021 1. Sports and Achilles Tendon Injuries Sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and thickening of the Achilles tendo 6/12/2023RN_Assignment 1.xlsx 3 pages 10 9 8 7 6 6 6 5 5 5 4 4 3 2 2 10 9 8 7 6 6 6 5 5 5 4 4 3 2 2 10 9 8 7 6 6 6 5 5 5 4 4 3 2 2 5.46666666666667 5.18222222222222 2.27644947719518 Mean Variance Standard Deviation (STDVP) Calculating Pearson Correlation Coefficient Calculating Pearson Correl 6/12/2023 Lesson 6: Strati±ed Sampling Lesson 6: Strati±ed Sampling Overview In Section 6.1, we discuss when and why to use strati±ed sampling. The estimate for mean and total are provided when the sampling scheme is strati±ed sampling. An example of using strati±ed sampling to compute the estimates as well as the standard deviation of the estimates are provided. Con±dence intervals for these estimates are then discussed. In Sections 6.2, the optimal allocation of sample size under di²erent conditions is given. Then we discuss post-strati±cation. It is important to note that the variance of estimates under post- strati±cation is di²erent from under strati±cation. In section 6.3, we use an example to illustrate that a strati±ed sample may not be better than simple random sample if the variable one strati±es on is not related to the response. At the end of section 6.3, we discuss strati±ed sampling for proportions. Lesson 6: Ch. 11.1-11.6 ofSamplingby Steven Thompson, 3rd edition Objectives Upon completion of this lesson you should be able to: know why and when to use strati±ed sampling, know how to estimate mean and total when strati±ed sampling is used, compute con±dence interval for these estimates, determine the optimal allocation of sample sizes, compute estimates when post-strati±cation is used, compute the variance for the estimates when post-strati±cation is used, and provide estimates for strati±ed sample for proportion. 6.1 - How to Use Strati±ed Sampling 6.1 - How to Use Strati±ed Sampling In strati±ed sampling, the population is partitioned into non-overlapping groups, called strata and a sample is selected by some design within each stratum. For example, geographical regions can be strati±ed into similar regions by means of some known variables such as habitat type, elevation or soil type. Another example might be to determine the proportions of defective products being assembled in a factory. In this case, sampling may be strati±ed by production lines, factory, etc. Want to read all 15 pages? Previewing 2 of 15 pages. Upload your study docs or become a member. Want to read all 15 pages? Previewing 3 of 15 pages. Upload your study docs or become a member. End of preview Want to read all 15 pages? Upload your study docs or become a member. #### Week 2 Assignment central tendency calculations#### Analysis Workbook and Checklist 2022#### 2-8.2powerpoint#### Week 2#### STA13 HW4 solution#### RNAssignment 1 Do Not Sell or Share My Personal Information We do not “sell” personal information that we collect directly from you, as “sell” is defined in the California Consumer Privacy Act, as amended (CCPA) or the Virginia Consumer Data Protection Act (VCDPA) or as “share” is defined under the CCPA. 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3841	https://ocw.mit.edu/courses/3-091sc-introduction-to-solid-state-chemistry-fall-2010/b61fb9c49126d9873b2644016348e605_MIT3_091SCF09_aln05.pdf	LN–5 3.091 – Introduction to Solid State Chemistry Lecture Notes No. 5 X-RAYS AND X-RAY DIFFRACTION Sources for Further Reading: 1. Azaroff, L.V., Introduction to Solids, McGraw-Hill, 1960. 2. Wert, C.A., and Thomson, R.M., Physics of Solids, McGraw-Hill, 1970. 3. Nuffield, E.W., X-Ray Diffraction Methods, Wiley, 1966. 4. Cullity, B.D., Elements of X-Ray Diffraction, Addison-Wesley, 1960. 1. HISTORICAL INTRODUCTION X-rays were discovered during the summer of 1895 by Wilhelm Röntgen at the University of Würtzburg (Germany). Röntgen was interested in the cathode rays (beams of electrons) developed in discharge tubes, but it is not clear exactly which aspects of cathode rays he intended to study. By chance he noticed that a fluorescent screen (ZnS + Mn++) lying on a table some distance from the discharge tube emitted a flash of light each time an electrical discharge was passed through the tube. Realizing that he had come upon something completely new, he devoted his energies to investigating the properties of the unknown ray “X” which produced this effect. The announcement of this discovery appeared in December 1895 as a concise ten page publication. The announcement of the discovery of X-rays was received with great interest by the public. Röntgen himself prepared the first photographs of the bones in a living hand, and use of the radiation was quickly adopted in medicine. In the succeeding fifteen years, however, very few fundamental insights were gained into the nature of 1 LN–5 X-radiation. There was some indication that the rays were waves, but the evidence was not clear-cut and could be interpreted in several ways. Then, at the University of Munich in 1912, Max von Laue performed one of the most significant experiments of modern physics. At his suggestion, Paul Knipping (who had just completed a doctoral thesis with Röntgen) and Walter Friedrich (a newly appointed assistant to Sommerfeld) directed a beam of X-rays at a crystal of copper sulfate and attempted to record the scattered beams on a photographic plate. The first experiment was unsuccessful. The result of a second experiment was successful. They observed the presence of spots produced by diffracted X-ray beams grouped around a larger central spot where the incident X-ray beam struck the film. This experiment demonstrated conclusively that X-radiation consisted of waves and, further, that the crystals were composed of atoms arranged on a space lattice. 2. ORIGIN OF X-RAY SPECTRA The interpretation of X–ray spectra according to the Bohr theory (LN-1) of electronic levels was first (and correctly) proposed by W. Kossel in 1920: the electrons in an atom are arranged in shells (K, L, M, N, corresponding to n = 1, 2, 3, 4, ..., etc.). Theory predicts that the energy differences between successive shells increase with decreasing n and that the electron transition from n = 2 to n = 1 results in the emission of very energetic (short wavelength) radiation (fig. 1), while outer shell transitions (say, M N outer shell transitions are associated with small ΔE, i.e. with the emission of radiation of long wave lengths L (x-rays) (x-rays) inner shell transitions are associated with large ΔE, the emission of radiation of short wave lengths n=4 K 3 2 1 Figure 1 X-rays are generated by inner shell electron transitions 2 LN–5 from n = 5 to n = 4) yield low energy radiation (long wavelength). For hydrogen, you recall, the wave number of the emitted radiation associated with a particular electron transition is given by the Rydberg equation: 1 1 R n2 n2 i f For “hydrogen-like” atoms with the atomic number Z (containing one electron only) the corresponding Rydberg equation becomes: 1 1 RZ2 n2 n2 i f From this relationship it is apparent that the energy difference associated with electron transitions increases strongly with the atomic number and that the wavelength of radiation emitted during such transitions moves with increasing Z from the 10–7 m range to the 10–10 m range (radiation now defined as X-rays). To bring about such inner shell transitions requires the generation of an electron vacancy: an electron must be removed, for example from the K shell (n = 1), of an atom. Such a vacancy is conveniently produced in an X-ray tube by an electron beam (generated by a heated filament which is made a cathode) impinging, after being subjected to an accelerating potential of several kV, into a target material made anode (fig. 2). The impinging electrons will transfer part of their energy to electrons of the x-rays (Kα) 1-50 kV e - (K shell) e - (from beam) M L x-rays cathode e- beam target material H2O cooled K vacuum x-rays target material (Cu for ex.) anode Figure 2 Generation of x-rays 3 LN–5 target material and result in electronic excitation. If the energy of the arriving electrons is high enough, some may knock out a K shell electron in the target and thus generate a vacancy. [It should be clear that a K → L excitation cannot take place since the L shell is filled: excitation must involve (n = 1) → (n = ∞).] When such a vacancy is generated, it can readily be filled by an electron from the L shell or the M shell of the same atom. These internal electron transitions give rise to the emission of “characteristic” X-radiation which, because of its short wavelength, has extremely high “penetrating” power. [Since an electron beam is used to generate X-rays, the X-ray tube has to be evacuated: to dissipate the energy flux arriving at the target, the anode support (onto which the target is mounted) is water-cooled.] Under standard operating conditions, the characteristic radiation emitted by the target comprises two sharp lines, referred to as Kα and Kβ lines (fig. 3). They are associated, respectively, with electron transitions from n = 2 to n = 1 and from n = 3 to n = 1. Emitted X-ray spectra were extensively studied by H.G.J. Moseley who established the relationship between the wavelength of characteristic radiation and the atomic number Z of the radiation emitting target material (fig. 4). Experimentally he found that the Kα lines for various target materials (elements) exhibit the relationship: K Z 1 2 (K Z2) Moseley’s empirical relationship (which reflects a behavior in agreement with the Rydberg equation) can be quantified. While the energy levels associated with outer electron transitions are significantly affected by the “screening” effect of inner electrons (which is variable and cannot as yet be determined from first principles), the conditions associated with X-ray generation are simple. Very generally, the screening effect of the innermost electrons on the nuclear charge is accounted for in an effective nuclear 4 70 LN–5 Kα Intensity (rel. units) energy Kβ n = 1 (K shell) n = 3 (M shell) λ Kβ = .6323x10 - 10 m n = 2 (L shell) λ Kα = 0.7101x10 - 10 m 10 An additional set of characteristic spectral lines (Lα ,Lβ) can be observed at longer wavelengths. These reult from transitions n = 3 to n = 2 and n = 4 to n = 2 λswl =f(V) .7101 wavelength (Ã) Spectrum of molybdenum (Mo) at 35 kV (schematic). Electronic transitions that give rise to character istic lines of X - ray spectra. The subscripts α, β, γ and δ designate the change in principle quantum number and are not related to intensity Kα Kβ Kγ Kδ Lα Lβ Lγ Mα Mβ Figure 3 Electronic transitions giving rise to characteristic X- ray spectra. charge (Z-σ) and the Rydberg equation assumes the form: 1 n 1 2 n2 R (Z )2 i f where σ = screening effect 5 LN–5 Atomic # (Z) 80 60 40 20 Considering the transition n2 → n1, screening of the Lα full nuclear charge is only provided by the one electron remaining in the Kα K shell. Thus it is possible to use the ν modified Rydberg Figure 4 Moseley relationship for Kα and Lα radiation equation, taking σ = 1. Accordingly, we have: K 2 1 2 1 1 2 R (Z 1)2 3 4 R (Z 1)2 where: R = Rydberg constant and Z = atomic number of the target material. [The minus sign (–) only reflects radiative energy given off by the system.] Similarly, for the characteristic Lα series of spectral lines (n = 3 to n = 2) we find, after removal of one L electron, that the screening of the electrons in the K shell and the remaining electrons in the L shell reduces the nuclear charge by 7.4 (empirical value). L 1 1 R (Z 7.4)2 5 R (Z 7.4)2 32 22 36 A second look at the X-ray spectrum of a Mo target, obtained with an electron accelerating potential of 35 kV (fig. 5), shows that the characteristic radiation (Kα, Kβ) appears superimposed on a continuous spectrum (continuously varying λ) of lower and varying intensity. This continuous spectrum is referred to as bremsstrahlung (braking radiation) and has the following origin. Electrons, impinging on the target material, may lose their energy by transferring it to orbiting electrons, as discussed above; in many instances, however, the electrons may come into the proximity of the force fields of target nuclei and, in doing so, will be “slowed down” or decelerated to a varying degree, 6 LN–5 Kα Kβ wave length (Å) k Volt 5 10 20 25 X-ray intensity continuous radiation characteristic radiation λswl = f(V) Figure 5 X-ray spectrum of Mo target as a function of applied Voltage. ranging from imperceptible deceleration to total arrest. The energy lost in this slowing down process is emitted in the form of radiation (braking radiation, or bremsstrahlung). This energy conversion, as indicated, can range from partial to complete (fig. 6). The e- (Ek1) e - (Ek2) hν Bremsstrahlung K L Ek1 = Ehν + Ek2 for Ek2 = 0, E(hν)max = Ek1 = λSWL = hν = eV ; λswl = hc/eV Figure 6 Origin of Bremsstrahlung; the continuous part of the X-ray spectrum incident electrons have an energy of e.V (electronic charge times accelerating Voltage) in the form of kinetic energy (mv2/2), and their total energy conversion gives rise to a 7 LN–5 Shortest Wavelength (SWL) – the cut-off of the continuous spectrum for decreasing values of λ (fig. 5). Analytically, we have: c hc eV hmax h ; SWL eV SWL From this relationship it is evident that the cut-off of the continuous spectrum toward decreasing λ’s (λSWL) is controlled by the accelerating potential (fig. 5). 3. THE “FINE STRUCTURE” OF CHARACTERISTIC X-RAYS It is customary to consider the characteristic X-ray spectral lines as discrete lines (Kα, Kβ, Lα, Lβ, etc.). In reality, they are not discrete since the electron shells involved in the associated electron transitions have energy sublevels (s, p, d orbitals). These sublevels give rise to a “fine structure” insofar as the Kα lines are doublets composed of Kα1 and Kα2 lines. Similarly, Lα, Lβ, etc., exhibit a fine structure. These considerations suggest that X-ray spectra contain information concerning the energetics of electronic states. Obviously, analysis of X-rays emitted from a target of unknown composition can be used for a quantitative chemical analysis. [This approach is taken routinely in advanced scanning electron microscopy (SEM) where X-rays, generated by the focused electron-beam, are analyzed in an appropriate spectrometer.] In fundamental studies it is also of interest to analyze soft (long λ) X-ray spectra. For example, take the generation of X-rays in sodium (Na). By generating an electron vacancy in the K shell, a series of Kα and Kβ lines will result. The cascading electron generates vacancies in the 2p level, which in turn can be filled by electrons entering from the 3s level (generation of “soft” X-rays). If the X-rays are generated in a Na vapor, the 3s → 2p transition will yield a sharp line; on the other hand, if X-rays are analyzed in sodium metal, the same transition results in the emission of a continuous broad band, about 30 Å in width. This finding confirms the existence of an energy band (discussed earlier). 8 LN–5 An analysis of the width and intensity distribution of the X-ray band provides experimental data concerning the energy band width and the energy state density distribution within the energy band (fig. 7). 2p 3s -5.01 -38.7 E (eV) 0 λ is fixed 2p 3s -38.7 0 λ is variable E (eV) Na (gaseous) Na (solid) Figure 7 Soft X-rays from 3s - 2p transitions in solid Na and Na vapor 4. USE OF X-RAYS FOR STRUCTURAL ANALYSIS The extensive use of X-rays for the analysis of atomic structural arrangements is based on the fact that waves undergo a phenomenon called diffraction when interacting with systems (diffracting centers) which are spaced at distances of the same order of magnitude as the wavelength of the particular radiation considered. X-ray diffraction in crystalline solids takes place because the atomic spacings are in the 10–10 m range, as are the wavelengths of X-rays. 5. DIFFRACTION AND BRAGG’S LAW The atomic structure of crystalline solids is commonly determined using one of several different X-ray diffraction techniques. Complementary structure information can also be obtained through electron and neutron diffraction. In all instances, the radiation used must have wavelengths in the range of 0.1 to 10 Å because the resolution (or smallest object separation distance) to which any radiation can yield useful information is about 9 LN–5 equal to the wavelength of the radiation, and the average distance between adjacent atoms in solids is about 10–10m (1 Å). Since there is no convenient way to focus X-rays with lenses and to magnify images, we do not attempt to look directly at atoms. Rather, we consider the interference effects of X-rays when scattered by the atoms, comprising a crystal lattice. This is analogous to studying the structure of an optical diffraction grating by examining the interference pattern produced when we shine visible light on the grating. (The spacing of lines on a grating is about 0.5 to 1 µm and the wavelength of visible radiation ranges from 0.4 to 0.8 µm.) In the optical grating the ruled lines act as scattering centers, whereas in a crystal it is the atoms (more correctly, the electrons about the atom) which scatter the incident radiation. The geometrical conditions which must be satisfied for diffraction to occur in a crystal were first established by Bragg. He considered a monochromatic (single wavelength) beam of X-rays with coherent radiation (X-rays of common wave front) to be incident on a crystal, as shown in fig. 8. Moreover, he established that the atoms which θ θ A B C d(hkl) incoming beam diffracted beam 1 2 2θ D Figure 8 Bragg's law, assuming the planes of atoms behave as reflecting planes. constitute the actual scattering centers can be represented by sets of parallel planes (in which the atoms are located) which act as mirrors and “reflect” the X-rays. In cubic systems the spacing of these planes, d(hkl) (see LN-4), is related to the lattice constant 10 LN–5 (a): d(hkl) a (1) h2 k2 l2 For constructive interference of the scattered X-rays (the appearance of a diffraction peak) it is required that the beams, scattered on successive planes, be “in phase” (have again a common wave front) after they leave the surface of the crystal. In terms of the beams labeled 1 and 2 in fig. 8 this requires that the distance AB + BC be equal to an integral number of wavelengths (λ) of the indicent radiation. Accordingly: AB BC n (n 1, 2, 3, ) Since AB = BC and sin AB [AB d(hkl) sin ] : d(hkl) n 2d(hkl) sin (2) This relation is referred to as Bragg’s Law and describes the angular position of the diffracted beam in terms of λ and d(hkl). In most instances of interest we deal with first order diffraction (n = 1) and, accordingly, Bragg’s law is: 2d(hkl) sin [We are able to make n = 1 because we can always interpret a diffraction peak for n = 2, 3, ... as diffraction from (nh nk nl) planes – i.e., from planes with one-nth the interplanar spacing of d(hkl).] If we consider fig. 8 as representative for a “diffractometer” set-up (fig. 11), we have a collimated beam of X-rays impinging on a (100) set of planes and at 2θ to the incident beam a detector which registers the intensity of radiation. For a glancing incident beam (small θ) the detector will register only background radiation. As θ increases to a value for which 2d sin θ = λ, the detector will register high intensity radiation – we have a 11 LN–5 diffraction peak. From the above it is evident that the diffraction angle (θ) increases as the interplanar spacing, d(hkl), decreases. The diffraction experiment as presently considered is intended to provide quantitative information on the volume (the lattice constant a) and shape characteristics (SC, BCC, FCC) of the unit cell. The intensity of diffraction peaks depends on the phase relationships between the radiation scattered by all the atoms in the unit cell. As a result, it happens quite often that the intensity of a particular peak, whose presence is predicted by Bragg’s law, is zero. (This is because Bragg’s law deals not with atom positions, but only with the size and shape of the unit cell.) For example, consider the intensity of the (100) diffraction peak of a crystal which has a BCC unit cell. The phase relationships show that the X-rays scattered at the top and bottom faces of the unit cell, (100) planes, interfere constructively, but are 180 out of phase with the X-rays scattered by the atom at the center of the unit cell. The resultant intensity is therefore zero. The rules which govern the presence of particular diffraction peaks in the different cubic Bravais lattices (SC, BCC and FCC) are given in Table I. TABLE I. Selection Rules for Diffraction Peaks in Cubic Systems Bravais Lattice Reflections Present Reflections Absent Simple Cubic All None Body-Centered Cubic (h+k+l) = even (h+k+l) = odd Face-Centered Cubic h,k,l unmixed h,k,l mixed (either all odd or all even) The rules given are strictly true only for unit cells where a single atom is associated with each lattice point. (Unit cells with more than one atom per lattice point may have their atoms arranged in positions such that reflections cancel. For example, diamond has an FCC Bravais lattice with two atoms per lattice point. All reflections present in diamond have unmixed indices, but reflections such as {200}, {222} and {420} are missing. The 12 LN–5 fact that all reflections present have unmixed indices indicates that the Bravais lattice is FCC – the extra missing reflections give additional information as to the exact atom arrangement.) A hypothetical diffraction experiment: A material is known to be of simple cubic structure; determine a, the lattice constant, by X-ray diffraction. In theory, the question may be answered by placing the crystal into a diffractometer, rotating it into all possible positions relative to the incident X-ray beam and recording all diffracting 2θ values. From the above we know that the smallest observed θ value must correspond to diffraction on {100} planes and also that d(100) = a. We may now use Bragg’s equation to determine a, the lattice constant: 2d sin 2a sin a 2 sin There are two simplifying assumptions in this problem: (1) we know the system is SC and (2) we are able, through rotation, to bring all planes present into diffraction conditions. 6. EXPERIMENTAL APPROACHES TO X-RAY DIFFRACTION In the context of this course we are interested in making use of X-ray diffraction for the purpose of (a) identifying (cubic) crystal systems, (b) determining the lattice constant, a, and (c) identifying particular planes or meaningful orientations. The possible approaches can, in principle, be identified through an examination of Bragg’s law. The Bragg condition for particular d(hkl) values can be satisfied by adjusting either one of two experimental variables: (a) λ, the wavelength of the X-ray beam used, or (b) θ, the orientation of the crystal planes relative to the incident X-rays. 13 LN–5 (a) Fixed θ, Variable λ: One means of satisfying Bragg’s law is to irradiate a stationary single crystal (θ fixed for all planes within the crystal) with an X-ray beam of “white” radiation, which contains the characteristic and continuous spectrum produced by an X-ray tube. (For λ variable we have the simultaneous exposure of a crystal to a range of λ values). Each set of planes will reflect (diffract) the particular λ which satisfies the Bragg condition for the fixed θ. The diffracted beams may conveniently be recorded with a Polaroid camera or, alternately, with an electronic imaging device. It is possible to analyze either the transmitted or the back-reflected X-rays. This experimental procedure is referred to as the Laue technique (fig. 9); it is mostly conducted in the (a) crystal film (b) X-rays X-rays film crystal Figure 9 Laue diffraction in (a) transmission and (b) back-reflection mode back-reflection mode. Note that the approach taken makes it possible to determine the values of θ for each reflection, but not the corresponding λ. Therefore, the technique cannot be used, for example, to determine lattice constants. However, it is very valuable if particular planes or crystal orientations are to be identified. (b) Fixed λ (Monochromatic X-Rays), Variable θ: The basic prerequisite for this approach is the availability of a monochromatic X-radiation of known wavelength (λ). Such radiation can be conveniently obtained by using a crystal (i.e., its diffracting property) as a filter or monochromator (fig. 10). Filter action is achieved by positioning the crystal in such a way that the unfiltered radiation emitted by the X-ray tube becomes incident at an angle, θ, on a set of low index planes which satisfy Bragg’s law 14 LN–5 Crystal monochromator x-rays "white" from tube λ Ka radiation Figure 10 Isolation of monochromatic radiation from target radiation for the highest intensity radiation (Kα) emitted. The condition of a fixed λ and variable θ is experimentally used in two techniques. Using a diffractometer (fig. 11), we place a θ 2θ crystal x-rays from generatior r-ray (being rotated into diffr. conditions) sample (ground to a powder) detector into the center of a rotating stage and expose it to a monochromator monochromatic X-ray beam. The sample is rotated into diffraction condition and the diffraction angle determined. In the Debye-Scherrer Figure 11 X-ray Diffractometer setup. method (fig.12) the sample is ground to a powder and placed (in an ampoule) into the center of a Debye-Scherrer camera. Exposed to monochromatic X-rays, in this way a large number of diffracted cone-shaped beams are generated such that the semiangles of the cones measure 2θ, or twice the Bragg angle for the particular diffracting crystallographic planes. The reason diffracted beams are cone-shaped is that the planes in question (within the multitude of randomly oriented grains) give rise to diffraction for any orientation around the incident beam as long as the incident beam forms the appropriate Bragg angle with these planes – thus there is a rotational symmetry of the diffracted beams about the 15 LN–5 specimen in powder form transmitted x-rays diffracted x-rays Kα radiation 2θ2 2θ1 2θ1 2θ2 2θ4 2θ opened filmstrip 0o 180o 2θ entrance filmstrip diffraction cones Figure 12 Debye-Scherrer powder diffraction setup and analysis direction of the incident beam. Those planes with the largest interplanar spacing have the smallest Bragg angle, θ. In a Debye–Scherrer arrangement, after exposing a powder of a crystalline material to monochromatic X-rays, the developed film strip will exhibit diffraction patterns such as indicated in fig. 12. Each diffraction peak (dark line) on the film strip corresponds to constructive interference at planes of a particular interplanar spacing [d(hkl)]. The problem now consists of “indexing” the individual lines – i.e., determining the Miller indices (hkl) for the diffraction lines: Bragg: 2d(hkl) sin ; d(hkl) h2 a k2 l2 2 4d2 sin2 ; d2 a2 (hkl) (hkl) (h2 k2 l2) 16 LN–5 Substitution and rearrangement of above yields: sin2 2 const. (h2 k2 l2) 4a2 Accordingly, we find that for all lines (θ values) of a given pattern, the relationship sin2 1 sin2 2 sin2 3 const. (h2 k2 l2)1 (h2 k2 l2)2 (h2 k2 l2)3 holds. Since the sum (h2 + k2 + l2) is always integral and λ2/4a2 is a constant, the problem of indexing the pattern of a cubic system is one of finding a set of integers (h2 + k2 + l2) which will yield a constant quotient when divided one by one into the observed sin2θ values. (Certain integers such as 7, 15, 23, etc. are impossible because they cannot be formed by the sum of three squared integers.) Indexing in step-by-step sequence is thus performed as follows: θ values of the lines are obtained from the geometric relationship of the unrolled film strip. Between the exit hole of the X-ray beam (2θ = 0) and the entrance hole (2θ = 180) the angular relationship is linear (fig. 12). The increasing θ values for successive lines are indexed θ1, θ2, θ3, etc., and sin2θ is determined for each. If the system is simple cubic we know that all planes present will lead to diffraction and the successive lines (increasing θ) result from diffraction on planes with decreasing interplanar spacing: (100), (110), (111), (200), (210), (211), (220), etc. From equation (3) above we recognize: sin2 1 sin2 2 sin2 3 sin2 4 sin2 5 const. 1 2 3 4 5 If the system is BCC, however, we know from the selection rules that only planes for which (h + k + l) = even will reflect. Thus: sin2 1 sin2 2 sin2 3 sin2 4 etc. const. 2 4 6 8 17 LN–5 [SC can be differentiated from BCC through the fact that no sum of three squared integers can yield 7, but 14 can be obtained from planes (321)]. For FCC systems, the selection rules indicate reflections on planes with unmixed h,k,l indices: sin2 3 1 sin2 4 2 sin2 8 3 const. After proper indexing, the constant is obtained: sin2 const. (h2 k2 l2) and the particular Bravais lattice is identified. The lattice constant of the unit cell is subsequently obtained, knowing the wavelength of the incident radiation: sin2 const. 2 (h2 k2 l2) 4a2 a2 2 (h2 k2 l2) 4 sin2 a (h2 k2 l2) 2 sin 18 MIT OpenCourseWare 3.091SC Introduction to Solid State Chemistry Fall 2009 For information about citing these materials or our Terms of Use, visit:
3842	https://psecommunity.org/wp-content/plugins/wpor/includes/file/2302/LAPSE-2023.9250-1v1.pdf	Citation: Donskoy, I. On the Existence and Applicability of Extremal Principles in the Theory of Irreversible Processes: A Critical Review. Energies 2022, 15, 7152. Academic Editors: Li Chen, Xiaoying Zhang and Feifei Qin Received: 12 September 2022 Accepted: 27 September 2022 Published: 28 September 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁl-iations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Review On the Existence and Applicability of Extremal Principles in the Theory of Irreversible Processes: A Critical Review Igor Donskoy Melentiev Energy Systems Institute, Siberian Branch of Russian Academy of Sciences, 664033 Irkutsk, Russia; donskoy.chem@mail.ru Abstract: A brief review of the development of ideas on extremal principles in the theory of heat and mass transfer processes (including those in reacting media) is given. The extremal principles of non-equilibrium thermodynamics are critically examined. Examples are shown in which the mechanical use of entropy production-based principles turns out to be inefﬁcient and even contradictory. The main problem of extremal principles in the theory of irreversible processes is the impossibility of their generalization, often even within the framework of a class of problems. Alternative extremal formulations are considered: variational principles for heat and mass transfer equations and other dissipative systems. Several extremal principles are singled out, which make it possible to simplify the numerical solution of the initial equations. Criteria are proposed that allow one to classify extremal principles according to their areas of applicability. Possible directions for further research in the search for extremal principles in the theory of irreversible processes are given. Keywords: thermodynamics; optimization; variational principles; heat and mass transfer; chemical kinetics 1. Introduction A thermodynamic analysis is one of the main methods for studying processes oc-curring in power and chemical engineering plants. The laws of thermodynamics make it possible to obtain a priori estimates and efﬁciency limits even in the absence of kinetic information about the systems under study. These abilities of thermodynamics are con-nected, ﬁrst of all, with the reducing problems to an equilibrium statement. Together with the postulate about the existence of a unique equilibrium, we obtain information about the method of ﬁnding it. Most of the information about the trajectory that led to this state is lost after achieving the equilibrium state. That is, the thermodynamic analysis makes it possible to ﬁnd equilibrium given an incomplete description of the system (for example, to ﬁnd the state of phase or chemical equilibrium without knowledge of the mechanism of mass transfer and chemical reactions). The possibility of working under a lack of in-formation about a system under interest is the main reason for the wide applicability of thermodynamic laws. Let’s assume that the subject of research is some physical-chemical system, the descrip-tion of which includes some basic conservation equations (usually, conservation of mass, energy, and charge). The system characteristics to be determined are naturally related to other variables through conservation equations and other constraints. In the case when the resulting system of equations and inequalities turns out to be incomplete or underde-termined, that is, if the number of variables is greater than the number of non-degenerate relations between them, then we need additional relations that may be quite arbitrary. Based on some principles that can be substantiated by physical intuition or probabilistic considerations, it is possible to redeﬁne and supplement the description of the system. The most natural choice is to deﬁne some extremal state corresponding to the extrema of criteria or their combinations, chosen for some reason. In this paper, we will call an extremal principle a rule for selecting solutions from a set compatible with given constraints. If the number of constraints is already equal Energies 2022, 15, 7152. Energies 2022, 15, 7152 2 of 23 to the number of variables, then the problem of ﬁnding an extremum degenerates. A feasible solution becomes extremal (simultaneously both maximum and minimum) due to its uniqueness. In practice, we more often deal with problems in which the set of constraints is less than the number of variables. In this case, the solution can be found based on the extremal principle. In this sense, the extremal principle reﬁnes information about the system by ﬁlling in or approximating unknown connections between variables (for example, inaccessible during measurements or unidentiﬁed, random-like ones). In this regard, the extremal principle is basically a regularization method for ill-posed problems . In some cases, the opposite direction is possible, when some constraints may be presented in the form of extremum conditions. Extremal principles in classical mechanics have, if not understandable, then at least a familiar form. In statics, this is the search for a minimum of potential energy; in dynamics, it is the principle of least action or its analogues. In the ﬁrst case, the extremal principle gives the problem of mathematical programming: the search for a minimum under given restrictions on the conﬁguration space. In the second case, the problem is formulated in a variational form: one needs to ﬁnd the extremal value of the integral, which depends on the trajectories of motion. The variational form can be preserved in the presence of dissipative forces by disregarding the temporal symmetry of the Lagrange function. When the kinetic energy is completely dissipated, the system goes into static equilibrium. The equations of mechanics, however, have several equivalent formulations, and the extremal formulation is just one of the options, which in some cases turn out to be more convenient than others. In other disciplines, extremal principles sometimes turn out to be a non-alternative way of formulating problems: ﬁrst of all, this, of course, applies to thermodynamics, although the same can be said, for example, about some areas of economics and information science. In thermodynamics, the extremal principles appear as an inevitable consequence of the rejection of a detailed microscopic description of the systems under study. The lack of information must be compensated, and to this end, extremal principles are used, built on the properties of special functions directly related to entropy. The extremal form of laws in theory of irreversible processes is often an attempt to extend the traditional equilibrium analysis given by Gibbs. There are a lot of extremal principles in this area, ranging from so formal that they cannot be explained by anything but formulas, and ending with so heuristic that it is impossible to formalize them. The situation is similar to the generation of new deﬁnitions and interpretations of entropy. It can be considered as some sort of blurring of the semantic ensembles during their evolution. The concept of the extremal principle in thermodynamics is not always clearly deﬁned. As a rule, this is not a mechanical principle that leads to a variational problem, but some sort of mathematical programming problem. The stability of thermodynamic equilibrium is related to the stability and uniqueness of the extremum of the corresponding thermo-dynamic function. The extremum conditions usually do not correspond to the solution of a variational problem, even if the principle is related to the entropy production. Instead, the problem is reduced to the mathematical programming problem, or to estimating the entropy production or quantities related to it. In some cases, the concept of “variational principle” is used because entropy is a statistical function of distribution functions, then the conditions for its extremum have a formal variational statement. The same logic can be, albeit limitedly, extended to entropy production. Variational formulations are usually not considered as a way to obtain the necessary equations in the theory of irreversible processes. More often, the entropy production is a selection criterion for solutions that meet the necessary conditions (conservation laws and boundary conditions), i.e., the solutions follow from the conservation equations, and the extremal principle serves to assess the stability of these solutions. In this sense, the extremal principles in the thermodynamics of irreversible processes are not a full-ﬂedged alternative statement of the problem. Extremal principles can be used to supplement the conservation equations in cases where there is uncertainty related to the multiple solutions or ill-posed conditions. Energies 2022, 15, 7152 3 of 23 Naturally, the question arises of the application validity of extremal principles in the thermodynamics of irreversible processes. Most of these principles do not have a strict theoretical justiﬁcation (at least for fairly complex problems). They are mainly in-ductive. The statement of principles acquires the features of empirical research when the researcher checks a set of hypotheses for compliance with the phenomena observed in a laboratory. At the same time, however, falsiﬁed principles are not always rejected: instead, the limits of their applicability are searched, which, however, do not always ﬁt into ordered schemes in the sense that it is not always possible to predict the working conditions of a particular principle. The existence of universal extremal principles in the theory of irreversible processes is quite a metaphysical hypothesis explicitly formulated in but implied earlier in other forms. Until now, this hypothesis has not been conﬁrmed. However, researchers continue to believe in the possibility of a positive solution to this problem. The grounds for such a belief are supported, in turn, by partial successes in this area, including the results of non-equilibrium thermodynamics (mainly for small deviations from equilibrium). Further progress, in my opinion, was much more modest: although some extremal principles were stated in several individual examples, they do not possess universality and general applicability comparable to the principles of equilibrium thermodynamics. Since the scope of extremal principles is boundless, we will consider, ﬁrst of all, the processes associated with dissipation. The presented review is an attempt to give an overview of the studies on extremal principles in the problems of heat and mass transfer of reacting media, as well as to critically analyze the issues of choosing the form of the extremal principle. Special attention is drawn to the problem of substantiation and generalization of extremal principles. Extremal principles for conservative systems are only mentioned for comparison. The review is organized as follows. Section 2 gives a recap of the extremal principles of equilibrium thermodynamics and their extensions related to kinetic restrictions on the attainability of equilibrium. Section 3 deals with the extremal principles related to entropy production. Section 4 is devoted to the extremal principles in the theory of heat and mass transfer processes that are not based on entropy or entropy production. Section 5 gives some generalizations of the material discussed above. 2. Equilibrium Thermodynamics Equilibrium thermodynamics is usually considered as a natural extension of the laws of statics. Thermodynamic potential functions are modiﬁed (extended) mechanical potential energy, the extremum of which corresponds to the equilibrium state. The equilibrium condition is the condition of the extremum of the thermodynamic function, which connects the thermal, mechanical, chemical, electromagnetic, and other interactions of the elements of the system under consideration. All of these interactions complete the description of the system with conservation equations. As a rule, the number of state variables in the physicochemical system is greater than the number of constraints imposed by conservation equations and boundary conditions. The extremal principle (entropy maximum or free energy minimum) has a constructive function, supplementing the system of equations with extremum conditions. It is possible to imagine a system in which the extremal principle will not be constructive: for example, in a one-component single-phase system with ﬁxed thermobaric parameters (pure inert gas in a closed volume under ambient conditions), the extremal principle will not provide additional information about the equilibrium state, since the set of feasible solutions to the constraint system degenerates into a single state. In mechanical systems without dissipation, the extreme principles of equilibrium thermodynamics will also be redundant. With the conservation of mechanical energy and without heat transfer, the entropy does not change because the equations of motion admit a unique solution at each moment (we will not touch on the discussion about the foundations of statistical mechanics yet). Energies 2022, 15, 7152 4 of 23 The extremal principles of equilibrium thermodynamics have not only a remarkable area of application, but also fairly clear-cut boundaries in this area. The subject of classic thermodynamics is equilibrium, so attempts to apply its apparatus to describe irreversible processes are not always correct. Although in some cases, the relations of equilibrium ther-modynamics are valid in a wider class of phenomena than only in states of ﬁnal equilibrium. For example, in distributed (inhomogeneous in space) and non-stationary (inhomogeneous in time) systems, equilibrium thermodynamic relations hold locally (although the choice of scales often requires additional research and justiﬁcation). One of the key properties of entropy is its concavity, so a fairly well-developed convex analysis is suitable for studying its extrema. For ideal physicochemical systems, the convex properties of thermodynamic functions signiﬁcantly improve the efﬁciency of using numerical methods to ﬁnd equilibria. In non-ideal systems, the thermodynamic functions can be non-convex, so the search for local equilibria may be quite a difﬁcult task even in relatively low dimensions. Another important property of entropy is its connection with the Lyapunov functions for the equations of dynamics of physical and chemical systems . In closed systems, the Clausius inequality itself can impose strong restrictions on their dynamics. Examples of the equilibrium thermodynamics applicability beyond the limits of the traditional range of problems are reviewed in . This extension is based on modiﬁed thermodynamic functions for non-equilibrium states introduced in . These functions contain terms proportional to the free energy needed to create additional conditions (exter-nal ﬁelds, thermostats, chemical membranes) in which this state will become equilibrium. The reversible process is constructed to reduce the problem so that the developed methods of equilibrium thermodynamics can be applied. Such a procedure makes it possible to present entropy as a function of composition and internal energy for systems consisting of locally equilibrium but globally nonequilibrium parts (subsystems). This interpolation (as a rule, implicitly) is widely used in non-equilibrium thermodynamics. The local equilibrium approximation will be discussed separately below. The formulation of additional relations is the addition of new information about the system . If the number of restrictions, including these additional relations, becomes equal to the number of variables, the extremal principle again loses its constructive function. For example, in the complete system of equations of chemical kinetics, the change in entropy is associated only with the reaction rates. Since the reaction rates are determined, there is no selection of states by the magnitude of the entropy. Then, the thermodynamic criteria serve only as a check for the correctness of the chemical kinetics equations (ﬁrst of all, the balance of the mechanism and the values of the kinetic coefﬁcients [7,8]). The papers [9–12] propose methods for simplifying the physical and chemical systems based on the contribution of individual processes to the total increase in entropy. The RCCE approach [13,14] and some others (for example, the entropy operator method [15,16]) are based on the approximation of differential equations for a fast subsystem of dynamic variables by equilibrium relations. In this case, the complexity of the complete system of equations (mainly associated with its numerical solution) is reduced by replacing it with an incomplete system of equations with a partial extremal principle, which in a simpler form gives information about the dynamic behavior. The extremal principle is an approximation, rather than an alternative formulation of the equations: it is valid only asymptotically, but in some cases, this is sufﬁcient for calculations. Naturally, the criteria by which the complete system is divided require additional analysis (for example, based on the spectrum of the Jacobi matrix [17,18]). The physical meaning of the approximation is, as in statistical physics, to suppress the high-frequency modes, whose behavior may become chaotic even for deterministic (not necessarily complex) systems. The extremal principle chooses the most stable conﬁguration of fast variables, smoothing out this uncertainty. Energies 2022, 15, 7152 5 of 23 3. Thermodynamics of Irreversible Processes In the previous section, we considered extremal principles based on entropy (or entropy-related quantities) as a system state function. In the thermodynamics of irreversible processes, along with entropy, its production, i.e., the rate of its change, plays an important role. This quantity does not have the entirety of the properties of the state function. As it was mentioned above, it is rather a qualitative criterion for choosing the direction of the processes. Although in some simple cases, the extrema of entropy production determine the stationary states of systems. An interesting way of representing entropy production as a characteristic of a complex dynamical system is given in [19,20]. However, this approach requires consideration of microscopic dynamics, which is out of scope of this review. It is necessary to mention the limits of the existence of entropy production. Entropy, as a measure of the probability of states, exists only for equilibrium systems (moreover, deviations from the mean are taken into account, with a certain weight, when calculating entropy). Therefore, in order to calculate changes in entropy, the procedure for constructing equivalent equilibrium systems mentioned in the previous section is required. Usually, the appropriate scales are chosen: if the characteristic spatial and temporal scales are signiﬁcantly larger than the molecular ones (that is, ﬂuctuations do not signiﬁcantly affect the values of the averages), then the entropy is considered as a function deﬁned at each point and each moment. In the general case, however, the entropy production can only be considered in a ﬁnite-difference sense, and the differential representation is only an approximation . The transition to continuous functions is another example of ﬁlling in missing information by choosing a convenient representation of variables. The derivative for the thermodynamic parameters exists only in an averaged sense, and its value is chosen with use of the implicit extremal principle. Namely, we choose the difference grid in order to minimize a weighted error which is a sum of the linearization error due to macroscopic nonlinearity, and the uncertainty of the derivative due to microheterogeneity. The chosen grid step can serve as a characteristic scale in the theory of irreversible processes. The ﬁnite-difference representation of the transport equations, usually considered a numerical approximation, is the result of applying the local equilibrium hypothesis. If the local equilibrium approximation becomes inapplicable, then statistical ap-proaches can be applied that use variable decomposition based on relaxation times (for example, the Fokker–Planck equation approach . The entropy production is extreme for stationary processes with a linear dependence of the variables’ change rate on the deviations of these variables from the equilibrium values. Then, the entropy production is a quadratic form in the variables, and its minimum corresponds to the equilibrium state when the variables take their equilibrium values. Prigogine’s theorem states that in the linear region, all variables, except for those speciﬁed through the boundary conditions, take equilibrium values, i.e., the production of entropy is extremal, namely, minimal under the existing restrictions. The extremal principle is equivalent, in this case, to the statement that the entropy production has the properties of a potential function, and the symmetry of the kinetic coefﬁcients follows from the properties of the Hessian matrix of this function in the vicinity of a stationary point . This area also includes classical hydrodynamic problems, in which the dissipation function in slow viscous ﬂow reaches a minimum in steady low-velocity ﬂows . Note that the main result of Prigogine’s theorem depends on the sequence of applying the operations of ﬁxing the boundary conditions and searching for an extremum [25,26]: in the simplest case, the minimum entropy production is zero. The extremal principles of the thermodynamics of irreversible processes are usually not variational principles. Formally, the entropy production is a functional of the ﬁelds of concentrations, temperatures, pressures, and other variables. However, contradictory results arise when applying the basics of the variational calculus to the extremum problem for such a functional: the Euler–Lagrange equations give conservation equations only under certain restrictions (either a modiﬁcation of the objective function or ﬁxing some variables) [27–30]. The Glansdorff–Prigogine principle , having a variational form is Energies 2022, 15, 7152 6 of 23 a generalization of the stability conditions to nonlinear problems giving the necessary stability criterion . The local potential method can be justiﬁed rather as a procedure for constructing numerical approximations than a way to obtain equations (it requires a subtle, if not teleological, variation technique). Biot and Gyarmati principles [33–35], in which the desired conservation equations appear explicitly as constraints on solutions, can be attributed to the same category (as well as similar principles proposed in [36,37]). If irreversible processes occur under conditions when it is impossible to neglect nonlinear dependencies, the second law of thermodynamics, in the form of inequality, again becomes the only justiﬁed extremal principle. Although the second law of thermodynamics, as mentioned above, signiﬁcantly limits the range of feasible solutions, the arbitrariness in the choice remains large. The existence of extremal principles consistent with the conservation equations and boundary conditions turns out in nonlinear dissipative systems to be a fortunate choice of variables rather than a consequence of their general properties. In this regard, variable transformations supporting an extremal form of a problem, are of interest. The minimum entropy production, even if it does not correspond to the phenomena observed in practice, often turns out to be desirable, for example, in thermal and chemical engineering. A decrease in entropy production may lead to an increase in the efﬁciency of devices (heat exchangers, engines, chemical reactors). Therefore, the search for conditions under which the entropy production turns out to be minimal is an important problem: ﬁnite time thermodynamics is devoted to its study [38–40]. It is necessary to discuss the principle of maximum entropy production. It is used, as a rule, not to obtain solutions, but to select them from a set of possible solutions compat-ible with boundary conditions and conservation equations. Historically, one of the ﬁrst principles of this kind is the Ziegler principle, which states that non-stationary processes proceed in such a way as to maximize the entropy production [41,42]. It is necessary to point out that the maximization of the entropy production does not follow the principle of maximum entropy, since the maximum value of entropy can be reached in many different ways. For example, the conversion of an oxygen-hydrogen mixture into water can go through slow oxidation or an explosion: the choice of a speciﬁc implementation depends on numerous factors, while the maximum entropy production is achieved during the explo-sion only. Attempts to substantiate the principle of maximum production begin with the works of Jaynes, where a similar concept is introduced, namely the principle of maximum caliber [5,43]. In the process of evolution, the statistical ensemble retains its phase volume, but the volume inside its hull grows. The linearized Boltzmann equation can demonstrate similar behavior near equilibrium [44,45]. The principle of maximum caliber proposes to select from all possible variants of evolution the one that allows obtaining the maximum volume of the ensemble hull. This option contains the least amount of information about the laws of dynamics (apparently, the linearized Boltzmann equation is an example of such over-reduced dynamics). On the one hand, this approach can give some upper bound for the entropy production consistent with the conservation equations. On the other hand, the actual system behavior may differ from this extremely dissipative one. As pointed out in , the maximum entropy production principle needs additional parameters that are difﬁcult to determine unambiguously (for example, integration limits). If dynamic equations are entirely unknown, then nothing prevents the system from moving to a state of ﬁnal equilibrium (thermal, chemical, and mechanical) during several molecular vibrations. If such a transition does not occur, then the principle should be clariﬁed and supplemented with restrictions on the dynamics. Because such constraints are often non-linear, the ex-tremal properties of the objective function may change. In , the existence of special metrics that deﬁne the trajectories as geodesics in the state space is assumed (for minimal entropy production principle, see ). However, the question of the speciﬁc form of such metrics for practically interesting cases remains open. Some authors assign the principle of maximum entropy production the role of a “hypothesis selection algorithm”, i.e., it implies an iterative cycle of reformulating the Energies 2022, 15, 7152 7 of 23 problem until the principle is fulﬁlled [49–51]. Others consider it more fundamental [52,53], although they do not give clear restrictions on its applicability. For example, in , a condition is put forward for the applicability of the principle of maximum entropy production: the system must be complex and far from equilibrium; if the system does not obey the principle, then it is not complex enough and not far from equilibrium. Thus, this principle can be interpreted rather as an attempt to apply thermodynamic criteria to the selection of solutions under the lack of information about the system. For example, some papers on the principle of maximum entropy production are aimed at studying the thermal regimes of planetary atmospheres [52–56]. Other attempts to justify the principle of maximum entropy production, sometimes involving methods from the humanities, can be found in . Note that most of the published works on the principle of maximum entropy production are devoted precisely to attempts to substantiate this principle, and not to its practical application. However, the situation with the principle of minimum entropy production is not better. Criticism of the principle of maximum entropy production is described in detail in [46,58]. First of all, the generalization of the second law of thermodynamics into dynamics is unjustiﬁed. Further, the choice of the process trajectory is ambiguous under conditions when the entropy production turns out to be essentially non-monotonic. In some papers, the principle of maximum entropy production is partially supported by some results of studies related to speciﬁc physicochemical systems (for example, mod-els of autocatalytic and enzymatic reactions [59,60], free convective ﬂow , planetary atmospheres from the above works, stellar statistics ). However, statements about the universality of the principle, in my opinion, are too exaggerated even for the classes of processes under consideration. It is possible to choose examples of autocatalytic reactions and free-convective ﬂows, in which neither the maximum nor the minimum of entropy production in the same formulation is fulﬁlled [63–65]. As pointed out in , it is difﬁcult to ﬁnd a phenomenon or process, the description of which with the help of the principle of maximum entropy production is uncontested and, at the same time, adequate. Naturally, for nonlinear systems, the search for a universal extremal principle is a practically hopeless task and attempts to ﬁnd ordered behavior can be started with the simplest (using the least amount of information about the system) hypotheses. Jaynes’s formalism and the hypothesis of molecular chaos are also, in a sense, extreme principles, which are rather difﬁcult to justify, but their application in different problems gives correct physical results. The predictive ability of the maximum entropy production principle remains uncertain. New varieties of entropy and entropy production are proposed, which may play the role of a state function in the theory of irreversible processes [43,67]. The diversity of approaches suggests that there are still no universal extreme principles in this area. It can be said that the pessimism of the works [2,68,69] has not yet been dispelled: their obituary tone was, apparently, quite appropriate. The extremal principles of thermodynamics of irreversible processes include the “construction law” on the optimization of system energy transfer , which, due to its non-strict formulation, depending on the situation, can be interpreted both in favor of maximum and minimum entropy production . This principle, as the name implies, is focused more on ﬁnding the optimal design (similar to the methods of thermodynamics in ﬁnite time), and is not a universal law of the organization of matter. Extremal principles based on the entropy production do not yet have clear limits of applicability, therefore, despite the obvious inconsistency, different authors analyze the same systems from the standpoint of maximum and minimum entropy production with different assumptions and close results . Note that in deterministic systems, each state and each trajectory are unique, therefore, the entropy production in each state and each region has a unique value, that is, both minimum and maximum under the existing restrictions. In this case, the extremal principle is redundant. We may abandon the deterministic description and replace part of the equations and restrictions with the extremum conditions of some new function. Then, for sufﬁciently complex systems, such Energies 2022, 15, 7152 8 of 23 a replacement can be carried out in several ways. Accordingly, the form of the objective function and the type of extremum may change depending on the replacement method. At the end of the section, it is necessary to mention the seemingly insurmountable problem of extremal principles in chemical kinetics. The previous section brieﬂy mentioned some approaches based on the thermodynamic estimates in the numerical simulation of chemical systems, but these approaches are patchy and woefully not universal. Direct analogies between dynamics and chemical kinetics (which may be expected from the “statics—equilibrium thermodynamics” analogy) do not work at all [73,74]. Attempts to apply extremal principles from the theory of irreversible processes to chemical reactions are still very limited. The relationship between the rate of a chemical reaction and its chemical afﬁnity can be essentially non-linear even for small deviations from equilibrium, and the interaction of numerous chemical reactions leads to the inapplicability of any simpliﬁ-cations. On the other hand, naive attempts to apply the principle of maximum entropy production to chemical transformations also do not lead to any reasonable results . However, in [76,77], the principle of maximum entropy production allowed to construct the trajectories of chemical systems with well-turned approximations for the entropy pro-duction in chemical reactions: the numerical results showed that the qualitative results are quite close to the direct kinetic calculations. Authors of [78–80] obtained interesting results on the thermodynamics of complex chemical systems by analyzing the dynamics of different functions from detailed kinetic calculations. Among these functions, the authors singled out a quadratic form for the second derivatives in a certain special norm (related to the thermodynamic properties of the reacting mixture), the integral of which, along the trajectory can be considered as the objective function of the extremal principle. At the same time, however, the extremal principle itself requires knowledge of the kinetic equations, i.e., the same amount of information is needed to write the functional as to write the kinetic equations. Similar functionals were considered in [81,82]. For stationary states in open systems, extremal principles were also proposed, for example, (using kinetic equations) and [84,85] (based on a special “thermodynamic” form of the chemical kinetics equations). Consideration of the chemical reaction as a diffusion process in a space of reaction coordinates was proposed in [86,87], so gradient law formalism can be applied to chemical kinetics problems. In [88,89], the Lyapunov functions for a chemical system were proposed, which are represented in terms of pair correlations for ﬂuctuations of reaction rates and concentrations: to write it, naturally, information about the reaction mechanism and kinetic coefﬁcients is needed. Three examples of the application of extremal principles to reactive media problems are considered in the Appendix A. 4. Extremal Principles for Dissipative Systems Not Directly Related to Thermodynamic Quantities As mentioned above, the earlier extremal principles in irreversible processes theory were related to works on viscous ﬂuid dynamics, where stationary laminar ﬂows sometimes obey the minimal dissipation principle [24,90,91]. The dissipation function, however, is directly related to the entropy production [23,92], so they will not be considered below. There are often attempts to associate extremal principles with entropy or entropy produc-tion, but this may be inappropriate, for example, if this connection requires unjustiﬁed redeﬁning entropy. The convenience of the extremal formulation led to the development of a theory for solving the inverse variational calculus problem . A sufﬁcient condition for the existence of a variational principle is the potentiality. For example, linear problems can easily be reduced to a potential form. However, this does not mean that the extremal principle does not exist for non-potential problems [94–97]. A variety of “recipes” for constructing extremal principles are proposed in [98–100]. Among them are the transformation of variables and the modiﬁcation of the functional space. Moreover, some problems admit the existence of several different extremal principles. Variational principles for mechanical Energies 2022, 15, 7152 9 of 23 systems with linear friction are proposed in [101–104]. The variational principles for the ﬂow of a viscous ﬂuid (with different limitations of applicability) are proposed in [105–110], including for turbulent ﬂows [111–114]. Some variants of the variational principles for diffusion and heat conduction are given in [115–118]. The variational estimates for the limit intensity of turbulent transport are obtained in [119,120]. The simplest way to formulate the extremal principle for an arbitrary equation is the method of least squares. Such formulations are used, for example, for the numerical solution of differential equations, in mechanics and chemical kinetics [121–123], and convective and random transport [124–126]. For this, however, it is necessary to know in advance the form of the equations: the extremal statement gives a simple computational algorithm. The extremal principles are sometimes for estimating transport coefﬁcients (for exam-ple, for turbulence transport [127–130] and transport in complex media [131,132]). Another area of the application of extremal principles is the numerical solution of heat and mass transfer problems: variational estimates allow one to choose grid parameters [133–137] or construct solutions in series [138,139]. There are a lot of papers on variational approaches in reaction-diffusion equations. Elliptic equations can often be presented in the variational form . First of all, these are the classical equations of chemical engineering (diffusion in catalytic reactors, thermal explosion theory, see), for which the variational statement allows us to obtain fairly good numerical estimates, even at low orders of approximation [141–147]. The propagation velocity of stationary chemical reaction waves also turns out to be related to variational estimates [148–152] (in some cases, they relate to entropy production [153,154]). In chemical kinetics, the variational principles based on the Pontryagin principle have been proposed, which allow one to ﬁnd the critical values of kinetic parameters and their sensitivity coefﬁcients . An important feature of the examples listed here is that the objective functions and functionals are by no means always directly related to entropy (or entropy production). Moreover, attempts to relate these functions to entropy may be completely unconvincing. For example, the variational principle for reaction-diffusion equations exists even in the approximation of an irreversible reaction (i.e., when chemical equilibrium does not exist in some sense). Friction intensity in mechanical systems determines the entropy production, but it is not determined by its extrema. When using explicit expressions for the friction forces, the entropy production is a deterministic value. Therefore, at each moment and on each interval, it has neither minima nor maxima. Of course, one can modify the deﬁnition of entropy or the deﬁnition of entropy in such a way that its extremum coincides with the desired solution. In this case, as already mentioned, the generality of the approach is usually lost. Arbitrariness cannot be introduced at such a fundamental level. We also note that the heat and mass transfer problems often admit the multiplicity of solutions compatible with the given conditions. The selection of solutions, in this case, occurs as a result of some evolution. For example, the stationary state is reached as the limit of the motion of a dynamical system from a given initial condition at large times. In the deterministic dissipative systems, the initial conditions determine an achievable stationary state. In stochastic dynamics, transition mechanisms lead to a distribution of characteristic residence times corresponding to different admissible states . System entropy (deﬁned in traditional way) and its derivatives do not necessarily determine the observed states. However, one can consider a probabilistic scheme with an ensemble of stationary states between which reversible transitions occur. These transitions may be represented as a pro-cess of transfer or transformation, allowing the application of equilibrium thermodynamics formalism. In this case, however, the thermodynamic functions must include information about the frequencies and energy characteristics of these transitions (trajectories). Again, modiﬁcations of the objective function are required [43,67,157,158]. Energies 2022, 15, 7152 10 of 23 5. Optimality and Illusion of Optimality The previous sections give a brief overview of the extremal principles that arise when describing equilibrium states and irreversible processes. As one can see, for almost every phenomenon it is possible to formulate, if not an exact extremal principle, then at least an asymptotically exact one approximating the behavior of variables in a limited range of conditions. Sometimes the extremal principle appears, in a sense, from a general law of nature (or lack of information about the system under study). In other cases, on the contrary, the extremal principle is rather an artiﬁcial construction that allows one to simplify calculations or give a new interpretation of known regularities. However, it is not always possible to unambiguously classify extreme principles. A metaphysical question arises: in what cases can an extremal principle be considered fundamental, and in what cases—artiﬁcial? It seems natural to make genealogical connections between extremal principles and establish which of them lead to valid regularities or formulas, and which are introduced heuristically or using untested hypotheses. However, sooner or later we will ﬁnd our-selves in the ﬁeld of axiomatics of physical theories. Since extremal principles are usually postulated, nothing prevents us from introducing new extremal principles, substantiating their necessity by the results obtained. Similarly, we cannot draw the line between those extremal principles which are the only formulation of the problem and those which have an alternative non-extremal statement. A more reasonable approach, in my opinion, is to estimate the amount of information that the extremal principle allows to obtain. An extremal principle that claims to be fundamental must be constructive. Many solutions must be possible that satisfy the existing restrictions. The principle application should highlight a narrow set of feasible solutions. Another informational advantage of the extremal principle should be the ability to generalize the problem for which it was initially formulated. The principle should give equations for describing processes and phenomena that would be difﬁcult or even impossible to obtain in another way (this remarkable property highlights the least action principle as a more efﬁcient way of formulating the mechanics’ equations). Finally, the solutions identiﬁed by the principle must correspond to the observed reality, at least within the limits of its applicability. Moreover, the limits of applicability are an integral part of the extremal principle (which is observed for equilibrium thermodynamics and linear non-equilibrium thermodynamics). An alternative set of requirements is proposed in (including possible relativistic extensions). If the application of the extremal principle requires a preliminary reduction (i.e., an artiﬁcial rejection of some information about the system) then the extremal principle loses its epistemological signiﬁcance (however, it can still be useful, for example, for numerical calculations). A wide range of tools that allow reducing differential and algebraic equations to an extremal form suggests that not all extremal principles are fundamental. The extremality of the phenomena observed in nature (understood in some sense) is not a property of these phenomena. It is a result of their interpretation (in some cases even “over-interpretation”). As mentioned above, the extremal principle, as a rule, serves to ﬁll the information gap about the system. It can be assumed, slightly changing Jaynes’s arguments, that the very existence of the extremal principle means that we do not have a sufﬁciently deﬁnite description of the system (in the sense of the parameters and restrictions necessary for the uniqueness of the solution). We are therefore forced, focusing on the features of the problem, to introduce the regularization of the problem based on some criteria. The choice of these criteria is not always unambiguous, and attempts to relate them to thermodynamic functions (or their derivatives) are not always justiﬁed. It is well-known that the stationary distribution of variables, the transfer of which occurs according to gradient laws (temperature, concentration, momentum in a continuous medium), in the simplest cases, obeys the condition of the minimum gradients squares sum. This fact has several interpretations, ranging from the “intolerance” of nature to Energies 2022, 15, 7152 11 of 23 discontinuities and ending with the “tendency” for the minimum production of entropy. On the other hand, a slight change in the system (adding source terms that lead the system to discontinuous or close to such solutions, which are observed, for example, during combustion) will modify the extremal principle, complicating its interpretation. The minimum entropy production hypothesis is not conﬁrmed even in the simplest case (it sufﬁces to consider a non-isothermal system). The extremal principle, which requires minimizing the gradients squares sum, turns out to be just an equivalent form of solving the stationary problem of diffusion transport under certain restrictions on the type of the equations. It has no predictive ability for more complex cases. Both formulations (differential and extremal) lead to the same solution in different ways: this argument is most often used to justify extreme principles, even if they cannot be generalized in any way. A signiﬁcant shortcoming of the extremal principles proposed in the thermodynamics of irreversible processes is their “structural instability”. Small changes in the formulation of the problem, the addition of new variables or transfer mechanisms, make the extremal principles unusable and force us to look for new formulations. This inconsistency may result from an overestimation of the extremal principles’ signiﬁcance for the theory of irreversible processes. Again, the solution of the complete heat and mass transfer equations gives a set of solutions for the desired variables without the application of extremal principles. Extremal principles appear in those cases when the formulation of the problem is incomplete: for example, the initial conditions are uncertain, or the connections between subsystems are unknown. If all of the initial data, boundary conditions and equations are known, there is no need for extremal principles. As in statistical physics, the extremal principle is not something intrinsic to the system under study: it emerges from the limitations of our knowledge of that system [160,161]. In other words, an extreme principle is a scientiﬁc way of expressing our ignorance of the description of physical systems. However, many different extremal principles proposed for similar systems express our ignorance in the way of description, and, ultimately, in the physical nature of phenomena. The problem of constructing consistent algorithms for choosing extremal principles in the theory of irreversible processes remains unsolved. Our selection of a system description always contains some arbitrariness, justiﬁed by the results usually obtained from the application of the chosen formalism. Accordingly, the problem of ambiguity in the choice of system boundaries and the extremal principle are closely connected. It can be assumed that the system boundaries determine the extremal principle (as, for example, in equilibrium thermodynamics, the form of the objective function depends on the conditions of the system’s interaction with the environment). In this case, the extremal principle should determine the system state through the boundary conditions in which it exists, in cases where the internal connections (limitations) in the system are not fully known. Of course, validation of the extremal principle should cover as many examples as possible. Then, the discrepancy between the observed phenomena and the consequences of the principle should serve as a stimulus for reformulating the problem. Ambiguity arises when new information about the system (e.g., empirical information) can be taken into account both in the form of an additional constraint (as, for example, in semi-empirical equilibrium models of chemical engineering processes [162,163]) and by modifying the objective function. In some cases, both methods give equivalent formulations, but in the general case (ﬁrst of all, when considering irreversible processes), there is no certainty in such equivalence. If the extremal principle is a hypothesis selection rule, then a one-to-one correspondence between the system under study and the extremal principle is implicitly assumed. This statement is not obvious. The discussion is not intended to refute the existing extremal principles in the ther-modynamics of irreversible processes. However, not every law of nature has to take the form of an extreme principle. Leonhard Euler believed that the world provided to us is “the best possible” since everything that happens in it follows the principles of minimum or maximum. However, with the same conﬁdence, it can be argued that our mathematical Energies 2022, 15, 7152 12 of 23 tools have developed enough to ﬁnd the principles of minimum or maximum in everything that happens. Sometimes extremality may be just a pattern that we deliberately look for in the systems under study. In this case, the discovered extreme principle does not necessarily give a broader view of the phenomenon; on the contrary, the search for extrema can lead to misconceptions, which, while retaining the usual form of the extremal principle, are basically local approximations. In the end, it is necessary to highlight prospects in the area of extremal principles in the theory of irreversible processes. On the one hand, breakthroughs are unlikely in this area, and one of the reasons is development of computational methods and techniques allowing to solve heat and mass transfer problems in a reasonable time (at least, approximately). Proposed principles have a narrow applicability (sometimes, constraints are so stiff that the principle becomes ad hoc procedure). On the other hand, progress in methods in inverse optimization and variational problem solving is remarkable, so we can believe that universal extremal principles for irreversible processes will be found in some form, not necessarily typical for non-equilibrium thermodynamics. Local approximations are not always useless: they may serve as a transition link towards more general theories. 6. Conclusions In this paper, extremal principles in the theory of irreversible processes (primarily, heat and mass transfer processes) are reviewed. The principles based on the extreme properties of entropy and entropy production are considered in detail. The role of extremal principles is discussed in connection with the choice of solutions from a set of formally admissible ones under conditions of uncertainty that arise due to the incomplete or incorrect formulation of the problem. Since uncertainty and incompleteness are associated with the available information about the system under study, then the unique solution is obtained by a priori statistical estimates (such as entropy and its modiﬁcations). In general, the substantiation of the entropy production-based principles is unsatisfactory. The deterministic macroscopic heat and mass transfer equations do not require additional principles for selecting solutions; at the same time, the proposed extremal principles of non-equilibrium thermodynamics, as a rule, do not result in kinetic equations (although in some cases the principles make it possible to estimate the coefﬁcients in approximations). Moreover, there are exact extremal (variational) principles for transfer equations that are not directly related to thermodynamic functions. The results of the review show that the use of the objective function in the form of entropy production limits the possibilities of applying extreme principles to heat and mass transfer processes. It is necessary to expand the toolkit of objective functions: one may look for generalizations among the exact extremal principles for transfer equations, instead of trying to associate a predeﬁned objective function with them. Funding: The research was carried out under State Assignment Project (no. FWEU-2021-0005) of the Fundamental Research Program of the Russian Federation 2021–2030 using the resources of the High-Temperature Circuit Multi-Access Research Center (Ministry of Science and Higher Education of the Russian Federation, project no 13.CKP.21.0038). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The author is grateful to V.A. Shamansky for the fruitful discussion on the manuscript subject. The author is also grateful to colleagues listed at t-invariant.org (accessed on 7 July 2022). Conﬂicts of Interest: The author declares no conﬂict of interest. Energies 2022, 15, 7152 13 of 23 Appendix A Example A1 Let us consider an isothermal diffusion-reaction equation in a porous medium of width L: Dc′′ −w = 0 (A1) Here D is diffusivity, c is reagent concentration, and w is reaction rate. Local entropy production is: σ = DR 2 (c′)2 c + A T w (A2) Here, R is gas constant, A is reaction afﬁnity. Equation (A2) is valid for an ideal mixture when the chemical potential of the reagent is: µ = µ0 + RT ln c Reversible reaction occurs with a rate: w = kc − k Keq (c0 −c) Here, k is the reaction rate constant, Keq in the equilibrium constant. Then the afﬁnity is: A RT = ln Keq + RT ln c c0 −c Extremal entropy production gives Euler–Lagrange equation: Dc′′ = w c0 c0 −c + A RT k + k Keq c + D 2c c′2 (A3) This equation is not equivalent to Equation (A1) because it contains some cumbersome terms. It can be shown, however, that Equation (A3) is valid near equilibrium, when A ≈0 and concentration distribution is close to uniform (so c’ ≈0). Comparison of numerical solutions is presented in Figure A1 (boundary conditions are c’(0) = 0 and c(L) = c0). It can be seen that the extremal entropy production principles overestimate the reaction rate (due to additional terms in the right part of Equations (A3) and (A4)). It is interesting that using the modiﬁed entropy production σ1 = σc we obtain the following equation: Dc′′ = w c0 c0 −c + A RT w + k + k Keq c (A4) Equation (A4) is closer to Equation (A1) due to the absence of a term proportional to gradient squared, but its applicability is still limited to the near-equilibrium area. Equation (A1) has the exact variational principle [140,164]. Solution of Equation (A1) minimizes a following integral: I = Z D 2 c′2 + k + k Keq c2 2 − k Keq c0c dx (A5) Integrand in Equation (A5) do not have direct connection to entropy production. Moreover, this principle cannot be used in non-isothermal conditions or Example A2 Let us consider the exothermic reaction in a ﬂat layer. The energy equation can be written as follows: λT′′ + Qw = 0 (A6) Here, λ is thermal conductivity, T is temperature, Q is reaction heat, w is reaction rate. Energies 2022, 15, 7152 14 of 23 Energies 2022, 15, 7152 14 of 24 Figure A1. Comparison of numerical solutions of Equations (1), (3), and (4). Values of the parame-ters: k = 0.01; Keq = 100; D = 10−5; L = 0.1; c0 = 1. Equation (A1) has the exact variational principle [140,164]. Solution of Equation (A1) minimizes a following integral: ( ) 2 2 0 2 2 eq eq D k c k I c k c c dx K K      = + + −          (A5) Integrand in Equation (A5) do not have direct connection to entropy production. Moreover, this principle cannot be used in non-isothermal conditions or Example A2 Let us consider the exothermic reaction in a flat layer. The energy equation can be written as follows: 0 T Qw + = (A6) Here, λ is thermal conductivity, T is temperature, Q is reaction heat, w is reaction rate. Using the combustion theory approximation, we can assume that the below ignition conditions reaction rate is quite low, so we can neglect the reagent depletion. Then Equa-tion (A6) gives the stationary temperature distribution under w = w(T). Local entropy production in the one-dimensional layer can be written as follows: ( ) 2 2 1 2 A T w T T    = + (A7) Let us introduce the generalized entropy production in a form σn = σTn, for given number n . Extremal entropy production conditions (Euler–Lagrange equations) give the following Equation for n = 0: ( ) 2 0 E T T w Q A T RT       − + − =     (A8) Here, E is the activation energy. Equation (A3) is deduced using approximate rela-tions: Figure A1. Comparison of numerical solutions of Equations (1), (3), and (4). Values of the parameters: k = 0.01; Keq = 100; D = 10−5; L = 0.1; c0 = 1. Using the combustion theory approximation, we can assume that the below igni-tion conditions reaction rate is quite low, so we can neglect the reagent depletion. Then Equation (A6) gives the stationary temperature distribution under w = w(T). Local entropy production in the one-dimensional layer can be written as follows: σ = 1 2 λ T2 T′2 + A T w (A7) Let us introduce the generalized entropy production in a form σn = σTn, for given number n . Extremal entropy production conditions (Euler–Lagrange equations) give the following Equation for n = 0: λT′′ −λ T T′2 + w Q −A E RT = 0 (A8) Here, E is the activation energy. Equation (A3) is deduced using approximate relations: ∂w ∂T ≈w E RT2 A = −∆Gr = −∆Hr + T∆Sr = Q + T∆Sr Equation (A3) implies that the entropy production is extremal for the low reaction afﬁnity and low sensitivity of the reaction rate with respect to temperature: RT E >> 1 + T∆Sr Q (A9) Moreover, there is an unwanted term proportional to gradient squared, which terms to zero only at the uniform temperature distribution. If n = 1 then we obtain equation: λT′′ + λ 2T T′2 −w QE RT + ∆Sr T + E R = 0 (A10) Equation (A10) contains a wrong sign of the reaction heat rate term. Again, the gradient squared term is present. Energies 2022, 15, 7152 15 of 23 If n = 2 then we have an equation closer to Equation (A6): λT′′ + Q −A E RT w = 0 (A11) This equation turns to Equation (A6) under condition Equation (A10). That is, the entropy production is minimal close to equilibrium as it was stated in Equation (A6) is a classical equation in combustion theory. It is convenient to transit to non-dimensional variables : ξ = x L; θ = E RT2 0 (T −T0); Ar = RT0 E ; Fk = QE RT2 0 L2 λ w(T0) Then, Equation (A6) can be written as follows: θ′′ + Fk exp θ 1 + Arθ = 0 (A12) If Ar is small (i.e., if E is large) then we can state the variational principle in a form of minimization of the following integral [147,168]: I = 1 Z 0 1 2 θ′2 −Fk exp θ dξ (A13) This principle is exact for Ar = 0, but the approximate principles can be proposed for other values . Interestingly, the stability condition for the solution of Equation (A12) is the local minimum condition of Equation (A13), and the critical condition is the inﬂection point existence [145,168]. This critical condition for Equation (A13) formally corresponds to the Glansdorff–Prigogine criterion, but only for the case of the generalized entropy production with n = 2. Moreover, condition Equation (A9) does not allow using the approximation Ar ≈0. Example A3 Let us consider catalytic heterogeneous reaction in the thin surface layer. Concentration and temperature at the distance δ is equal to the constant values c0 and T0. Then, we can calculate the stationary diffusion and heat ﬂows near the surface: j = D δ (c0 −c)q = λ δ (T0 −T) Here, D is diffusivity, T is surface temperature, and c is surface concentration of reagent. Reversible reaction occurs at the surface with a following reaction rate: w = k1c −k2(c0 −c) Then, we can equate the diffusion ﬂow and reaction rate, j = w, which gives an equation for the surface reagent concentration: c = c0 1 + Da Keq 1 + Da 1 + 1 Keq Here, Da is the diffusion Damkohler number, Keq is the equilibrium constant. In a limit of large Keq we obtain the usual diffusional kinetics formula . Note, that the kinetic constants are functions of temperature, k1 = k1(T) and k2 = k2(T), and they relate to each other by mass action law: k1 k2 = exp −∆Gr 0 RT = Keq Energies 2022, 15, 7152 16 of 23 Then, the stationary surface temperature can be found: T −T0 = D λ Qc0Da 1 + 1 Keq   1 + Da Keq 1 + Da 1 + 1 Keq − 1 1 + Keq   Here, Da and Keq are temperature-dependent. Using non-dimensional variables, we can write: θ = θad Le Da 1 + 1 Keq   1 + Da Keq 1 + Da 1 + 1 Keq − 1 1 + Keq   Here, θad is the non-dimensional adiabatic temperature. Introducing the Damkohler number for the ground temperature T0 and assuming Ar = 0, we obtain: θ = θad Le Da0eθ 1 + e−sθ" 1 + Da0eθ(1−s) 1 + Da0eθ1 + e−sθ − 1 1 + esθ # (A14) Here, s is the relation of the chemical reaction afﬁnity and activation energy (s = −A0/E). Typical behavior of the solutions of Equation (A14) is presented in Figure A2. Energies 2022, 15, 7152 17 of 24 Figure A2. Solutions of Equation (9) at Le = 1, θad = 10, s = 10. In the range of Da0 values from 0.0011 to 0.041, the solution becomes ambiguous. That is, at lower values of Da0, the surface temperature differs little from the ground tempera-ture, but with an increase in Da0 (with an increase in the reaction rate or with an increase in diffusion resistance), new solutions appear. The upper (high-temperature) and lower (low-temperature) branches of the solution are stable. The system may end up in one of the stationary states, depending on the initial conditions. In the range of multiple solu-tions, the entropy production at a fixed Da0 is minimal on the low-temperature branch and maximum on the high-temperature branch. Entropy production for a given system cannot be non-extremal, since non-extremal stationary states are unstable, but it is impos-sible to predict a priori in which state of the two extrema the system stays without infor-mation about its dynamics. Solutions may have a more complex form, with intermediate stable stationary states: in this case, extreme estimates turn out to be less plausible . In unstable flows (for example, during turbulence), the Damkohler number can fluc-tuate, so it becomes possible to switch between the branches of the solution. In this case, knowing the distribution of the Damkohler numbers over observation times and the char-acteristic fluctuation frequencies, one can estimate the average surface temperature by representing the stationary states as a two-component ensemble. For each solution shown in Figure A1, the entropy production can be calculated. Moreover entropy production can be calculated even for those temperatures that cannot Figure A2. Solutions of Equation (9) at Le = 1, θad = 10, s = 10. In the range of Da0 values from 0.0011 to 0.041, the solution becomes ambiguous. That is, at lower values of Da0, the surface temperature differs little from the ground temperature, but with an increase in Da0 (with an increase in the reaction rate or with an increase in diffusion resistance), new solutions appear. The upper (high-temperature) and lower (low-temperature) branches of the solution are stable. The system may end up in one of the stationary states, depending on the initial conditions. In the range of multiple solutions, the entropy production at a ﬁxed Da0 is minimal on the low-temperature branch and maximum on the high-temperature branch. Entropy production for a given system cannot be non-extremal, since non-extremal stationary states are unstable, but it is impossible to predict a priori in which state of the two extrema the system stays without information about its dynamics. Solutions may have a more complex form, with intermediate stable stationary states: in this case, extreme estimates turn out to be less plausible . In unstable ﬂows (for example, during turbulence), the Damkohler number can ﬂuc-tuate, so it becomes possible to switch between the branches of the solution. In this case, Energies 2022, 15, 7152 17 of 23 knowing the distribution of the Damkohler numbers over observation times and the char-acteristic ﬂuctuation frequencies, one can estimate the average surface temperature by representing the stationary states as a two-component ensemble. For each solution shown in Figure A1, the entropy production can be calculated. More-over, entropy production can be calculated even for those temperatures that cannot be solu-tions of Equation (A14). An example of such a calculation is shown in Figure A3: it can be seen that Equation (A14) has three stationary solutions at 0.11 (stable), 4.33 (unstable), and 9.96 (stable). In this case, the entropy production has no local extrema and increases monotonically with the temperature (the markers highlight the stationary temperature values). 52 18 of 24 Figure A3. Calculation results for Da0 = 0.01: (a) violation of equality (9); (b) entropy production. Maximum entropy production principle state that a high-temperature stationary state has a higher probability of observing; minimum entropy production principle state the opposite. These extremal principles are contradictory and unsatisfactory in the pre-sented case. Critical values of Da0 (i.e., values that correspond to change in the number of roots) are solutions of the following equation : 0 0 Da   =  Variations of entropy production do not give the correct stability condition. How-ever, we can estimate one of the critical values (ignition boundary) using an approxima-tion cs ≈ c0, which is applicable in low-temperature conditions. Then temperature can be calculated from the following equation: 0 ad Da e Le   = (A15) Its solution can be found using the Lambert W-function: 0 adDa W Le     = −     The complex in parentheses cannot be less than −e−1, and for θad = 10 and Le = 1 the critical value of Da0 is 0.037 (which is close enough to 0.041). Figure A3. Calculation results for Da0 = 0.01: (a) violation of equality (9); (b) entropy production. Maximum entropy production principle state that a high-temperature stationary state has a higher probability of observing; minimum entropy production principle state the opposite. These extremal principles are contradictory and unsatisfactory in the presented case. Critical values of Da0 (i.e., values that correspond to change in the number of roots) are solutions of the following equation : ∂Da0 ∂θ = 0 Variations of entropy production do not give the correct stability condition. 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3843	https://study.com/skill/learn/how-to-calculate-the-total-sum-of-squares-within-and-between-ssw-and-ssb-explanation.html	How to Calculate the Total Sum of Squares Within and Between (SSW and SSB) \| Statistics and Probability \| Study.com Log In Sign Up Menu Plans Courses By Subject College Courses High School Courses Middle School Courses Elementary School Courses By Subject Arts Business Computer Science Education & Teaching English (ELA) Foreign Language Health & Medicine History Humanities Math Psychology Science Social Science Subjects Art Business Computer Science Education & Teaching English Health & Medicine History Humanities Math Psychology Science Social Science Art Architecture Art History Design Performing Arts Visual Arts Business Accounting Business Administration Business Communication Business Ethics Business Intelligence Business Law Economics Finance Healthcare Administration Human Resources Information Technology International Business Operations Management Real Estate Sales & Marketing Computer Science Computer Engineering Computer Programming Cybersecurity Data Science Software Education & Teaching Education Law & Policy Pedagogy & Teaching Strategies Special & Specialized Education Student Support in Education Teaching English Language Learners English Grammar Literature Public Speaking Reading Vocabulary Writing & Composition Health & Medicine Counseling & Therapy Health Medicine Nursing Nutrition History US History World History Humanities Communication Ethics Foreign Languages Philosophy Religious Studies Math Algebra Basic Math Calculus Geometry Statistics Trigonometry Psychology Clinical & Abnormal Psychology Cognitive Science Developmental Psychology Educational Psychology Organizational Psychology Social Psychology Science Anatomy & Physiology Astronomy Biology Chemistry Earth Science Engineering Environmental Science Physics Scientific Research Social Science Anthropology Criminal Justice Geography Law Linguistics Political Science Sociology Teachers Teacher Certification Teaching Resources and Curriculum Skills Practice Lesson Plans Teacher Professional Development For schools & districts Certifications Teacher Certification Exams Nursing Exams Real Estate Exams Military Exams Finance Exams Human Resources Exams Counseling & Social Work Exams Allied Health & Medicine Exams All Test Prep Teacher Certification Exams Praxis Test Prep FTCE Test Prep TExES Test Prep CSET & CBEST Test Prep All Teacher Certification Test Prep Nursing Exams NCLEX Test Prep TEAS Test Prep HESI Test Prep All Nursing Test Prep Real Estate Exams Real Estate Sales Real Estate Brokers Real Estate Appraisals All Real Estate Test Prep Military Exams ASVAB Test Prep AFOQT Test Prep All Military Test Prep Finance Exams SIE Test Prep Series 6 Test Prep Series 65 Test Prep Series 66 Test Prep Series 7 Test Prep CPP Test Prep CMA Test Prep All Finance Test Prep Human Resources Exams SHRM Test Prep PHR Test Prep aPHR Test Prep PHRi Test Prep SPHR Test Prep All HR Test Prep Counseling & Social Work Exams NCE Test Prep NCMHCE Test Prep CPCE Test Prep ASWB Test Prep CRC Test Prep All Counseling & Social Work Test Prep Allied Health & Medicine Exams ASCP Test Prep CNA Test Prep CNS Test Prep All Medical Test Prep College Degrees College Credit Courses Partner Schools Success Stories Earn credit Sign Up Copyright How to Calculate the Total Sum of Squares Within and Between (SSW and SSB) High School Statistics Skills Practice Goutham Thiagarajan Instructors Goutham Thiagarajan Goutham graduated from the University of Virginia with a BS in Engineering Science (Concentrations in BME and CS). While at UVA, he worked as a tutor for other undergraduate students in probability and statistics. View bio Example SolutionsPractice Questions How to Calculate the Total Sum of Squares Within and Between (SSW and SSB) Step 1: For each group of data, calculate the mean. Step 2: Subtract the group mean from each data point that belongs to that group for all data points. Square and sum all these differences to get the SSW. Step 3: Calculate the grand mean by taking the mean of all data points regardless of group. If all the groups have the same size, then the grand mean is also the mean of the group means. Step 4: Subtract the grand mean from each group mean and square each of these differences. Multiply each squared difference by the number of observations in the corresponding group. Add these numbers to get the SSB. What is a one-way ANOVA and What are Total Sum of Squares Within and Between (SSW and SSB)? One-way ANOVA: A technique used to determine whether or not the population means of multiple samples (usually at least 3) are the same. It computes an F-statistic, which compares the variance within the samples and the variance between the samples. This F-statistic is dependent on the SSW and the SSB. SSW: The SSW is the sum of squared differences between a value and its sample mean for all values. SSB: The SSB is the sum of squared differences between a value and the grand mean, the mean of all values regardless of sample, for all values. Let's use these steps and definitions to calculate the SSW and SSB for two different sets of data. Examples of Calculating Total Sum of Squares Within and Between (SSW and SSB) Example 1: A botanist wants to determine how soil salinity affects plant growth. He conducts an experiment by growing the same species of plant in three different concentrations of salt for a month and measures each plant's height in inches. His results can be seen in the table below: \| 5% salt \| 10% salt \| 15% salt \| --- \| 8 \| 5 \| 2 \| \| 7 \| 4 \| 4 \| \| 5 \| 4 \| 7 \| \| 10 \| 5 \| 2 \| \| 6 \| 6 \| 3 \| Calculate the SSW and SSB for this data. Step 1: For each grouping of data, calculate the mean. For the 5% group: x¯5%=8+7+5+10+6 5=7.2 For the 10% group: x¯10%=5+4+4+5+6 5=4.8 For the 15% group: x¯15%=2+4+7+2+3 5=3.6 Step 2: Subtract the group mean from each data point that belongs to that group for all groups. Square and sum all these differences to get the SSW. S S W=(8−7.2)2+(7−7.2)2+(5−7.2)2+(10−7.2)2+(6−7.2)2+(5−4.8)2+(4−4.8)2+(4−4.8)2+(5−4.8)2+(6−4.8)2+(2−3.6)2+(4−3.6)2+(7−3.6)2+(2−3.6)2+(3−3.6)2=34.8 Step 3: Calculate the grand mean by taking the mean of all data points regardless of group. Since all the groups have the same size, we can take the mean of the group means. μ=7.2+4.8+3.6 3=5.2 Step 4: Subtract the grand mean (5.2) from each group mean: For the 5% salt group: 7.2−5.2=2.0 For the 10% salt group: 4.8−5.2=−0.4 For the 15% salt group: 3.6−5.2=−1.6 Square each of these differences: (2.0)2=4.0(−0.4)2=0.16(−1.6)2=2.56 Multiply each squared difference by the number of data points in the group: 4.0×5=20.0 0.16×5=0.8 2.56×5=12.8 Sum these products: 20.0+0.8+12.8=33.6 In conclusion, the SSW for this data set is 34.8 and the SSB is 33.6. Example 2: A psychologist wants to determine how well people can recall a series of numbers based on the time they are given to look at the numbers. She splits 9 people into 3 groups of 3 and each group is given 1 minute, 3 minutes, or 5 minutes to remember a series of 10 numbers. She records how many correct numbers each person in each group recalled and her data is below: \| 1 minute \| 3 minutes \| 5 minutes \| --- \| 6 \| 7 \| 9 \| \| 4 \| 6 \| 10 \| \| 4 \| 8 \| 8 \| Calculate the SSW and SSB for this data. Step 1: For each grouping of data, calculate the mean. For the 1-minute group: x¯1 m i n u t e=6+4+4 3≈4.67 For the 3-minute group: x¯3 m i n u t e s=7+6+8 3=7 For the 5-minute group: x¯5 m i n u t e s=9+10+8 3=9 Step 2: Subtract the group mean from each data point that belongs to that group for all groups. Square and sum all these differences to get the SSW. S S W=(6−4.67)2+(4−4.67)2+(4−4.67)2+(7−7)2+(6−7)2+(8−7)2+(9−9)2+(10−9)2+(8−9)2=6.67 Step 3: Calculate the grand mean by taking the mean of all data points regardless of group. Since all the groups have the same size, we can take the mean of the group means. μ=4.67+7+9 3=6.89 Step 4: Subtract the grand mean (6.89) from each group mean: For the 1-minute group: 4.67−6.89=−2.22 For the 3-minute group: 7−6.89−5.2=0.11 For the 5-minute group: 9−6.89=2.11 Square each of these differences: (−2.22)2=4.93(0.11)2=0.0121(2.11)2=4.45 Multiply each squared difference by the number of data points in the group: 3×4.93=14.79 3×0.0121=0.0363 3×4.45=13.35 Sum these products: 14.79+0.363+13.35=28.18 In conclusion, the SSW for this data set is 6.67 and the SSB is 28.18. Get access to thousands of practice questions and explanations! Create an account Table of Contents How to Calculate the Total Sum of Squares Within and Between (SSW and SSB) What is a one-way ANOVA and What are Total Sum of Squares Within and Between (SSW and SSB)? Examples of Calculating Total Sum of Squares Within and Between (SSW and SSB) Example 1 Example 2 Test your current knowledge Practice Calculating the Total Sum of Squares Within and Between (SSW and SSB) Recently updated on Study.com Videos Courses Lessons Articles Quizzes Concepts Teacher Resources Signs, Symptoms & Causes of Bloat in Dogs Fushimi Inari Taisha Shrine \| Location, Gates & History Kennedy Terminal Ulcer \| Overview & Treatment Lewis Dodgson in Jurassic Park \| Role, Traits & Quotes Charles Martel & the Battle of Tours \| Biography & Legacy Rock of Ages Hymn \| History, Author & Meaning Iron III Oxide \| Formula, Properties & Molar Mass Circumcircle Definition, Properties & Examples Louisiana's Demographics \| Racial Groups & Examples Externalizing Behaviors \| Causes, Consequences & Examples How to Solve Trigonometric Equations for X Education 107: Intro to Curriculum, Instruction, and... 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3844	https://www.scribd.com/document/519552633/Geometry-Cheat-Sheet-3d-Shape-Formulas	Geometry Cheat Sheet 3d Shape Formulas \| PDF \| Area \| Volume Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 100%(10)100% found this document useful (10 votes) 22K views 2 pages Geometry Cheat Sheet 3d Shape Formulas This document provides formulas for calculating the volume and surface area of various 3D shapes. It defines key 3D shape elements such as faces, edges, and vertices. Formulas are given for … Full description Uploaded by MT AI-enhanced description Go to previous items Go to next items Download Save Save Geometry Cheat Sheet 3d Shape Formulas For Later Share 100%100% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Geometry Cheat Sheet 3d Shape Formulas For Later You are on page 1/ 2 Search Fullscreen Name Date GEOMETRY QUICK GUIDE 5: 3D SHAPE FORMULAS 3D SHAPES All 3d shapes can be described in terms of their faces, vertices and edges. Face - a flat or curved surface Edge - line where 2 faces meet Vertex - point where 3 or more edges meet CUBE Volume = s 3 Surface area = 6 s 2 where s is the length of one side CUBOID (RECTANGULAR PRISM) Volume = ℓ x w x h Surface area = 2 ℓ h 2 ℓ w 2wh where ℓ = length, w = width, h = height PYRAMIDS Volume of a general pyramid = 1 / 3 A h where A = base area and h = height REGULAR TETRAHEDRON Volume = b 3 /6√2 Surface area = √3 b 2 SQUARE PYRAMID Volume = 1 / 3 s 2 h Surface area = s 2 2sh PRISMS Volume of any prism = Ah Surface area of a closed prism = 2A + (h x p) where A = base area, h = height, p = base perimeter TRIANGULAR PRISM Volume = A ℓ or ½ b h ℓ Surface area = bh + 2 ℓs + ℓb SPHERES Volume = 4 / 3 п r 3 Surface area = 4 п r 2 RIGHT CYLINDER Volume = п r 2 h Surface area = 2 п r (r + h) RIGHT CIRCULAR CONE Volume = 1 / 3 п r 2 h Surface area = п r (r + s) h s b b s A h r h r r h s ℓ w h h A p b ℓ h s p h A A face vertex edge Unlock this document Upload a document to download this document or subscribe to read and download. Upload to download Start your 30 day free trial or Unlock the next 2 pages after an ad adDownload to read ad-free Name Date GEOMETRY QUICK GUIDE 5: 3D S HAPE FORMULAS 3D SHAPES All 3d shapes can be described in terms of their faces, vertices and edges. • Face - a flat or curved surface • Edge - line where 2 faces meet • Vertex - point where 3 or more edges meet CUBE Volume = s 3 Surface area = 6 s 2 where s is the length of one side CUBOID (RECTANGULAR PRISM) Volume = ℓ x w x h Surface area = 2 ℓ h 2 ℓ w 2wh where ℓ = length, w = width, h = height PYRAMIDS Volume of a general pyramid = 1 / 3 A h where A = base area and h = height REGULAR TETRAHEDRON Volume = b 3 /6√2 Surface area = √3 b 2 SQUARE PYRAMID Volume = 1 / 3 s 2 h Surface area = s 2 2sh PRISMS Volume of any prism = Ah Surface area of a closed prism = 2A + (h x p) where A = base area, h = height, p = base perimeter TRIANGULAR PRISM Volume = A ℓ or ½ b h ℓ Surface area = bh + 2 ℓs + ℓb SPHERES Volume = 4 / 3 п r 3 Surface area = 4 п r 2 RIGHT CYLINDER Volume = п r 2 h Surface area = 2 п r (r + h) RIGHT CIRCULAR CONE Volume = 1 / 3 п r 2 h Surface area = п r (r + s) h s b b s A h r h r r h s ℓ w h h A p b ℓ h s p h A A face vertex edge Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Geometry Cheat Sheet 3 3d Shapes 100% (4) Geometry Cheat Sheet 3 3d Shapes 2 pages Geometry Formula Sheet 33% (3) Geometry Formula Sheet 2 pages Cubes: Geometry Formula Sheet - 3D Shapes 100% (2) Cubes: Geometry Formula Sheet - 3D Shapes 2 pages Basic Geometric Figures: Geometry No ratings yet Basic Geometric Figures: Geometry 59 pages MEASUREMENTS English 100% (1) MEASUREMENTS English 22 pages Understanding Quadrilaterals CBSE Class 8 Worksheet 0% (1) Understanding Quadrilaterals CBSE Class 8 Worksheet 4 pages Angle Properties Cheat Sheet 100% (2) Angle Properties Cheat Sheet 1 page Geometry Math Reference Sheet Cheat Sheet For Olympiad Use 100% (3) Geometry Math Reference Sheet Cheat Sheet For Olympiad Use 2 pages Geometry Cheat Sheet 3d Shape Formulas - Compress No ratings yet Geometry Cheat Sheet 3d Shape Formulas - Compress 2 pages Properties & Volumes No ratings yet Properties & Volumes 21 pages CON3483 Ch5 Mensuration No ratings yet CON3483 Ch5 Mensuration 36 pages Something Solid Par No ratings yet Something Solid Par 5 pages Cubes: Geometry Formula Sheet - 3D Shapes No ratings yet Cubes: Geometry Formula Sheet - 3D Shapes 2 pages Geometry Formulas 100% (2) Geometry Formulas 21 pages Compound Shapes Area Worksheet 0% (2) Compound Shapes Area Worksheet 2 pages Holiday Homework Class 8TH (Mathematics) 100% (1) Holiday Homework Class 8TH (Mathematics) 6 pages Notes: Chapter 7: Basic Angle Properties 50% (2) Notes: Chapter 7: Basic Angle Properties 2 pages Geometry Cheat Sheet Final - PDF 100% (2) Geometry Cheat Sheet Final - PDF 2 pages Plane & Solid Geometry Handouts 100% (4) Plane & Solid Geometry Handouts 14 pages Circle Theorems Year 10 Drilling No ratings yet Circle Theorems Year 10 Drilling 10 pages Geometry Formulas All Gathered On One Easy Cheat Sheet 100% (2) Geometry Formulas All Gathered On One Easy Cheat Sheet 1 page Area and Perimeter 2D Shapes 50% (6) Area and Perimeter 2D Shapes 3 pages Geometry Formulas 2D 3D Perimeter Area Volume 75% (4) Geometry Formulas 2D 3D Perimeter Area Volume 2 pages Chapter 5 - Area and Volume 60% (5) Chapter 5 - Area and Volume 58 pages Geometry Cheat Sheet 4 2d Shapes Formulas 100% (3) Geometry Cheat Sheet 4 2d Shapes Formulas 2 pages 13-2 Surface Area of Prisms and Pyramids Worksheet PDF No ratings yet 13-2 Surface Area of Prisms and Pyramids Worksheet PDF 2 pages Frequency Tables and Histograms Notes No ratings yet Frequency Tables and Histograms Notes 6 pages Area and Perimeter of Composite Shapes Centers No ratings yet Area and Perimeter of Composite Shapes Centers 6 pages Practice Algebra Test 100% (2) Practice Algebra Test 3 pages Algebraic Fractions - Addition and Subtractions - Solving Equations With Algebraic Fractions - Higher - KS4 - by - Hassan Lakiss PDF 100% (1) Algebraic Fractions - Addition and Subtractions - Solving Equations With Algebraic Fractions - Higher - KS4 - by - Hassan Lakiss PDF 14 pages Volume and Surface Area of 3D Shapes: Key Points 0% (1) Volume and Surface Area of 3D Shapes: Key Points 5 pages Volume and Surface Area Questions 100% (1) Volume and Surface Area Questions 7 pages Chapter 11 Multiple Choice Questions With Answers PDF 100% (2) Chapter 11 Multiple Choice Questions With Answers PDF 3 pages Cone - Pyramid - Cylinder - Sphere Volume and Surface Area Worksheet No ratings yet Cone - Pyramid - Cylinder - Sphere Volume and Surface Area Worksheet 2 pages To Draw Their Projections Means F.V, T.V. & S.V. 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3845	https://www.metric-conversions.org/length/miles-to-kilometers.htm	Miles to Kilometers (miles to km) Miles to Kilometers (miles to km) conversion calculator 1mi = 1.60934km Note: Fractional results are rounded to the nearest 1/64. For a more accurate answer please select "decimal" from the options above the result. Note: You can increase or decrease the accuracy of this answer by selecting the number of significant figures required from the options above the result. Note: For a pure decimal result please select "decimal" from the options above the result. km to miles (Swap units) Miles to Kilometers (miles to km) formula Kilometers = Miles 1.6093445 1 Miles to Kilometers calculation Kilometers = Miles 1.6093445 Kilometers = 1 1.6093444978926 Kilometers = 1.60934 Is there a simple way to convert Miles to Kilometers in my head? Converting miles to kilometers in your head may seem like a daunting task, but with a simple trick, you can easily estimate the conversion without the need for a calculator. The key is to remember that 1 mile is approximately equal to 1.6 kilometers. By keeping this conversion factor in mind, you can quickly estimate the distance in kilometers. To convert miles to kilometers, simply multiply the number of miles by 1.6. For example, if you want to convert 10 miles to kilometers, you can estimate it by multiplying 10 by 1.6, which gives you 16 kilometers. Similarly, if you have a distance of 50 miles, you can estimate it to be around 80 kilometers. If you need a more precise conversion, you can use a slightly more accurate conversion factor of 1.60934. This will give you a more accurate estimate, but it may require a bit more mental calculation. However, for most everyday purposes, the approximation of 1.6 is sufficient. How long is 1 Mile? 1 Mile - distance walked in 15-20 minutes by an average person 2.5 Miles - Approximate length of Central Park, NY 3.1 Miles - The furthest distance that be seen on earth to the horizon 7900 Miles - the approximate distance from the North to South Pole How long is 1 Kilometer? 1 Kilometer - The distance travelled in 9 to 12 minutes walking at normal pace 4 Kilometers - the length of Central Park, NY 5 Kilometers - The furthest distance that be seen on earth to the horizon 12,713.6 Kilometers - the distance from the North to South Pole Some common Miles to Kilometers conversions 1 mile in km = 1.6093km 1 mile to km = 1.6093km 2 miles in km = 3.2187km 4 miles in km = 6.4374km 5 miles in km = 8.0467km 8 miles in km = 12.875km 10 miles in km = 16.0934km What is a mile? A mile is a unit of length commonly used in the United States and some other countries. It is equal to 5,280 feet or 1,760 yards. The word "mile" is derived from the Latin word "mille," meaning thousand, as it originally represented the distance covered in 1,000 paces by a Roman legionary. A mile is equivalent to 1760yds or 5280ft. The mile is commonly used in the United States for measuring long distances, such as road distances and race distances. It is also used in the aviation and maritime industries for navigation purposes. However, in most other countries, the metric system is used, and the kilometer is the preferred unit for measuring long distances. What is a kilometer? A kilometer is a unit of length in the metric system, commonly used to measure distances. It is equal to 1,000 meters or approximately 0.621 miles. The prefix "kilo" in kilometer denotes a factor of 1,000, making it a larger unit compared to a meter. This unit is widely used around the world, especially in countries that have adopted the metric system. To put it into perspective, a kilometer is roughly equivalent to 3,281 feet or 39,370 inches. In terms of everyday objects, it is approximately the distance covered in a 10-15 minute walk or the length of a typical city block. Kilometers are commonly used to measure longer distances, such as the length of a road or the distance between cities. The use of kilometers as a unit of measurement offers several advantages. It provides a standardized and consistent way to measure distances, making it easier to communicate and compare measurements across different regions and countries. Additionally, the decimal-based nature of the metric system simplifies calculations and conversions between different units of length. How do I use the Fibonacci sequence to convert from Miles to Kilometers? The Fibonacci sequence is a mathematical sequence in which each number is the sum of the two preceding ones. This sequence has been found to have various applications in different fields, including unit conversions. To use the Fibonacci sequence to convert from miles to kilometers, we can assign each Fibonacci number to represent a specific conversion factor. For instance, let's consider the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The difference between two adjacent numbers in the sequence provides a good estimation of the relationship between miles and kilometers and this becomes a better estimate as the sequence progresses. For instance, 5 miles is approximately 8 kilometers (in fact 8.046). Similarly, 8 miles is approximately 13 kilometers (in fact 12.874). This is because the closed-form expression of the sequence is 1.61803 which is very close to the conversion factor of Miles to Kilometers which is 1.60934. Miles to Kilometers table Starting value Increment Accuracy Miles Kilometers 0mi 0.00000km 1mi 1.60934km 2mi 3.21869km 3mi 4.82803km 4mi 6.43738km 5mi 8.04672km 6mi 9.65607km 7mi 11.26541km 8mi 12.87476km 9mi 14.48410km 10mi 16.09344km 11mi 17.70279km 12mi 19.31213km 13mi 20.92148km 14mi 22.53082km 15mi 24.14017km 16mi 25.74951km 17mi 27.35886km 18mi 28.96820km 19mi 30.57755km Miles Kilometers 20mi 32.18689km 21mi 33.79623km 22mi 35.40558km 23mi 37.01492km 24mi 38.62427km 25mi 40.23361km 26mi 41.84296km 27mi 43.45230km 28mi 45.06165km 29mi 46.67099km 30mi 48.28033km 31mi 49.88968km 32mi 51.49902km 33mi 53.10837km 34mi 54.71771km 35mi 56.32706km 36mi 57.93640km 37mi 59.54575km 38mi 61.15509km 39mi 62.76444km Miles Kilometers 40mi 64.37378km 41mi 65.98312km 42mi 67.59247km 43mi 69.20181km 44mi 70.81116km 45mi 72.42050km 46mi 74.02985km 47mi 75.63919km 48mi 77.24854km 49mi 78.85788km 50mi 80.46722km 51mi 82.07657km 52mi 83.68591km 53mi 85.29526km 54mi 86.90460km 55mi 88.51395km 56mi 90.12329km 57mi 91.73264km 58mi 93.34198km 59mi 94.95133km Miles Kilometers 60mi 96.56067km 61mi 98.17001km 62mi 99.77936km 63mi 101.38870km 64mi 102.99805km 65mi 104.60739km
3846	https://www.youtube.com/watch?v=jfu1dCDV1aE	Uniform Motion Problems \| Application of Linear Equation in One Variable \| Precalculus Jonas' Class Notes 11600 subscribers 24 likes Description 1877 views Posted: 8 Oct 2020 Uniform Motion Problems Objects that move at a constant speed are said to be in uniform motion. When the average speed of an object is known, it can be interpreted as its constant speed. For example, a bicyclist traveling at an average speed of 25 miles per hour is in uniform motion. If an object moves at an average speed (rate) r, the distance d covered in time t is given by the formula 𝑑=𝑟×𝑡 That is, rate × time = distance. Example 1: Tanya, who is a long-distance runner, runs at an average speed of 8 kilometers per hour (kph). Two hours after Tanya leaves your house, you leave in your Honda and follow the same route. If your average speed is 40 kph, how long will it be before you catch up to Tanya? How far will each of you be from your home? Example 2: A father and daughter leave home at the same time in separate automobiles. The father drives to his office, a distance of 24 km, and the daughter drives to school, a distance of 28 km. They arrive at their destinations at the same time. What are their average rates, if the father’s average rate is 12 kph less than his daughter’s? Transcript: [Music] hello everybody welcome back to our discussion on the application of linear equation in one variable for this video i am going to talk about the uniform motion problem the objects that move at a constant speed are said to be in uniform motion when the average speed of an object is known it can be interpreted as its constant speed for example a bicyclist traveling at an average speed of 25 miles per hour is in uniform motion okay so if the given is in average speed it is the assumption of the problem that the motion is uniform and if it's a b non-uniform motion the motion is not i mean the speed is not decreasing or nor it is increasing a uniform lung [Music] hindi nagbabago on canyon speed okay so if an object moves at an average speed r the distance d covered in time t is given by the formula d is equal to r times t that's distance is equal to rate times time or rate times time is equal to distance and we shall use this formula to answer algebra problems involving uniform motion hey let us illustrate this one by answering this example tanya who is a long distance runner runs at an average speed of 8 kilometers per hour two hours after tanya leaves your house you live in your honda and follow the same route if your average speed is 40 kilometers per hour how long will it be before you catch up to tanya how far will each of you be from your home okay if you take a look at the problem understand it the unknown here is the time you travel from your house until you catch up with anya okay so [Music] uh depending on your perspective and asap so non-problem nato jung designation nothing for time t can be the time of tanya starting from house where you catch up or when you started leaving your house until you catch up with tanya dependen okay so in our in this example from your house until you catch up with and yeah okay so this one time your time your time to travel but this one i mean t is the time you travel from your house until when you catch up with tanya okay if t is the time you are from your house to anya and we know that tanya traveled for two hours before you started catching up two hours okay that is why our representation for the time tanya travel is t plus two hey the time you traveled plus two hours just like what we did in our previous examples a gagawa taiwan table but this time if a uniform motion problem and table nothing ion the first column is rate or the speed or the velocity and the second column is the time and the third column is the distance [Music] okay so the rate of tanya is 8 kilometers per hour while the rate your rate is 40 kilometers per hour okay representation we have d plus two uh that's t okay the distance is just a product of rate and time that's eight multiplied by t plus two uh detournament is 40 times two hours after tanya leaves your house you live in your honda and follow the same route okay if your average speed is 40 kilometers per hour how long will it be before you catch up to tanya okay that means you already traveled you know for the same distance let's say you catch up with tanya the distance traveled by tanya is equal to the distance you traveled in this case we are going to equate the distances together so that's eight multiplied by t plus two is equal to 40 t okay now let us solve for t so that's 80 plus 2 is equal to 40 t okay so 40 minus 80 that's okay so let's do 40 minus 80 is equal to this is 16 right equals 16 hey that's so 14 minus 8 is 32 t is equal to 16. divide both sides by 32 so 16 divided by 32 is 0.5 okay that's 0.5 hours okay that's the time because balika is a representation what is the time your time to travel from your house until when you catch up with tanya so [Music] how far will each of you be from your home okay that is asking for the distance your distance since distance during this time you can use distance nitanya or your distance whatever okay so the distance i got it into the line of 40 times t 40 d is equal to 40 times 0.5 so this is 20 kilometers so this is in hours okay so we can conclude that the time uh the time you travel from your home until you catch up with tanya 0.5 hours okay so the distance from your home until that point is 20 kilometers this is another example on uniform motion problem a father and daughter leave home at the same time in separate automobiles the father drives to his office a distance of 24 kilometers and the daughter drives to school a distance of 28 kilometers they arrive at their destination at the same time what are their average rates if the father's average rate is 12 kilometers per hour less than his daughters okay so is their average rate okay average rate is 12 kilometers per hour less than his daughter so let us be the rate of the daughter and based from that statement average of the father is 12 kilometers per hour less than so we have r minus 12 the rate of the father okay all right so let nothing detox [Music] daughters rate fathers rate okay a since marinating representation let us create a table the author and father first column the distance they traveled the distance they traveled together daughter traveled for a distance of 28 kilometers beta was in father and distance to his office is 24 kilometers okay so anger were nothing representation young time okay so based from our equation in equation now then distance is equal to rate times time we can actually solve for time the time is equal to distance over rate the detail the time is distance over rate so this is 28 divided by r as a father and a mind that's 24 divided by r minus 12. hey that's distance over rate song and a sabito what is the relationship between the motion of the daughter and the motion of the mother exactly that they arrive at their destinations at the same time ibizabihin their times are equal okay so we are going to equate the time of travel of the daughter and the time of trouble of the other that's 28 over r is equal to 24 over r minus 12 okay so this is rational we are going to eliminate the denominator by multiplying the lcd the lcd is r times r minus 12 so r times r minus 12 so cancel your d2 so that's 28 multiplied by r minus 12 equals minus 12 so that's 24 r the result for r that's 28 r minus 28 times 12 is 336 is equal to 24 r so 28 r minus 24 r is equal to [Music] 336 so that's 4 r is equal to 336 divide both sides by 4 that is 84. so you go back to your representation r is the daughter's rate okay so this is in k ph that is the rate of the daughter and the father's rate is r minus 12 that's 84 minus 12 and that is equal to 72 kilometers per hour okay so these are the unknowns the average rate of the father and the average rate of the daughter and your conclusion should be the average rate of the daughter is 84 kilometers per hour while the average rate of the father is 72 kilometers per hour that answers the problem thank you
3847	https://www.youtube.com/watch?v=Ex2VhqHgRdw&pp=0gcJCb4JAYcqIYzv	Find a Point Closest to a Curve sumchief 2650 subscribers 8 likes Description 90 views Posted: 7 Jun 2025 Using Calculus we find the Closet Point on a Curve y=x^3 to the point (4,4) the point on the Curve is (x,x^3) To minimise we use the Vector \|PQ\| and minimise it by taking its derivative. But then we need to solve a Quintic, so we adopt the Newton Raphson Method using the iterative formula and setting x_0=4 . After we solve this we find the Equation of the Normal to the Tangent at this point, although it is an estimate it is very close. sumchief iteration alevel maths newtonraphsonmethod derivative calculus calculus1 calculus2 geometry 1 comments Transcript: So in this question, we've got a curve y = x cubed and there is a point q with coordinates 4a 4. We want to find the coordinates that q is closest to the curve y = x cubed. So where is that going to be? Well, it's going to be approximately here. But we don't know where those coordinates are. Now what we can say is that this point here wherever P is there will be a tangent and then a normal reaching to the point Q. So that we can say so what we want to do is we first of all want to find out what the relationship between this and P is. So if we know Q is at 4a 4 then we know that P is at now as this curve is Y = X cubed X will just have some value X which we need to find and the Y coordinate will be X cubed. Now if we can find the distance from that then we can work with that and hopefully find the coordinates of where that may be. Let's find a way of doing that. Now what we can do is we can use the distance formula in terms of vectors. So if we turn PQ into a vector and find the magnitude of that vector that will give us our distance. So to do that we need our formula which will be the square root of X. So the point P to Q. So be X - 4. So difference between the x is squared. And here we'll have the ycoordinates x cubed and 4. We also squared that. And that's all in terms of the square root. So now we need to try and get a value for x out of this. And that's looking pretty intimidating at the moment. But what we can do now if we square both sides, we'll get rid of the square root. That'll be a good start. And then we'll have x - 4^ 2 + x cub - 4 all squared. So that's got rid of the square root. And this side we'll have this magnitude squared. So it's basically squared both sides. So now what we want to do is find the point where it's closest to the curve. So ideally we want to minimize this. So if we can minimize it, basically to do that we'll need to take the derivative and set it to to zero. So what we ideally want is the derivative of this magnitude which in this case is in terms of x and then take the derivative of that and set it to zero and then hopefully we can find that as a minimum. So let's do that. Let's take the derivative of both sides. So we take the derivative of this. We have just bring the power to the front and then we've got PQ and drop the power by one. Obviously that's a one. Take the derivative of this. So here we'll have two times whatever's in the brackets. And then by the chain rule we need the derivative of what's in the brackets. In this case is just one. I'm just going to write it there just so as you can see. And then the same here. Bring this power to the front. Then we got x cub - 4. And then by the chain rule, we need the derivative of what's inside of here, which is 3x^ 2. Okay. Now we can see here now we've got 2 22 2. So we can cancel those. So now we've got pq equals x - 4. That two and the one will cancel out by that. This one, the two will cancel out. And then we'll have 3x^2 x cub - 4. Okay. And we want to set that to zero. Okay. Right. Let's multiply this out and see where it takes us. So now we've got x - 4 3x^2 x cub is 3x 5 and 4 that that's -2x^2 and that's zero. So it looks like we've got a quintic. So to solve this is going to take some doing. So if I just rewrite this and put the uh p the powers of x in. So I start with 3 x 5 - 12x^2 + x - 4 is zero. I can't see anywhere of any obvious solutions. Sometimes you can plug in a zero or a one, it'll be obvious that that's one of the solutions and then maybe long division can help you get to some more. But there's if I plug in a zero or a one, it's not going to do me any favors. So what we need is a numerical method to try and find x here at its minimum. So the best one I can think of at the moment for this would be the Newton Rafson method. So Newton Rafson method or Newton's method. So the formula for that is it's an iterative formula. So what we have we have x n + 1 = xn - f of xn over the derivative of xn. And that will lead us to some sort of value for our x where it will keep getting closer and closer and closer. And then once we get to enough decimal places and we're happy with that, then we can declare that as our xcoordinate. So let's clear this off the board and let's set up this Newton's method to try and solve this quintic. Okay. Right. Let's set up Newton's method with all the building blocks that we need to solve that. So let's write it all out and then we'll pick a value for our starting point xn. So then we've got xn and then we need to subtract this fraction here. So it's the ratio between the function and its derivative. So f of xn that will be that in terms of xn. So then I've got -2 x n^ 2 + xn - 4. And then derive divide that by its derivative. Well, let's just do each one at a time. So that's 15 x n 4 12 that's going to be time 2. So that's 24 x n and then derivative of that is just one. So now we need a value of x to start off our iteration. So we need the initial value. So that's going to be x0. That's our starting point. Well, as q is at 4, 4, we know four is going to be reasonably close as it x coordinate. So we'll use our iteration first point as x is four. Then what we got to do is plug those values into here and see where it takes us. So then our first iteration will be x1 which will equal x n. That's going to be x0, isn't it? So that's going to be 4 - 3 4 ^ 5 - 12 4^ 2 + 4 - 4 and then we divide that plugging in four for all of these. 15 4 24 4 + 1. Okay, so plug that into your calculator. you'll get approximately 3.231 231. Let's write that there. So that's the first one. So you can see it's gone quite a long way from four. So this is by no means the end of our iterations. So then you need to keep going and going and going. So what I'd recommend is that what you type in your calculator, you do set up a four and have that as your answer. And then what you need to do is set up a formula in your calculator. Right? So here's your that will set up four as our answer. So answer minus 3 answer to the^ of 5 - 12 answer squared plus answer - 4. And then here do this one. 15 answer to the 4 - 24 answer and then + one. And then you can just keep pressing equals equals equals equals or execute execute whichever calculator you've got and then it will iterate and iterate and you can watch this number get smaller and smaller. So as you go through the system your x1 that's my x1 there. So if I write a few down x2 would be approximately 2.634 634. Then you get 2.186 1.875. So that's a few so far. Now you notice these are still got quite big jumps. So we need to keep going and keep going and keep going until the first three or four decimal places don't change anymore. And then what you'll do is you'll eventually flatten out the result to 1 6 2 6 04 and that will be x like 20 or something like that if you keep going. It's probably bottom out just before that. If you keep going you'll notice it doesn't change anymore from there. So that then will be a suitable value for our coordinates of P. So P will be at 1.62604 comma. Now that's going to be X cubed. So it's the value there. So now we can plug that in the calculator. Cube that and we'll get approximately 4.299. So therefore P is at 1.626 4.299. So P is at 1.626 626 comma 4.299. Now that looks pretty reasonable to me. So as x is 4, it's going to be in a little bit and y is 4. So therefore here it's going to be a little bit higher. So it's going to be 4.299. So that looks reasonable. So now let's try and find the equation of the normal to the tangent of this curve y= x cubed involving these coordinates. So the equation of the normal is going to be a straight line. So it's going to be the form of y = mx plus c. So let's box all these calculations off to be separate. So if that's our coordinate, we can find m because we've got two coordinates. We've got 44 and we've got 1.626 4.299. So m equals change in y over change in x. So change in y that's going to be four minus 4.299 and change in x is also going to be 4 - 1.626 and that will come out as an approximation of approximately minus 0.126. So let's call that approximately minus 1/8. So that will be easier to work with than that, but it's going to be reasonably close. So then what we can do is can set that equal to our m. And then we can plug that into here and see if we can find out what we've got. So y = -18. So that's - x over 8 + c. Now we know some coordinates of of that normal. We know x is 4, y is 4. So plug that in and maybe we can get c. So that will then lead us to 4 = -4 over 8 + c. Well, if that's minus a half, c must be 9 / 2. So c = 9 / 2. So now we've got the equation of our normal. So we've got y = - x / 8 + 9 / 2. And if you want to bring all your x's on one side, multiply everything by 8. So then we'll have 8 y + x and then times that by 8, we get 9 x is 72 / 2 is 36. So that's the equation of our normal. That one there would be 8 y + x = 36. So that's a way of solving that. And I like this method here to solve a quintic to get an approximation. Just can give us an exact value, but I think to about six or eight decimal places, we'll be fine for this example here. Okay.
3848	https://www.youtube.com/watch?v=q89YrFmyHY4	Hydraulic Grade Line and Energy Grade Line Fluid Matters 37300 subscribers 527 likes Description 39984 views Posted: 25 Oct 2018 MEC516/BME516 Fluid Mechanics, Chapter 3 Control Volume Analysis, Part 11: A discussion of the Hydraulic Grade Line and Energy Grade Line. This information will be helpful for Lab 4. All of the videos in the course and a copy (pdf) of this fluid mechanics presentation can be downloaded at: Course Textbook: F.M. White and H. Xue, Fluid Mechanics, 9th Edition, McGraw-Hill, New York, 2021. fluidmechanics #fluiddynamics 0:00 Introduction 0:11 Overview 2:30 Definition of "Head" 4:29 Hydraulic Grade Line (HGL) and Energy Grade Line (EGL) 11:15 Example: Inviscid Flow Through a Venturi Meter 15:40 Example: Real (Viscous) Flow Through a Venturi Meter 18:06 Video Demonstration: Venturi Flow Meter 20:34 Example: Venturi Meter 23:10 Example: HGL and EGL for a Piping System 31 comments Transcript: Introduction this is chapter 3 control volume analysis part 11 in this video I'm going Overview to discuss the concepts of hydraulic grade line an energy grade line what these are are graphical representations of the energy components in their Bernoulli equation and over here on the right I've shown an example now this example I'm going to discuss in substantial detail in the coming slides but just to give you a sort of a brief overview what I draw a grade line an energy grade line R as there are graphs of the energy components where the horizontal axis is distance so distance through the piping system and the vertical axis is the energy content of the flow and so in this way you can see by plotting these lines you can get some insight into the energy transformations that happen through the piping system so in this video what I'm going to do is I'm going to start by defining what the hydraulic grade line is what the energy grade line is as well and then I'm going to apply these concepts to a couple of examples I'm going to start with simple frictionless Bernoulli type flows so remember that the Bernoulli equation applies to flows that are inviscid have no viscosity no energy losses they're not completely realistic flows and then I'm going to apply these concepts to real flows so flows with head losses flows with pressure losses down the pipe I'm going to show how you can apply it to flow across a valve and even flow through a pump where energy is added so in other words we're going to apply the concepts of hydraulic grade line and energy grade line to flows that are described by the steady flow energy equation that you learned about in a previous video now you'll notice that in the textbook these concepts are presented quite early in Chapter three hydraulic grade line and energy grade line are and something like section 3.2 but I've delayed the presentation of these concepts until late in the chapter because I think it makes more sense once you understand the steady flow energy equation before we go on to talk about Definition of "Head" the concepts of hydraulic grade line and energy grade line I just want to do a little bit of review I want to talk about the Bernoulli equation again and in particular I want to talk about this form of the Bernoulli equation in this form as you may recall from a previous video each of the terms has the units of height in other meters or feet and we call this head now recall that head is the energy content of the flow per unit weight of the flow so energy content in joules so Newton meters divided by the weight of the flow Newton's annual of course the Newton's cancel out and you just end up with meters so we can represent the energy content of the flow as a height and this height we call head so the first term v squared upon 2 G of course that's the kinetic energy per unit weight we call this the velocity head again expressed as a height P upon gamma is the flow work done by pressure sometimes I call it pressure energy and this would be P upon gamma would be the pressure energy per unit weight of the fluid called pressure head again in meters and then Z is just a measure of the elevation the potential energy of the flow so we call it of course per unit weight so we call it elevation head or just head now in a Bernoulli flow the kinetic energy plus the pressure energy plus the potential energy all add up to a constant along a streamline remember Bernoulli's equation applies to frictionless flows or inviscid flows flows without any viscosity so there's no loss so the energy content along a streamline remain constant so now we can get to the Hydraulic Grade Line (HGL) and Energy Grade Line (EGL) definitions of hydraulic grade line and energy grade line and what I've done here is I've reproduced the form of their Bernoulli equation that we're going to use in this form each of the terms has units of meters and can be interpreted as a head as I mentioned in the introduction hydraulic grade line an energy grade line our graphical representations of the energy components of the Bernoulli equation and so here on the right hand side what we're looking at is is flow in a pipe and what we're plotting here is on the horizontal we're plotting of course the distance along the pipe and then the vertical axis is the head or the energy content of the flow per unit weight now the hydraulic grade line is defined as a plot of the height of the pressure head plus the elevation head so Z which is the elevation hat or the potential energy of the flow per unit weight is plotted relative to some arbitrary reference point so we can just arbitrarily pick some point that we call z equals 0 a datum but the pressure head the p upon gamma is plotted relative to the center line of the pipe so you can see over here a point one right there we use the center line of the pipe because in these types of flows we're representing the flow in a one dimensional approach so there's really only one coordinate that's the distance along the pipe and so the pressure in the pipe is taken is the pressure on the center line of the pipe as a suitable representative value it's interesting to note and and perhaps helpful in terms of understanding this that the hydraulic grade line is the height that a fluid would rise to if you had a bossam atour attached to the pipe at that location and for that reason it's sometimes called piezo metric headline I mentioned this because it's maybe easy to remember the definition that way that hydraulic grade line is the height that the pazham être would rise to if you had one attached to the pipe and the diagram here over here shows this at point one you can see that the hydraulic grade line passes through the top of the fluid in the pedometer at point one and the same thing over here at point two so the hydraulic grade line at Point 1 is is Z 1 plus P 1 upon gamma and the hydraulic grade line at Point 2 is Z 2 plus P 2 upon gamma in contrast the energy grade line is a plot of the total energy content so it's really a plot of their Bernoulli constant so it's the sum of the velocity head the pressure head and the elevation head for an inviscid flow or a frictionless flow like you would assume when we're applying Bernoulli's equation the energy content on a stream line is constant and so the energy grade line is a horizontal line we have no energy losses due to friction due the viscosity and we have no energy additions for example due to a pump in the system so the energy grade line at point one and point two and in fact all points in between is just a constant it's equal to Z 1 P 1 upon gamma plus v1 squared upon 2 G and that's the same all the way across because the energy grade lines are constant in this kind of flow it's not very helpful it's not it doesn't give you very much insight into what's going on you'll find and we're going to discuss this later in this presentation that energy grade line is a more useful descriptor for real flows those with viscous losses pumps in other words for flows where the flow can be described using the steady flow energy equation that you learned about in a previous presentation in Chapter three it'll ask a question why does the hydraulic grade line here we're talking about is the hydraulic grade line why does it increase in the flow direction you might want to pause the video at that point and just think about this I'm going to give the answer in just a second the answer is because the difference between the energy grade line and the hydraulic grade line is the velocity head so you can see here the difference of 0.1 between the hydraulic grade line and the total amount of energy is just the velocity head the kinetic energy per unit weight and the same things true over here at point two that's the difference between the energy grade line and the hydraulic grade line and what's interesting is in this flow you can see that the pipe diameter is gradually getting bigger in the flow direction that means that the velocity is going to be decreasing the average velocity will be decreasing because of continuity so you'll have less kinetic energy in the flow as you move downstream and so the hydraulic grade line will approach the energy grade line basically what's happening is you're converting kinetic energy into pressure energy you're converting velocity head into pressure head as the flow moves downstream so I hope that makes sense and so that's the definition of energy grade line and I draw a great line and what we're going to do now is move on and do some examples the first example I'm going to Example: Inviscid Flow Through a Venturi Meter consider is inviscid or frictionless flow through a venturi meter and you've seen this kind of device before in some of the other videos in Chapter three a venturi meter can be used to measure the flow rate in a pipe basically you have a constricted section in the pipe here and because of continuity the velocity of the fluid in the pipe will increase in the as the cross-sectional area decreases and from the pressure difference between 0.2 and point one you can deduce the average velocity in the pipe and the volume flow rate so what we're going to do now is look at this kind of device that you're already familiar with and plot the hydraulic grade line and the energy grade line just to illustrate how this graphical approach works as it mentioned the heights of these curves in both cases represents the various energy contents of the flow per unit weight as we discussed on the previous slide the hydraulic grade line corresponds to the height that a pedometer would rise if there was one attached to the pipe so it's Z plus P upon gamma the elevation head plus the pressure head and here at Point one you can see this is the point that a prisoner would rise to Z measured relative to some arbitrary reference plus p 1 upon gamma now I draw a grade line here remains constant until we get to the section of the pipe where the cross sectional area decreases and when the cross-sectional area decreases the average velocity of the flow increases because of continuity you're going to have more kinetic energy which means you must have less pressure head so you gain velocity head at the expense of pressure head and so the pressure at point two decreases we have a minimum pressure at the throat and indeed a pedometer would rise to a much lower height because of the low pressure the energy grade line here for this type of flow since we're considering frictionless flow is just a constant it's the summation of the elevation had the pressure had and the velocity had if we assume there's no losses and the difference between the hydraulic grade line and the energy grade line is just the velocity head and so we have a small difference here at the inlet and when the velocity increases you get a maximum velocity at the throat we get the maximum difference between the energy grade line and the hydraulic grade line now point out that this is a ideal inviscid flow and so if this if the pipe diameter at 1 was the same as the pipe diameter 3 it had the same average velocity and so the pressure would decrease toward the throat and then you'd get complete recovery so the height of the hydraulic grade line at Point 1 would equal the height of the hydraulic grade line at Point 3 in an ideal frictionless flow in this type of flow you'd have no pressure losses of course in reality as a fluid flows downstream you will get pressure losses so when real flows the pressure at Point 3 will be somewhat lower than the pressure at Point 1 and the hydraulic grade line won't recover completely so the pedometer will have a smaller vertical deflection downstream and we're going to talk about that in some of the next examples which consider non-ideal flows so next we're going to consider actual flow through a venturi meter so unlike the previous example this is a real viscous flow and this is actually a preview of what you're going to see in lab number 4 which is the venturi meter Example: Real (Viscous) Flow Through a Venturi Meter lab so here's a picture of the apparatus from lab number 4 there's a water flow from left to right and in this experiment there's a piece of Plexiglas on the side of the venturi flow meter so that you can see the cross sectional area you can see the cross sectional area gets reduced gradually to a minimum here at what's called the throat which is the point of maximum velocity and then as the flow moves downstream you can see that the cross sectional area of the flow increases in the velocity will decrease in this region there's also seven pedometers here and so we can actually make a measurement of the hydraulic grade line see how high the fluid rises in those pedometers and compare that to what Bernoulli equation would predict for an ideal inviscid flow and that's one of the objectives of the lab but it's nice today to show this in in order to explain the concepts of hydraulic grade line and energy grade line for actual flows this is the experimental setup for lab number four you can see the venturi flow meter at the bottom with the flow from left to right so on the left hand side there's this blue water tank and there's a supply of water continuously to that tank and the pipe that you see up here is actually an overflow pipe so there's water coming into the pipe and so the water level in this tank will be somewhere up around here and they'll be a continuous overflow you can see the venturi meter they're attached to the bottom of the water tank there are as I mentioned seven pedometers and in this case I've added a little bit of blue dye to the flow just so you can see the heights of the pedometers when you do the experiment for lab four there won't be any dye and there's a valve here just a hand valve on the outlet that allows me to control the flow rate through the system Video Demonstration: Venturi Flow Meter next I'm going to show a short video presentation of lap number 4 the venturi flow meter and after I've shown the this video we're going to look at the analysis of the hydraulic grade line and the energy grade line for this actual flow now before I start the video you'll notice here that there's no flow through the venturi at the moment this valve over here is closed and so all of the pedometers here have fluid at the same height they would be at equilibrium with the fluid in the tank and so that represents the fluid level in the tank at the start so now let's just watch the video so at the beginning here I'm just opening that valve to let some flow through and you'll notice the pedometer levels changing dramatically now you can see it's pretty much a steady state so you're seeing the steady-state hydraulic grade line finally what I'm gonna do is just turn the flow back down turn it off and there it is with the flow off and we're back to the original situation so now there's no flow through the venturi flow meter and all of the pedometers are back to the original level the height of the level in the adjacent tank and you can see that because I drew a little bit of flow the height of the water level in the tank is decreased slightly so next what we're going to do is analyze the results of this I'm going to describe the hydraulic grade line and energy grade lines using those azam etre results this is a screenshot from the Example: Venturi Meter experiment showing roughly the steady-state pazham etre levels and in this experiment the water level in the tank would be approximately here and so you can see the pedometers are all lower than the water level in the tank the heights of each parameter indicate the local pressure so it's P upon gamma or gamma would be the gamma of water and P would be the local pressure and so we can get the hydraulic grade line by just simply joining the fluid levels in the tops of the pedometers and you can see that the minimum pressure here there the minimum in the hydraulic grade line is at the throat of the venturi meter where you have the maximum velocity so what's happened is you've gained kinetic energy you've gained velocity head at the expense of pressure head I also point out something interesting an in contrast to the ideal flow and eventually that we talked about earlier you don't get complete recovery of the pressure head so the heights of the Azha meters for the first and the last pedometer are not the same even though the cross sectional areas of the venturi meter are the at those locations this is an indication that you've got some losses in the system some energy losses so pressure is decreasing in the downstream direction you're losing some pressure head because of well viscous losses and because of turbulence and so now when we draw the energy grade line on the diagram you can see this slopes downward in the flow direction this is because of energy losses associated with viscosity and turbulence at the tank side here the velocity is approximately zero and so the energy grade line and the hydraulic grade line coincide remember the difference between the hydraulic grade line and the energy grade line is the velocity head v squared upon 2 g which is zero at the tank and it's a maximum to get the maximum difference where the velocity head is a maximum which is at the throat here Example: HGL and EGL for a Piping System okay I'm going to end with a discussion of hydraulic grade line and energy grade line for a what I'll call a complex piping system so what we have here is a large tank that's feeding flow into a pipeline and this is not drawn to scale of course we would have might have a very long pipeline to the point where you need to increase the pressure and the pipeline using a pump in order to get the required flow rate and then most piping systems would have many valves I've just drawn one valve here and the pipe ends in a nozzle and a high speed fluid jet and so the task here is to draw the hydraulic grade line and energy grade line for this type of system and so here they are the upper line the dashed red line is the energy grade line and the lower solid red line is the hydraulic grade line as in all the previous cases the height of the hydraulic grade line above the center line of the pipe so above the center line the height here is just the pressure head P upon gamma and of course we have some arbitrary datum here our elevation head which really doesn't play a role in this case because we have a horizontal piping system now in all cases as before the difference between the energy grade line and the hydraulic grade line is just the velocity at V squared upon 2 G and so back in the tank here at the tank where the velocity is about zero or very low the hydraulic grade line and energy grade lines are the same now in this case both the energy grade line and the hydraulic grade line sloped downward in the flow direction because of energy losses as I mentioned these energy losses are because of viscous of viscous friction viscosity and because of turbulence now if you had a long piping system you wouldn't have enough head from the tank to drive the flow so you'd need a pump and the pump as we learned in previous videos adds energy so we would have a the head of the pump being added to the flow at that point and moving further downstream here most piping systems have valves you probably have a valve on each side of the pump in order to isolate the pump I've just drawn one here when flow passes through a valve it passes through a complex path and not a smooth pass path through the pipe and so there's some energy losses associated with that associated with turbulence and so you'd have some energy losses that head loss from the valve and finally at the end of the pipe the nozzle converts all the remaining pressure head into velocity head and so they hydraulic grade line Falls all the way to the center of the pipe so it's zero at that point relative to the center line of the pipe this is about as complex as these hydraulic grade line and energy grade line problems yet if you understand this one you've probably got a handle on this so as I've done in many of these videos I'm gonna end by just showing something that's completely unrelated to the current topic but kind of cool and interesting and related to fluid mechanics this is some high-speed photography showing incredible complexities of the flow associated with striking a simple match it uses something called salir in photography which reveals the very small density variations in the fluid it's about two and a half minutes long and it's really I think a fascinating video so hang in there till the end it's gets more interesting as the video goes on so at this point I'll just keep quiet and play the video because I think it speaks for itself you
3849	https://www.pearson.com/channels/general-chemistry/textbook-solutions/mcmurry-8th-edition-9781292336145/ch-6-ionic-compounds-periodic-trends-and-bonding-theory/which-ion-has-a-larger-atomic-radius-cu-or-cu2-explain-your-reasoning	My Courses Chemistry General Chemistry Organic Chemistry Analytical Chemistry GOB Chemistry Biochemistry Intro to Chemistry Biology General Biology Microbiology Anatomy & Physiology Genetics Cell Biology Physics Physics Math College Algebra Trigonometry Precalculus Calculus Business Calculus Statistics Business Statistics Social Sciences Psychology Health Sciences Personal Health Nutrition Business Microeconomics Macroeconomics Financial Accounting Product & Marketing Agile & Product Management Digital Marketing Project Management AI in Marketing Programming Introduction to Python Microsoft Power BI Data Analysis - Excel Introduction to Blockchain HTML, CSS & Layout Introduction to JavaScript R Programming CalculatorsAI ToolsStudy Prep BlogStudy Prep Home Ch.6 - Ionic Compounds: Periodic Trends and Bonding Theory All textbooksMcMurry 8th EditionCh.6 - Ionic Compounds: Periodic Trends and Bonding TheoryProblem 52 Chapter 6, Problem 52 Which ion has a larger atomic radius, Cu+ or Cu2+? Explain your reasoning. Verified step by step guidance 1 Step 1: Understand that atomic radius is the distance from the center of an atom's nucleus to its outermost electron. In ions, the atomic radius is affected by the number of electrons and protons. Step 2: Recognize that Cu+ and Cu2+ are ions of the same element, copper. Cu+ has lost one electron, while Cu2+ has lost two electrons. Both ions have the same number of protons. Step 3: Recall that the more protons an ion has, the stronger the pull on the electrons (due to the positive charge of the protons). This pull is called the effective nuclear charge. Step 4: Understand that when an ion loses electrons and becomes positively charged (like Cu+ and Cu2+), the effective nuclear charge increases. This means the remaining electrons are pulled closer to the nucleus, decreasing the atomic radius. Step 5: Conclude that since Cu2+ has lost more electrons than Cu+, it has a higher effective nuclear charge and therefore a smaller atomic radius than Cu+. Verified video answer for a similar problem: This video solution was recommended by our tutors as helpful for the problem above. Video duration: 1m Play a video: Was this helpful? Key Concepts Here are the essential concepts you must grasp in order to answer the question correctly. Atomic Radius Atomic radius refers to the distance from the nucleus of an atom to the outermost shell of electrons. It is influenced by the number of electron shells and the effective nuclear charge experienced by the outer electrons. Generally, atomic radius increases down a group in the periodic table and decreases across a period due to increased nuclear charge pulling electrons closer. Recommended video: Guided course 02:02 Atomic Radius Ionization and Charge When an atom loses electrons to form a cation, its charge increases, which can affect its size. A higher positive charge results in a greater effective nuclear charge, pulling the remaining electrons closer to the nucleus. Therefore, cations with higher charges typically have smaller radii compared to their neutral atoms or cations with lower charges. Recommended video: Guided course 01:19 Ionization Energy Comparison of Cu+ and Cu2+ Copper can exist in multiple oxidation states, notably +1 (Cu+) and +2 (Cu2+). Cu+ has lost one electron, while Cu2+ has lost two. The loss of an additional electron in Cu2+ increases the effective nuclear charge on the remaining electrons, resulting in a smaller atomic radius for Cu2+ compared to Cu+, making Cu+ the larger ion. Recommended video: Guided course 02:11 Oxyacid Strength Comparison Related Practice Textbook Question Which atom or ion in the following pairs would you expect to be larger?(c) Cr3+ or Cr6+ 559 viewsTextbook Question Order the following ions from smallest to largest: Sr2+, Se2-, Br-, Rb+. 1757 viewsTextbook Question Order the following ions from smallest to largest: Mg2+, O2-, F-, Na+. 1574 viewsTextbook Question The following ions all have the same number of electrons: Ti4+, Sc3+, Ca2+, S2-. Order them according to their expected sizes, and explain your answer. 1149 views
3850	https://www.geeksforgeeks.org/maths/how-does-a-positive-slope-differ-from-a-negative-slope/	How does a positive slope differ from a negative slope? Slope defines the steepness of a line and the way in which it tilts. The positive slope differs from the negative slope by the inclination of the line. In this article, we will explore how the positive slope differs from the negative slope. Table of Content What is Slope? The slope of a line defines the nature of steepness or incline of the particular line and it is defined as the ratio of change in the vertical measurement which is called the rise to the change in the horizontal measurement which is called the run. Mathematically, the slope m is calculated using the formula: m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} Here, Δy stands for change in the y coordinate and Δx for change in the x coordinate. What is a Positive Slope? Positive slope can be defined as the orientation of a line on a graph in which the line moves upward from left to right. This means that as the value of x-coordinate increases that of y-coordinate also increases. The slope of this line is positive because the change in ‘y’, which is equal to the ‘rise’, is greater than zero and so is the change in ‘x’, which is the ‘run’. Example: In the real world, such as a road that tilts up a hill or the development of a plant from seed can be described with a positive value of the slope if the height increases together with time or distance. Characteristics of Positive Slope Some of the characteristics of positive slopes are as follows: What is a Negative Slope? This is a trend line that inclines downwards on the graph paper and this is what is often referred to as the negative slope. Here, for the same change in the x-coordinate, the y-coordinate decreases which gives a negative value for the slope. The negative sign is the slope of the line to suggest that two variables are inversely related to each other. Example: A negative slope can be demonstrated in practical applications, such as a loss in the value of a car over time, or a downward hill on which distance is measured as it increases, while elevation decreases. Characteristics of Negative Slope Some of the characteristics of negative slopes are as follows: Difference between Positive and Negative Slope The table below represents how a positive slope differs from a negative slope. \| Aspects \| Positive Slope \| Negative Slope \| --- \| Direction of Line \| Ascends from left to right \| Descends from left to right \| \| Relationship between variables \| Direct relationship (both increases) \| Inverse relationship (one increases, one decreases) \| \| Practical implications \| Indicates growth or increase in value \| Indicates decline or decrease in value \| \| Angle with X-axis \| Acute angle with positive x-axis \| Obtuse angle with the positive x-axis \| \| Impact on Functions Behavior \| Increases the output value for increasing input in linear functions \| Decreases the output value for increasing input in linear functions \| \| Significance in Economics \| Represents positive correlation, such as demand increasing with price \| Represents negative correlation, such as supply decreasing as price increases \| Aspects Positive Slope Negative Slope Direction of Line Ascends from left to right Descends from left to right Relationship between variables Direct relationship (both increases) Inverse relationship (one increases, one decreases) Practical implications Indicates growth or increase in value Indicates decline or decrease in value Angle with X-axis Acute angle with positive x-axis Obtuse angle with the positive x-axis Impact on Functions Behavior Increases the output value for increasing input in linear functions Decreases the output value for increasing input in linear functions Significance in Economics Represents positive correlation, such as demand increasing with price Represents negative correlation, such as supply decreasing as price increases Conclusion The distinctions between the positive and negative slope are important for the students studying mathematics as it underpins the ways of interpreting the linear equation, learning about the variables’ correlation, and applying these concepts in the problem-solving process. From the simple task of determining the steepness of a line on a graph or the rate of changes in a real-life situation, the slope is a key component of the mathematician’s toolbox. Read More, T Explore Maths Basic Arithmetic What are Numbers? Arithmetic Operations Fractions - Definition, Types and Examples What are Decimals? 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3851	https://gchem.cm.utexas.edu/data/section2.php?target=ka-kb-constants.php	back to Tables Listing Acid/Base Ionization Constants Please note: Although these values are common (and published) values associated with the substances given, there are many possible sources for these values. Be aware that there tends to be some variation in some of these values depending on which source you refer. One should always know which source to use. These are established reference values in many books - but be aware of slight differences when working problems. Having said all that, we (UT Chemistry) will give you the values that we use for exams. The student should know that these values will be given on an exam. Ka values for some common acids \| Name \| Formula \| Ka \| pKa \| --- --- \| \| acetic acid \| CH3COOH \| 1.8 × 10-5 \| 4.74 \| \| acrylic acid \| CH2CHCOOH \| 5.6 × 10-5 \| 4.25 \| \| benzoic acid \| C6H5COOH \| 6.4 × 10-5 \| 4.19 \| \| boric acid \| H3BO3 \| 5.8 × 10-10 \| 9.24 \| \| butanoic acid \| CH3CH2CH2COOH \| 1.5 × 10-5 \| 4.82 \| \| chloroacetic acid \| CH2ClCOOH \| 1.4 × 10-3 \| 2.85 \| \| chlorous acid \| HClO2 \| 1.2 × 10-2 \| 1.92 \| \| formic acid \| HCOOH \| 1.8 × 10-4 \| 3.74 \| \| hydrocyanic acid \| HCN \| 6.2 × 10-10 \| 9.21 \| \| hydrofluoric acid \| HF \| 6.3 × 10-4 \| 3.20 \| \| hydrogen peroxide \| H2O2 \| 2.4 × 10-12 \| 11.62 \| \| hypobromous acid \| HBrO \| 2.0 × 10-9 \| 8.70 \| \| hypochlorous acid \| HClO \| 3.5 × 10-8 \| 7.46 \| \| hypoiodous acid \| HIO \| 2.0 × 10-11 \| 10.70 \| \| lactic acid \| CH3CH(OH)COOH \| 1.38 × 10-4 \| 3.86 \| \| nitrous acid \| HNO2 \| 4.0 × 10-4 \| 3.40 \| \| phenol \| C6H5OH \| 1.6 × 10-10 \| 9.80 \| \| propanoic acid \| CH3CH2COOH \| 1.3 × 10-5 \| 4.89 \| \| trichloroacetic acid \| CCl3COOH \| 2.2 × 10-1 \| 0.66 \| Kb values for some common bases \| Name \| Formula \| Kb \| pKb \| --- --- \| \| ammonia \| NH3 \| 1.8 × 10-5 \| 4.74 \| \| aniline \| C6H5NH2 \| 4.3 × 10-10 \| 9.37 \| \| dimethylamine \| (CH3)2NH \| 5.4 × 10-4 \| 3.27 \| \| ethylamine \| C2H5NH2 \| 5.6 × 10-4 \| 3.25 \| \| hydrazine \| NH2NH2 \| 1.7 × 10-6 \| 5.77 \| \| hydroxylamine \| NH2OH \| 1.1 × 10-8 \| 7.96 \| \| methylamine \| CH3NH2 \| 4.38 × 10-4 \| 3.36 \| \| nicotine \| C10H14N2 \| 1.0 × 10-6 \| 6.00 \| \| piperidine \| C5H10NH \| 1.3 × 10-3 \| 2.89 \| \| propylamine \| C3H7NH2 \| 3.7 × 10-4 \| 3.43 \| \| pyridine \| C5H5N \| 1.8 × 10-9 \| 8.74 \| \| triethylamine \| (C2H5)3N \| 1.0 × 10-3 \| 3.00 \| \| trimethylamine \| (CH3)3N \| 6.5 × 10-5 \| 4.19 \| Ka values for some polyprotic acids \| Name \| Formula \| Ka \| pKa \| --- --- \| \| arsenic acid \| H2AsO4 \| 1) 5.0 × 10-3 2) 9.3 × 10-8 3) 3.0 × 10-12 \| 1) 2.30 2) 7.03 3) 11.52 \| \| carbonic acid \| H2CO3 \| 1) 4.3 × 10-7 2) 5.6 × 10-11 \| 1) 6.37 2) 10.25 \| \| citric acid \| H3C6H5O6 \| 1) 8.4 × 10-4 2) 1.8 × 10-5 3) 4.0 × 10-6 \| 1) 3.08 2) 4.74 3) 5.40 \| \| oxalic acid \| H2C2O4 \| 1) 6.5 × 10-2 2) 6.1 × 10-5 \| 1) 1.19 2) 4.21 \| \| phosphoric acid \| H3PO4 \| 1) 7.5 × 10-3 2) 6.2 × 10-8 3) 4.8 × 10-13 \| 1) 2.12 2) 7.21 3) 12.32 \| \| sulfuric acid \| H2SO4 \| 1) ~100 2) 1.2 × 10-2 \| 1) -2 2) 1.92 \| \| sulfurous acid \| H2SO3 \| 1) 1.5 × 10-2 2) 1.0 × 10-7 \| 1) 1.82 2) 7.00 \|
3852	https://www.powerthesaurus.org/disjointed/antonyms	DISJOINTED Antonyms: 462 Opposite Words & Phrases Log in Feedback Help Center Dark mode AboutPRO MembershipExamples of SynonymsTermsPrivacy & Cookie Policy Antonyms for Disjointed 462 opposites of disjointed- words and phrases with opposite meaning Lists synonyms antonyms definitions sentences thesaurus Words Phrases Idioms Parts of speech Adjectives Verbs Nouns Tags disconnect incoherent unbound connectedverb unitedverb coherentadj. logicaladj. organizedadj.verb unifiedverb orderedadj.verb all in one consistentadj. continuousadj. joinedverb thoroughly thought throughadj. efficientadj. arrangedverb cohesive orderlyadj. incorporatedverb interconnected methodicaladj. systematizedadj. closely associated closely linkedadj. closely connectedadj. closely associated with consolidatedverb Log in Power Thesaurus ✌️ Less advertisements, more content and additional features Log in AdvertisementsEnjoy Ad-Free with PRO! AboutPRO MembershipExamples of SynonymsTermsPrivacy & Cookie Policy Power Thesaurus © 2025
3853	https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem?srsltid=AfmBOop4EIP5Xi5otZAanZW208PdW79kcSX8lXY9hp1ZyshCrnlArxPh	Art of Problem Solving Pythagorean Theorem - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Pythagorean Theorem Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Pythagorean Theorem The Pythagorean Theorem states that for a right triangle with legs of length and and hypotenuse of length we have the relationship . This theorem has been known since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually. The Pythagorean Theorem is one of the most frequently used theorems in geometry, and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem. This is generalized by the Pythagorean Inequality and the Law of Cosines. Contents 1 Proofs 1.1 Proof 1 1.2 Proof 2 1.3 Proof 3 1.4 Proof 4 1.5 Proof 5 2 Pythagorean Triples 3 Problems 3.1 Introductory 3.1.1 Problem 1 3.1.2 Problem 2 4 External Links 4.1 Videos Proofs In these proofs, we will let be any right triangle with a right angle at , and we use to denote the area of triangle . Proof 1 Let be the foot of the altitude from . , , are similar triangles, so and . Adding these equations gives us Proof 2 Let be the foot of the altitude from . Since , , are similar right triangles, and the areas of similar triangles are proportional to the squares of corresponding side lengths, But since triangle is composed of triangles and , , so . Proof 3 Consider a circle with center and radius . Since and are perpendicular, is tangent to . Let the line meet at and , as shown in the diagram: Evidently, and . By considering the Power of a Point with respect to , we see Proof 4 Consider a square of side length a+b. Inside it, place four congruent right triangles—each with legs a and b and hypotenuse c—arranged so that their right angles occupy the corners of the large square and their hypotenuses form a smaller, tilted square at the center. The area of the large square is (a+b)2=a 2+2 a b+b 2. The area can also be expressed as the sum of the four triangles, each with area 1 2 a b, totaling 2 a b, plus the inner square with area c 2. Setting these equal gives: Simplifying, Proof 5 Consider the right triangle with a right angle at . Represent the vectors in an inner product space equipped with the dot product . Because is a right angle, the vectors and are orthogonal: By definition of the norm induced by the inner product, and the hypotenuse vector is Calculate the squared length of : Using orthogonality , this reduces to Since , the length of the hypotenuse, we conclude which is the Pythagorean Theorem. This proof is highly abstract and depends on familiarity with vector spaces and inner products, moving far beyond the classical Euclidean framework into the realm of modern algebraic geometry. Pythagorean Triples Main article: Pythagorean triple A Pythagorean triple is a of positive integers such that . All such triples contain numbers which are side lengths of the sides of a right triangle. Among these, the Primitive Pythagorean triples, are those in which the three numbers are coprime. A few of them are: Note that is the only Pythagorean triple that consists of consecutive integers. Any triple created by multiplying all three numbers in a Pythagorean triple by a positive integer is Pythagorean. In other words, if is a Pythagorean triple it follows that ) will also form a Pythagorean triple for any positive integer constant . For example, Also note that one easy way to find Pythagorean triples is as follows. Choose any odd number . Find . Find and . Your Pythagorean triple is , , and . Problems Introductory 2020 AMC 8 Problem 18 2023 AMC 10A Problem 11 2007 AMC 12A Problem 10 2006 AIME I Problem 1 Problem 1 Rectangle is inscribed in a semicircle with diameter as shown in the figure. Let and let What is the area of Solution Problem 2 A rhombic dodecahedron is a solid with congruent rhombus faces. At every vertex, or edges meet, depending on the vertex. How many vertices have exactly edges meet? Solution External Links Wikipedia: Pythagorean Theorem Cut-the-Knot: 122 Proofs of the Pythagorean Theorem Videos Proving the Pythagorean Theorem Using the Pythagorean Theorem Part 1 Using the Pythagorean Theorem Part 2 Pythagorean Triple Warning! Power of Pythagorean Triples Retrieved from " Categories: Geometry Theorems Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
3854	https://www.cordeliauys.co.uk/the-letdown-reflex-1	General 3 — Cordelia Uys 0 Skip to Content Cordelia Uys Home About me My Resources Contact Consultations Introducing Solids Breastfeeding Preparation Facebook Groups Parenting Course NCT Drop-in Blossom Antenatal Open Menu Close Menu Cordelia Uys Home About me My Resources Contact Consultations Introducing Solids Breastfeeding Preparation Facebook Groups Parenting Course NCT Drop-in Blossom Antenatal Open Menu Close Menu Home About me My Resources Contact Consultations Introducing Solids Breastfeeding Preparation Facebook Groups Parenting Course NCT Drop-in Blossom Antenatal The let-down reflex by Cordelia Uys, Breastfeeding Counsellor April 2020 The anatomy of your breasts There are several ducts on the inside of your nipples which branch out into your breasts; at the end of the ducts are a multitude of alveoli containing milk producing cells. These cells take nutrients from your blood and turn them into milk. The milk ducts then transport milk to the tip of your nipples, where milk is released from several openings called milk duct orifices. This network of ducts looks similar to a tree, with the trunk being the nipple, the branches the milk ducts and the leaves the alveoli. Oxytocin and Prolactin The two main hormones involved in milk production are oxytocin and prolactin. Oxytocin, often called the hormone of love, causes the contractions of orgasm, birth and lactation. It promotes feelings of love, bonding, calm and well-being in both you and your baby. Prolactin is the hormone that creates milk; it induces calmness and mothering behaviours. Prolactin levels are highest in your body between 2am and 4am, which is why mothers often notice they produce more milk at night, and why breastfeeding (or pumping) at night in the early months is so crucial to establishing and maintaining your milk supply. The let-down reflex When your baby suckles at your breasts, your brain releases oxytocin, which travels in your blood down to your breasts, where it makes the muscles surrounding the milk producing cells contract and push the milk out. This process is called the milk ejection reflex, or the let-down reflex, as milk is ‘let-down’ from the breasts. The removal of milk from your breasts – by your baby or by pumping - then tells your body to make more prolactin. Milk production is reliant on supply and demand: the more often your baby feeds (or the more often you pump), and the more milk is removed, the more milk you will make. The Feedback Inhibitor of Lactation (FIL) and the importance of frequent milk removal Whenever your breasts get hard and full, a protein known as the Feedback Inhibitor of Lactation, builds up in your breasts, telling them to stop making milk. This means milk production slows down when your breasts are fuller and speeds up when your breasts are softer. This is why regular milk removal is so important for maintaining a good milk supply. A baby will typically breastfeed at least 8-12 times in 24 hours; in fact, many babies will breastfeed more than 12 times in 24 hours. A mother who is exclusively pumping should aim to pump at least 8 times in 24 hours because regular milk removal helps promote good milk production and protect her milk supply. If milk is not removed very regularly either by breastfeeding or pumping, a mother's breasts will get the message to gradually produce less and less milk. What does the let-down reflex feel like? Some women don’t feel the let-down reflex at all, others feel a tingling or tightening sensation in their breasts, or a sudden feeling of fullness, and a few women actually find the let-down reflex quite uncomfortable or even painful. Women usually experience two let-downs during a feed, but many are only aware of the first one. This tends to occur after the baby has been suckling for a minute or two. If you are one of those mothers who don’t feel their let-down, the way you’ll know it’s happening is that you will see, and possibly hear, your baby swallowing faster, and you may notice milk dripping from your other breast. Uterine contractions In the first week or so of your baby’s life, you will often feel uterine contractions at the same time as the let-down reflex. This is because the oxytocin causing the muscles around your milk producing cells to contract also causes your uterus to contract back down to its pre-pregnancy size. This is one of the beneficial effects of breastfeeding: your uterus shrinks back down faster. These contractions can feel quite painful, especially with second and subsequent babies, and some women choose to take paracetamol to help with the pain. Paracetamol is compatible with breastfeeding. Breathing in through your nose and blowing long breaths out through your mouth, as during birth contractions, can also work well. Oxytocin – the ‘shy’ hormone When you’re breastfeeding your baby, having a let-down reflex is rarely something you need to think about, as your baby’s presence is all your brain needs to release oxytocin, but when mothers are pumping, and especially if they’re doing so when away from their baby, or when they are anxious about their milk production, or indeed feeling worried about anything else, it can be harder to get the milk flowing. Oxytocin is linked to loving feelings, and although a mother might like her pump and be pleased to have it, few women actually love their pump. In addition, oxytocin is often called the ‘shy’ hormone, as it’s harder to get a release of oxytocin if you aren’t feeling safe, calm and unobserved. Imagine being told you needed to have an orgasm within a set amount of time, especially if you weren’t in the mood. It would be even harder if someone you didn’t feel comfortable with were sitting there watching you! Once mothers are used to pumping, and especially when mums are pumping at the same time, or times, every day, their breasts become accustomed to the process and having a let-down isn’t usually a problem, but if you are new to pumping, there are some steps you can take to help with the process. How to encourage the let-down reflex Warmth and massage Firstly, apply warmth to your breasts - pouring comfortably hot water into a clean nappy makes a great warm compress – and then gently massage or stroke your breasts. It’s even better if someone you feel comfortable with can rub your shoulders and/or feet. If your baby isn’t in front of you, have a photo of him or her to look at, and a piece of clothing or muslin that smells of them to sniff. Try to relax Try to relax as much as possible, for example by listening to music you enjoy, or some hypnobirthing tapes. Some mothers find practising mindfulness or imagining their favourite place really helpful. Watching a TV programme that puts you in a good mood can work wonders. One mother who had to do a lot of double pumping at the beginning of her breastfeeding journey, used to watch an episode of Friends during each of her pumping sessions. Pain prevents oxytocin release Please remember that pain impedes the release of oxytocin, so it’s essential that your breast shields fit comfortably, and that the vacuum on your pump isn’t set too high. Start with the vacuum low and the cycles high, then once the milk starts to flow, turn the cycles half way down, and the vacuum up so that the suction is firm but totally comfortable. If you put the vacuum up too high, you’ll produce less milk and you might also cause nipple damage. Try not to stare at the pump sets Rather than sitting and staring at the pump sets, some mothers choose to cover the pump sets with baby socks, so that they can’t see how much milk is coming, and therefore don’t obsess about whether or not they’re producing enough milk. Pumping bra When double pumping, it’s recommended to either buy or make yourself a pumping bra that will hold the pump sets in place. You can customise an old exercise bra you don’t mind sacrificing, by cutting holes in it where the pump sets need to go. Make sure the bra is comfortable and doesn’t dig in anywhere or push the pump sets too tightly into your breasts as this can cause blocked ducts. A pumping bra will also allow you to massage and compress your breasts while you pump. This is known as hands-on pumping and it helps both with the let-down reflex and also in removing more milk from your breasts. Oxytocin is triggered in many ways By having your baby in skin to skin By seeing, hearing and smelling your baby Sometimes by hearing someone else’s baby By thinking of your baby By breastfeeding By pumping Some women even experience a let-down when witnessing a kind act, such as seeing someone help an elderly person cross the road, or when seeing something they find adorable, such as a video of puppies playing. Things that help oxytocin to flow Feeling warm Feeling calm, safe and relaxed Not feeling observed Not feeling pressured or anxious Not being in pain Gently stroking or massaging your breasts and nipples Having your shoulders and/or feet massaged Seeing a photo of your baby Practising mindfulness Listening to a hypnobirthing tape Listening to music you enjoy Watching a TV programme or film that is fun and light-hearted Imprinting on your pump One last thing to point out, is that when a mother has become used to a particular brand of pump, it can sometimes be hard for her to get a let-down reflex for a different brand. It’s as if her brain has imprinted on that brand and she’s bonded with it. In fact, if a mother has been exclusively pumping for a while, and has been separated from her baby while pumping, sometimes she might need to think about her pump to get a let-down when she starts breastfeeding her baby. This is another reason why it’s a good idea to look at a photo of your baby while pumping, rather than staring at your pump. If you would like, or need, to change the brand of pump you’re using, it can help to spend a week alternating between the two types of pumps so that your brain learns to accept the new brand. Email: cordeliauys@gmail.com Phone: +447770325579 London, UK I’m particularly delighted that all the photos of parents and babies on my website are of families that I’ve worked with. I would like to thank all the parents who allowed me to use their photos and feedback on my website, and a special thank you to the photographer, Nitin Sachania for allowing me to use a number of his photos.
3855	https://ejrnm.springeropen.com/articles/10.1186/s43055-023-01153-3	Advertisement Role of 18F-fluorodeoxyglucose positron emission tomography/computed tomography in the detection of recurrence and peritoneal metastasis from ovarian cancer in correlation with cancer antigen-125 tumor marker levels Egyptian Journal of Radiology and Nuclear Medicine volume 55, Article number: 9 (2024) Cite this article 1199 Accesses 2 Citations Metrics details Abstract Background The chronic nature of ovarian cancer and disease recurrence has a considerable impact on the assessment of follow-up strategies and treatment planning for both oncologists and radiologists. It is imperative to conduct adequate follow-up in ovarian cancer to detect and treat recurrence as early as possible. Presently, surveillance of patients with this malignancy involves the combination of serial CA-125 assay and diverse imaging procedures, yet normal CA-125 levels cannot entirely rule out disease relapse. PET/CT provides whole-body functional imaging that does not necessities contrast injection, and allows for precise diagnosis and restaging of patients with suspected ovarian cancer recurrence, thereby strongly impacting disease management decisions. Our study aims to evaluate the value of FDG-PET/CT as a follow-up imaging tool in detecting and localizing recurrence of ovarian cancer, in conjunction with CA-125 tumor markers. Results In our study, it was demonstrated that recurrent disease manifested in FDG-PET/CT in 24 cases, with 9 of those cases exhibiting CA-125 levels within the normal range. There were two instances of false negative results and one instance of false positive results in FDG-PET/CT. Additionally, three cases were found to be free of disease relapse in FDG-PET/CT and exhibited normal CA-125 levels throughout the follow-up period (true negative). The prevalence of disease recurrent sites was 12% for local recurrence, 60% for peritoneal metastasis, 64% for nodal deposits and 28% for distant metastatic disease. The accuracy of FDG-PET/CT was 88.8%, with a sensitivity of 91.3% and specificity of 75%. Furthermore, FDG-PET/CT showed a positive predictive value of 95.5% and a negative predictive value of 60.3%. Conclusions PET/CT imaging provides a comprehensive and functional view of the entire body, which can accurately diagnose and restage cases with ovarian cancer recurrence. This approach plays a critical role in identifying peritoneal carcinomatosis and is considered a more dependable method than CA-125 tumor markers for detecting and monitoring ovarian cancer recurrence. Additionally, PET/CT imaging has the potential to decrease the number of second-look laparotomies and can thus significantly impact the management plan. Background Ovarian cancer is the most lethal of all gynecologic cancers, ranking fourth among all fatal diseases in women. It represents a difficult medical condition that poses significant diagnostic challenges in its early stages and is associated with a high incidence of 2-year relapse of early and advance stages following initial treatment . It has been reported that the main reason for ovarian cancer's high fatality rate is the occurrence of peritoneal recurrence, which occurs frequently because of concealed metastasis . The early detection and accurate localization of ovarian cancer recurrent disease is crucial as it aids in determining the feasibility of secondary surgery distinguishing cases with optimal curative cytoreduction from the palliative approach . The CA-125 transmembrane glycoprotein is crucial for evaluating therapy response and ovarian cancer recurrence. However, its specificity is limited due to increased levels of colorectal cancer and inflammation. Elevated CA-125 levels indicate disease existence, but not the location or quantity of metastatic foci 44 "), [5 The prognostic value of 18F-FDG PET/CT in monitoring chemotherapy in ovarian cancer both at initial diagnosis and at recurrent disease. Clin Nucl Med 43(10):735–738")]. Computed tomography (CT) and magnetic resonance imaging (MRI) are frequently utilized imaging techniques in the detection of ovarian cancer recurrence. However, a specific challenge arises in traditional diagnostic imaging when these techniques are unable to reveal findings, particularly for small implanted tumors on the visceral surface since the primary way of metastasis is through the peritoneal spread rather than the parenchymal route. However, there is still a suspicion of tumor recurrence due to elevated CA-125 levels. Several reports have then advocated that FDG-PET/CT can reveal lesions otherwise missed on CT alone in recurrent ovarian carcinoma, of particular assistance in identifying possible extra-abdominal metastases [6, 7]. FDG-PET/CT integrates the anatomical architecture of tissues with the metabolic activity of cells, effectively fusing anatomical and functional imaging into a single scan. It possesses a special significance in accurately diagnosing ovarian cancer recurrence. It allows for a precise evaluation of disease recurrence, which in turn enables efficient restaging of the illness optimizing the treatment strategies . The potential risks and limitations of second-look surgery have raised questions about the value of positive FDG-PET as a noninvasive alternative, particularly for peritoneal metastasis evaluation. The baseline PET/CT also plays a crucial role in therapy monitoring that could early implicate neoadjuvant therapy protocols and prevent ineffective therapy in non-responders . Our research was conducted to evaluate the effectiveness of PET/CT as a subsequent imaging modality for identifying and localizing instances of ovarian cancer recurrence, while simultaneously utilizing CA-125 tumor markers as a result aiding in better management aimed to improve quality of life. Methods Study population This prospective cross-sectional study has been granted approval by the ethical and scientific committees at our facility. The study involved twenty-seven adult female patients their age above 18 years and had been histopathologically confirmed to have previously managed ovarian cancer. Patients were referred from the oncology department at our institution during the period from December 2020 to December 2022, due to suspicion of recurrence. The initial stage at the time of diagnosis and the result of cytoreductive surgery to the magnitude of residual implants have been known as the most influential prognostic factors for relapse in individuals afflicted with ovarian cancer. Each patient had formally given their written consent after receiving adequate information. All of the patients underwent a PET/CT scan using an 18F-FDG tracer and a 24-slice CT scanner (Discovery IQ 5-ring, GE health care). Adult female patients who were pathologically proved to have ovarian cancer following initial treatment of either surgery or chemotherapy or a combination of both owing to clinical suspicion of recurrent ovarian malignancy or elevated CA-125 levels were included in our study. Any patient who has been referred for initial staging without having any primary treatment, patients who histopathologically proved to have another malignancy, individuals with uncontrolled blood glucose levels exceeding 150 mg/dl and patients with incomplete clinical data and laboratory result were excluded from our study. Methodology Patient preparations All patients provided relevant clinical history and informed consent after receiving a detailed explanation of the imaging procedure. Patients were asked to bring all related previous investigations, histopathological results and CA-125 levels. Patients were abstained from consuming food and drinks, except for water, for a minimum of 4–6 h before undergoing examination, they were instructed to stay hydrated and avoid intense physical activity 24 h before the scan, and those with comorbidities may take medications on the day of the scan, except for diabetes medications which must be taken at least 6 hours before the procedure. Before administering (18F-FDG), the physical mass and blood sugar level were assessed, with a requirement of blood sugar levels below 140 mg/Dl. Diabetic patients with high blood glucose levels may use their insulin medications. If not controlled, they were rescheduled after normalizing their blood sugar with their endocrinologist. In the case of contrast injection, kidney function tests are performed with serum creatinine levels less than 1.5mg/dl and GFR above 45. A large IV cannula was used to deliver the prescribed dosage and probable iodized contrast substance. FDG-PET/CT image acquisition: The selected cases were administered 18F-FDG based on their physical mass, with a usual dose of 7.5 MBq/kg. Following the injection, the cases were instructed to remain in quiet resting rooms for approximately 40 minutes and were given oral hydration. PET/CT images were obtained using a GE Discovery IQ machine, with imaging beginning 40 minutes after injection of FDG-radioactive substance and including a low-dose non-contrast CT for attenuation correction and anatomical localization, followed by a CT examination and injection of contrast (if required), and concluding with a 25-minute PET scan. Axial, sagittal and coronal PET images were created using a standard iterative algorithm with attenuation correction based on CT scan data, and a quantity of nonionic iodinated contrast (Omnipaque 350) was dispensed at a dose of 1 ml/kg in case of contrast administration. Image analysis Axial and multi-planar images of CT, PET and fused PET/CT images were analyzed for ovarian carcinoma recurrence by an oncology consultant radiologist (more than 10-year experience) and a nuclear medicine consultant (5-year experience in PET/CT), The inter-observer agreement was measured and found to be 0.88, indicating almost perfect agreement, and the final diagnosis was reached by consensus with subsequent anonymization of results. The qualitative analysis involved visual examination of regions with increased FDG uptake in a focused manner. This was to identify local pelvic recurrence, peritoneal metastatic disease, metastatic lymphadenopathy or hematogenous metastatic disease. Lesions with increased FDG uptake were considered abnormal if substantially greater than background soft tissue activity. Semiquantitative analysis was done by recording the maximum SUV of all patients. ROIs were placed over abnormal focal uptake for analysis and compared with soft tissue background. SUV is defined as the radioactive concentration in tissue or lesion in megabecquerels per gram divided by the injected dose in megabecquerels and the patient's body weight in grams. The liver, lung and bone marrow typically have SUVs ranging from 0.5 to 2.5. Neoplastic growths often have a higher SUV than 2.5–3.0. Data analysis An initial PET/CT assessment and CA-125 levels were done in all patients histopathologically proved to have managed ovarian cancer with clinical suspicion of recurrence. A subsequent follow-up PET/CT and CA-125 tumor marker levels were performed to assess PET/CT observations after treatment. Additional follow-up PET/CT and CA-125 levels were done in 17 cases. Positive cases received the standard chemotherapy regimen (paclitaxel 175 mg/m2 intravenous and carboplatin AUC 5 to AUC 6 intravenous) both on day 1 and the cycle repeats every 21 days and re-assessment done every 3 months. Due to advanced disease stage of our included cases, second-look surgery was only done in seven positive cases, comparing observations to histopathological results. The evaluation of treatment response was additionally carried out in our investigation utilizing PET/CT and CA-125 tumor. PET/CT imaging follow-up response Guided by the PERCIST criteria published by the Journal of Nuclear Medicine 2009 , the target lesion that was measured for intensity during pre- and post-treatment scans did not necessarily remain the same. In the case of progressive disease, the presence of any new lesion or an increase of greater than 30% in SUVmax of the hottest lesions indicates progression. In contrast, a reduction of 30% or more in SUVmax of the sum of the hottest lesions at baseline and follow-up indicates a partial response. Complete response is defined as the normalization of FDG uptake of all deposits, with FDG uptake similar to that of the surrounding background. Stable disease is characterized by the absence of partial, complete or progressive disease. CA-125 response This was achieved as guided by NACB ovarian cancer panel recommendation : CA-125 responders were identified as individuals who had a minimum of a 50% reduction in CA-125 levels in comparison with their initial measurement. Those who did not respond to treatment were classified as non-responders, as indicated by a CA-125 level increase of twofold or more from the baseline value. The assessment was conducted to evaluate the correlation between 18F-FDG-PET/CT results and CA-125 tumor marker levels. The outcomes of this study were divided into the following categories: If the PET/CT findings in the subsequent PET/CT follow-up exhibited an increase or decrease in morphology/metabolic activity in alignment with the clinical indications of progression or regression, even in the absence of a histological diagnosis, the concordant response of CA-125 levels following the management of ovarian cancer or the findings from histopathology was deemed to be indicative of a true positive outcome. True negative outcomes were established in instances where no recurrence was detected through follow-up PET/CT or clinical monitoring. False negative results were reported when PET/CT results were normal, but relapse was distinguished by tissue biopsy, imaging or clinical follow-up. False positive outcomes were observed when lesions detected by PET/CT were proved to be benign by histopathology or other imaging techniques. Statistical analysis The statistical package for the Social Sciences (SPSS) version 28 (IBM Corp., Armonk, NY, USA) was utilized to code and enter data. Quantitative data were summarized using mean, standard deviation, median, minimum and maximum, while categorical data were summarized using frequency (count) and relative frequency (percentage). The nonparametric Mann–Whitney test (Chan, 2003a) was employed for comparisons between quantitative variables, and the Spearman correlation coefficient (Chan, 2003b) was used for correlations between quantitative variables. Standard diagnostic indices, including sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV) and diagnostic efficacy, were calculated as described by Galen (1980). Statistical significance was considered at p-values less than 0.05. Results A sum of 27 individuals whose ages varied from 20 to 70 years possessing a mean age of 56 years were histopathologically proved as ovarian cancer postoperative. Out of the total, 96% of the patients underwent operative intervention and received chemotherapy, while only 4% underwent surgical treatment solely as a primary approach. Epithelial serous carcinoma was the main pathological subtype of our cases (Table 1). They were directed to our institution and experienced an FDG-PET/CT scan for suspected recurrence on clinical grounds or for routine monitoring. For every focal abnormal uptake of radiotracer, qualitative and semiquantitative SUVmax was assessed. Guided by CT the localization of focal FDG uptake was ascertained in the following manner: 12% of our cases exhibited local/operative bed recurrence, 60% had peritoneal metastasis, 64% of cases demonstrated nodal deposits, and 28% of cases presented with distant metastatic disease with overlapping of findings. Estimated SUVmax showed significant values compared to the physiological uptake rates observed at various sites of recurrent disease (Table 2). Cancer antigen 125 (CA-125) levels of included cases showed a range of 2–1700 U/ml with a mean value of 150.59 U/ml, and nine of our included cases showed CA-125 levels below normal range (normal level 35 U/ml) Table 3. The initial PET/CT and CA-125 levels were correlated with follow-up, and PET/CT scans were found to be true positive in 77.7% of cases; the PET/CT findings were in accordance with the follow-up CA-125 tumor markers and subsequent follow-up PET/CT results, with seven cases being confirmed through histopathology. True negative results were detected in 11.7% of cases that remained disease-free during follow-up. A false positive result was observed in 3.7% of cases, where one patient showed a healed stress fracture with complete metabolic remission, initially regarded as a metastatic deposit. False negative results were found in 7.4% of cases, with PET/CT scan missed small peritoneal lesions in one case that were later detected through a second-look surgery. In another case, non-metabolically active pulmonary nodules were missed by PET/CT but detected through conventional CT and were regressed following chemotherapy treatment (considered metastatic). The statistical analysis has indicated that FDG-PET/CT possesses sensitivity, specificity, NPV, PPV and accuracy of 91.3%, 75%, 60.3%, 95.4% and 88.89%, respectively (Table 4). Among the 24 positive cases that were diagnosed with recurrent disease on PET/CT, 9 of them exhibited CA-125 levels within the normal range of <35 U/mL. CA-125 tumor markers have a sensitivity of 62.5% and an accuracy of 64%. Following the administration of chemotherapy to patients with positive PET/CT results, subsequent evaluation of treatment response was carried out through follow-up PET/CT and CA-125 tumor markers at 6 months for all cases and 1 year for 17 cases (table 5). Correlation between follow-up semiquantitative PET parameters and CA-125 levels was done to detect treatment response. The results showed that a significant association was only observed between CA-125 values and semiquantitative PET parameters (P < 0.05) in the third follow-up. Figure 1 provides a visual representation of these findings. The graph showed the correlation between the 3rd follow-up semiquantitative PET parameters (SUVmax) values and CA-125 tumor maker levels The PERCIST criteria were utilized to classify patients as either responders (partial metabolic response and complete metabolic response) or non-responders (stable metabolic disease and progressive metabolic disease) with a split of 22.2% and 66.6%, respectively, as illustrated in Table 6 and Figure 2. While results for CA-125 tumor marker responders were only 12.0% and non-responders were 28%, the rest was nonsignificant value change. The graph shows the treatment response detected by PET/CT FDG-PET/CT has proved to be effective in detecting and assessing treatment responses for patients with peritoneal metastasis, thereby guiding their management. This is supported by a significant P-value of 0.002, as depicted in Figures 3, 4, 5, 6. A 66-year-old female patient, having a history of ovarian mucinous carcinoma with positive peritoneal metastasis underwent TAH and BSO 2 years ago. CA-125 tumor marker level was within the normal range of about 16.9 U/ml. PET/CT was requested on a clinical base which revealed low-grade metabolically active abdomino-pelvic peritoneal and omental sheets and soft tissue lesions; associated with mild ascites. Non-contrast CT, fused PET/CT and PET images from up to down SUVmax of liver and of mediastinal blood pool = 2.5 and 2, respectively. a Large peritoneal lesion was seen over the left lumbar measuring 6.5 × 2.9 with SUVmax was 4.7 . b Other sizable deep pelvic hypodense peritoneal lesion SUVmax was 3.4 Follow-up PET/CT for the same case was done 6 months later after receiving chemotherapy showed a favorable therapeutic response, tumor marker was also declining compared to her baseline level CA-125 = 7.2 U/ml. Non-contrast CT, fused PET/CT and PET images from up to down with SUVmax of liver and of mediastinal blood pool = 2.5 and 2, respectively, showed morphological and metabolic regressive course of abdomino-pelvic peritoneal and omental sheets and soft tissue lesions with 38.3% SUVmax decline of the hottest lesion, a the left lumbar lesion measured 3.9 × 2.2 cm with estimated SUVmax = 2.9 b while SUVmax on deep pelvic lesion 2.9 A 62-year-old female patient, having a history of TAH and BSO 3 years ago for a pathologically proven case of ovarian endometrioid grade II carcinoma with positive peritoneal metastasis. The patient presented with an elevated CA-125 tumor marker 150.4 U/ml. A total body PET/CT was requested. Contrast CT, fused PET/CT and PET images from up to down SUVmax of liver and mediastinal blood pool = 3 and 2.5, respectively, revealed metabolically active abdominal abdomino-pelvic peritoneal and omental sheets and nodules associated with moderate ascites. a Largest omento-peritoneal lesion seen deep to the anterior abdominal wall measuring 2.7cm with SUVmax = 6.1 and b other one seen along peritoneal reflection of the right side of pelvis measuring 19mm with SUVmax = 6.7 Follow-up PET/CT was done 6 months after receiving chemotherapy and showed poor therapeutic response. Tumor markers showed further increased levels of CA-125 by 205 U/ml. Non-contrast CT, fused PET/CT and PET images with SUVmax of liver and mediastinal blood pool = 3 and 2.5 showed morphological and metabolic progressive course marked by an increase of 38.8% of the hottest lesion. a Peri-hepatic abdomino-pelvic peritoneal (b) and omental nodules and sheets; associated with moderate ascites; c the most sizable lesion seen underneath the anterior abdominal wall at the level of iliac crest measuring 3.8 cm with SUVmax = 9.3 FDG-PET/CT was also able to detect, follow up and evaluate the treatment response guiding the patient management in cases with nodal and distant metastasis with significant P values =0.012 and 0.025 for nodal and distant metastatic disease, respectively (Figures 7, 8, 9, 10, 11). A 59-year-old female patient, presented with a history of ovarian cancer treated with surgery pathologically proven high-grade serous carcinoma followed by postoperative chemotherapy remained disease free for 3 years. Suspected local recurrence was detected by conventional imaging despite a normal CA-125 tumor marker level (13.5 U/ml). A second-look laparotomy and excision was done for multiple deposits at the anterior abdominal wall scar, vagina, lymph nodes, serosal deposits allover bowel and ileocecal; all confirmed by histopathology. A whole-body PET/CT was requested for re-evaluation, SUVmax of liver and mediastinal blood pool = 2.3 and 1.9, respectively. a Non-contrast CT, fused PET/CT and PET images showed residual multiple abdomino-pelvic serosal and peritoneal deposits. b PET images showed a metabolically active most sizable peritoneal lesion at the gastro splenic region measuring 5 cm achieving SUVmax of 11.8. c Metabolically active left inguinal LN measuring 1.7 cm achieving SUVmax of 2.9 The patient started chemotherapy and a further follow-up PET/CT assessment was done 6 months later and showed poor therapeutic response with increased metabolic activity by 37.2% of the hottest lesion. A low-dose CT, fused PET/CT and PET images with SUVmax of liver and mediastinal blood pool = 2.3 and 1.9, respectively. a Newly developed multiple metabolically active pelvic peritoneal metastatic nodules. b Increased number and metabolic activity of metastatic pelvic lymph nodes largest right external iliac 3.7 cm with SUVmax 16.2. c and d Patient also developed supra-diaphragmatic metabolically active mediastinal nodal lesions and low-grade metabolically active pulmonary nodules, note also peri-hepatic peritoneal nodule at last PET image The patient changed the line of chemotherapy treatment and a further follow-up PET/CT was done which showed a good response to therapy. Coronal PET images showed a pre-treatment multiple supra- and infra-diaphragmatic metabolically active metastatic lesions and b a remarkable regressive course of supra- and infra-diaphragmatic metabolically active metastatic lesions A 64-year-old female patient, presented with a history of ovarian cancer treated with surgery pathologically proven high-grade serous carcinoma followed by postoperative chemotherapy remained disease free for 1 year. Axillary lump and excisional biopsy were done one year later pathologically proven metastatic carcinoma of ovarian origin. Persistent mildly elevated CA-125 tumor marker level (47.9 U/ml) was detected, so PET/CT was requested for re-assessment. Non-contrast CT, fused PET/CT and PET images with SUVmax of liver and of mediastinal blood pool = 2.9 and 2, respectively, showed metabolically active metastatic supra- and infra-diaphragmatic nodal metastatic lesions together with moderate ascites and low-grade metabolically active peritoneal, mesenteric and omental nodules and sheets. a Metabolically active right axillary lymph node. b Metabolically active upper abdominal lymph nodes. c Metabolically active omental soft tissue nodules underneath the anterior abdominal wall SUVmax = 4.7. d Metabolically active pelvic external iliac lymph node A 60-year-old female patient, presented with a history of ovarian cancer treated with surgery pathologically proven ovarian serous adenocarcinoma followed by postoperative chemotherapy remained disease free for 2 years. A recent elevated CA-125 tumor marker level (660U/ml) was detected, so PET/CT was requested for re-assessment. Contrast CT, fused PET/CT and PET images SUVmax of liver and of mediastinal blood pool = 2.7 and 2.2, respectively, showed multiple metabolically active metastatic lymph nodes in the following groups. a Bilateral supra- and infra-clavicular lymph node groups. b Left presternal, multiple superior mediastinal, carinal, infra-carinal and right hilar lymph nodes. The largest and most active one measures 3 × 3.5 cm with SUVmax = 11.8. c Multiple abdominal lymph nodes the most sizable one seen at porto-caval lymph node group measures 6.7 × 5.2cm with SUVmax = 9.6. d Moderate ascites with metabolically active herniated omental soft tissue sheets Discussion Around 70% to 80% of people who have been diagnosed with epithelial ovarian cancer will encounter a recurrence of the disease, even after undergoing the most effective surgical procedures and initial treatment . Traditionally, the CA-125 has been examined at regular intervals, although there has been controversy about the clinical efficacy of utilizing CA-125 progression alone as a trigger for beginning second-line chemotherapy. It is prudent to postpone treatment in asymptomatic patients with CA-125 advancement and small volume disease or no radiological indications of recurrence [13, 145 ")]. In this study, PET/CT was found to be a precise method for detecting suspected ovarian cancer recurrence, with positive scans observed in all cases with elevated CA-125 levels, and even in most patients with normal CA-125 levels. In our study PET/CT performed efficiently in very low-level change of CA-125, Additionally, FDG-PET/CT proved to be a much more sensitive and accurate method in the early detection of ovarian cancer recurrence compared to serum CA-125 tumor marker levels. FDG-PET/CT had a sensitivity of 91% and an accuracy of 88.8%, whereas the serum CA-125 tumor marker had a sensitivity and accuracy of only 63% and 64%, respectively. These findings are supported by similar studies conducted by Dragosavac , Gu , Khiewvan and colleagues, which all demonstrate the superior sensitivity of PET/CT over serum CA-125 (91.3% vs. 62%, 99% vs. 72% and 80% vs. 64%, respectively). In contrast to previous research findings, they noted that PET/CT demonstrated a lower level of sensitivity, measuring 58.2%. Furthermore, their study did not identify any statistically significant disparities in the diagnostic precision of FDG-PET, CT and the combined FDG-PET and CT modalities . Our study found that even though nine individuals had CA-125 levels within the normal range, disease recurrence was still present due to clinical suspicion, which was revealed by FDG-PET/CT. This led us to conclude that a normal CA-125 level should not be used as evidence to rule out tumor recurrence. Sun et al have mentioned that combining FDG-PET/CT imaging with serum CA-125 can complement and validate each other, significantly improving the diagnostic efficacy of ovarian cancer recurrence and metastasis . This provides a crucial reference for the early identification of ovarian cancer recurrence and metastasis. It has been stated that the transcoelomic route is the most common way of spreading and is responsible for peritoneal metastatic spread in ovarian cancer . Conversely, our study revealed that nodal recurrence, predominantly in the abdomino-pelvic region, was the most frequently encountered site of disease relapse, with a prevalence of 64%. The observations of Dragosavac and colleagues are consistent with our findings, as they also reported that lymph nodes were the primary site of recurrent disease in their study . The average SUVmax of the metastatic lesions was notably higher than the physiological uptake in various areas of recurrent disease at the initial assessment PET/CT scans. Through our research, we discovered that PET/CT can successfully detect metastatic lesions, even in small-sized lymph nodes guided by elevation of their metabolic activity, surpassing other imaging methods. This corresponds with the findings of Yuan Y et al, who highlighted that FDG-PET/CT is more precise than CT and MR imaging in detecting lymph node metastasis in ovarian cancer patients . Manganaro and colleagues have also reported that PET/CT imaging has demonstrated superior accuracy than other imaging techniques in the identification of small lymph node deposits . In our study, the peritoneal metastatic deposits were found to be the subsequent most common sites for recurrent illness with 60% prevalence. It was found that FDG-PET/CT can effectively evaluate peritoneal sheets and ascites, thereby distinguishing nodular peritoneal lesions from the gut. This, in turn, can prove beneficial in reducing the frequency of laparotomies, as pointed out by Rubini et al . PET/CT has been shown to have an increased likelihood of detecting metastatic lesions in the mesentery and peritoneum as previously reported . However, it is important to acknowledge that the presence of minute peritoneal seeding, measuring less than 5 mm, has been recognized as a significant contributing factor to the occurrence of false negative outcomes, as shown in our study. This highlights the importance of considering the limitations of imaging techniques, as peritoneal tumor spread may not always be detectable with these methods. The advantage provided by PET/CT is the ability to conduct functional imaging of the entire body in a single session. This capability enables the detection of concealed lesion sites. In our study, PET/CT was able to detect unexpected lesion sites like supra-diaphragmatic lymph node metastases, with a P value of 0.001 and accurately identified distant metastases in various organs such as the liver, spleen, lungs, pleura and bone marrow. In the context of post-chemotherapy management of recurrent cases, FDG-PET/CT was found to be effective in monitoring and assessing the response of different sites of disease relapse with a special emphasis on peritoneal metastasis, with a significant P value of 0.002. In alignment with the findings of previous research which stated that PET/CT is a better choice with the added benefit of being able to guide future treatments by using metabolic response data derived from SUV change of 20 and 55% from baseline after the first and third cycles, respectively, making FDG-PET an early predictor of treatment response . We found that FDG-PET/CT is more sensitive and accurate than serum CA-125 in monitoring the treatment response in the follow-up of recurrent cases. CA-125 is only closely related to PET/CT findings at the third follow-up. This supports previous research that reported a gap between PET response and CA-125 response . Highlighting the potential importance of PET as an early scanning tool for evaluating tumor response. The limitation of our study is the quite small size of the sample. Additionally, some of our cases were not confirmed through a second-look laparotomy and pathological examination owing to the advanced stage of the disease. Conclusions In our study, all cases of patients with elevated CA-125 levels displayed positive PET/CT findings. Furthermore, a majority of patients with normal CA-125 levels exhibited a positive PET/CT finding. As a result, we emphasize that PET/CT provides a comprehensive functional imaging approach that does not necessitate contrast injection, enabling accurate diagnosis and restaging of ovarian cancer recurrence, potentially influencing patient management. Additionally, PET/CT serves as a more dependable tool for detecting and monitoring therapy response than CA-125 tumor markers. Availability of data and materials The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. Abbreviations 18 Fluorine Area under the curve Estimated glomerular filtration rate Fluorodeoxyglucose National Academy of Clinical Biochemistry The National Comprehensive Cancer Network Positron emission tomography/computed tomography Standard deviation Maximum standardized uptake value References Kemppainen J, Hynninen J, Virtanen J, Seppänen M (2019) PET/CT for evaluation of ovarian cancer. Seminars in nuclear medicine. WB Saunders, pp 484–492 Google Scholar Nakamura K, Yoshikawa N, Mizuno Y, Ito M, Tanaka H, Mizuno M, Kajiyama H (2021) Preclinical verification of the efficacy and safety of aqueous plasma for ovarian cancer therapy. Cancers 13(5):1141 Article CAS PubMed PubMed Central Google Scholar Nougaret S, Addley HC, Colombo PE, Fujii S, Al Sharif SS, Tirumani SH, Reinhold C (2012) Ovarian carcinomatosis: how the radiologist can help plan the surgical approach. Radiographics 32(6):1775–1800 Article PubMed Google Scholar Parkash J, Bansro V, Chhabra GS, Mujahid Z (2023) Rising CA-125 (cancer antigen 125) levels: Cancer recurrence or a vaccine reaction? Cureus. Article PubMed PubMed Central Google Scholar Rubello D, Marzola MC, Colletti PM (2018) The prognostic value of 18F-FDG PET/CT in monitoring chemotherapy in ovarian cancer both at initial diagnosis and at recurrent disease. 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Eur J Nucl Med Mol Imaging 48:1998–2008 Article CAS PubMed Google Scholar Download references Acknowledgements We thank Dr. Mohamed Samy (associate professor of radiology at national cancer institute) for interpretation of FDG-PET/CT images. Funding The authors state that this work has not received any funding. Author information Authors and Affiliations Tahrir General Hospital, Giza Behind Poliomyelitis Institution, Egyptian Ministry of Health & Population, Nasouh Pasha St., Imbaba, 3854574, Giza, Egypt Ghada Ali Elsayed Faculty of Medicine, El-Demerdash Hospital, Ain Shams University, 56, Ramsis St., El Abbasia, 4390042, Cairo, Egypt Randa Hussien Abdullah, Remon Zaher Elia & Khaled Sayed Ahmed Search author on:PubMed Google Scholar Search author on:PubMed Google Scholar Search author on:PubMed Google Scholar Search author on:PubMed Google Scholar Contributions RZE and RHA reviewed the images. GAE and KSA analyzed and interpreted the patient data. GAE wrote the manuscript and RHA reviewed it. All authors have read and approved the manuscript. Corresponding author Correspondence to Ghada Ali Elsayed. Ethics declarations Ethics approval and consent for participation Approval of the ethical committee of the ‘Radiology department, Faculty of Medicine, Ein-Shams University’ was granted before conducting this prospective study; Reference number: not applicable. Local institutional review board approval was granted before conducting this prospective study, and written informed consent was obtained from all patients. Consent for publication All patients included in this research gave written informed consent to publish the data contained within this study. If the patients were less than 16 years old, deceased or unconscious when consent for publication was requested, written informed consent for the publication of these data was given by their parents or legal guardians. Competing interests The authors declare that they have no competing interests. Additional information Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Reprints and permissions About this article Cite this article Elsayed, G.A., Abdullah, R.H., Elia, R.Z. et al. Role of 18F-fluorodeoxyglucose positron emission tomography/computed tomography in the detection of recurrence and peritoneal metastasis from ovarian cancer in correlation with cancer antigen-125 tumor marker levels. Egypt J Radiol Nucl Med 55, 9 (2024). Download citation Received: 30 August 2023 Accepted: 19 November 2023 Published: 03 January 2024 DOI: Share this article Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 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3856	https://www.mathworks.com/matlabcentral/answers/213028-equate-the-parameters-of-two-equations-in-symbolic	equate the parameters of two equations in symbolic - MATLAB Answers - MATLAB Central Skip to content MATLAB Answers MATLAB Help Center Community Learning Get MATLABMATLAB Sign In My Account My Community Profile Link License Sign Out Contact MathWorks Support Visit mathworks.com Search Answers Answers Help Center Answers MathWorks MATLAB Help Center MathWorks MATLAB Answers File Exchange Videos Online Training Blogs Cody MATLAB Drive ThingSpeak Bug Reports Community MATLAB Answers File Exchange Cody AI Chat Playground Discussions Contests Blogs More Communities Treasure Hunt People Community Advisors Virtual Badges About Home Ask Answer Browse MATLAB FAQs More Contributors Recent Activity Flagged Content Manage Spam Help You are now following this question You will see updates in your followed content feed. You may receive emails, depending on your communication preferences. equate the parameters of two equations in symbolic Follow 3 views (last 30 days) Show older comments Elricon 22 Apr 2015 Vote 0 Link × Direct link to this question Cancel Copy to Clipboard ⋮ Vote 0 Link × Direct link to this question Cancel Copy to Clipboard Commented:Star Strider on 22 Apr 2015 Accepted Answer:Star Strider Open in MATLAB Online Hello, I have two equations, one with known parameters and one with unknown. I would like to ask for a simple way to solve the system that equates the parameters for the two equations Theme Copy e.g. eq. 1 = (a2 - a1 - a3 + 1)x1 + (2a1 - a2 - 3)x2 + (3 - a1)x3 eq. 2 = 0.957x1 - 0.253x2 + 0.119x3 Of course in this simple case it is easy to see that a1=2.043, a2=1.339 and a3=0.177. One manual way that I have found to work is to create two sym vectors that v1 and v2 that contain the parameters of x1,x2 and x3 and then use the solve function Theme Copy p=solve(y==b) However this method solves the solution into a structure p.a1, p.a2 and p.a3 which is not convient (I would like to have it a vector) And second I have to manually create the sym vectors v1 and v2 which again is not convenient when I have to repeat the same thing for different equations of different orders. Any ideas to work around this problem? thank you in advance 0 Comments Show -2 older comments Hide -2 older comments Sign in to comment. Sign in to answer this question. Accepted Answer Star Strideron 22 Apr 2015 Vote 2 Link × Direct link to this answer Cancel Copy to Clipboard ⋮ Vote 2 Link × Direct link to this answer Cancel Copy to Clipboard Open in MATLAB Online You can easily create the vector yourself in the format you choose from these results: Theme Copy syms a1 a2 a3 x1 x2 x3 eq1 = (a2 - a1 - a3 + 1)x1 + (2a1 - a2 - 3)x2 + (3 - a1)x3; eq2 = 0.957x1 - 0.253x2 + 0.119x3; [C1,T] = coeffs(eq1, [x1, x2, x3]); [C2,T] = coeffs(eq2, [x1, x2, x3]); [a1, a2, a3] = solve(C1 == C2, [a1, a2, a3]) producing: Theme Copy a1 = 2881/1000 a2 = 603/200 a3 = 177/1000 0 Comments Show -2 older comments Hide -2 older comments Sign in to comment. More Answers (1) Elricon 22 Apr 2015 Vote 0 Link × Direct link to this answer Cancel Copy to Clipboard ⋮ Vote 0 Link × Direct link to this answer Cancel Copy to Clipboard Hi Star Strider, the coeffs function was what I was looking for! Thank you! 1 Comment Show -1 older comments Hide -1 older comments Star Strideron 22 Apr 2015 × Direct link to this comment Cancel Copy to Clipboard ⋮ Link× Direct link to this comment Cancel Copy to Clipboard My pleasure! Sign in to comment. Sign in to answer this question. FEATURED DISCUSSION ###### MATLAB EXPO 2025 Registration is Now Open! November 12 – 13, 2025 Registration is now open for MathWorks annual virtual event MATLAB EXPO 2025... Devin in MATLAB EXPO 0 View Post See Also MATLAB Answers vpasolve finds no solutions except the null solution 3 Answers Merging three table data or array data in column wise in alternative sequence 0 Answers solve many independent equation in a run 1 Answer Entire Website MultiParEig File Exchange [B1, B2, B3,... Bn] = repeat2combine(dim, A1, A2, A3... 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3857	https://www.physio-pedia.com/Diaphragmatic_Breathing_Exercises?lang=en	Diaphragmatic Breathing Exercises - Physiopedia Search Search Search Toggle navigation p Physiopedia p Physiopedia About News Contribute Courses PAI Resources Shop Contact Login p Physiopedia About News Contribute Courses PAI Resources Shop Contact p+ Contents Editors Categories Cite Contents loading... Editors loading... Categories loading... When refering to evidence in academic writing, you should always try to reference the primary (original) source. That is usually the journal article where the information was first stated. In most cases Physiopedia articles are a secondary source and so should not be used as references. Physiopedia articles are best used to find the original sources of information (see the references list at the bottom of the article). If you believe that this Physiopedia article is the primary source for the information you are refering to, you can use the button below to access a related citation statement. Cite article Diaphragmatic Breathing Exercises Jump to:navigation, search Original Editor- Ahmed EssamTop Contributors - Ahmed Essam, Stella Constantinides, Kim Jackson, Lilian Ashraf, Hing Long Yip, Khloud Shreif, Ewa Jaraczewska, Oyemi Sillo and Lucinda hampton [x] Contents 1 Introduction 2 Aims 3 Physiological Effect 4 Positioning 4.1 Supine position 4.2 Sitting position 5 Precautions 6 Indication 7 References Introduction[edit \| edit source] Diaphragmatic breathing is a type of breathing exercise that helps strengthen your diaphragm, an important muscle that helps you breathe as it represents 80% of breathing. This breathing exercise is also sometimes called belly breathing or abdominal breathing. . Two phases of breathing When the diaphragm is functioning effectively in its role as the primary muscle of inspiration, ventilation is efficient and the oxygen consumption of the muscles of ventilation is low during relaxed (tidal) breathing. When a patient relies substantially on the accessory muscles of inspiration, the mechanical work of breathing (oxygen consumption ) increases and the efficiency of ventilation decreases. Controlled breathing techniques, which emphasise diaphragmatic breathing are designed to improve the efficiency of ventilation, decrease the work of breathing, increase the excursion of the diaphragm, and improve gas exchange and oxygenation.Also breathing from the diaphragm generate intra-abdominal pressure for control of posture and you can accomplish slow respiration. Aims[edit \| edit source] To mobilize secretions and teach breathing control. To teach effective coughing and remove secretions. To teach relaxation. To teach postural awareness. To mobilize thorax and shoulder girdle. Physiological Effect[edit \| edit source] Diaphramatic breathing have many physiological effects. Its effect on the respiratory system is: Using the diaphragm consciously during respiration increases the lung capacity Improve the efficacy of oxygen ventilation through decreasing the respiratory rate and increasing the tidal volume Improving alveolar ventilation by reducing alveolar dead space and increasing the arterial oxygen saturation Improving blood oxygen levels Other physiological effects include: Activating the parasympathetic nervous activity while suppressing the sympathetic nervous activity Improving core muscle stability. Helps with relaxation, lowering the harmful effects of the stress hormone cortisol on your body. Increased efficiency of venous return Lower your blood pressure Coping with the symptoms of post-traumatic stress disorder (PTSD). Positioning[edit \| edit source] Supine position[edit \| edit source] Lie on your back on a flat surface (or in bed) with your knees bent. You can use a pillow under your head and your knees for support if that's more comfortable. Place one hand on your upper chest and the other on your belly, just below your rib cage. Breathe in slowly through your nose, letting the air in deeply, towards your lower belly. The hand on your chest should remain still, while the one on your belly should rise. Tighten your abdominal muscles and let them fall inward as you exhale through pursed lips. The hand on your belly should move down to its original position. You can also practice this sitting in a chair, with your knees bent and your shoulders, head, and neck relaxed. Practice for five to 10 minutes, several times a day if possible. Sitting position[edit \| edit source] Sit up straight in a chair lengthen the distance between your navel and sternum. Keep your shoulders relax. Keep the pelvis in neutral position (Sit on your sitting bones). Place your hands at either side of your lower ribs. Breath in slowly through your nose. As you inhale feel your ribs expanding outwards and upwards. During inhalation is generated expansion of the trunk in three directions front , sides and back. Breath out from your nose. As you exhale feel your lower ribs moving inwards . Precautions[edit \| edit source] Never allow a patient to force expiration. Expiration should be relaxed or lightly controlled. Forced expiration only increases turbulence in the airways, leading to bronchospasm and increased airway restriction. Do not allow a patient to take a highly prolonged expiration. This causes the patient to gasp with the next inspiration. The patient's breathing pattern then becomes irregular and insufficient. Do not allow the patient to initiate inspiration with accessory muscles and upper chest. Allow the patient to perform deep breathing for only three or four inspirations and expirations at a time to avoid hyperventilation. Indication[edit \| edit source] Post -operative pain Airway obstruction (COPD, asthma) Sleep apnea Atelectasis Restriction of breathing due to musculoskeletal abnormality or obesity Central nervous system deficit Neurological patient with muscle weakness. Surgical procedure such as thoracic or abdominal surgeries. References[edit \| edit source] ↑ 1.01.11.2Russo MA, Santarelli DM, O’Rourke D. The physiological effects of slow breathing in the healthy human. Breathe. 2017 Dec 1;13(4):298-309. ↑ 2.02.12.2Patrick Mckeown. The breathing cure. City of Publication: OxyAt Books, Year.2021 ↑ 3.03.1Zisi D, Chryssanthopoulos C, Nanas S, Philippou A. The effectiveness of the active cycle of breathing technique in patients with chronic respiratory diseases: A systematic review. Heart & Lung. 2022 May 1;53:89-98. ↑ 4.04.1Örün D, Karaca S, Arıkan Ş. The Effect of Breathing Exercise on Stress Hormones. BibTeXEndNoteRefManRefWorks ↑ 5.05.1Hamasaki H. Effects of diaphragmatic breathing on health: a narrative review. Medicines. 2020 Oct 15;7(10):65. ↑Hsu SL, Oda H, Shirahata S, Watanabe M, Sasaki M. Effects of core strength training on core stability. Journal of physical therapy science. 2018;30(8):1014-8. ↑Guitonneau J, Jouvion AX, Paul F, Trappier T, De Brier G, Thefenne L. Is Physiotherapy useful for post-traumatic stress disorder in military personnel?. Annals of Physical and Rehabilitation Medicine. 2017 Sep 1;60:e55. ↑Yokogawa M, Kurebayashi T, Ichimura T, Nishino M, Miaki H, Nakagawa T. Comparison of two instructions for deep breathing exercise: non-specific and diaphragmatic breathing. Journal of physical therapy science. 2018;30(4):614-8. BibTeXEndNoteRefManRefWorks ↑jivan sharma. Diaphragmatic Breathing Technique. Available from: [last accessed 3/4/2020] ↑Mendes LP, Moraes KS, Hoffman M, Vieira DS, Ribeiro-Samora GA, Lage SM, Britto RR, Parreira VF. Effects of diaphragmatic breathing with and without pursed-lips breathing in subjects with COPD. Respiratory care. 2019 Feb 1;64(2):136-44. ↑Courtney R. Breathing retraining in sleep apnoea: A review of approaches and potential mechanisms. Sleep and Breathing. 2020 Dec;24:1315-25. ↑Westerdahl E, Lindmark B, Eriksson T, Hedenstierna G, Tenling A. Deep-breathing exercises reduce atelectasis and improve pulmonary function after coronary artery bypass surgery. Chest. 2005 Nov 1;128(5):3482-8. ↑Chuter TA, Weissman C, Mathews DM, Starker PM. Diaphragmatic breathing maneuvers and movement of the diaphragm after cholecystectomy. Chest. 1990 May 1;97(5):1110-4. • Retrieved from " Categories: Exercise Therapy Respiratory Disease - Interventions Respiratory Get Top Tips Tuesday and The Latest Physiopedia updates Email Address [x] I give my consent to Physiopedia to be in touch with me via email using the information I have provided in this form for the purpose of news, updates and marketing. HP Yes please It's free, and you can unsubscribe any time. Privacy policy. Our Partners The content on or accessible through Physiopedia is for informational purposes only. Physiopedia is not a substitute for professional advice or expert medical services from a qualified healthcare provider. 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Cite article Breathing Exercises Jump to:navigation, search Original Editor - Lucinda hampton Top Contributors - Lucinda hampton , Prashna Singh , Stella Constantinides , Robin Amel , Kim Jackson , Samuel Winter , Jess Bell and Rishika Babburu Contents 1 Introduction 2 Diaphragmatic Breathing 3 PursedDiaphragmatic Breathing Exercises - Physiopedia . Cite article Diaphragmatic Breathing Exercises Jump to:navigation, search Original Editor - Ahmed Essam Top Contributors - Ahmed Essam , Stella Constantinides , Kim Jackson , Lilian Ashraf , Hing Long Yip , Khloud Shreif , Ewa Jaraczewska , Oyemi Sillo and Lucinda hampton Contents 1 Introduction 2 AimsSuperior Scapula and Cervicogenic Headaches - Physiopedia of Upper Trapezius 5.1 Posture 5.2 Improper Scapula Function 5.2.1 Shoulder Dyskinesia 5.2.2 Shoulder Hike 5.3 Cervical issues 6 Treatment 6.1 Diaphragmatic Breathing Retraining 6.2 Postural Training 6.3 Thoracic Manipulations 6.4 Exercise Therapy 6.4.1 Cueing 6.5 Addressing Antagonists 7 Summary 8Diaphragmatic hernia - Physiopedia and patient's health state. The main techniques of this area of intervention are incentive spirometry, active cycle of breathing, and thoracic expansion exercises. References[edit \| edit source] ↑ 1.0 1.1 1.2 Radiopedia Diaphragmatic Hernia Available: Related online courses Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction – Plus Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction Understand the impact of mechanical ventilation on diaphragm function and how to overcome it Start course 1-1.5 hours - - - - Powered by Physiopedia Course instructor Rina Pandya Dr. Rina is an excellent course presenter and has •The Effect of Posture on the Diaphragm – Plus The Effect of Posture on the Diaphragm Learn how positioning in ICU and poor posture can negatively impact breathing mechanics Start course 1-1.5 hours - - - - Powered by Physiopedia Course instructor Rina Pandya Dr. Rina is an excellent course presenter and has experience with a range of conditions •Introduction to Myofascial Pain – Plus . Diaphragm Anatomy and Differential Diagnosis 0.5-1 hour 719 Presented by Rina Pandya More info. Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction 1-1.5 hours 504 Presented by Rina Pandya More info. Femoroacetabular Impingement (FAI) 1.5-2 hours 695 Presented by Rina Pandya More info •Biomechanics of the Hip – Plus More info. Diaphragm Anatomy and Differential Diagnosis 0.5-1 hour 719 Presented by Rina Pandya More info. Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction 1-1.5 hours 504 Presented by Rina Pandya More info. Femoroacetabular Impingement (FAI) 1.5-2 hours 695 Presented •Anatomy of the Hip – Plus and Headache Webinar 1 hour 172 Presented by Rina Pandya More info. Diaphragm Anatomy and Differential Diagnosis 0.5-1 hour 719 Presented by Rina Pandya More info. Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction 1-1.5 hours 504 Presented by Rina Pandya More info. Femoroacetabular • Related online courses Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction – Plus Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction Understand the impact of mechanical ventilation on diaphragm function and how to overcome it Start course 1-1.5 hours - - - - Powered by Physiopedia Course instructor Rina Pandya Dr. Rina is an excellent course presenter and has • The Effect of Posture on the Diaphragm – Plus The Effect of Posture on the Diaphragm Learn how positioning in ICU and poor posture can negatively impact breathing mechanics Start course 1-1.5 hours - - - - Powered by Physiopedia Course instructor Rina Pandya Dr. Rina is an excellent course presenter and has experience with a range of conditions • Introduction to Myofascial Pain – Plus . Diaphragm Anatomy and Differential Diagnosis 0.5-1 hour 719 Presented by Rina Pandya More info. Diaphragmatic Breathing and Ventilator-Induced Diaphragmatic Dysfunction 1-1.5 hours 504 Presented by Rina Pandya More info. Femoroacetabular Impingement (FAI) 1.5-2 hours 695 Presented by Rina Pandya More info • Close suggested results
3858	https://puzzleaday.wordpress.com/2021/06/13/deleting-digits/	Deleting Digits \| Puzzle a Day Puzzle a Day A new puzzle, riddle, game or interesting tidbit posted every day MenuSkip to content Home About Search Search for: Deleting Digits June 13, 2021 April 28, 2021 / Puzzling You are presented with the number 12345678910111213….96979899100. This number has been formed by writing the first one hundred positive integers after one another. Your challenge is to delete 100 digits from this number with the goal of having the largest possible number remaining. Scroll down for a clue and further down for the answer. Clue:The lengthy number presented at the start contains 192 digits. Your challenge was to delete 100 of these digits leaving a 92 digit long number. In order to create the largest possible number, it is imperative to have as many digit 9s as possible at the front of the number. Scroll down for the answer. Answer:999997859606162…..979899100 The lengthy number presented at the start contains 192 digits. Your challenge was to delete 100 of these digits leaving a 92 digit long number. In order to create the largest possible number, it is imperative to have as many digit 9s as possible the front of the number. The first 9s that appear in the counting numbers are: 9, 19, 29, 39 and 49. By removing all the digits up to 49 except the 9s we are left with a number starting with 999995051…We can’t remove any more digits after the first five 9s to leave a 9 or 8 as the next digit. This is because we would have removed too many digits. We can however remove the digits between 50 and the ‘5’ from 57 to leave 7 as our 6 th digit. We now have the number 999997585960….. There is one more digit to remove to total 100 removed digits and it is clear we need to remove the ‘5’ from 58. This leaves us with the answer 999997859606162…..979899100 Share this: Click to share on X (Opens in new window)X Click to share on Facebook (Opens in new window)Facebook Like Loading... Related General Puzzle, maths, Number Puzzle puzzle, puzzles Post navigation ← Don’t Be Sour If You Can’t Solve This Puzzle Change A Letter → Leave a comment Cancel reply Δ My new book, The Encyclopedia of Science Oddities,is available on Amazon. It features the most mind-blowing facts in all of science—in an encyclopedic format. Check it out here. My book,The World's First Book of TRUE Lateral Thinking Puzzles, is available on Amazon. Every puzzle in the book actually happened! The puzzles are all new & each answer contains a fascinating, factual backstory. Check it out here. My book,The Ultimate Book of Geography Oddities, is now available on Amazon. Prepare to be taken on a whirlwind journey through more than 1,000 of the world’s most surprising, unbelievable and downright bizarre geographical curiosities. Check it out here. Follow Blog via Email Enter your email address to follow this blog and receive notifications of new posts by email. Email Address: Follow Recent Posts A Matchstick Puzzle A Curious Animal Sequence A Quick Geometry Puzzle Three Rebus Puzzles A Cube Puzzle Categories Chess Codes Do It Yourself games General Knowledge General Puzzle Geography How many… Illusion Interactive Job Interview Questions Language Lateral thinking Letter Puzzle Logic magic Matchstick maths Murder mystery My Creations Number Puzzle Number Tricks Only Connect Paradox Picture puzzle Raven's Matrices Rebus Riddles Science Spatial Tests Uncategorized Video wordle Blog at WordPress.com. Comment Reblog SubscribeSubscribed Puzzle a Day Join 316 other subscribers Sign me up Already have a WordPress.com account? Log in now. Puzzle a Day SubscribeSubscribed Sign up Log in Copy shortlink Report this content View post in Reader Manage subscriptions Collapse this bar Loading Comments... Write a Comment... Email (Required) Name (Required) Website %d Design a site like this with WordPress.com Get started
3859	https://education.ufl.edu/patterson/files/2020/05/ManyakBaumannManyak2018.pdf	289 Morphological Analysis Instruction in the Elementary Grades: Which Morphemes to Teach and How to Teach Them Patrick C. Manyak, James F. Baumann, Ann-Margaret Manyak Teaching the meanings of common affixes and steps for inferring the meanings of affixed words enhances students’ word learning and fosters their interest in and attention to words. T he students in Ann-Margaret’s (third author) third-grade class are mingling purposefully. They each have a word in their hand that fea-tures a common prefix, a common suffix, or both a prefix and a suffix, and they are trying to identify a group of peers whose words all share the same base word (e.g., redo, doable, overdo, undoable). When the students have found their groups, they sit down together and talk about each of the words, identify-ing the prefixes and suffixes and working out the word meanings. The conversations are lively and demonstrate the students’ knowledge of the af-fixes (“I have undoable. It has the prefix un-, so that means not, and the suffix -able, which means can. So, undoable means something that can’t be done”). After a few minutes, the groups quickly share their analysis of their words with the class, and then the class launches into a rousing game of Affix Jeopardy, a review activity that fosters intense group discus-sion of prompts such as “A team that doesn’t lose a single game is .” By the end of the game, the class has spent 30 minutes highly engaged in analyzing affixed words. The activities described in this opening vignette represent instruction in morphological analysis (MA). We view MA as the process of using affixes (prefixes and suffixes), base words, and word roots to infer the meanings of words. In the elementary grades, instruction in MA typically includes teach-ing students commonly occurring affixes and word roots and a strategy for using knowledge of these word parts to construct meanings for unfamiliar words. Considerable research has demonstrated that instruction in MA contributes to word recog-nition, spelling, and vocabulary knowledge (Ash & Baumann, 2017; Bowers, Kirby, & Deacon, 2010; Carlisle, 2010; Goodwin & Ahn, 2013). Furthermore, MA is an important dimension of vocabulary in-struction in the Common Core State Standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). In The Reading Teacher, several authors have described the importance of and general strategies for teaching MA (Baumann, Ware, & Edwards, 2007; Goodwin, Lipsky, & Ahn, 2012; Kieffer & Lesaux, 2007). In this article, we provide teachers with fur-ther support in this area by addressing which spe-cific affixes to teach in the upper elementary grades and describing a comprehensive approach for teach-ing affixes. In the following section, we discuss several key principles from research on MA that guided our de-velopment of a multidimensional approach to affix instruction. Next, we briefly discuss the research projects in which we refined and evaluated this FEATURE ARTICLE The Reading Teacher Vol. 72 No. 3 pp. 289–300 doi:10.1002/trtr.1713 © 2018 International Literacy Association Patrick C. Manyak is an associate professor of literacy education at the University of Wyoming, Laramie, USA; email pmanyak@uwyo.edu. James F. Baumann is the Chancellor’s Chair for Excellence in Literacy Education Emeritus at the University of Missouri, Columbia, USA; email baumannj@missouri.edu. Ann-Margaret Manyak is a third-grade teacher in the East Grand School District, Grand County, CO, USA; email amanyak@egsd.org. 290 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org approach with diverse elementary school students. We then describe the development of a list of affix-es (and a similar list of Latin and Greek word roots) that provides a potential scope and sequence for af-fix instruction in grades 3–5. Finally, we present the set of instructional activities that we designed to teach the meanings of affixes and the strategic use of affix knowledge to infer word meanings. Key Principles From Research on MA Instruction Literacy researchers have long rec-ognized the importance of mor-phology in students’ language and literacy development (Anglin, 1993; Nagy & Anderson, 1984; White, Power, & White, 1989). For ex-ample, Nagy and Anderson dem-onstrated that beginning in third grade, approximately 60% of words that students encounter in texts are constructed of derivational morphemes (affixes and roots). Highlighting the importance of morphologically derived words in vocabulary development, Nagy and Anderson stated, For each word learned there are more than three derived words with meanings recognizably re-lated to that of the base, and at least two of these involve fairly transparent relation-ships. This demonstrates that the ability to utilize morphological relatedness among words puts a stu-dent at a distinct advantage in dealing with unfamiliar words. (p. 323) More recent studies and reviews of research have continued to underscore the importance of MA and have provided general guidelines for effec-tive MA instruction (Baumann, Edwards, Boland, Olejnik, & Kame’enui 2003; Baumann et al., 2002; Bowers et al., 2010; Carlisle, 2010; Goodwin & Ahn, 2013; Goodwin et al., 2012; Kieffer & Lesaux, 2007; Nagy, Berninger, & Abbott, 2006). Further, sev-eral studies have focused specifically on affix in-struction (Baumann et al., 2002, 2003; Graves & Hammond, 1980; White, Power, & White, 1989). These studies have demonstrated that teaching affixes and the use of these affixes to construct word meanings can improve students’ knowledge of the taught affixes (White, Power, & White, 1989) and use of these affixes to infer the meanings of untaught vocabulary words (Baumann et al., 2002, 2003; Graves & Hammond, 1980). Our reading of previous re-search in the area of MA led us to identify three principles that informed our approach to teaching affixes. First, although our primary interest is in teach-ing MA to enhance students’ ability to infer word meanings, instruction in word parts also contributes to word reading (Carlisle, 2010). In particular, in-struction focused on using mor-phemes, the smallest meaning-ful units in words, to read words with multiple parts has proven effective with upper elementary students (Bowers et al., 2010; Goodwin & Ahn, 2013). Thus, we were conscious of the benefit of providing students with at least some guided practice in us-ing morphemes to read affixed words. Second, MA instruction should have several goals. Carlisle (2010) found that MA in-struction in research interven-tions typically addressed one or more of four different objectives: (1) awareness of the morphological structure of words; (2) meanings of specific affixes and roots; (3) analysis of how a word’s morphemes contribute to its meaning, grammatical function, or spelling; and (4) strategies for using MA to infer word meanings. Although our affix instruc-tion addressed all of these objectives, it focused on the meanings of high-utility affixes and a strategy for us-ing MA to infer word meanings. In addition, we also introduced unfamiliar words that contained the target affixes, thus expanding students’ general vocabulary. Third, affixes and base words differ in terms of their semantic transparency (Carlisle & Katz, 2006). In simple terms, the meaning of an affixed word can be more easily inferred (e.g., dishonest) or less eas-ily inferred (e.g., discard) from its parts. In designing lessons to introduce elementary students to MA, we PAUSE AND PONDER ■ How might the lists of target affixes and roots presented in this article cause you to rethink the affixes and roots that you select to teach? ■ How could the lists in this article help your school provide more systematic instruction in high-value affixes and roots? ■ How do you teach affixes? What are the strengths and weakness of this instruction? How could the instructional approaches described in this article enhance your current instruction? ■ If you are a primary-grade teacher, how have you responded to standards within the English Language Arts strand of the Common Core State Standards that specify knowledge and use of common affixes in grades K–2, and how might this article change your instruction in this area? 291 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org intentionally selected semantically transparent words for instruction, believing that these words would best reinforce students’ knowledge of the affixes and sup-port their use of MA to infer word meanings. Finally, in addition to using these specific insights from MA research, we also sought to apply gen-eral guidelines for effective literacy teaching (e.g., Pressley, 2006) when developing MA instruction. In particular, when designing lessons, we sought to bal-ance explicit instruction and highly participatory ac-tivities, foster student engagement, prompt students to engage in metalinguistic talk, and provide ongo-ing review of taught meanings and strategies. These principles have guided Patrick’s (first au-thor) and Jim’s (second author) research and devel-opment of affix instruction for over a decade. In the following section, we describe the two recent re-search projects in which we refined and tested the affix instruction we share in this article. Our Research on Teaching Morphemic Analysis Patrick, Jim, and colleagues conducted a large, feder-ally funded three-year research project involving the design and implementation of a multifaceted, com-prehensive vocabulary instructional program (MCVIP) in fourth- and fifth-grade classrooms, including sev-eral mixed English learner and native English speak-er classes (Baumann et al., 2013). MCVIP instruction focused on several key components of vocabulary instruction, including the teaching of word-learning strategies. As a part of this strategy instruction, the team drew on Jim’s prior research on MA (Baumann et al., 2002, 2003) to develop a set of explicit lessons for teaching the meanings of common affixes, Latin and Greek word roots, and a morphological strategy for inferring word meanings (Baumann, Edwards, Boland, & Font, 2012). To assess the MA instruction, the MCVIP team constructed the Morphemic Analysis Assessment (MAA), a 53-item test that assessed students’ abil-ity to segment words into individual morphemes, match taught affixes and roots to their meanings, and select the best meanings for low-frequency af-fixed words not included in MCVIP lessons. MCVIP MA instruction produced large, statistically signifi-cant pretest–posttest (fall to spring) gains on the MAA at each research site for each year of the study. At the conclusion of the MCVIP research, Patrick and Ann-Margaret, a third-grade teacher in a small town in Colorado, initiated the Vocabulary and Language Enhancement (VALE) project in Ann- Margaret’s classroom. As one part of this project, they further developed a multidimensional approach to teaching affixes. During the 2016–2017 school year, Patrick administered a 42-item version of the MAA (the original test without the items assessing Latin and Greek word roots) to Ann-Margaret’s students in September and in May. A paired-sample t-test indi-cated that pretest–posttest gains were statistically significant, with an accompanying extremely large effect size of 2.38 (Cohen’s d statistic; d > 0.8 is con-sidered large). In addition, a comparison between students who scored lower on the MAA pretest and those who scored higher on the MAA pretest indi-cated no significant differences between these two groups’ pretest–posttest growth. Thus, all of the stu-dents, regardless of their initial performance on the MAA, responded positively to the affix instruction. It is important to note that neither the MCVIP nor VALE projects included a control group. Thus, it is possible that other factors may have contributed to the students’ pretest–posttest growth on the MAA or that other forms of instruction may have resulted in even greater growth. However, given that the MAA as-sessed knowledge and skills that were closely aligned with MA instruction, we believe it is highly likely that this instruction contributed centrally to the students’ pretest–posttest gains. In addition, Patrick’s qualitative observations of the MA instruction documented con-sistently high student engagement in the lessons, a so-phisticated level of student discourse related to word parts and their meanings, and students’ enthusiasm for locating words that included the taught affixes. Finally, although we have no direct evidence that the gains that students in both projects made on the MAA contributed to more general outcomes such as accelerated growth in vocabulary knowledge, we believe that teaching students to break apart words by and build words with morphemes, master the meanings of common affixes, and analyze how af-fixes affect word meanings constitutes a valuable component of comprehensive vocabulary instruc-tion, one that prompts students to engage in careful analysis of words and provides them with tools to better infer meanings of unfamiliar words. Which Morphemic Elements to Teach Affixes Table 1 contains a list of affixes that we recommend teaching in grades 3–5. It contains 41 affixes, with 14, 16, and 11 listed for instruction in grades 3, 4, and 292 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org Table 1 Affixes for Instruction at Grades 3, 4, and 5 Family Grade 3 Grade 4 Grade 5 Meaning Example words Not prefixes dis- not, opposite dislike, disobey, disagree un- not, opposite unhappy, unlock, unafraid in- not, opposite incorrect, invisible, inappropriate im- not, opposite impossible, impolite, impatient non- not, opposite nonfiction, nonstop, nonliving il- not, opposite illegal, illogical, illegible ir- not, opposite irregular, irresponsible Position prefixes pre- before pretest, preheat, preschool post- after postgame, postwar, postseason mid- middle midnight, midday, midair inter- between intercity, interstate, interact intra- among intrastate, intracellular fore- before foresee, foretell, forewarn trans- across transatlantic, transnational, transplant Over/under prefixes over- more than, too much overheat, overwork, overpriced super- over, high, big, extreme superheat, superstar, supermarket under- low, too little undersea, underachiever, undercook sub- under, below subset, subtitle, subcommittee Against prefixes anti- against antifreeze, antiwar, antidiscrimination counter- against, opposite counterclockwise, counterargument Bad prefixes mis- bad, wrong misspell, misunderstand, misbehave mal- bad, wrong malnutrition, maltreat, malformed Number prefixes uni- one unicycle, unicolor, unicellular mono- one monorail, monotone, monoplane bi- two bicycle, biweekly, biplane tri- three tricycle, triangle, trimotor Other useful prefixes re- again, back rewrite, rebuild, rearrange de- take away, from deice, debug, defrost co- with, together coauthor, coequal More and most suffixes -er more of something taller, smarter, warmer -est most of something tallest, smartest, warmest Person who suffixes -er person who teacher, writer, banker -or person who sailor, actor, explorer -ist person who artist, guitarist, nutritionist -ee person who employee, trainee, attendee (continued) 293 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org 5, respectively. The list emanates from Jim’s prior research on teaching MA as a vocabulary-learning strategy to upper elementary students. The mor-phemes in this table result from several syntheses of existing affix lists and a more recent systematic analysis of which morphemes might be taught and when. We give details of the development of this list in Table 2. Family Grade 3 Grade 4 Grade 5 Meaning Example words Other useful suffixes -ful full of useful, joyful, cheerful -ness state or quality of weakness, illness, careless -ly like, full of clearly, costly, carefully -y like, full of lengthy, chilly, wealthy -less without hopeless, worthless, careless -able can be, worthy doable, workable, knowledgeable Table 1 Affixes for Instruction at Grades 3, 4, and 5 (continued) Table 2 Development of the Affix and Latin/Greek Root Lists Jim initially assembled the most commonly included affixes on lists prepared by noted scholars and organized them into affix families, which were used in two intervention studies with fourth and fifth graders (Baumann et al., 2002, 2003). He expanded the lists of high-frequency affixes (Baumann, Font, Edwards, & Boland, 2005) that were taught to fifth graders in a yearlong vocabulary study (Baumann et al., 2007). Later, he conducted a more thorough analysis that resulted in 35 morphemes being included in the word-learning strategies component of the MCVIP research (Baumann et al., 2013). For this article, Jim conducted a more systematic analysis, as follows: 1. Returning to the most noteworthy lists of morphemes scholars have recommended for instruction and adding potential candidates to the MCVIP list. This resulted in a more extensive list of 92 items (69 affixes and 23 word roots) that were potential candidates for instruction in grades 3–5. 2. Analyzing the 92 morphemes to (a) designate those that were on White, Sowell, and Yanagihara’s (1989) empirical list of the most frequently occurring affixes; (b) include frequency ranks for each affix and root from Becker, Dixon, and Anderson-Inman’s (1980) rank-ordered list of 6,531 morphographs (generally synonymous with morpheme), which they constructed by analyzing 25,782 words from school texts; (c) identify which affixes and roots Templeton (2004) recommended be taught in grades 3 and 4, grades 5 and 6, or grades 7+; and (d) listing the most frequently occurring example words for each morpheme (e.g., unhappy for the prefix un-, teacher for the suffix -er, television for the root tele). 3. Engaging in an analysis of a tabular display of the information from the preceding point for all 92 morphemes to determine which merited instruction, and if so, at which grade level. This analysis involved examining all data points and applying an empirical-theoretical-experiential analysis process, as per the following example. Not prefix family example: The prefix un- was on White, Sowell, and Yanagihara’s (1989) list, it had a rank of 9 on Becker et al.’s (1980) analysis, and Templeton (2004) recommended that it be taught to students in grades 3 and 4; thus, we list un- at grade 3. In comparison, the prefix im- was also on White, Sowell, and Yanagihara’s list and had Templeton’s grades 3–4 designation, but Becker et al. gave it a lower frequency rank (38), so we designate it for instruction in grade 4. The prefix il-, also on White, Sowell, and Yanagihara’s list, had a lower frequency yet (88), so we recommend it be taught in grade 5. Although the quantitative data were highly influential in judging which affixes should be taught and when, there were exceptions. For example, the prefix a- (meaning not or without) had a relatively high rank (25), but the potential instructional words that fit our strict definition were relatively low-frequency and not necessarily conceptually accessible for elementary students (e.g., asymptomatic, asepsis). Therefore, we do not recommend that this affix be taught in grades 3–5. 294 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org Jim’s primary objective when selecting affixes to teach was to identify those affixes that, when taught well, enabled students to infer the meanings of as many novel words containing the target affixes as possible. Jim used four criteria for selecting affixes for instruction: 1. Teach affixes that meet a strict definition of prefix or suffix. A strict definition (Stotsky, 1978) means that an affix must be attached to a base word, as in dislike, unfair, hopeless, and teacher. This excludes words that have absorbed or assimilated prefixes (e.g., accept, erase). 2. Teach affixes that have consistent, concrete meanings. For example, the prefix dis- consis-tently means not, and the suffix -ful typically means full of. 3. Teach affixes that have the highest frequen-cy. This gives students the potential to learn the greatest number of new words through instruction in MA. For example, teaching dis- and un- leads to learning scores of new words, whereas teaching the less frequent prefixes dys- and hypo- does not. 4. When possible, organize affixes into seman-tic groups or families. Grouping affixes into semantic categories provides students with a mnemonic for remembering the meanings of related affixes. We offer several guidelines and qualifications when selecting affixes to teach from Table 1. First, although we believe that our list of affixes provides an excellent starting point for planning affix in-struction in grades 3–5, the list includes only sug-gested affixes to teach. Therefore, teachers should make final decisions about what affixes to teach and when by drawing from their knowledge and experience, their local curriculum, and local or na-tional standards. Second, teachers should adjust the recommend-ed grade levels for instruction up or down as needed or even differentiate within a single class, given stu-dents’ developmental levels in related literacy skills such as spelling. For instance, Ann-Margaret’s third graders were high performing; therefore, she taught many of the affixes identified on our list as fourth- or fifth-grade targets. However, regardless of when affixes are initially taught, we consider it essential that teachers provide cumulative review and re-teaching of all target affixes. Third, note that our affix list does not include inflectional suffixes: plurals (e.g., cats, bushes) and tense inflections (e.g., walks, walked, walking). These are important aspects of MA, but we assume that these will have been taught previously in grades 1 and 2. Similarly, although we not address instruc-tion in compound words in this article, we recom-mend such instruction beginning in late first grade or in second grade. Fourth, given that the frequency of prefixed words in reading materials increases greatly in third grade (White, Power, & White, 1989), we chose grade 3 as a starting point for focused affix instruction and thus for our grade-level affix lists. However, the Common Core State Standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) identified knowl-edge of frequently occurring affixes as a standard for grades K–2, and clearly, students in these grades encounter affixed words and thus may benefit from instruction in common affixes. For affix instruction in grades K–2, we recommend selecting a small num-ber of target prefixes and suffixes that have concrete meanings and are present in common words with meanings accessible to younger students. For exam-ple, working with a team of second-grade teachers, Patrick selected the following 12 affixes for instruc-tion at the second-grade level: un-, dis-, and in- (not prefix family); over- and under- (place prefix family); mis- (bad prefix family); re- (other useful prefixes); -er and -est (more and most suffix family); -er, -or, and -ist (person who suffix family). These affixes have clear primary meanings and generate a num-ber of words with meanings that are accessible to most primary students (e.g., unhappy, dishonest, incor-rect, overcook, underwater, misbehave, redo, writer, actor, artist). With regard to instruction, we believe that the explicit instruction, guided practice, and engag-ing review strategies that we describe later in this article would be appropriate or could easily be mod-ified for the primary grades. Finally, our list of affixes is not exhaustive; in-stead, it represents those affixes that we believe are most appropriate for instruction in grades 3–5 and that will enable students to figure out the meanings of as many new words as possible (or re-inforce their understandings of affixed words that they know incompletely). In the More to Explore sidebar at the end of this article, we list resources that teachers can use to identify additional, low-er frequency affixes and word roots, should they wish to extend MA to additional morphemes. 295 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org Latin and Greek Word Roots Jim’s development of a systematic list of mor-phemes to teach in the upper elementary grades extended beyond affixes to Latin and Greek word roots. Although this article focuses on affix in-struction, we include his list of 22 Latin and Greek word roots for instruction (see Table 3). To select the roots for this list, Jim followed the four affix selection criteria outlined previously (and the spe-cific procedures described in Table 2). Thus, he selected the word roots listed in Table 3 based on frequency and consistent meanings; when pos-sible, he placed them into semantic families. Our experience and research (White, Power, & White, 1989) have suggested that formal instruction in roots is beneficial for students at and above grade 4, so we recommend that fourth- and fifth-grade teachers begin teaching the word roots listed as they continue with instruction in affixes. Although teaching word roots is similar to teaching affixes, we refer readers to additional sources for specific descriptions of word root instruction (Baumann et al., 2007, 2012). How to Teach Affixes We now describe and illustrate the four activi-ties that constituted the multidimensional VALE affix instruction. This instruction resulted from a three-year process in which Patrick and Ann- Margaret refined and enhanced the MCVIP MA lessons in her third-grade class. The majority of VALE affix instruction took place during an eight- week period. Ann-Margaret began each week with a PowerPoint lesson that provided an explic-it introduction to one of the affix families and an Table 3 Latin and Greek Word Roots for Instruction at Grades 4 and 5 Family Root Meaning Example words Look and light roots scope to look at telescope, microscope, kaleidoscope vis, vid to see or watch vision, video, visibility photo light photograph, photocell, photon Communication roots dict to speak or say predict, dictator, dictaphone script, scribe write scribble, transcribe, manuscript phon/phone sound telephone, headphone, symphony graph to write or draw biography, autograph, paragraph aud/audi to hear audience, audible, auditorium Build or break roots rupt break eruption, interrupt, bankrupt fract break fracture, fraction, refract struct build construct, structure, destruct Movement roots tract drag, pull tractor, subtract, distract mot, mov move motion, remote, demote port carry export, import, portable Other useful roots bio life biology, biofuel, symbiotic tele far telescope, television, telegram geo earth geology, geography, geode therm heat thermometer, thermostat, hypothermia micro small, tiny microscope, microwave, microchip astr star astronomy, astronaut, astrobiologist path(y) feeling, suffering sympathy, empathy, telepathy phobia fear zoophobia, hydrophobia, acrophobia 296 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org experience with the word-part strategy. Following each introductory lesson, she used three exten-sion and review activities for the remainder of the week. Explicit Instruction in Affixes and the Word-Part Strategy Each week of VALE affix instruction began with ex-plicit teaching of an affix family and guided prac-tice in using the word-part strategy to infer word meanings. We used a consistent six-step sequence of activities to introduce each prefix family and a set of similar activities to introduce the suffix fami-lies. Here, we illustrate these six steps using exam-ples from the first prefix family that Ann-Margaret taught, the not prefix family. 1. Introduction: Present and discuss a chart that includes each prefix, its meanings, and exam-ple words (see Figure 1). 2. Analyze words: Explain how the target affixes affect word meanings (“When you see a not family prefix, simply say not before the rest of the word. For example, when you see unhappy, you say not happy.”). Ask students to explain the meanings of a series of words containing the target affix (“What does incorrect mean? What does dishonest mean?”). 3. Examine affixed and pseudo-affixed words: Explain that some words that begin with the prefix letters do not actually contain the prefix: We know that unhappy begins with the prefix un-. We can test this by saying not before the base word and checking if what we say makes sense. In this case, “not happy” makes sense. The word uncle also begins with un-, but these are just the letters u-n and not a prefix. We know this because when we say “not cle,” it doesn’t make sense. Look at these two words (unkind and uniform) and test each of them. Which one has a not prefix? How do you know? 4. Practice building words: Present a slide that has a column of prefixes and a column of base words and ask students to build specific words using one of the prefixes and one of the base words. (“Who can use one of the prefixes and one of the base words to build a word that means not kind?”) 5. Quiz: Show a simple fill-in-the-blank quiz that prompts students to provide common words that include the target affixes. (“The mo-ment I broke the dishes, I wished I could .” [disappear]) 6. Collection challenge: Challenge students to find words that include the target affixes and add them to a wall chart, with the enticement that the class will play a game of Affix Jeopardy when students have added a certain number of words to the chart. The initial PowerPoint lesson also demonstrated how the use of morphemes can help a reader decode polysyllabic words. Ann-Margaret returned to this focus again in the suffix lessons, where students were prompted to put slashes between prefixes, bases, and suffixes in affixed words such as unfor-givable and then to use the morphemic elements to read the words. The second PowerPoint lesson in-troduced the word-part strategy (see Figure 2), and each subsequent lesson guided students in applying the four steps of this strategy to infer an unfamiliar word presented initially in the context of a sentence (for an example of the first three steps in this prac-tice, see Figure 3). Extension and Review Activities Following each week’s explicit introductory lesson, Ann-Margaret engaged students in three extension and review activities. These activities fostered ac-tive student participation, heightened the class’s in-terest in affixed words, and provided opportunities for students to apply MA to construct word mean-Figure 1 Slide Introducing the Not Prefix Family Note. The color figure can be viewed in the online version of this article at 297 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org ings. Here we describe these extension and review activities. Collecting Affixes. Each PowerPoint lesson that Ann- Margaret taught concluded with a challenge to find and chart words that included the week’s affixes. For example, the lesson on the place prefix family concluded with the following charge: Your challenge is to look and listen for words that be-gin with the place prefix family prefixes. If you read or hear one, then write it down and share it with Mrs. Manyak. If your word uses a place prefix, you will add the word to the chart and share it with the class. When the class has collected 12 words, we will be ready for more Affix Jeopardy! To start the affix collection, Ann-Margaret put up a poster-paper chart with columns for each of the week’s affixes (see Figure 4). When a student found a word that they believed included a target affix, they shared it with Ann-Margaret to confirm that it was an appropriate example. If so, the student added the word to the affix chart. Ann-Margaret then called the class’s attention to the word and discussed its meaning. The Collecting Affixes ac-tivity made the students aware of affixed words throughout the day and enabled Ann-Margaret to discuss meanings of new words containing the tar-get affixes. Word Family Grouping. Ann-Margaret used Word Family Grouping to establish teams for games of Affix Jeopardy. The activity prompted students to analyze the word parts in sets of related words and to use MA to construct word meanings. Word Family Grouping began with each student receiving a word card with a word belonging to one of four base-word families (for an example, see Table 4). Figure 2 Slide Outlining the Word-Part Strategy Note. The color figure can be viewed in the online version of this article at Figure 3 Guided Practice With the Word-Part Strategy Note. The color figure can be viewed in the online version of this article at Figure 4 Not Prefix Family Wall Chart Note. The color figure can be viewed in the online version of this article at Table 4 Families for Word Family Grouping happier unhappy happiness happiest unhappily redo undo doable overdo undoable overuse reuse useful useless unusable thoughtful thoughtless rethink unthinkable overthink 298 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org The students identified the base word in their word and found the other students whose words included the same base word. Once students had gathered in their groups, they identified the affix-es in their words and discussed each word’s mean-ing. Ann-Margaret then put a list of all of the words on the document camera and asked the groups to divide each of their words into prefix, base, and suffix. Following the groups’ directions, she put slashes between these elements (e.g., un\|do\|able). She also asked the group to explain the meaning of each word (e.g., “something that can’t be done”). The groups received 100 points for the upcoming Affix Jeopardy game if they correctly segment-ed each of their words and explained each word meaning. Patrick’s qualitative observations of Word Family Grouping indicated that students were highly en-gaged in peer teaching, metalinguistic talk, and thoughtful word analysis throughout the activity. Furthermore, the groups rarely stumbled when seg-menting words or explaining the word meanings to the rest of the class. Affix Jeopardy. During the period of affix instruc-tion, Ann-Margaret typically led the class in a game of Affix Jeopardy once a week. Affix Jeopardy constituted an engaging review of target affixes and affixed words. In Ann-Margaret’s class, stu-dents participated in four- or five-member teams. The teams took turns selecting a column and value on the Affix Jeopardy table (the bottom row counted for 100 points, the second-to-last row 200 points, and so on). Ann-Margaret revealed the prompt, and the teams briefly discussed their answer. Ann-Margaret established that she could call on any member of the team to respond, so the teams had to ensure that each member was prepared to answer. If a team did not answer cor-rectly, the other teams had a chance to respond. If none of the teams knew the word, Ann-Margaret would then give a hint, often providing the base word, and the groups would have another chance to guess. The Jeopardy games reviewed previously taught affixes. Thus, the first game focused exclusively on words that included the not prefix family. Table 5 shows a Jeopardy board used at the end of affix in-struction that reviews various affixes studied. Each of the Jeopardy boards included common words at the 100- and 200-point rows (e.g., unhappy, redo) and less familiar words at the higher point rows (e.g., ungrateful, precaution). Consequently, the games reviewed the target affixes using well-known ex-ample words and also introduced students to less familiar words that employed these affixes. The less familiar words often prompted students to en-gage in serious MA as they strove to work out an-swers by combining the target prefixes with a va-riety of relevant base words. Many of their guesses represented plausible nonwords (e.g., “precareful” for precaution). On such occasions, Ann-Margaret Table 5 Affix Jeopardy Board Other useful prefixes Not family prefixes Place family prefixes Prefix + suffix I can never say cinnamon correctly. I always ___ it. A solution to a problem that is not logical is . The football team was at the 50-yard line, or . Something that can’t be replaced is . When a sentence sounds awkward, good writers try to ___ it. I don’t like to wear formal clothes. I like to dress in an ___ way. To take caution ahead of time Something that just doesn’t help is . To spell something wrong Something that is not perfect The trip was four hours long, so after two hours, we were ___ there. Something that can be used again is . To remove ice from a car windshield Someone who doesn’t tell the truth is . After the game, the angry coach refused to give a ___ interview. The boy’s story about seeing an alien spaceship was . To behave in the wrong way I always want to be the first to ___ my presents on Christmas. To not get paid enough Someone who doesn’t show a lot of respect is . 299 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org highlighted the good thinking that the group had done and that their guess, although not an actual word, indeed called to mind the meaning of the word in question. Collectively, the PowerPoint lessons, Collecting Affixes, Word Family Grouping, and Affix Jeopardy took up relatively little class time over an eight-week period. However, through these activities, students received explicit instruction in affix meanings and guided practice with the word-part strategy, inde-pendently hunted for words with the target affixes, collaboratively analyzed affixed words, and encoun-tered unfamiliar words that included the target af-fixes. Student engagement remained high across these activities and, as we described previously, the students demonstrated tremendous growth on an assessment of MA. Conclusion Given that affixed words proliferate in reading texts beginning in third grade (Nagy & Anderson, 1984; White, Power, & White, 1989), teaching students the meanings of common affixes and the strategic use of affix knowledge to infer novel word meanings can play a role in students’ vocabulary development (Baumann et al., 2002, 2003, 2013). In this article, we shared a principle- and evidence-based list of tar-get affixes for grades 3–5, along with a set of Latin and Greek word roots to teach in grades 4 and 5. We also described a set of instructional activities, developed over several years of implementation and refinement, that produced robust learning and a high level of student engagement. Although affix instruction should not be the sole focus of MA in-struction in the elementary grades (Goodwin et al., 2012), we have found that teaching the meanings of commonly occurring affixes and a strategy for analyzing the meanings of affixed words promoted students’ interest in words and provided them tools for independent word learning. Thus, we encourage teachers to plan for and implement affix instruc-tion as a valuable component of comprehensive vocabulary instruction. NOTES Preparation of this article was partially supported by a grant (R305A090163) from the Institute of Education Sciences, U.S. Department of Education. Any opinions, findings, and con-clusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the Institute of Education Sciences or the U.S. Department of Education. REFERENCES Anglin, J.M. (1993). Vocabulary development: A morphological analysis. Monographs of the Society for Research in Child Development, 58(10, Serial No. 238). Ash, G.E., & Baumann, J.F. (2017). Vocabulary and reading comprehension: The nexus of meaning. In S.E. Israel (Ed.), Handbook of research on reading comprehension (2nd ed., pp. 377–405). New York, NY: Routledge. Baumann, J., Blachowicz, C., Bates, A., Cieply, C., Manyak, P., Peterson, H., … Graves, M. (2013). The development of a comprehensive vocabulary instruction program for nine- to eleven-year-old children using a design experiment approach. In T. Plomp & N. Nieveen (Eds.), Educational design research: Part B. Illustrative cases (pp. 23–47). Enschede, The Netherlands: SLO. Retrieved from nl/publications/edr/contents/ Baumann, J.F., Edwards, E.C., Boland, E., & Font, G. (2012). Teaching word-learning strategies. In E.J. Kame’enui & J.F. Baumann (Eds.), Vocabulary instruction: Research to practice (2nd ed., pp. 139–166). New York, NY: Guilford. Baumann, J.F., Edwards, E.C., Boland, E.M., Olejnik, S., & Kame’enui, E.J. (2003). Vocabulary tricks: Effects of instruction in morphology and context on fifth-grade students’ ability to derive and infer word meanings. American Educational Research Journal, 40(2), 447–494. 10.3102/00028312040002447 Baumann, J.F., Edwards, E.C., Font, G., Tereshinski, C.A., Kame’enui, E.J., & Olejnik, S. (2002). Teaching morphemic and contextual analysis to fifth-grade students. Reading Research Quarterly, 37(2), 150–176. Baumann, J.F., Font, G., Edwards, E.C., & Boland, E. (2005). Strategies for teaching middle-grade students to use word-TAKE ACTION! 1. If you teach grades 3–5, select a relevant prefix family from the grade-level list and prepare an explicit lesson that includes the using the six steps described in the article. If you teach grades K–2, select a small number of prefixes with concrete meanings and prepare an explicit lesson based on but modifying the six steps where appropriate. 2. Prepare a lesson that introduces the four steps in the word-part strategy and guides students in applying the steps to infer the meanings of less familiar affixed words (e.g., predawn, rearrange). 3. Teach the prefix family and word-part strategy lessons and conclude by challenging students to identify words that feature the target prefixes and add them to a class chart. 4. Prepare and play a game of Affix Jeopardy to review the taught prefixes and introduce students to a few challenging words containing those affixes. 5. Discuss with your colleagues the possibility of imple-menting consistent schoolwide MA instruction using the grade-level lists of affixes and roots and the teach-ing activities presented in this article. 300 Feature Article The Reading Teacher Vol. 72 No. 3 November/December 2018 literacyworldwide.org part and context clues to expand reading vocabulary. In E.H. Hiebert & M.L. Kamil (Eds.), Teaching and learning vocabulary: Bringing research to practice (pp. 179–205). Mahwah, NJ: Erlbaum. Baumann, J.F., Ware, D., & Edwards, E.C. (2007). “Bumping into spicy, tasty words that catch your tongue”: A formative experiment on vocabulary instruction. The Reading Teacher, 61(2), 108–122. Becker, W.C., Dixon, R., & Anderson-Inman, L. (1980). Morphographic and root word analysis of 26,000 high-frequency words (Technical Report No. 1980-1). Eugene: Follow Through Project, College of Education, University of Oregon. Bowers, P.N., Kirby, J.R., & Deacon, S.H. (2010). The effects of morphological instruction on literacy skills: A systematic review of the literature. Review of Educational Research, 80(2), 144–179. Carlisle, J.F. (2010). Effects of instruction in morphological awareness on literacy achievement: An integrative review. Reading Research Quarterly, 45(4), 464–487. 10.1598/RRQ.45.4.5 Carlisle, J.F., & Katz, L. (2006). Effects of word and morpheme familiarity on reading of derived words. Reading and Writing, 19(7), 669–693. Goodwin, A.P., & Ahn, S. (2013). A meta-analysis of morphological interventions in English: Effects on literacy outcomes for school-age children. Scientific Studies of Reading, 17(4), 257– 285. Goodwin, A., Lipsky, M., & Ahn, S. (2012). Word detectives: Using units of meaning to support literacy. The Reading Teacher, 65(7), 461–470. Graves, M.F., & Hammond, H.K. (1980). A validated procedure for teaching prefixes and its effect on students’ ability to assign meaning to novel words. In M.L. Kamil & A.J. Moe (Eds.), Perspectives on reading research and instruction: Twenty-ninth yearbook of the National Reading Conference (pp. 184–188). Washington, DC: National Reading Conference. Kieffer, M.J., & Lesaux, N. (2007). Breaking down words to build meaning: Morphology, vocabulary, and reading comprehension in the urban classroom. The Reading Teacher, 61(2), 134–144. Nagy, W.E., & Anderson, R.C. (1984). How many words are there in printed school English? Reading Research Quarterly, 19(3), 304–330. Nagy, W., Berninger, V.W., & Abbott, R.D. (2006). Contributions of morphology beyond phonology to literacy outcomes of upper elementary and middle school students. Journal of Educational Psychology, 98(1), 134–147. 10.1037/0022-0663.98.1.134 National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for English language arts and literacy in history/ social studies, science, and technical subjects. Washington, DC: Authors. Pressley, M. (2006). Reading instruction that works: The case for balanced teaching (3rd ed.). New York, NY: Guilford. Stotsky, S.L. (1978). Teaching prefixes in the elementary school. The Elementary School Journal, 78(4), 278–283. https:// doi.org/10.1086/461113 Templeton, S. (2004). The vocabulary–spelling connection: Orthographic development and morphological knowledge at the intermediate grades and beyond. In J.F. Baumann & E.J. Kame’enui (Eds.), Vocabulary instruction: Research to practice (pp. 118–138). New York, NY: Guilford. White, T.G., Power, M.A., & White, S. (1989). Morphological analysis: Implications for teaching and understanding vocabulary growth. Reading Research Quarterly, 24(3), 283– 304. White, T.G., Sowell, J., & Yanagihara, A. (1989). Teaching elementary students to use word-part clues. The Reading Teacher, 42(4), 302–308. MORE TO EXPLORE The following websites provide access to a variety of lists of prefixes, suffixes, and word roots for those interested in expanding instruction beyond the lists included in this article: ■ ■“English Language Roots” provided by PrefixSuffix. com: php?navblks=1011110 ■ ■Prefix, suffix, and root dictionaries by Eugene M. McCarthy: ■ ■“Common Prefixes, Suffixes, and Root Words” by Jessica DeForest: roots/gre_rts_afx1.htm ■ ■Dictionary of affixes by Michael Quinion: org
3860	https://www.moorparkcollege.edu/sites/moorparkcollege/files/media/pdf_document/2020/chem_1b_expt6.pdf	Name: _____ Section: ___ Chemistry M01B Laboratory Manual Page 34 Experiment #6 – Colorimetric Determination of Co+2 Although volumetric and gravimetric mass analyses are commonly used, spectroscopy is the technique most often used for modern chemical analysis. Spectroscopy is the study of electromagnetic radiation emitted or absorbed by a given chemical species. Since the quantity of radiation absorbed or emitted can be related to the quantity of the absorbing or emitting species present, this technique can be used for quantitative analysis. There are many spectroscopic techniques available from X-rays, ultraviolet, infrared, and visible light, to name a few. We will consider one form here which is based on the absorption around visible light. If a liquid is colored, it is because some component of the liquid absorbs visible light. In a solution, the greater the concentration of the light-absorbing substance, the more light absorbed and the more intense the color of the solution. The quantity of light absorbed by a substance can be measured using a spectrophotometer. The instrument consists of: (1) a source that emits wavelengths of light in a measurable region (i.e. visible light which has wavelengths 400 to 700 nm); (2) a monochromator which selects a given wavelength of light; (3) a sample holder for the solution being measured; and (4) a detector which compares the intensity of incident light Io to the intensity of light after it has passed through the sample I. When a beam of light passes through a substance, some of the energy is often absorbed by the substance. This causes a decrease in the intensity of the transmitted beam. The ratio I / Io is called the transmittance, T, a measure of the fraction of light that passes through the sample holder (or cuvette) which contains the absorbing solution. The amount of light absorbed by the solution is given by the absorbance, A, where: A = –log (I / Io) = –log T (1) Io I b Absorbing Solution of concentration c T = I/Io A = -log (I/Io) = -log T %T = 100% T The distance, b, the light travels through the solution (in cm) and the concentration, c, of the absorbing species (in mol / L) are represented in the schematic above. A beam of parallel radiation with an intensity is shown before (Io) and after (I) it has passed through a layer of solution with a measured thickness at a certain concentration. The Beer-Lambert law is the basis for using spectroscopy in quantitative analysis which relates absorbance (A) to the concentration of the absorbing solution (c) and the path length of the cuvette (b). That is: Name: ____ Section: ___ Chemistry M01B Laboratory Manual Page 35 A = ε b c (2) where ε is the molar absorptivity or the molar extinction coefficient (in L / mol·cm). Each pure substance has its own unique extinction coefficient. Note that during the experiment, the same cuvette should be used for all measurements. With the same cuvette, the path length (b) and the extinction coefficient (ε) remain constant. Therefore, we can mathematically say that ε b = k (constant). If we write the concentration (c) as M for molarity, our new equation becomes: A = k M (3) Once absorption values for different concentrations are obtained, a Beer’s law plot of absorbance (vertical axis) versus concentration (horizontal axis) is made. A best-fitting line of the data points is constructed, from which you can determine your equation in slope-intercept form A = (ε b) c + 0 or A = k M + 0. By measuring the path length of your cuvette, the extinction coefficient can then be calculated. In this experiment, there will be three basic tasks to accomplish using the spectrophotometer. First, students will collectively determine the wavelength at which 0.100 M Co(NO3)2 will absorb best. Next, a standard absorbance curve from which the extinction coefficient can be calculated will be constructed. Finally, an unknown Co(NO3)2 solution will be analyzed for concentration determination. Procedure Part I. Prepare a standard set of Co(NO3)2 solutions Using a buret containing the stock solution of 0.100 M Co(NO3)2 and a 10.00 mL volumetric flask, prepare the following solutions of Co2+ by dilution: 0.0800 M, 0.0600 M, 0.0400 M, 0.0200 M. Part II. Calibrate the Spectrovis If you are the second user of the Spectrovis, you may skip Part II of this procedure and start at Part III Step 1 using the solutions that you prepared in Part I. 1. Plug the USB cable from the Spectrovis into the PC. 2. Open Logger Pro on the computer. 3. Under Experiment choose Calibrate then Spectrometer 1. 4. Allow the 90 s warmup. 5. Fill the cuvette (up to about ¾) with the blank solution (blank is just deionized water for this experiment). Insert the cuvette into the Spectrovis. Make sure the clear sides are the ones in the light path. 6. Select Finish Calibration. 7. When OK becomes available, select it. 8. The machine is now calibrated and there is no need to re-calibrate unless you exit out of Logger Pro. Name: ____ Section: ___ Chemistry M01B Laboratory Manual Page 36 Part III. Determination of Maximum Absorbance Wavelength. To obtain a spectrum and determine λmax: 1. Discard the blank solution and use the same cuvette for the next steps. 2. Fill your cuvette (up to about ¾) with the stock 0.100 M Co2+ solution. 3. Insert the cuvette into the Spectrovis. 4. Select the green button on the computer screen to start collection. 5. Once the spectrum is stabilized (usually takes a second), click stop. 6. To determine the wavelength where absorbance is maximum (λmax), move your cursor over the peak and write down the λ (the reading should be down in the lower left hand corner). Part IV. Standard Calibration Curve 1. Click the Configure Spectrometer button . 2. Choose the Absorbance vs Concentration option. 3. ON the bottom drop down menu, change from single 10 nm into individual wavelengths. 4. Choose the λmax you determined from the previous step. 5. Click OK. 6. If one or two pop-up window/s appear/s, choose Erase and Continue AND/OR “No” on the next pop-up window. 7. Click collect (green button). 8. Fill (up to about ¾) your cuvette with the stock (0.100 M) Co2+ solution and insert the cuvette into the Spectrovis. 9. When the absorbance reading stabilizes (usually takes a second), click KEEP. 10. On the pop-up window, enter the concentration of the sample and click OK. DO NOT CLICK STOP. You will click STOP in step 13. 11. Discard the solution in the cuvette. 12. Repeat steps 8 through 11 for each of the standard solutions that you prepared. Also, repeat step 8 through 11 for deionized water (a 0.000 M solution of Co2+). 13. When you have measured ALL of these solutions, Click STOP to end data collection. 14. Click Linear Fit . If a pop-up window appears, choose the “latest run” option. 15. Write down the slope of the line (what is the slope equal to?). Part V. Determination of Unknown [Co2+] 1. Fill the cuvette with your unknown solution. 2. Insert the cuvette into the Spectrovis. 3. Click collect. If a pop-up window appears, choose “store latest run.” 4. Write down the absorbance (shown on the bottom left corner of the screen). 5. Calculate the concentration of your unknown solution using Beer’s Law. Name: _____ Section: ___ Chemistry M01B Laboratory Manual Page 37 Data and Calculations Wavelength of maximum absorbance is __. [Co+2] Volume of 0.100 M Co(NO3)2 Stock Solution ABSORBANCE 0.1000 M mL 0.0800 M mL 0.0600 M mL 0.0400 M mL 0.0200 M mL 0.0000 M mL UNKNOWN # ABSORBANCE Path length of the cuvette: ___ Using Microsoft Excel, plot a graph of absorbance (y) verses concentration (x). Using the graph plotted from your data and the path length of the cuvette, calculate the extinction coefficient. Extinction Coefficient: __ Solve for the concentration on your unknown solution… (a) [Co+2] (read from graph) (b) [Co+2] (calculate from line equation and slope value) SHOW CALCULATIONS: Name: ____ Section: ___ Chemistry M01B Laboratory Manual Page 38 Post-Lab Questions: Colorimetric of Co+2 1. Calculate the transmittance of a solution if its absorbance is 0.352. 2. Calculate the absorbance of a solution if the transmittance is 0.647. 3. The following absorbance values for four solutions with known MnO4– concentrations were measured using a spectrophotometer: Solution [MnO4–] Absorbance 1 0.700 x 10–4 M 0.175 2 1.00 x 10–4 M 0.250 3 2.00 x 10–4 M 0.500 4 3.50 x 10–4 M 0.875 A. Using Microsoft Excel, plot a graph of Absorbance vs. Concentration of MnO4–. Write the trendline linear equation from the plotted graph. B. Determine the slope of the graph and include its units. C. Determine the concentration of an unknown MnO4– sample whose absorbance is 0.780. Name: ______ Section: ___ Chemistry M01B Laboratory Manual Page 39 D. Using the graph paper, below, construct a graph of Absorbance vs Concentration of MnO4–. Draw a linear trendline and determine the equation of the line that you drew. How does this compare to the graph that you made using Excel?
3861	https://genchem.chem.umass.edu/chem112/112_Experiment_5.htm	\| \| \| Experiment 5 - Polyprotic Acids Determining Ka's Using pH Titration Curves \| \| \| \| \| \| Before going to Lab: 1. Check out the sample pre-lab quiz on main Web Site or:- Click Here 2. Either use your computer to follow the procedure or print one:- Click Here Bringing your computer to Lab See either Experiment 1, or 2 for guidelines on this. \| \| \| \| Introduction: pH Titration Curves 'Idealized': To date the equivalence point of an acid base reaction has been determined using an indicator. In this experiment we are going to monitor the changes in pH that occurs during the titration of a weak polyprotic acid with a strong base. At the equivalence point one should expect to see a dramatic change in pH as the solution goes from acidic to strongly basic. Depicted on the left is an idealized pH titration curve for a weak diprotic acid. The first thing that you should notice is that there are two regions where we see a significant pH change. These, if you wish, correspond to two separate titrations. Titration 1 is the reaction of the first proton with the base (in this case sodium hydroxide). H2X(aq) + NaOH(aq) = NaHX(aq) + H2O(l) The second titration corresponding to the reaction of the second proton with sodium hydroxide NaHX(aq) + NaOH(aq) = Na2X(aq) + H2O(l) So, in essence, titrations of a weak polyprotic acid with a strong mono protic base are a combination of a number of titrations depending on the number of acidic protons on the polyprotic acid. The overall reaction is the sum of the two titration's H2X(aq) + 2 NaOH = Na2X(aq) + 2 H2O(l) In determining the quantity of the acid or the molarity of the acid, we are normally just interested in the final equivalence point. In a pH titration plot, this is determined by finding the point of inflection on the final area where we see a significant rise in pH (This can be approximated by determining the midpoint.) However, this plot contains some other interesting features. First off, if we look at the area corresponding to the first titration, it should come as no surprise that its equivalence point corresponds to the addition of exactly 1/2 the volume of NaOH required to reach the final equivalence point. The real neat point comes at the 1/2 way point of each titration. Let us focus on the Titration 1. At the 1/2 way point, the concentration of H2X(aq) remaining in the solution is equal to 1/2 the initial concentration of H2X! The concentration of NaHX(aq) produced is also numerically equal to 1/2 the initial concentration of H2X! So what, you may ask. Let's focus for a moment on the acid equilibrium associated with the acid that we are dealing with in titration 1. H2X(aq) + H2O(l) ® HX- + H3O+ Ka = [H3O+][HX-]/[H2X] or written in another way [H3O+] = Ka{[H2X]/[HX-]} using the concentrations that we know for H2X and HX- (=NaHX) at the 1/2 way point we get [H3O+] = Ka{1/2[H2X]initial/[1/2H2X]initial} [H3O+] = Ka From the graph we can determine the pH at this point and since pH=-log10[H3O+], we can determine [H3O+] at this point and thus obtain the Ka for this equilibrium. Neat! Since this is a polyprotic acid, this corresponds to Ka1. Guess what you can determine from the pH at the midpoint of the second titration. This information can be used to help identify the acid in question since Ka for a large number of polyprotic acids are known. The first acid that you will be following today is citric acid which is an acid that falls into the idealized category. You should see three areas where the pH undergoes significant changes and should be able to determine the three Ka values for citric acid and compare the result to the three known values given below. H3C6H5O7(aq) + H2O(l) <=>H2C6H7O7- + H3O+ Ka1 = 7.4x10-3 @ 25oC H2C6H5O7- + H2O(l) <=>HC6H6O72- + H3O+ Ka2 = 1.7x10-5 @ 25oC HC6H5O72- + H2O(l) <=>C6H5O73- + H3O+ Ka3 = 4.0x10-7 @ 25oC pH Titration Curves 'The Real World': In reality, many polyprotic acids only show one discernable equivalence point! The vast majority of the time, this corresponds to final equilibrium. If this is the case then, all the other equivalence points can be determined by knowing what type of polyprotic acid one is dealing with, i.e., diprotic or tri protic. For a triprotic acid, the other two equivalence points should correspond to 1/3 and 2/3 of the volume of the base required to reach the final one and thus one can still determine the Ka values. Note, however, I did say the vast majority of the time. How one knows how to determine whether the observed equivalence point equals the removal of the final proton I leave for you to explore! [Hint: what is the pH at the 1/2 equivalence point of a titration of a polyprotic acid equal to] The second acid that you will be looking at in this lab is phosphoric acid, a triprotic acid whose Ka values are given below. H3PO4(aq) + H2O(l) <=>H2PO4- + H3O+ Ka1 = 7.5x10-3 @ 25oC H2PO4- + H2O(l) <=>HPO42- + H3O+ Ka2= 6.2x10-8 @ 25oC HPO42- + H2O(l) <=>PO43- + H3O+ Ka3= 3.6x10-13 @ 25oC Experimental Procedure: Your TA will also demonstrate the best set up for this experiment. 1. Using a graduated cylinder, place ~ 20mL of the ~0.02M citric acid into a small beaker. If necessary add distilled water such that the tip of the pH probe is covered. 2. Fill your buret with the ~0.02M NaOH solution. Record the exact molarity of this solution. Record the initial buret reading. Remember that this corresponds to 0.00mL of NaOH added. 3. Record the initial pH of the Citric acid. 4. Carefully add the NaOH recording the volume of NaOH required to effect a pH change of 0.2. Continue this process until the pH reaches 12. 5. Plot a graph of 'pH' versus 'Volume of NaOH" added and from this graph determine: 1. The Ka values for citric acid. 2. The exact concentration of the citric acid. 6. Repeat steps one through five using the ~0.02M phosphoric acid and determine the Ka values for phosphoric acid and the exact molarity of the phosphoric acid solution. In the discussion portion of your write up, be sure to address the correlation between the Ka values that you obtained and those given to you in this procedure. Address any unusual problems that you encountered. \|
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3863	https://physics.stackexchange.com/questions/574726/finding-constraint-relation-between-pulley	homework and exercises - Finding constraint relation between pulley - Physics Stack Exchange Join Physics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Finding constraint relation between pulley [closed] Ask Question Asked 5 years, 1 month ago Modified5 years, 1 month ago Viewed 339 times This question shows research effort; it is useful and clear -2 Save this question. Show activity on this post. Closed. This question is off-topic. It is not currently accepting answers. Homework-like questions and check-my-work questions are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. If you intend to modify your question, please read the links above carefully before editing. Note that answers with complete solutions may be deleted! Closed 5 years ago. Improve this question I have to find out the relation between F and M, if rope is pulled with constant velocity V. Please help, these types of multiple pulley messes up my constraint equation. Thanks. homework-and-exercises newtonian-mechanics constrained-dynamics Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Aug 21, 2020 at 14:14 Tony Stark 1,618 1 1 gold badge 8 8 silver badges 26 26 bronze badges asked Aug 21, 2020 at 13:20 RajakrRajakr 734 1 1 gold badge 5 5 silver badges 9 9 bronze badges 2 Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation.ACuriousMind –ACuriousMind♦ 2020-08-21 14:14:55 +00:00 Commented Aug 21, 2020 at 14:14 The weight of M is supported by four segments of rope, each with the same tension (assuming you can ignore the mass and friction of the pulleys). The two fixed pulleys on the right have no significant effect.R.W. Bird –R.W. Bird 2020-08-21 14:20:35 +00:00 Commented Aug 21, 2020 at 14:20 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. In constraint relations what I like to is to measure the distance of block and pulleys from the wall above - inertial frame. Next I try to correlate the distances to the length of the string which is constant. This means add or subtract the distances from each other until you get the length of the string . Further I like to use the fact that all points of a string must have the same acceleration - string is inextensible. Note : I am denoting the distance from letters corresponding to the first letter colours shown in figure. The constraint relation here will be: b + 2(r-g) + (b-g) y +(y-p) + (w-p) =length of string Note what I am doing I am simply going from left to right,taking one part of string at a time and using the distances of pulleys in some way or the other to calculate the length of that part of string. Further note that y and p are constant - fixed pulleys. Further you may differentiate these to find velocities and accelerations of pulleys. If you are not familiar with these problems,then try with simple problems first. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Aug 21, 2020 at 14:06 answered Aug 21, 2020 at 13:28 Tony StarkTony Stark 1,618 1 1 gold badge 8 8 silver badges 26 26 bronze badges 2 Sir I am familiar with it the method used for solving. I am confused with this problem Rajakr –Rajakr 2020-08-21 13:36:54 +00:00 Commented Aug 21, 2020 at 13:36 @Rajakr Think from my perspective. I understand that you find it confusing but I do not know what exactly do you find it confusing about it?Tony Stark –Tony Stark 2020-08-21 13:40:33 +00:00 Commented Aug 21, 2020 at 13:40 Add a comment\| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions homework-and-exercises newtonian-mechanics constrained-dynamics See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Related 0Problem in constraint equations 0Force acting on a massive pulley 1Mechanics Pulley system 2Finding the acceleration constraint of multiple pulleys 0What does v v actually represent in pulley problems? 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3864	https://brainly.com/question/12520414	[FREE] The sides AB , BC , and AC of \triangle ABC are tangent to a circle at points P , Q , and R - brainly.com 3 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +66,7k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +33,7k Ace exams faster, with practice that adapts to you Practice Worksheets +7,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified The sides A B, BC, and A C of △A BC are tangent to a circle at points P, Q, and R respectively. Find A P, PB, BQ, QC, CR, and R A if A B=10 cm, BC=12 cm, and C A=5 cm. 2 See answers Explain with Learning Companion NEW Asked by zainmakada • 04/20/2019 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 58723052 people 58M 4.3 10 Upload your school material for a more relevant answer AP = RA = 1.5cm BP=BQ=8.5 cm QC=CR=3.5cm Given : AB = 10 cm, BC = 12 cm, and CA = 5 cm. Explanation The diagram is attached below. We know that the tangent lines from a point are equal Let AP = RA = x BP=BQ=y QC=CR=z We know AB=10 AP+BP=10 x+y=10 BQ+QC=12 y+z=12 CR+RA=5 z+x=5 WE know that AB+BC+CA=10+12+5=27 A P+BP+BQ+QC+CR+R A=27 x+x+y+y+z+z=27 2(x+y+z)=27 x+y+z=13.5 we know that x+y=10, Replace it in above equation x+y+z=13.5 R e pl a ce x+y w i t h 10 10+z=13.5 z=3.5 x+y+z=13.5 y+z=12 x+12=13,5 x=1.5 x+y+z=13.5 z+x=5 5+y=13.5 y=8.5 Learn more : brainly.in/question/17213757 Answered by lisboa •4.1K answers•58.7M people helped Thanks 10 4.3 (15 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 58723052 people 58M 4.2 9 Calculus-Based Physics - Jeffrey W. Schnick Physics for K-12 - Sunil Singh Surveying and Mapping - Christian Tiberius Upload your school material for a more relevant answer In triangle ABC, with the given side lengths, the lengths of the segments are: AP = 1.5 cm, PB = 8.5 cm, BQ = 8.5 cm, QC = 3.5 cm, CR = 3.5 cm, RA = 1.5 cm. Explanation To find the lengths of segments AP, PB, BQ, QC, CR, and RA in triangle ABC with given tangent points to the incircle, we proceed as follows: Let: AP = x PB = y BQ = y QC = z CR = z RA = x From the problem, we know the side lengths: AB = 10 cm BC = 12 cm CA = 5 cm Using the relationship between the segments: From triangle AB: A P+PB=A B x+y=10 (1) From triangle BC: BQ+QC=BC y+z=12 (2) From triangle CA: CR+R A=C A z+x=5 (3) Now, we have three equations: x+y=10 y+z=12 z+x=5 We can solve these equations step by step: From (1), express y in terms of x: y=10−x Substitute y into (2): (10−x)+z=12 z=12−10+x z=2+x Substitute z into (3): (2+x)+x=5 2 x+2=5 2 x=5−2 2 x=3 x=1.5 Now substitute x back to find y and z: From (1): y=10−1.5=8.5 From the new z expression: z=2+1.5=3.5 Therefore, we have: AP = RA = x = 1.5 cm PB = BQ = y = 8.5 cm QC = CR = z = 3.5 cm In summary: AP = 1.5 cm PB = 8.5 cm BQ = 8.5 cm QC = 3.5 cm CR = 3.5 cm RA = 1.5 cm Examples & Evidence This problem is an application of the properties of tangent lines from a point to a circle, and using algebra to solve a system of equations that represent the relationship between the lengths of the triangle's sides and the segments where the tangents touch the circle. The solution uses the property that tangents drawn from an external point to a circle are equal, and sets up equations based on this principle, allowing for the calculation of each tangent length based on the given side lengths of the triangle. Thanks 9 4.2 (14 votes) Advertisement Community Answer This answer helped 1137177621 people 1137M 5.0 38 Answer: AP = RA = 1.5 cm PB = BQ = 8.5 cm QC = CR = 3.5 cm Step-by-step explanation: The distance from the vertex to the two nearest tangent points is the same. If we say the distance AP = RA = x, then PB = BQ = 10-x, and QC = CR = 5 -x. Since we know BQ +QC = 12 We can substitute the above expressions involving x to find what x is. (10 -x) +(5 -x) = 12 15 -2x = 12 x = (15 -12)/2 = 1.5 This tells us ... AP = RA = 1.5 cm PB = BQ = 8.5 cm QC = CR = 3.5 cm Answered by sqdancefan •61.9K answers•1.1B people helped Thanks 38 5.0 (37 votes) 1 Advertisement ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. New questions in Mathematics Solve the given radical equation. Check all proposed solutions. 3 x+28=x+6 For the quadratic function defined, (a) write the function in the form P(x)=a(x−h)2+k, (b) give the vertex of the parabola, and (c) graph the function. P(x)=x 2−6 x−3 Simplify.5⋅7 What are the additive and multiplicative inverses of h(x)=x−24? A. additive inverse: j(x)=x+24; multiplicative inverse: k(x)=x−24 1 B. additive inverse: j(x)=x−24 1; multiplicative inverse: k(x)=−x+24 C. additive inverse: j(x)=−x+24; multiplicative inverse: k(x)=x−24 1 D. additive inverse: j(x)=−x+24; multiplicative inverse: $k(x)=x+24 What expression is equivalent to x 5⋅x 5? 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3865	https://allen.in/maths/prism	NCERT SolutionsCBSE NotesCBSE Exam HomeMathsPrism Prism - 3D Shape Prisms are three-dimensional shapes of geometry with similar polygonal bases of different shapes and lateral surfaces, which are generally parallelograms or rectangles. Every prism has its own volume and surface area based on the number of edges in the bases of the prism. Here, we will explore different properties of these prisms along with some of the most important formulas, such as the volume of a prism or the area of a prism that can help in understanding this important geometrical shape and its uses in real life. 1.0Definition of Prism A prism is a polyhedron consisting of two identical, congruent, and parallel polygonal bases. The lateral faces linking the bases are parallelograms. The bases are arranged so that the prism's height is the perpendicular distance between them. The form of the polygonal bases classifies the prism into a triangular prism, rectangular prism, pentagonal prism, or hexagonal prism. 2.0Properties of a Prism All prisms possess several key properties common to every type, which include: Faces: A prism will always have two congruent, identical bases and a series of parallelogram-shaped lateral faces. The amount of lateral faces is equal to the number of sides on the polygonal base. Edges: The faces meet on the edges of a prism. Three types of edges occur on a prism: edges of lateral faces and bases. The total number of edges of a prism is 3n, given that n refers to the number of sides contained within the base polygon. Vertices: A prism has vertices where the edges intersect. The vertices are twice the number of vertices of the base polygon. Height: The height (or altitude) of the prism is defined as the perpendicular distance between its two bases. Basically, it is a measure of how tall a prism is. This height plays an important role in calculating the prism's volume and surface area. 3.0Types of Prisms As mentioned earlier, prisms come in various forms and shapes depending on the number of edges of the polygonal base. Here are some key types of these prisms: Triangular Prism: As the name suggests, the base of this type is triangular with 5 faces, of which 2 are triangular bases, and 3 are rectangular lateral faces. It has 9 edges and 6 vertices. The volume of any given triangular prism can be found by simply multiplying the area of the base and height of the prism. V= Base Area × Height Here, the Base is the triangle, so the base area may be found using the formula: Base Area =21× Base × Height The surface area of a triangular prism can be found by summing the areas of the two triangular bases and three rectangular lateral faces. Mathematically, it can be expressed as: Area =2× Base Area of Triangle + Perimeter of Triangle × Height of Prism Rectangular Prism (Cuboid): A rectangular prism has 6 faces (base and lateral sides), all in rectangular shape, giving it a cuboidal shape. It has 12 edges with 8 vertices. The volume of a rectangular prism is the same as the volume of a cuboid, meaning it is the product of the length(l), width(w) & height(h) of the prism. Mathematically, it can be written as: Volume of Rectangular Prism =l×w×h , The surface area of a rectangular prism may be calculated using the following formula: Lateral Surface area of Rectangular Prism =2lw+2lh+2wh Square Prism: Just like a rectangular prism, a square prism has 6 faces, of which 2 are square bases and 4 are rectangular lateral faces. It also has 12 edges and 8 vertices. Volume of a square prism can be calculated using the formula for the base area of the prism, i.e square (a2), which is: Volume of Square Prism = Base Area × Height , The surface area of a square prism is calculated with the help of the following formula: Surface Area = 2 × Base Area + Perimeter of Base × Height Trapezoidal Prism: A prism with bases in the shape of a trapezoid is considered a trapezoidal prism. It has 6 faces, with 2 trapezoidal bases and 4 rectangular lateral faces. The volume of a Trapezoidal prism is calculated using the formula of the base area of the prism, i.e a trapezium (½ × (a + b) × h1): Volume = Base Area × Height , The lateral surface area of a trapezoidal prism is calculated using the formula: Surface Area = 2 × Base Area + Sum of Areas of 4 Rectangular Faces Pentagonal Prism: This type of prism has a pentagon as its base with 7 faces (2 pentagonal bases and 5 rectangular lateral faces). It also contains 15 edges and 10 vertices. The volume of a pentagonal prism is the product of its base area and the perpendicular distance between these two bases, or the height. This can be expressed as: V= Base Area × Height Here, the base is pentagonal and can be calculated using the formula: Base Area of Pentagonal Base =41×5(5+25)×s2, The surface area of the pentagonal prism can be calculated by summing up the 2 pentagonal base areas and the area of the 5 rectangular lateral faces: A=2× Base Area + Perimeter of Triangle × Height of Prism Hexagonal Prism: In this prism, hexagon is present as the base with 8 faces (2 hexagonal bases and 6 rectangular lateral faces). It also has 18 edges & 12 vertices. The volume of a hexagonal prism is the product of the area of the base and the height: V= Base Area × Height The base is hexagonal in shape; hence, the base area is written as: Base Area =233×s2, The surface area of the hexagonal prism is the sum of 2 base areas and 6 rectangular faces. A=2× Base area + Perimeter of Triangle × Height of Prism 4.0Solved Examples Problem 1: A triangular prism has a triangular base where the base and the height of the triangle are 5cm & 12 cm. The height (or length) of the prism is 10 cm. Calculate the volume of the triangular prism. Solution: given the base and height of the triangle are 5cm and 12 cm, and the length or height of the prism is 10cm. Here, the base area for the triangular base is, Base Area =21×5×12=30 cm2 Now, the volume of the prism, V=30×10=300 cm3 Problem 2: A hexagonal prism has a side length of 4 cm for its hexagonal base and a height (length of the prism) of 15 cm. Calculate the volume of the hexagonal prism. Solution: Given the side of the hexagonal base is 4 cm and the length of the prism is 15cm, Base Area =233×s2 Base Area =233×42=243 cm2 Now, the volume for the given prism, V= Base Area × Height V=243×15=3603 cm3 Problem 3: A triangular prism has a base of 10 cm and a height of 6 cm for the triangular base. The height of the prism is 12 cm. If the two sides of the triangle are 8 cm and 10 cm, calculate the surface area of the triangular prism. Solution: Given that the base of the triangular base is 10cm, the height is 6cm. The other two sides are 8cm and 10cm. The length of the prism is 12cm. The Base area of the triangular base: Base Area =21× Base × Height Base Area =21×10×6=30 cm2 Now, the surface area of the prism: A=2× Base Area + Perimeter of Triangle × Height of Prism Perimeter of triangle = 8 + 10 + 10 = 28cm A=2×30+28×12=396 cm2 Table of Contents 1.0Definition of Prism 2.0Properties of a Prism 3.0Types of Prisms 4.0Solved Examples Frequently Asked Questions The shape of a prism is determined by the shape of its polygonal base. Yes, a prism can have any polygonal base, which means it can have as many sides as the polygon allows. The height of a prism is the perpendicular distance between its two bases, influencing both its volume and surface area. Prisms have edges that form the perimeter of the polygonal base and edges that connect the corresponding vertices of the two bases. Join ALLEN! (Session 2025 - 26) Choose class Choose your goal Preferred Mode Choose State
3866	https://math.stackexchange.com/questions/1781733/proof-that-the-period-of-sinx-is-2-pi	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Proof that the Period of $\sin(x)$ is $2\pi$. Ask Question Asked Modified 4 years, 5 months ago Viewed 6k times 8 $\begingroup$ As I was walking through campus today, I had an interesting question pop into my head: How can we prove that the period of $\tan(x)$ is $\pi$ rather than $2\pi$? The answer to this was extremely straightforward: We start off with $$\tan(x) = \tan(x + T) = {\tan(x) + \tan(T) \over 1 - \tan(x) \tan(T)}$$ to give us $$-\tan^2(x)\tan(T) = \tan(T)$$ $$0 = \tan(T) + \tan^2(x)\tan(T)$$ $$0 = \tan(T)[1 + \tan^2(x)]$$ $$\implies \tan(T) = 0\;\;\;\;\;\text{and}\;\;\;\;1 + \tan^2(x) = 0 \implies \text{No real solution for any $x\in\mathbb{R}$}$$ Which for $\tan(T) = 0 \implies {\sin(T) \over \cos(T)} = 0 \implies \sin(T) = 0$, we have $T = 0, \pi \implies T = \pi$ to show that the period of $\tan(x)$ is $\pi$ if we desire a nontrivial answer. But I got stuck trying to do the same with $\sin(x)$. I tried: $$\sin(x) = \sin(x + T) = \sin(x)\cos(T) + \sin(T)\cos(x)$$ $$\implies \sin(x)[1 - \cos(T)] = \sin(T)\cos(x)$$ $$\implies \tan(x) = {\sin(T) \over 1 - \cos(T)}$$ But I got stuck here. I'm not sure how to isolate a single trig function in terms of $T$. I Googled this proof, but everyone either uses Taylor Expansions, Euler's Formula, or calculus. But I'm looking for an argument I could present to someone with knowledge of trigonometry and no more. Any ideas? algebra-precalculus trigonometry proof-writing Share edited May 12, 2016 at 14:17 Decaf-MathDecaf-Math asked May 12, 2016 at 1:03 Decaf-MathDecaf-Math 4,74211 gold badge2626 silver badges4545 bronze badges $\endgroup$ 3 $\begingroup$ Hah. That's probably what's wrong. Thanks! $\endgroup$ Decaf-Math – Decaf-Math 2016-05-12 01:10:26 +00:00 Commented May 12, 2016 at 1:10 2 $\begingroup$ How are you defining $\sin(x)$ and $\cos(x)$? In your development of the periodicity of $\tan(x)$, you used knowledge of the tangent function at $\pi$ and presumably at all values less than $\pi$. So, it seems to be a circular argument. My suggestion is to start from a definition of the function. $\endgroup$ Mark Viola – Mark Viola 2016-05-12 01:12:18 +00:00 Commented May 12, 2016 at 1:12 $\begingroup$ If $\sin (x+T)=\sin x$ for ALL $x$, and $\cos T \ne 0,$ then from your last line, $\sin x$ is a constant function. A similar argument, if $\tan T\ne 0 $,applies in the second equation regarding the period of $\tan x.$ $\endgroup$ DanielWainfleet – DanielWainfleet 2016-05-12 02:13:20 +00:00 Commented May 12, 2016 at 2:13 Add a comment \| 4 Answers 4 Reset to default 5 $\begingroup$ If $T$ is such that for all $x\in \mathbb{R} $ we have $\sin(x+T) =\sin(x) $, then in particular, setting $x=0$, we have $$\sin T =0$$ So $T=k\pi$ with $k \in \mathbb{Z} $. To conclude, we then need to check that $\sin(x+2\pi) =\sin(x)$ using your formula above (and that $\pi$ is not a period, by plugging $x=-\pi/2$ for example). Share edited May 12, 2016 at 1:23 answered May 12, 2016 at 1:17 Joel CohenJoel Cohen 9,47411 gold badge3333 silver badges4242 bronze badges $\endgroup$ 1 $\begingroup$ I forgot that this was possible: We're assuming that it holds for any $x\in\mathbb{R}$, so we're allowed to choose $x=0$ in particular and solve for $T$ that way. It's not the first thing in my mind because it feels too similar to proof by example which isn't valid. $\endgroup$ Decaf-Math – Decaf-Math 2016-05-12 01:33:27 +00:00 Commented May 12, 2016 at 1:33 Add a comment \| 3 $\begingroup$ Use the sum formulas $$\sin(x+T)=\sin x \cos T+\cos x\sin T$$ and the fact that $\sin (2\pi)=0$ and $\cos (2\pi)=1$ gives you a period of $2\pi$. Conversely is $\sin (x+T)=\sin x$ for all $x$ then $\sin T=0$ so $T=k\pi$ and one can easily see that $k$ must be even. Share edited May 12, 2016 at 1:38 answered May 12, 2016 at 1:26 Rene SchipperusRene Schipperus 40.6k22 gold badges3838 silver badges8484 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ If $\sin(x+T)=\sin x,$ using Prosthaphaeresis Formula, $$2\sin\dfrac T2\cos\left(x+\dfrac T2\right)=0$$ as $\cos\left(x+\dfrac T2\right)$ is dependent on $x,$ $\cos\left(x+\dfrac T2\right)=0$ won't give a constant value of $T$ So, we need $\sin\dfrac T2=0\iff\dfrac T2=n\pi$ where $n$ is any integer $\implies T=?$ Share answered May 12, 2016 at 16:07 lab bhattacharjeelab bhattacharjee 280k2020 gold badges213213 silver badges337337 bronze badges $\endgroup$ Add a comment \| 0 $\begingroup$ From it follows that there must exist an integer $n_0$ such that $T = 2\cdot n_0 \cdot \pi$. Since $0, it follows that $0<2\cdot n_0 \cdot \pi < 2 \cdot \pi$, so $0. Thus, $n_0$ is an integer between $0$ and $1$, which is a contradiction. Share answered Apr 6, 2021 at 20:09 JeanJean 1 $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions algebra-precalculus trigonometry proof-writing See similar questions with these tags. 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3867	https://www.jogc.com/article/S1701-2163(17)30317-1/pdf	Neuroimaging Findings of Congenital Toxoplasmosis, Cytomegalovirus, and Zika Virus Infections: A Comparison of Three Cases - Journal of Obstetrics and Gynaecology Canada Skip to Main ContentSkip to Main Menu Login to your account Email/Username Your email address is a required field. E.g., j.smith@mail.com Password Show Your password is a required field. Forgot password? [x] Remember me Don’t have an account? Create a Free Account If you don't remember your password, you can reset it by entering your email address and clicking the Reset Password button. 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Ok Infectious DiseasesVolume 39, Issue 12p1150-1155 December 2017 Download Full Issue Download started Ok Neuroimaging Findings of Congenital Toxoplasmosis, Cytomegalovirus, and Zika Virus Infections: A Comparison of Three Cases Heron Werner, PhD Heron Werner, PhD Affiliations Department of Radiology, Clínica de Diagnóstico por Imagem (CDPI), Rio de Janeiro, Brazil Search for articles by this author 1 ∙ Pedro Daltro, PhD Pedro Daltro, PhD Affiliations Department of Radiology, Clínica de Diagnóstico por Imagem (CDPI), Rio de Janeiro, Brazil Search for articles by this author 1 ∙ Tatiana Fazecas, MD Tatiana Fazecas, MD Affiliations Department of Radiology, Clínica de Diagnóstico por Imagem (CDPI), Rio de Janeiro, Brazil Search for articles by this author 1 ∙ Mohammad Zare Mehrjardi, MD Mohammad Zare Mehrjardi, MD Affiliations Department of Radiology, Shohada Tajrish Hospital, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran Section of Neuroimaging, Division of Clinical Research, Climax Radiology Education Foundation, Tehran, Iran Search for articles by this author 2,3 ∙ Edward Araujo Júnior, PhD Edward Araujo Júnior, PhD Correspondence Corresponding Author: Dr. Edward Araujo Júnior, Department of Obstetrics, Paulista School of Medicine, Federal University of São Paulo (EPM-UNIFESP), São Paulo, Brazil araujojred@terra.com.br Affiliations Department of Obstetrics, Paulista School of Medicine, Federal University of São Paulo (EPM-UNIFESP), São Paulo, Brazil Search for articles by this author 4araujojred@terra.com.br Affiliations & Notes Article Info 1 Department of Radiology, Clínica de Diagnóstico por Imagem (CDPI), Rio de Janeiro, Brazil 2 Department of Radiology, Shohada Tajrish Hospital, School of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran 3 Section of Neuroimaging, Division of Clinical Research, Climax Radiology Education Foundation, Tehran, Iran 4 Department of Obstetrics, Paulista School of Medicine, Federal University of São Paulo (EPM-UNIFESP), São Paulo, Brazil Publication History: Received March 22, 2017; Accepted May 10, 2017; Published online August 2, 2017 Footnotes: Competing interests: None declared. DOI: 10.1016/j.jogc.2017.05.013 External LinkAlso available on ScienceDirect External Link Copyright: © 2017 The Society of Obstetricians and Gynaecologists of Canada/La Société des obstétriciens et gynécologues du Canada. Published by Elsevier Inc. All rights reserved. Get Access Outline Outline Abstract Key Words Abbreviations References Article metrics Related Articles Share Share Share on Email X Facebook LinkedIn Sina Weibo Add to Mendeley bluesky Add to my reading list More More Get Access Cite Share Share Share on Email X Facebook LinkedIn Sina Weibo Add to Mendeley Bluesky Add to my reading list Set Alert Get Rights Reprints Download Full Issue Download started Ok Previous articleNext article Show Outline Hide Outline Abstract Key Words Abbreviations References Article metrics Related Articles Abstract Résumé Abstract Abstract Objective Toxoplasmosis, cytomegalovirus (CMV), and Zika virus (ZIKV) are among the common infectious agents that may infect the fetuses vertically. Clinical presentations of these congenital infections overlap significantly, and it is usually impossible to determine the causative agent clinically. The objective was the comparison of neuroimaging findings in three fetuses who underwent intrauterine infection by toxoplasmosis, CMV, and ZIKV. Methods Three confirmed cases of congenital toxoplasmosis, CMV, and ZIKV infections were included in the study over 7 months prospectively. Prenatal ultrasound, fetal brain MRI, and postnatal neuroimaging (CT or MRI) were performed on all of the included cases and interpreted by an expert radiologist. Results The mean GA at the time of prenatal imaging was 34.5±3.5 weeks. The main neuroimaging findings in congenital toxoplasmosis were randomly distributed brain calcifications and ventricular dilatation on ultrasounds (US), as well as white matter signal change on fetal brain MRI. The main neuroimaging findings of congenital CMV infection included microcephaly, ventriculomegaly, and periventricular calcifications on US, as well as pachygyria revealed by fetal MRI. The case of congenital ZIKV infection showed microcephaly, ventriculomegaly, and periventricular calcifications on ultrasound, as well as brain atrophy and brain surface smoothness on fetal MRI. Conclusion Although the neuroimaging findings in congenital infections are not pathognomonic, in combination with the patient history may be suggestive of one of the infectious agents, which will guide the management strategy. Résumé Objectif Toxoplasma gondii, le cytomégalovirus (CMV) et le virus Zika (ZIKV) font partie des agents infectieux courants qui peuvent infecter les fœtus de façon verticale. Les signes cliniques de ces infections congénitales se chevauchent de façon importante, d'où l'impossibilité, la plupart du temps, de déterminer l'agent en cause par des méthodes cliniques. L'objectif était de comparer les résultats de neuroimagerie de trois fœtus ayant contracté la toxoplasmose, le CMV et le ZIKV en milieu intra-utérin. Méthodologie Trois cas confirmés d'infection congénitale de Toxoplasma gondii, de CMV et de ZIKV ont été étudiés dans le cadre de cette étude prospective de sept mois. Chacun des sujets a subi une échographie prénatale, une IRM du cerveau fœtal et une neuroimagerie postnatale (TDM ou IRM), et les résultats ont été interprétés par un radiologiste expert. Résultats L'AG moyen au moment de l'imagerie prénatale était de 34,5 semaines ± 3,5 semaines. Le cas de toxoplasmose congénitale présentait principalement des calcifications cérébrales aléatoirement distribuées et une dilatation ventriculaire à l'échographie, ainsi qu'une variation du signal de la substance blanche du cerveau fœtal à l'IRM. Le cas d'infection congénitale au CMV présentait notamment une microcéphalie, une ventriculomégalie et des calcifications périventriculaires à l'échographie, ainsi qu'une pachygyrie à l'IRM fœtale. Le cas d'infection congénitale au ZIKV, quant à lui, présentait une microcéphalie, une ventriculomégalie et des calcifications périventriculaires à l'échographie, ainsi qu'une atrophie cérébrale et un cerveau lisse à l'IRM fœtale. Conclusion Même si les résultats de neuroimagerie dans ces cas d'infections congénitales ne sont pas pathognomoniques, leur association aux antécédents du patient peut donner une idée de l'agent infectieux en cause, ce qui guidera la prise en charge. Key Words Toxoplasmosis cytomegalovirus Zika virus ultrasound magnetic resonance imaging Abbreviations CMV (cytomegalovirus) ZIKV (Zika virus) CTX (congenital toxoplasmosis) CCID (congenital cytomegalic inclusion disease) CZS (congenital Zika syndrome) US (ultrasound) Get full text access Log in, subscribe or purchase for full access. Get Access References 1. Malinger, G. ∙ Werner, H. ∙ Rodriguez Leonel, J.C. ... Prenatal brain imaging in congenital toxoplasmosis Prenat Diagn. 2011; 31:881-886 Crossref Scopus (55) PubMed Google Scholar 2. van der Knaap, M.S. ∙ Vermeulen, G. ∙ Barkhof, F. ... Pattern of white matter abnormalities at MR imaging: use of polymerase chain reaction testing of Guthrie cards to link pattern with congenital cytomegalovirus infection Radiology. 2004; 230:529-536 Crossref Scopus (162) PubMed Google Scholar 3. Malinger, G. ∙ Lev, D. ∙ Zahalka, N. ... Fetal cytomegalovirus infection of the brain: the spectrum of sonographic findings AJNR Am J Neuroradiol. 2003; 24:28-32 PubMed Google Scholar 4. Doneda, C. ∙ Parazzini, C. ∙ Righini, A. ... Early cerebral lesions in cytomegalovirus infection: prenatal MR imaging Radiology. 2010; 255:613-621 Crossref Scopus (98) PubMed Google Scholar 5. Nunes, M.L. ∙ Carlini, C.R. ∙ Marinowic, D. ... Microcephaly and Zika virus: a clinical and epidemiological analysis of the current outbreak in Brazil J Pediatr (Rio J). 2016; 92:230-240 Crossref Scopus (90) PubMed Google Scholar 6. Carvalho, F.H. ∙ Cordeiro, K.M. ∙ Peixoto, A.B. ... Associated ultrasonographic findings in fetuses with microcephaly because of suspected Zika virus (ZIKV) infection during pregnancy Prenat Diagn. 2016; 36:882-887 Crossref Scopus (63) PubMed Google Scholar 7. Zare Mehrjardi, M. ∙ Keshavarz, E. ∙ Poretti, A. ... Neuroimaging findings of Zika virus infection: a review article Jpn J Radiol. 2016; 34:765-770 Crossref Scopus (45) PubMed Google Scholar 8. Moore, C.A. ∙ Staples, J.E. ∙ Dobyns, W.B. ... Characterizing the Pattern of Anomalies in Congenital Zika Syndrome for Pediatric Clinicians JAMA Pediatr. 2017; 171:288-295 Crossref Scopus (719) PubMed Google Scholar 9. Zare Mehrjardi, M. ∙ Poretti, A. ∙ Huisman, T.A. ... Neuroimaging findings of congenital Zika virus infection: a pictorial essay Jpn J Radiol. 2017; 35:89-94 Crossref Scopus (41) PubMed Google Scholar 10. Zare Mehrjardi, M. Neuroimaging findings of Zika virus infection: Emphasis of the emerging global threat Jpn J Radiol. 2017; 35:87-88 Crossref Scopus (6) PubMed Google Scholar 11. Hohlfeld, P. ∙ MacAleese, J. ∙ Capella-Pavlovski, M. ... Fetal toxoplasmosis: ultrasonographic signs Ultrasound Obstet Gynecol. 1991; 1:241-244 Crossref Scopus (72) PubMed Google Scholar 12. Antsaklis, A. ∙ Daskalakis, G. ∙ Papantoniou, N. ... Prenatal diagnosis of congenital toxoplasmosis Prenat Diagn. 2002; 22:1107-1111 Crossref Scopus (46) PubMed Google Scholar 13. Werner, H. ∙ Sodré, D. ∙ Hygino, C. ... First-trimester intrauterine Zika virus infection and brain pathology: prenatal and postnatal neuroimaging findings Prenat Diagn. 2016; 36:785-789 Crossref Scopus (36) PubMed Google Scholar 14. Farkas, N. ∙ Lev, D. ∙ Schweiger, A. ... The importance of prenatal neuroimaging in prediction of developmental outcome of fetuses infected with cytomegalovirus Harefuah. 2010; 149:45-48 PubMed Google Scholar 15. Boppana, S.B. ∙ Fowler, K.B. ∙ Vaid, Y. ... Neuroradiographic findings in the newborn period and long-term outcome in children with symptomatic congenital cytomegalovirus infection Pediatrics. 1997; 99:409-414 Crossref Scopus (237) PubMed Google Scholar Figures (4)Figure Viewer Article metrics Related Articles Figure 1 Neuroimaging findings of a 33-week fetus with congenital toxoplasmosis. (A) Transabdominal ultrasound in the axial view showing cystic dilatation of the posterior horns of the lateral ventricles (∗), and the cortical nodular foci (arrow). (B) MRI T2-weighted in sagittal views demonstrating multiple annular lesions of varying size with heterogeneous signal intensity, predominantly hyperintense (arrows). Note diffusely distributed edema throughout the parenchyma. Note hepatomegaly (black arrow). (C) MRI T2-weighted (left) and T1-weighted with fat saturation (right) in axial view showing the supratentorial ventricular dilatation (). (D). Postnatal MRI showing the ventriculomegaly and diffuse hyperintensity throughout the parenchyma, representing myelination disorder or diffuse white matter edema. Figure 2 Neuroimaging findings of a 32-week fetus with congenital cytomegalovirus infection. (A) Transabdominal fetal head ultrasound (lest) with 3-D reconstruction in the rendering mode (right) showing microcephaly, and periventricular calcifications (arrows). (B) Transvaginal fetal US clearly demonstrating the periventricular calcifications (arrows). (C) Transabdominal US in the axial plane showing mild ventriculomegaly. An intraventricular septation is present, but it is difficult to be visualized (arrow). Note the abnormal development of the Sylvian fissure (arrowhead). (D) T2-weighted MRI in axial, coronal, and sagittal planes showing ventriculomegaly, abnormal development of the Sylvian fissure, and pachygyria (arrows). (E) Postnatal CT demonstrating ventricular dilatation and periventricular calcifications (arrow). Figure 3 Neuroimaging findings of a 37-week fetus with congenital Zika syndrome. (A) Transabdominal axial (left) and transvaginal sagittal (right) ultrasound showing microcephaly and parenchymal calcifications (arrow). (B) T2-weighted MRI in axial view showing microcephaly, cortical atrophy, relative smoothness of the brain surface (arrow), and asymmetric colpocephaly (). (C) T2-weighted MRI in coronal and sagittal views showing the smoothness of the brain surface (arrows). (D) 3-D sagittal reconstruction from head CT (left), and the corresponding 3-D printed model on thermoplastic acrylonitrile butadiene styrene (ABS) (right). Note collapsed appearance of the skull, and the redundant occipital fold (arrows). (E) T1-weighted MRI in axial view (left) showing multiple hyperintense foci (arrows) at the corticomedullary junction in correspondence to the calcifications (arrow) seeing on the CT axial image (right). H. 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3868	https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas?srsltid=AfmBOoqy2mFzG0dLQAPIucTqUBiuLzO5a9vQeuYij5X9c54FG0uhHqe9	Art of Problem Solving Vieta's Formulas - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Vieta's Formulas Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Vieta's Formulas In algebra, Vieta's formulas are a set of results that relate the coefficients of a polynomial to its roots. In particular, it states that the elementary symmetric polynomials of its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests/tournaments. Contents 1 Statement 2 Proof 3 Problems 3.1 Introductory 3.2 Intermediate 4 Advanced 5 See also Statement Let be any polynomial with complex coefficients with roots , and let be the elementary symmetric polynomial of the roots. Vieta’s formulas then state that This can be compactly summarized as for some such that . Proof Let all terms be defined as above. By the factor theorem, . We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients. When expanding the factorization of , each term is generated by a series of choices of whether to include or the negative root from every factor . Consider all the expanded terms of the polynomial with degree ; they are formed by multiplying a choice of negative roots, making the remaining choices in the product , and finally multiplying by the constant . Note that adding together every multiplied choice of negative roots yields . Thus, when we expand , the coefficient of is equal to . However, we defined the coefficient of to be . Thus, , or , which completes the proof. Problems Here are some problems with solutions that utilize Vieta's quadratic formulas: Introductory 2005 AMC 12B Problem 12 2007 AMC 12A Problem 21 2010 AMC 10A Problem 21 2003 AMC 10A Problem 18 2021 AMC 12A Problem 12 Intermediate 2017 AMC 12A Problem 23 2003 AIME II Problem 9 2008 AIME II Problem 7 2021 Fall AMC 12A Problem 23 2019 AIME I Problem 10 Advanced 2020 AIME I Problem 14 See also Polynomial Retrieved from " Categories: Algebra Polynomials Theorems Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
3869	https://math.stackexchange.com/questions/2220837/turning-a-solid-of-revolution-into-a-function-of-x-and-y	calculus - Turning a solid of revolution into a function of $x$ and $y$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Turning a solid of revolution into a function of x x and y y Ask Question Asked 8 years, 5 months ago Modified1 month ago Viewed 1k times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I have been exploring solids of revolutions. I am trying to find different ways of expressing them to calculate their areas and volumes. For example, if a revolve the function y=x 1 2 y=x 1 2 around the x x-axis, it will form a solid. I have therefore added another dimension, or variable. How would I define this solid in terms of a function of x x and y y that results in z z? This would allow to me integrate the multivalued function with iterated integrals. Thank You calculus solid-of-revolution Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Nov 8, 2017 at 22:39 Teddy38 3,367 2 2 gold badges 14 14 silver badges 33 33 bronze badges asked Apr 6, 2017 at 13:44 DiffusionDiffusion 5,921 1 1 gold badge 12 12 silver badges 31 31 bronze badges 1 1 Here's a MathJax tutorial :)Shaun –Shaun♦ 2017-04-06 14:11:23 +00:00 Commented Apr 6, 2017 at 14:11 Add a comment\| 5 Answers 5 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. convert meridian y=x−−√y=x to an assemblage of meridians x 2+y 2−−−−−−√=r=f(z)x 2+y 2=r=f(z) So to find corresponding volume from y=2+sin x y=2+sin⁡x change it to form r=2+sin z r=2+sin⁡z so that V=π∫z 2 z 1 r 2 d z=π∫r 2 r 1 r 2 d z V=π∫z 1 z 2 r 2 d z=π∫r 1 r 2 r 2 d z depending on whether you like to express r r in terms of z z or z z in terms of r r respectively. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Apr 9, 2017 at 11:34 NarasimhamNarasimham 42.5k 7 7 gold badges 46 46 silver badges 112 112 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. I think you want Pappus's (2 n d 2 n d) Centroid Theorem: the volume of a planar area of revolution is the product of the area A and the length of the path traced by its centroid R, i.e., 2πR. When composite areas are involved, the centroid is the weighted sum of the component centroids. The bottom line is that the volume is given simply by V=2 π R A V=2 π R A. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Apr 7, 2017 at 15:50 Cye WaldmanCye Waldman 8,236 2 2 gold badges 18 18 silver badges 37 37 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. I will solve your example using two variables and make some statements at the end about the general process. You start with y=x 1/2 y=x 1/2 and you rotate about the x x-axis. You do this in a way that the distance from the x x-axis to your graph is constant. Let's call this distance (the height of your graph) r r s.t. r=x 1/2 r=x 1/2. We now have a solid that has been rotated into the z z-plane, where every y z y z-plane contains the circular cross section of the solid. The area of a circle is π r 2 π r 2. This means the area of the circle that lies on the plane x=1 x=1 is π∗1 1/2 2=π π∗1 1/2 2=π. By repeating this process for every value of x x, we get the function A=π x 1/2 2=x π A=π x 1/2 2=x π s.t. A A represents the area of the circle associated with each value of x x. By integrating A A, you get the sum of the area under the curve (sum of areas of the circles), and hence the volume V V: V=∫x π d x V=∫x π d x Plugging in bounds will get you the volume of the solid between your choice of x x values. For this method, in general, you will want to: 1.) Identify the radius of the circle, making note of the axis of revolution. For example, if you rotated y=x 1/2 y=x 1/2 along the y y-axis, your radius would be y 2 y 2, obtained by arranging y y in terms of x x. 2.) Write out Area as a function of r r and the axis of revolution (reducing the problem to two variables). 3.) Integrate along the axis of revolution to find volume. I should note that this is only one method of finding the volume of solids of revolution, which is known as the disc method. There exists a slight modification of this method known as the washer method for hollow solids, and a shell method which sums the surface area of several cylinders. More complex methods, usually involving all three dimensions also exist. By request, one such way is as follows: Here is the method I think you want to know about We understand that when we rotate a graph about the x-axis, we get circles on the x z x z-plane of radius y=f(x)y=f(x). This graph of this circle can thus be denoted using r 2=y 2+z 2 r 2=y 2+z 2. But what does r r equal? In fact, r r here is still y=x 1/2 y=x 1/2 from the disc method, even though y y is now variable. This is because y=x 1/2 y=x 1/2 only when z=0 z=0. Now that we have our relations, we seek to find a order of integration to find the volume of the solid. I chose to again, first find the area of the circle and then integrate along the x x-axis. Let's consider how we would normally approach integrating a 2-dimensional function with height y y and length x x. Normally, we opt to define y y in terms of x x and integrate the function y=f(x)y=f(x) along the range of x x that we wish to find the area for. We can apply this same method to the y z y z plane circle. We know y 2+z 2=x 1/2 2 y 2+z 2=x 1/2 2. This is equivalent to y=±x−z 2−−−−−√y=±x−z 2. By the nature of the graph, we end up with two functions, and hence two integrals, which we will look at later. For now, we consider what range of z z we wish to integrate these functions along. This range is the range for which the circle is defined, or the range where x−z 2−−−−−√x−z 2 is real. This is: −r−r to +r+r, which is −x 1/2−x 1/2 to +x 1/2+x 1/2. We now have the following 2 integrals: ∫x 1/2−x 1/2 x−z 2−−−−−√d z∫−x 1/2 x 1/2 x−z 2 d z ∫x 1/2−x 1/2−x−z 2−−−−−√d z∫−x 1/2 x 1/2−x−z 2 d z Because the latter of the two is negative, the physical area of the circle is ∫x 1/2−x 1/2 x−z 2−−−−−√d z−∫x 1/2−x 1/2−x−z 2−−−−−√d z∫−x 1/2 x 1/2 x−z 2 d z−∫−x 1/2 x 1/2−x−z 2 d z which is equivalent to 2∫x 1/2−x 1/2 x−z 2−−−−−√d z 2∫−x 1/2 x 1/2 x−z 2 d z We have now found the area of the circle. Finally, we integrate this area along the range of x x for which we wish to find the volume of the solid of revolution. Moving the coefficient to the beginning of the expression, we get: V=2∫x 2 x 1∫x 1/2−x 1/2 x−z 2−−−−−√d z d x V=2∫x 1 x 2∫−x 1/2 x 1/2 x−z 2 d z d x where x 1 x 1 and x 2 x 2 are real constants. In this case, the d A d A you speak of is d z d x d z d x. Our integral can be generalized as ∬R f(x,z)d A∬R f(x,z)d A In fact, by rearranging the order of integration, you can change the variables that d A d A represents. This is because the most general formula of volume is V=∭S d V V=∭S d V Because the first integral is the integral of 1 1, this integral is usually left out. Theory-wise, however, this form is nice because it shows you that different equations for volume of the solid are really just rearranged orders of integration of this equation (often accompanied by simplifications done through geometrical observation or other tricks, such as Jacobian coordinate changes, but this is another topic). Apologies for misinterpreting your question the first time, and I hope this helped! EDIT: Added another method as requested by OP Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Apr 9, 2017 at 10:54 answered Apr 7, 2017 at 3:51 SmallFishSmallFish 85 1 1 silver badge 8 8 bronze badges 3 My answer is below: @UniS20 Diffusion –Diffusion 2017-04-07 15:42:48 +00:00 Commented Apr 7, 2017 at 15:42 Hey, I'll leave a new answer in a bit! Sorry about the late response, I've had a rough 2 days...SmallFish –SmallFish 2017-04-09 04:48:46 +00:00 Commented Apr 9, 2017 at 4:48 Sorry I decided to edit my comment...SmallFish –SmallFish 2017-04-09 10:54:36 +00:00 Commented Apr 9, 2017 at 10:54 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Thank you for your input. I already knew about the disc method for finding the volume of solids of revolution. You did do a good job of going in depth while explaining it to me. However I seek 3-dimensional methods, perhaps the ones you referred to at the end of your comment. I am writing an exploration about solids of revolution and need various ways of computing their volume (and surface area but that's for another time). I will try to explain to you my methods and hopefully you can further help me, but please excuse my formatting as I am relatively new to this website. While learning about volumes in Cartesian coordinates. I recognized a pattern of finding the area, denoted dA, and then integrate the function by double integrals, where dA became dxdy. As such, it would theoretically become base (dA) times height(the function). And that is what the disc method is doing. Unfortunately, the majority of examples were volumes under a surface, which is not what I am trying to accomplish. I am trying to integrate my solid of revolution by finding its 3-dimensional equation. Solid of Revolution equation For a function of the form: y=f(x), the corresponding equation that I came up with for rotating it around the x-axis is f(x)2=z 2+y 2 f(x)2=z 2+y 2 I used the circle equation because every point on the original function would become a circle on the yz-plane when revolved around the x-axis. You can correct me if this formula is wrong. From this point is elementary to solve for z and thus, have a function of the form f(x,y). I have graphed the function ln(x) revolved around the x-axis here. As you see, I plugged ln(x) into the equation and solved for z. Now, my question becomes, what kind of integral do I need to integrate this function, hence find its volume. I was thinking using the dA method to integrate the individual circle and then integrate over x? But that does seem very similar to the disc method, only that we are using the 3D Cartesian equation. Similarly we could use polar coordinates to find the area of individual circles. I was also thinking about using iterated integrals, double or triple (which do you think would work?), to integrate the solid with respect to x,y and z. Thanks you very much for reading this and helping me, as I know the formatting is mediocre. Ps: I had to write an "answer" because my post was too long for a comment. @UniS20 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jun 12, 2020 at 10:38 CommunityBot 1 answered Apr 7, 2017 at 15:42 DiffusionDiffusion 5,921 1 1 gold badge 12 12 silver badges 31 31 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. To revolve an equation around the x axis, replace y y by y 2+z 2−−−−−−√y 2+z 2. In your example, y=x−−√y=x (x≥0 x≥0) turns to y 2+z 2−−−−−−√=x−−√y 2+z 2=x or y 2+z 2=x y 2+z 2=x. This is a cone of revolution. From the circle x 2+y 2=r 2 x 2+y 2=r 2, you obtain the sphere x 2+y 2+z 2=r 2 x 2+y 2+z 2=r 2. And for the shifted circle x 2+(y−2)2=1 x 2+(y−2)2=1, y 2+z 2−4 y 2+z 2−−−−−−√+4=1−x 2 y 2+z 2−4 y 2+z 2+4=1−x 2 or 16(y 2+z 2)=(x 2+y 2+z 2+3)2 16(y 2+z 2)=(x 2+y 2+z 2+3)2, which is the equation of a torus. Notice that for curves such that y y changes sign, you obtain self-intersecting surfaces. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Aug 20 at 10:20 answered Aug 20 at 10:15 Yves DaoustYves Daoust 1,524 1 1 silver badge 13 13 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus solid-of-revolution See similar questions with these tags. 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3870	https://brainly.com/question/33589188	[FREE] Pipetting and Accuracy in Measurement 1. What was the mass of the water you pipetted? 2. What was the - brainly.com 1 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +29,8k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +26,6k Ace exams faster, with practice that adapts to you Practice Worksheets +5,6k Guided help for every grade, topic or textbook Complete See more / Biology Expert-Verified Expert-Verified Pipetting and Accuracy in Measurement What was the mass of the water you pipetted? What was the volume of the water you pipetted? Explain why the density of water is 0.998 g/ml and not 1.0 g/ml. 1 See answer Explain with Learning Companion NEW Asked by nairo2839 • 06/15/2023 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 3350344 people 3M 0.0 0 Upload your school material for a more relevant answer Pipetting and Accuracy in Measurement:When pipetting, accuracy in measurement is critical to obtaining reliable results. Hence the density of water is 0.998 g/ml and not 1.0 g/ml. The most common sources of error are the users, pipette, and environmental variables, such as humidity and temperature. Measurements can be made in volumes ranging from a few microliters to several milliliters, and they must be precise and accurate.The mass of water that was pipetted: To answer this question, additional information is required. Without the information on how much water was pipetted, it is impossible to determine its mass.Volume:The pipetted volume of water must also be determined to determine its mass. The volume pipetted should be indicated on the pipette.Explain why the density of water is 0.998 g/ ml and not 1.0 g/ml:The density of water varies with temperature. The density of water at 4°C is 1.0 g/ml. As temperature rises, water expands, and its density decreases. At 20°C, the density of water is 0.998 g/ml. As the temperature of water rises above 4°C, it becomes less dense, but as the temperature decreases below 4°C, water becomes more dense. Hence the density of water is 0.998 g/ml and not 1.0 g/ml.Answer more than 100 words:Inaccuracies in measurement can lead to inaccurate results. As a result, when pipetting, accuracy in measurement is critical to obtaining reliable results. The most common sources of error are the users, pipette, and environmental variables, such as humidity and temperature. Measurement can be done in volumes ranging from a few microliters to several milliliters, and it must be precise and accurate. When pipetting, ensure that the pipette is correctly calibrated and that the user's technique is correct. Temperature and humidity should be kept constant, and the lab's guidelines must be followed when pipetting. The mass of water that was pipetted can be calculated by multiplying its volume by its density. To calculate the mass, the pipetted volume of water must also be determined, and the volume pipetted should be indicated on the pipette. The density of water varies with temperature. The density of water at 4°C is 1.0 g/ml. As temperature rises, water expands, and its density decreases. At 20°C, the density of water is 0.998 g/ml. As the temperature of water rises above 4°C, it becomes less dense, but as the temperature decreases below 4°C, water becomes more dense. Hence the density of water is 0.998 g/ml and not 1.0 g/ml. To know more about Measurement visit; brainly.com/question/28913275 SPJ11 Answered by ketyzalem •13.4K answers•3.4M people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) This answer helped 3350344 people 3M 0.0 0 Upload your school material for a more relevant answer To calculate the mass of water pipetted, multiply the volume by its density. The density of water is 0.998 g/mL due to its temperature-dependent behavior, reaching a density of 1.0 g/mL at 4°C. Accurate pipetting technique and conditions are vital for reliable results. Explanation Mass of the Water You Pipetted: To determine the mass of the water you pipetted, you need to know the volume of water you dispensed and use the formula: Mass=Volume×Density If you pipetted, for example, 10 mL of water, the mass would be: Mass=10 mL×0.998 g/mL=9.98 g Volume of the Water You Pipetted: The volume of the water pipetted is simply the reading you get from the pipette. For example, if you pipetted to the 10 mL mark on a pipette, then the volume is 10 mL. This value should be noted, as the volume is essential for calculating mass. Density of Water: The density of water is often stated as 0.998 g/mL instead of 1.0 g/mL due to its temperature dependence. At 4°C, water reaches its maximum density of approximately 1.0 g/mL. However, as the temperature increases, water molecules move faster, causing the water to expand and thus lowering its density. Thus, at room temperature (around 20°C), the density is measured at 0.998 g/mL. Changes in temperature and other conditions, like impurities in the water, can also affect density. This is why scientists state the density of water more precisely rather than generalizing it as 1.0 g/mL. In summary, accurate measurements while pipetting are crucial. Always check your pipette for calibration and ensure that environmental factors are controlled for the best results. Examples & Evidence For example, if you pipetted 20 mL of water at room temperature, the mass would be calculated as 20 mL x 0.998 g/mL = 19.96 g. This illustration helps to understand how changes in volume directly affect mass according to density. The stated density of water as 0.998 g/mL at 20°C can be found in standard chemistry textbooks and resources, confirming the temperature effect on density. Thanks 0 0.0 (0 votes) Advertisement nairo2839 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Biology solutions and answers Community Answer 1 When a food handler can effectively remove soil from equipment using normal methods, the equipment is considered what? Community Answer 1 Gene got Medicare before he turned 65 and enrolled into a Medicare Advantage plan. He calls in February the month before his 65th birthday and is unhappy with his current plan. On the date of the call, what can Gene do about his coverage? Community Answer 3 3. Which plant food must be transported to the serving site at 41F or below? A-chopped celery B-died tomatoes C-sliced cucumbers D-shredded carrots Community Answer 50 A food worker is putting chemicals into clean spray bottles, what must a food worker include on the each spray bottle? Community Answer 2 In the word search below are the names of several pieces of lab equipment. As you find each piece of equipment, record its name on the list. There are only 13 words out of the listBunsen burner,Pipestem triangle, Evaporating dish, Beaker, Utility clamp,Iron ring, Mortar and pestle, Crucible and cover, Gas bottle, Saftey goggles,Corks, Watch glass, Erlenmeyer flask, Wire gauze, Pipet, Buret,Triple beam balance, Test tube rack, Funnel, Scoopula,Well plate, Wire brush,File,Wash bottle, Graduated cylinder,Thanks Community Answer 5.0 2 Which of the following is NOT approved for chemical sanitizing after washing and rinsing? Quaternary ammonium Chlorine lodine Detergent Community Answer 5.0 2 which objective lens will still remain in focus when placed at the longest working distance from the specimen? Community Answer 5.0 9 How should food workers protect food from contamination after it is cooked? O a. Refrigerate the food until it is served O b. Use single-use gloves to handle the food O c. Apply hand sanitizer before handling the food O d. Cover the food in plastic wrap until it is served Community Answer 33 Suppose a one-year old child is playing with a toy near an electrical out-let. He sticks part of the toy into the outlet. He gets shocked, becomes frightened, and begins to cry. For several days after that experience, he shows fear when his mother gives hi.m the toy and he refuses to play with it. What are the UCS? U CR? CS? CR? New questions in Biology Identify the different parts of the respiratory system from the following list: nasal cavity, trachea, alveoli, larynx, bronchi, nose, pharynx, bronchiole. Identify the different parts of the respiratory system from the following list: nasal cavity trachea alveoli larynx bronchi nose pharynx bronchiole How many systems can you spot in this puzzle? \| C \| \| D \| \| V \| U \| S \| R \| D \| T \| O \| A \| C \| V \| T \| I \| w \| --- --- :--- :--- :--- :--- :--- :--- \| I \| \| O \| \| U \| R \| H \| E \| F \| K \| N \| V \| A \| N \| J \| E \| Q \| \| R \| \| B \| \| N \| I \| A \| D \| A \| C \| A \| z \| F \| E \| V \| H \| U \| \| C \| \| H \| \| I \| N \| W \| R \| D \| F \| V \| X \| V \| I \| S \| C \| C \| \| U \| \| N \| R \| \| A \| D \| S \| D \| R \| T \| I \| T \| V \| N \| D \| S \| \| L \| \| J \| A \| \| R \| C \| Q \| A \| S \| T \| C \| T \| K \| E \| C \| U \| \| A \| K \| \| L \| \| Y \| C \| W \| E \| S \| U \| E \| R \| U \| R \| F \| M \| \| T \| U \| \| U \| F \| \| B \| S \| E \| D \| A \| C \| E \| S \| V \| T \| L \| \| O \| F \| \| C \| R \| \| O \| G \| O \| A \| C \| E \| L \| K \| O \| H \| A \| \| R \| E \| \| S \| P \| \| I \| R \| A \| T \| O \| R \| Y \| T \| U \| I \| T \| \| Y \| D \| \| U \| D \| \| P \| S \| E \| R \| T \| H \| B \| N \| S \| A \| E \| \| K \| S \| M \| \| E \| D \| \| F \| A \| D \| E \| N \| G \| T \| J \| S \| L \| \| L \| S \| R \| \| J \| H \| \| T \| Y \| U \| J \| H \| N \| K \| F \| A \| E \| \| I \| N \| T \| \| E \| G \| \| U \| M \| E \| N \| T \| A \| R \| Y \| O \| K \| \| M \| U \| S \| \| C \| U \| \| L \| A \| R \| V \| G \| B \| A \| X \| X \| S \| 1. Who introduced the first fish ponds in the Philippines?a. Chinese b. Japanese c. Malay d. Americans 2. Where did the practice of fish culture begin?a. China b. Philippines c. Indonesia d. Thailand 3. An individual specializes in the practice of aquaculture, which involves the breeding, rearing, and harvesting of aquatic organisms such as fish, shellfish, and aquatic plants.a. Fish Caretaker b. Aquaculturist c. Research Officer d. Fish Wharf Operator 4. Their role revolves around managing and overseeing the operations of a fish wharf, which is a facility where fishing vessels unload their catch, and where fish are sorted, processed, and distributed.a. Fish Caretaker b. Aquaculturist c. Research Officer d. Fish Wharf Operator 5. An individual or entity that owns and operates a fish farm or aquaculture facility.a. Fish Farm Owner b. Fish Handler c. Fish Distributor d. Fish Trader Which statement describes one way that RNA differs from DNA? A. One provides energy, and the other carries genetic information. B. One is a protein, and the other is a nucleic acid. C. They contain different five-carbon sugars and a different nitrogenous base. D. They are both carbohydrates, but one is a polysaccharide. Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
3871	https://en.wikipedia.org/wiki/Nuclear_export_signal	Jump to content Nuclear export signal العربية فارسی Français Galego 한국어 日本語 Português 中文 Edit links From Wikipedia, the free encyclopedia Amino acid sequence causing a protein to be exported from the nucleus to the cytoplasm \| \| \| --- \| \| \| This article is missing information about non-CRM1 pathway (NESbase has a few unconventional ones but how do they work?). Please expand the article to include this information. Further details may exist on the talk page. (April 2019) \| A nuclear export signal (NES) is a short target peptide containing 4 hydrophobic residues in a protein that targets it for export from the cell nucleus to the cytoplasm through the nuclear pore complex using nuclear transport. It has the opposite effect of a nuclear localization signal, which targets a protein located in the cytoplasm for import to the nucleus. The NES is recognized and bound by exportins. NESs serve several vital cellular functions. They assist in regulating the position of proteins within the cell. Through this NESs affect transcription and several other nuclear functions that are essential to proper cell function. The export of many types of RNA from the nucleus is required for proper cellular function. The NES determines what type of pathway the varying types of RNA may use to exit the nucleus and perform their function and the NESs may effect the directionality of molecules exiting the nucleus. Structure [edit] Computer analysis of known NESs found the most common spacing of the hydrophobic residues to be LxxxLxxLxL, where "L" is a hydrophobic residue (often leucine) and "x" is any other amino acid; the spacing of these hydrophobic residues may be explained by examination of known structures that contain an NES, as the critical residues usually lie in the same face of adjacent secondary structures within a protein, which allows them to interact with the exportin. Ribonucleic acid (RNA) is composed of nucleotides, and thus, lacks the nuclear export signal to move out of the nucleus. As a result, most forms of RNA will bind to a protein molecule to form a ribonucleoprotein complex to be exported from the nucleus. Eukaryotic Linear Motif resource defines the NES motif for exportin within a single entry, TRG_NES_CRM1_1. The single-letter amino acid sequence pattern of NES, in regular expression format, is: ``` ([DEQ].{0,1}[LIM].{2,3}[LIVMF][^P]{2,3}[LMVF].[LMIV].{0,3}[DE])\| ([DE].{0,1}[LIM].{2,3}[LIVMF][^P]{2,3}[LMVF].[LMIV].{0,3}[DEQ]) ``` In the above expression, LIMVF are all hydrophobic residues, while DEQ are hydrophilic aspartic acid, glutamic acid, and glutamine. In human language, this is an extension of the "common pattern" that includes hydrophilic residues surrounding it as well as slight variations in the length of xxx and xx fragments seen above. Mechanism [edit] Nuclear export first begins with the binding of Ran-GTP (a G-protein) to exportin. This causes a shape change in exportin, increasing its affinity for the export cargo. Once the cargo is bound, the Ran-exportin-cargo complex moves out of the nucleus through the nuclear pore. GTPase activating proteins (GAPs) then hydrolyze the Ran-GTP to Ran-GDP, and this causes a shape change and subsequent exportin release. Once no longer bound to Ran, the exportin molecule loses affinity for the nuclear cargo as well, and the complex falls apart. Exportin and Ran-GDP are recycled to the nucleus separately, and guanine exchange factor (GEF) in the nucleus switches the GDP for GTP on Ran. Chemotherapy [edit] The process of nuclear export is responsible for some resistance to chemotherapy drugs. By limiting a cell's nuclear export activity it may be possible to reverse this resistance. By inhibiting CRM1, the export receptor, export through the nuclear envelope may be slowed. Survivin is a NES that inhibits cellular apoptosis. It interacts with the mitotic spindles during cellular division. Due to the usually rapid proliferation of tumour cells, survivin is more expressed during the presence of cancer. The level of survivin correlates to how resistance to chemotherapy a cancerous cell is and how likely that cell is to replicate again. By producing antibodies to target the NES survivin, apoptosis of cancerous cells can be increased. Examples [edit] NES signals were first discovered in the human immunodeficiency virus type 1 (HIV-1) Rev protein and cAMP-dependent protein kinase inhibitor (PKI). The karyopherin receptor CRM1 has been identified as the export receptor for leucine-rich NESs in several organisms and is an evolutionarily conserved protein. The export mediated by CRM1 can be effectively inhibited by the fungicide leptomycin B (LMB), providing excellent experimental verification of this pathway. Other proteins of various functions have also been experimentally inhibited of the NES signal such as the cyto-skeletal protein actin, which functions include cell motility and growth. The use of LBM as a NES inhibitor proved successful for actin resulting in accumulation of the protein within the nucleus, concluding universal functionality of NES throughout various protein functional groups. Regulation [edit] Not all NES substrates are constitutively exported from the nucleus, meaning that CRM1-mediated export is a regulated event. Several ways of regulating NES-dependent export have been reported. These include masking/unmasking of NESs, phosphorylation and even disulfide bond formation as a result of oxidation. The binding of NES to the export receptor of a protein gives the universal export function of NES an individually specified activation of export to each protein. Studies of specified NES amino acid sequences for particular proteins show the possibility of blocking the NES activation of one protein with an inhibitor for that amino acid sequence while other proteins of the same nucleus remain unaffected. NESbase [edit] NESbase is a database of proteins with experimentally verified leucine-rich nuclear export signals (NES). The verification is performed by, among others, Technical University of Denmark Center for Biological Sequence Analysis and University of Copenhagen Department of Protein Chemistry. Every entry in its database includes information whether nuclear export signals were sufficient for export or if it was only mediated transport by CRM1 (exportin). References [edit] ^ Fukuda, Makoto; Asano, Shiro; Nakamura, Takahiro; Adachi, Makoto; Yoshida, Minoru; Yanagida, Mitsuhiro; Nishida, Eisuke (November 1997). "CRM1 is responsible for intracellular transport mediated by the nuclear export signal". Nature. 390 (6657): 308–311. Bibcode:1997Natur.390..308F. doi:10.1038/36894. ISSN 0028-0836. PMID 9384386. S2CID 4420607. ^ Li, Zhengguo; Kearse, Michael G.; Huang, Chuan (2019-01-02). "The nuclear export of circular RNAs is primarily defined by their length". RNA Biology. 16 (1): 1–4. doi:10.1080/15476286.2018.1557498. ISSN 1547-6286. PMC 6380329. PMID 30526278. ^ la Cour T, Kiemer L, Mølgaard A, Gupta R, Skriver K, Brunak S (June 2004). "Analysis and prediction of leucine-rich nuclear export signals". Protein Eng. Des. Sel. 17 (6): 527–36. doi:10.1093/protein/gzh062. PMID 15314210. ^ "ELM - Detail for TRG_NES_CRM1_1". elm.eu.org. Retrieved 10 April 2019. ^ El-Tanani, Mohamed; Dakir, El-Habib; Raynor, Bethany; Morgan, Richard (2016-03-14). "Mechanisms of Nuclear Export in Cancer and Resistance to Chemotherapy". Cancers. 8 (3): 35. doi:10.3390/cancers8030035. ISSN 2072-6694. PMC 4810119. PMID 26985906. ^ Fukuda, Makoto; Asano, Shiro; Nakamura, Takahiro; Adachi, Makoto; Yoshida, Minoru; Yanagida, Mitsuhiro; Nishida, Eisuke (1997-11-20). "CRM1 is responsible for intracellular transport mediated by the nuclear export signal". Nature. 390 (6657): 308–311. Bibcode:1997Natur.390..308F. doi:10.1038/36894. ISSN 0028-0836. PMID 9384386. S2CID 4420607. ^ Wada, Atsushi; Fukuda, Makoto; Mishima, Masanori; Nishida, Eisuke (1998-03-16). "Nuclear export of actin: a novel mechanism regulating the subcellular localization of a major cytoskeletal protein". The EMBO Journal. 17 (6): 1635–1641. doi:10.1093/emboj/17.6.1635. ISSN 0261-4189. PMC 1170511. PMID 9501085. ^ Rowe, Thomas C.; Ostrov, David; Dawson, Jana L.; Pernazza, Danielle; Lawrence, Nicholas J.; Sullivan, Daniel M. (2013-11-15). "Targeting The Nuclear Export Signal In Multiple Myeloma". Blood. 122 (21): 1925. doi:10.1182/blood.V122.21.1925.1925. ISSN 0006-4971. ^ Tanja la Cour; Ramneek Gupta; Kristoffer Rapacki; Karen Skriver; Flemming M. Poulsen; Søren Brunak (2003). "NESbase version 1.0: a database of nuclear export signals". Nucleic Acids Research. 31 (1): 393–396. doi:10.1093/nar/gkg101. PMC 165548. PMID 12520031. External links [edit] Eukaryotic Linear Motif resource motif class TRG_NES_CRM1_1 NESbase Retrieved from " Categories: Cell biology Short linear motifs Cell signaling Molecular genetics Hidden categories: Articles with short description Short description matches Wikidata Articles to be expanded from April 2019
3872	https://www.scribd.com/document/410251804/A-V-Akopyan-A-A-Zaslavsky-Geometry-of-Conics-1-pdf	A. V. Akopyan, A. A. Zaslavsky - Geometry of Conics PDF \| PDF Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 623 views 140 pages A. V. Akopyan, A. A. Zaslavsky - Geometry of Conics PDF Document Information Uploaded by Abdullapoor RamReddy AI-enhanced title Go to previous items Go to next items Download Save Save A.V._Akopyan,_A._A._Zaslavsky-_Geometry_of_Conic... 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3873	https://www.youtube.com/watch?v=ZnkuI3PB_UA	Factoring a Negative Leading Coefficient Cole's World of Mathematics 46100 subscribers 118 likes Description 14709 views Posted: 25 Aug 2016 This video goes through one example and explains why factoring out a negative leading coefficient can be helpful when you need to completely factor a polynomial expression. 6 comments Transcript: okay today we're going to take a look at a little trick of factoring out a negative all right this will come in handy if you've got a polynomial expression and the leading term has a negative coefficient and the directions say factor completely you're going to want to probably go ahead and factor out a negative so that it will set it up nice for the rest of the problem when you encounter a trinomial you'll be able to easily Factor it all right so in this example right here I have polynomial by Leading term has a negative coefficient all right so that's I'm gonna that's gonna tell me okay when I do this I'm going to take out a negative factor all right now I'm going to go ahead and Factor like normal I'm going to look at the 3 and the 12 and the 15. all right the greatest common factor that I can take out of all of those would be a three but I'm going to choose to make it negative because I want to get the negative 3 out of that first term all right then I'm going to take a look at all of my variables all right the largest number of variables that I can take out is going to be an X so my greatest common factor I'm going to take out it's going to be a negative 3x so I'm going to take that negative 3x X out and then I'm going to see what's left all right in this first term I've got a negative 3 I'm taking that out I've got x to the third I'm going to let me take out one of those it's going to leave me with an x squared all right middle term I've got a 12 and I'm going to take out a negative 3 that's going to leave me with a negative 4. all right I've got two x's and I take out 1 that's going to leave me with 1. all right and you can always double check yourself all right if I were to distribute this back out I should be able to remultify this and get what I started with so negative 3x times a negative 4X is going to give me that positive 12x squared okay so I did do that right now on this last term I've got a 15x I'm going to factor out the negative 3 that's going to leave me with a negative 5 and then X taking out the X then I will not have any variables right there okay now directions almost always say factor completely so now I have taken out the greatest common factor but I can't necessarily stop I need to check this inside here and see if it's something that can be factored further all right it turns out to be a nice little trinomial and in this case yes I can use guess and check and Factor this if I would not have taken the negative out I would have had a negative leading coefficient right there and then it would have been hard to factor this trinomial all right so by taking that negative out see it's just a little trick it helps you to factor this trinomial a little bit easier all right now hopefully factoring the trinomial of guessing check is something that you've already done so it's not my intention of teaching that in this lesson okay a trinomial will factor into two binomials this one has a nice little leading coefficient of one so I just need an X and an X because x times x is x squared you're going to find by guessing check find two numbers that multiply together to equal negative 5 but add to get Negative 4. that turns out to be a positive one then negative five so plus one and minus 5. 1 times negative 5 gives me that negative 5. checking that middle term I get a negative 5x plus a 1x will give me the negative 4X right there all right so um just a little trick there all right leading coefficient on a polynomial is negative go ahead and factor out a negative so that if you have to continue factoring it's going to make the factoring probably a whole lot easier if you like the video and the little tricks you think is going to help you go ahead and give me a like and of course you can always subscribe to the channel I would appreciate that too thanks
3874	https://math.stackexchange.com/questions/2203573/a-solution-to-a-xb-c-yb-xc-a-yc-xa-b-y-0	ordinary differential equations - A solution to $a_x(b-c)_y+b_x(c-a)_y+c_x(a-b)_y=0$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more A solution to a x(b−c)y+b x(c−a)y+c x(a−b)y=0 a x(b−c)y+b x(c−a)y+c x(a−b)y=0 Ask Question Asked 8 years, 6 months ago Modified8 years, 6 months ago Viewed 79 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. Let a,b,c∈k[x,y]a,b,c∈k[x,y], where k k is a field of characteristic zero. Further assume that each monomial in each of {a,b,c}{a,b,c} has odd degree. I wish to find a solution to: a x(b−c)y+b x(c−a)y+c x(a−b)y=0 a x(b−c)y+b x(c−a)y+c x(a−b)y=0, where x x denotes the partial derivative with respect to x x and y y denotes the partial derivative with respect to y y. The only solution I have found is a=b=c a=b=c. Is it the unique solution to that equation? Any help will be appreciated. ordinary-differential-equations polynomials partial-derivative Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Mar 26, 2017 at 8:55 user237522user237522 7,247 3 3 gold badges 16 16 silver badges 34 34 bronze badges Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. How about a b c=y+x=y−x=y a=y+x b=y−x c=y ? Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Mar 26, 2017 at 10:31 quasiquasi 61.3k 3 3 gold badges 46 46 silver badges 105 105 bronze badges 1 You are right..this is also a solution.user237522 –user237522 2017-03-26 15:13:29 +00:00 Commented Mar 26, 2017 at 15:13 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Can you clearly specify what function is known and what is unknown. If the three functions a(x,y),b(x,y),c(x,y)a(x,y),b(x,y),c(x,y) are all unknown, chose arbitrary functions for b(x,y)b(x,y) and c(x,y)c(x,y) which then become known and you get a PDE to be solved for a(x,y)a(x,y). So, they are an infinity of solutions. A lot are easy to find. For example, if we arbitrary chose {b(x,y)=x+y c(x,y)=x y{b(x,y)=x+y c(x,y)=x y the PDE is (1−x)a x+(y−1)a y=y−x(1−x)a x+(y−1)a y=y−x which solution from method of characteristics is a(x,y)=x+y+F((x−1)(y−1))a(x,y)=x+y+F((x−1)(y−1)). This gives a set of solutions : ⎧⎩⎨b(x,y)=x+y c(x,y)=x y a(x,y)=x+y+F((x−1)(y−1)){b(x,y)=x+y c(x,y)=x y a(x,y)=x+y+F((x−1)(y−1)) with any differentiable function F F. One can do the same with other choice of functions b(x,y)b(x,y) and c(x,y)c(x,y) leading to other solutions of the initial problem. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Mar 26, 2017 at 10:36 JJacquelinJJacquelin 68.6k 4 4 gold badges 40 40 silver badges 94 94 bronze badges 3 Thank you very much! Indeed, the three functions are all unknown. Please notice that I assumed that each of {a,b,c}{a,b,c} has all monomials of odd degree, so your answer with c=x y c=x y is not relevant.user237522 –user237522 2017-03-26 15:08:20 +00:00 Commented Mar 26, 2017 at 15:08 Ho! Sorry, I didn't noticed it. So my whole answer isn't relevant. You should not accept it.JJacquelin –JJacquelin 2017-03-26 18:17:20 +00:00 Commented Mar 26, 2017 at 18:17 It's ok, thanks for trying to help! Any adjustments of your answer to the special case are welcome.user237522 –user237522 2017-03-26 20:30:28 +00:00 Commented Mar 26, 2017 at 20:30 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions ordinary-differential-equations polynomials partial-derivative See similar questions with these tags. 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3875	https://www.newadvent.org/summa/1016.htm	\| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- \| \| Home \| Encyclopedia \| Summa \| Fathers \| Bible \| Library \| \| \| \| \| ABCDEFGHIJKLMNOPQRSTUVWXYZ \| Home > Summa Theologiae > First Part > Question 16 Question 16. Truth Does truth reside in the thing, or only in the intellect? Does it reside only in the intellect composing and dividing? The comparison of the true to being The comparison of the true to the good Is God truth? Are all things true by one truth, or by many? The eternity of truth The unchangeableness of truth Article 1. Whether truth resides only in the intellect? Objection 1. It seems that truth does not reside only in the intellect, but rather in things. For Augustine (Soliloq. ii, 5) condemns this definition of truth, "That is true which is seen"; since it would follow that stones hidden in the bosom of the earth would not be true stones, as they are not seen. He also condemns the following, "That is true which is as it appears to the knower, who is willing and able to know," for hence it would follow that nothing would be true, unless someone could know it. Therefore he defines truth thus: "That is true which is." It seems, then, that truth resides in things, and not in the intellect. Objection 2. Further, whatever is true, is true by reason of truth. If, then, truth is only in the intellect, nothing will be true except in so far as it is understood. But this is the error of the ancient philosophers, who said that whatever seems to be true is so. Consequently mutual contradictories seem to be true as seen by different persons at the same time. Objection 3. Further, "that, on account of which a thing is so, is itself more so," as is evident from the Philosopher (Poster. i). But it is from the fact that a thing is or is not, that our thought or word is true or false, as the Philosopher teaches (Praedicam. iii). Therefore truth resides rather in things than in the intellect. On the contrary, The Philosopher says (Metaph. vi), "The true and the false reside not in things, but in the intellect." I answer that, As the good denotes that towards which the appetite tends, so the true denotes that towards which the intellect tends. Now there is this difference between the appetite and the intellect, or any knowledge whatsoever, that knowledge is according as the thing known is in the knower, whilst appetite is according as the desirer tends towards the thing desired. Thus the term of the appetite, namely good, is in the object desirable, and the term of the intellect, namely true, is in the intellect itself. Now as good exists in a thing so far as that thing is related to the appetite—and hence the aspect of goodness passes on from the desirable thing to the appetite, in so far as the appetite is called good if its object is good; so, since the true is in the intellect in so far as it is conformed to the object understood, the aspect of the true must needs pass from the intellect to the object understood, so that also the thing understood is said to be true in so far as it has some relation to the intellect. Now a thing understood may be in relation to an intellect either essentially or accidentally. It is related essentially to an intellect on which it depends as regards its essence; but accidentally to an intellect by which it is knowable; even as we may say that a house is related essentially to the intellect of the architect, but accidentally to the intellect upon which it does not depend. Now we do not judge of a thing by what is in it accidentally, but by what is in it essentially. Hence, everything is said to be true absolutely, in so far as it is related to the intellect from which it depends; and thus it is that artificial things are said to be true a being related to our intellect. For a house is said to be true that expresses the likeness of the form in the architect's mind; and words are said to be true so far as they are the signs of truth in the intellect. In the same way natural things are said to be true in so far as they express the likeness of the species that are in the divine mind. For a stone is called true, which possesses the nature proper to a stone, according to the preconception in the divine intellect. Thus, then, truth resides primarily in the intellect, and secondarily in things according as they are related to the intellect as their principle. Consequently there are various definitions of truth. Augustine says (De Vera Relig. xxxvi), "Truth is that whereby is made manifest that which is;" and Hilary says (De Trin. v) that "Truth makes being clear and evident" and this pertains to truth according as it is in the intellect. As to the truth of things in so far as they are related to the intellect, we have Augustine's definition (De Vera Relig. xxxvi), "Truth is a supreme likeness without any unlikeness to a principle": also Anselm's definition (De Verit. xii), "Truth is rightness, perceptible by the mind alone"; for that is right which is in accordance with the principle; also Avicenna's definition (Metaph. viii, 6), "The truth of each thing is a property of the essence which is immutably attached to it." The definition that "Truth is the equation of thought and thing" is applicable to it under either aspect. Reply to Objection 1. Augustine is speaking about the truth of things, and excludes from the notion of this truth, relation to our intellect; for what is accidental is excluded from every definition. Reply to Objection 2. The ancient philosophers held that the species of natural things did not proceed from any intellect, but were produced by chance. But as they saw that truth implies relation to intellect, they were compelled to base the truth of things on their relation to our intellect. From this, conclusions result that are inadmissible, and which the Philosopher refutes (Metaph. iv). Such, however, do not follow, if we say that the truth of things consists in their relation to the divine intellect. Reply to Objection 3. Although the truth of our intellect is caused by the thing, yet it is not necessary that truth should be there primarily, any more than that health should be primarily in medicine, rather than in the animal: for the virtue of medicine, and not its health, is the cause of health, for here the agent is not univocal. In the same way, the being of the thing, not its truth, is the cause of truth in the intellect. Hence the Philosopher says that a thought or a word is true "from the fact that a thing is, not because a thing is true." Article 2. Whether truth resides only in the intellect composing and dividing? Objection 1. It seems that truth does not reside only in the intellect composing and dividing. For the Philosopher says (De Anima iii) that as the senses are always true as regards their proper sensible objects, so is the intellect as regards "what a thing is." Now composition and division are neither in the senses nor in the intellect knowing "what a thing is." Therefore truth does not reside only in the intellect composing and dividing. Objection 2. Further, Isaac says in his book On Definitions that truth is the equation of thought and thing. Now just as the intellect with regard to complex things can be equated to things, so also with regard to simple things; and this is true also of sense apprehending a thing as it is. Therefore truth does not reside only in the intellect composing and dividing. On the contrary, the Philosopher says (Metaph. vi) that with regard to simple things and "what a thing is," truth is "found neither in the intellect nor in things." I answer that, As stated before, truth resides, in its primary aspect, in the intellect. Now since everything is true according as it has the form proper to its nature, the intellect, in so far as it is knowing, must be true, so far as it has the likeness of the thing known, this being its form, as knowing. For this reason truth is defined by the conformity of intellect and thing; and hence to know this conformity is to know truth. But in no way can sense know this. For although sight has the likeness of a visible thing, yet it does not know the comparison which exists between the thing seen and that which itself apprehends concerning it. But the intellect can know its own conformity with the intelligible thing; yet it does not apprehend it by knowing of a thing "what a thing is." When, however, it judges that a thing corresponds to the form which it apprehends about that thing, then first it knows and expresses truth. This it does by composing and dividing: for in every proposition it either applies to, or removes from the thing signified by the subject, some form signified by the predicate: and this clearly shows that the sense is true of any thing, as is also the intellect, when it knows "what a thing is"; but it does not thereby know or affirm truth. This is in like manner the case with complex or non-complex words. Truth therefore may be in the senses, or in the intellect knowing "what a thing is," as in anything that is true; yet not as the thing known in the knower, which is implied by the word "truth"; for the perfection of the intellect is truth as known. Therefore, properly speaking, truth resides in the intellect composing and dividing; and not in the senses; nor in the intellect knowing "what a thing is." And thus the Objections given are solved. Article 3. Whether the true and being are convertible terms? Objection 1. It seems that the true and being are not convertible terms. For the true resides properly in the intellect, as stated (Article 1); but being is properly in things. Therefore they are not convertible. Objection 2. Further, that which extends to being and not-being is not convertible with being. But the true extends to being and not-being; for it is true that what is, is; and that what is not, is not. Therefore the true and being are not convertible. Objection 3. Further, things which stand to each other in order of priority and posteriority seem not to be convertible. But the true appears to be prior to being; for being is not understood except under the aspect of the true. Therefore it seems they are not convertible. On the contrary, the Philosopher says (Metaph. ii) that there is the same disposition of things in being and in truth. I answer that, As good has the nature of what is desirable, so truth is related to knowledge. Now everything, in as far as it has being, so far is it knowable. Wherefore it is said in De Anima iii that "the soul is in some manner all things," through the senses and the intellect. And therefore, as good is convertible with being, so is the true. But as good adds to being the notion of desirable, so the true adds relation to the intellect. Reply to Objection 1. The true resides in things and in the intellect, as said before (Article 1). But the true that is in things is convertible with being as to substance; while the true that is in the intellect is convertible with being, as the manifestation with the manifested; for this belongs to the nature of truth, as has been said already (Article 1). It may, however, be said that being also is in the things and in the intellect, as is the true; although truth is primarily in things; and this is so because truth and being differ in idea. Reply to Objection 2. Not-being has nothing in itself whereby it can be known; yet it is known in so far as the intellect renders it knowable. Hence the true is based on being, inasmuch as not-being is a kind of logical being, apprehended, that is, by reason. Reply to Objection 3. When it is said that being cannot be apprehended except under the notion of the true, this can be understood in two ways. In the one way so as to mean that being is not apprehended, unless the idea of the true follows apprehension of being; and this is true. In the other way, so as to mean that being cannot be apprehended unless the idea of the true be apprehended also; and this is false. But the true cannot be apprehended unless the idea of being be apprehended also; since being is included in the idea of the true. The case is the same if we compare the intelligible object with being. For being cannot be understood, unless being is intelligible. Yet being can be understood while its intelligibility is not understood. Similarly, being when understood is true, yet the true is not understood by understanding being. Article 4. Whether good is logically prior to the true? Objection 1. It seems that good is logically prior to the true. For what is more universal is logically prior, as is evident from Phys. i. But the good is more universal than the true, since the true is a kind of good, namely, of the intellect. Therefore the good is logically prior to the true. Objection 2. Further, good is in things, but the true in the intellect composing and dividing as said above (Article 2). But that which is in things is prior to that which is in the intellect. Therefore good is logically prior to the true. Objection 3. Further, truth is a species of virtue, as is clear from Ethic. iv. But virtue is included under good; since, as Augustine says (De Lib. Arbit. ii, 19), it is a good quality of the mind. Therefore the good is prior to the true. On the contrary, What is in more things is prior logically. But the true is in some things wherein good is not, as, for instance, in mathematics. Therefore the true is prior to good. I answer that, Although the good and the true are convertible with being, as to suppositum, yet they differ logically. And in this manner the true, speaking absolutely, is prior to good, as appears from two reasons. First, because the true is more closely related to being than is good. For the true regards being itself simply and immediately; while the nature of good follows being in so far as being is in some way perfect; for thus it is desirable. Secondly, it is evident from the fact that knowledge naturally precedes appetite. Hence, since the true regards knowledge, but the good regards the appetite, the true must be prior in idea to the good. Reply to Objection 1. The will and the intellect mutually include one another: for the intellect understands the will, and the will wills the intellect to understand. So then, among things directed to the object of the will, are comprised also those that belong to the intellect; and conversely. Whence in the order of things desirable, good stands as the universal, and the true as the particular; whereas in the order of intelligible things the converse of the case. From the fact, then, that the true is a kind of good, it follows that the good is prior in the order of things desirable; but not that it is prior absolutely. Reply to Objection 2. A thing is prior logically in so far as it is prior to the intellect. Now the intellect apprehends primarily being itself; secondly, it apprehends that it understands being; and thirdly, it apprehends that it desires being. Hence the idea of being is first, that of truth second, and the idea of good third, though good is in things. Reply to Objection 3. The virtue which is called "truth" is not truth in general, but a certain kind of truth according to which man shows himself in deed and word as he really is. But truth as applied to "life" is used in a particular sense, inasmuch as a man fulfills in his life that to which he is ordained by the divine intellect, as it has been said that truth exists in other things (Article 1). Whereas the truth of "justice" is found in man as he fulfills his duty to his neighbor, as ordained by law. Hence we cannot argue from these particular truths to truth in general. Article 5. Whether God is truth? Objection 1. It seems that God is not truth. For truth consists in the intellect composing and dividing. But in God there is not composition and division. Therefore in Him there is not truth. Objection 2. Further, truth, according to Augustine (De Vera Relig. xxxvi) is a "likeness to the principle." But in God there is no likeness to a principle. Therefore in God there is not truth. Objection 3. Further, whatever is said of God, is said of Him as of the first cause of all things; thus the being of God is the cause of all being; and His goodness the cause of all good. If therefore there is truth in God, all truth will be from Him. But it is true that someone sins. Therefore this will be from God; which is evidently false. On the contrary, Our Lord says, "I am the Way, the Truth, and the Life" (John 14:6). I answer that, As said above (Article 1), truth is found in the intellect according as it apprehends a thing as it is; and in things according as they have being conformable to an intellect. This is to the greatest degree found in God. For His being is not only conformed to His intellect, but it is the very act of His intellect; and His act of understanding is the measure and cause of every other being and of every other intellect, and He Himself is His own existence and act of understanding. Whence it follows not only that truth is in Him, but that He is truth itself, and the sovereign and first truth. Reply to Objection 1. Although in the divine intellect there is neither composition nor division, yet in His simple act of intelligence He judges of all things and knows all things complex; and thus there is truth in His intellect. Reply to Objection 2. The truth of our intellect is according to its conformity with its principle, that is to say, to the things from which it receives knowledge. The truth also of things is according to their conformity with their principle, namely, the divine intellect. Now this cannot be said, properly speaking, of divine truth; unless perhaps in so far as truth is appropriated to the Son, Who has a principle. But if we speak of divine truth in its essence, we cannot understand this unless the affirmative must be resolved into the negative, as when one says: "the Father is of Himself, because He is not from another." Similarly, the divine truth can be called a "likeness to the principle," inasmuch as His existence is not dissimilar to His intellect. Reply to Objection 3. Not-being and privation have no truth of themselves, but only in the apprehension of the intellect. Now all apprehension of the intellect is from God. Hence all the truth that exists in the statement—"that a person commits fornication is true"—is entirely from God. But to argue, "Therefore that this person fornicates is from God", is a fallacy of Accident. Article 6. Whether there is only one truth, according to which all things are true? Objection 1. It seems that there is only one truth, according to which all things are true. For according to Augustine (De Trin. xv, 1), "nothing is greater than the mind of man, except God." Now truth is greater than the mind of man; otherwise the mind would be the judge of truth: whereas in fact it judges all things according to truth, and not according to its own measure. Therefore God alone is truth. Therefore there is no other truth but God. Objection 2. Further, Anselm says (De Verit. xiv), that, "as is the relation of time to temporal things, so is that of truth to true things." But there is only one time for all temporal things. Therefore there is only one truth, by which all things are true. On the contrary, it is written (Psalm 11:2), "Truths are decayed from among the children of men." I answer that, In one sense truth, whereby all things are true, is one, and in another sense it is not. In proof of which we must consider that when anything is predicated of many things univocally, it is found in each of them according to its proper nature; as animal is found in each species of animal. But when anything is predicated of many things analogically, it is found in only one of them according to its proper nature, and from this one the rest are denominated. So healthiness is predicated of animal, of urine, and of medicine, not that health is only in the animal; but from the health of the animal, medicine is called healthy, in so far as it is the cause of health, and urine is called healthy, in so far as it indicates health. And although health is neither in medicine nor in urine, yet in either there is something whereby the one causes, and the other indicates health. Now we have said (Article 1) that truth resides primarily in the intellect; and secondarily in things, according as they are related to the divine intellect. If therefore we speak of truth, as it exists in the intellect, according to its proper nature, then are there many truths in many created intellects; and even in one and the same intellect, according to the number of things known. Whence a gloss on Psalm 11:2, "Truths are decayed from among the children of men," says: "As from one man's face many likenesses are reflected in a mirror, so many truths are reflected from the one divine truth." But if we speak of truth as it is in things, then all things are true by one primary truth; to which each one is assimilated according to its own entity. And thus, although the essences or forms of things are many, yet the truth of the divine intellect is one, in conformity to which all things are said to be true. Reply to Objection 1. The soul does not judge of things according to any kind of truth, but according to the primary truth, inasmuch as it is reflected in the soul, as in a mirror, by reason of the first principles of the understanding. It follows, therefore, that the primary truth is greater than the soul. And yet, even created truth, which resides in our intellect, is greater than the soul, not simply, but in a certain degree, in so far as it is its perfection; even as science may be said to be greater than the soul. Yet it is true that nothing subsisting is greater than the rational soul, except God. Reply to Objection 2. The saying of Anselm is correct in so far as things are said to be true by their relation to the divine intellect. Article 7. Whether created truth is eternal? Objection 1. It seems that created truth is eternal. For Augustine says (De Lib. Arbit. ii, 8) "Nothing is more eternal than the nature of a circle, and that two added to three make five." But the truth of these is a created truth. Therefore created truth is eternal. Objection 2. Further, that which is always, is eternal. But universals are always and everywhere; therefore they are eternal. So therefore is truth, which is the most universal. Objection 3. Further, it was always true that what is true in the present was to be in the future. But as the truth of a proposition regarding the present is a created truth, so is that of a proposition regarding the future. Therefore some created truth is eternal. Objection 4. Further, all that is without beginning and end is eternal. But the truth of enunciables is without beginning and end; for if their truth had a beginning, since it was not before, it was true that truth was not, and true, of course, by reason of truth; so that truth was before it began to be. Similarly, if it be asserted that truth has an end, it follows that it is after it has ceased to be, for it will still be true that truth is not. Therefore truth is eternal. On the contrary, God alone is eternal, as laid down before (I:10:3. I answer that, The truth of enunciations is no other than the truth of the intellect. For an enunciation resides in the intellect, and in speech. Now according as it is in the intellect it has truth of itself: but according as it is in speech, it is called enunciable truth, according as it signifies some truth of the intellect, not on account of any truth residing in the enunciation, as though in a subject. Thus urine is called healthy, not from any health within it but from the health of an animal which it indicates. In like manner it has been already said that things are called true from the truth of the intellect. Hence, if no intellect were eternal, no truth would be eternal. Now because only the divine intellect is eternal, in it alone truth has eternity. Nor does it follow from this that anything else but God is eternal; since the truth of the divine intellect is God Himself, as shown already (Article 5). Reply to Objection 1. The nature of a circle, and the fact that two and three make five, have eternity in the mind of God. Reply to Objection 2. That something is always and everywhere, can be understood in two ways. In one way, as having in itself the power of extension to all time and to all places, as it belongs to God to be everywhere and always. In the other way as not having in itself determination to any place or time, as primary matter is said to be one, not because it has one form, but by the absence of all distinguishing form. In this manner all universals are said to be everywhere and always, in so far as universals are independent of place and time. It does not, however, follow from this that they are eternal, except in an intellect, if one exists that is eternal. Reply to Objection 3. That which now is, was future, before it (actually) was; because it was in its cause that it would be. Hence, if the cause were removed, that thing's coming to be was not future. But the first cause is alone eternal. Hence it does not follow that it was always true that what now is would be, except in so far as its future being was in the sempiternal cause; and God alone is such a cause. Reply to Objection 4. Because our intellect is not eternal, neither is the truth of enunciable propositions which are formed by us, eternal, but it had a beginning in time. Now before such truth existed, it was not true to say that such a truth did exist, except by reason of the divine intellect, wherein alone truth is eternal. But it is true now to say that that truth did not then exist: and this is true only by reason of the truth that is now in our intellect; and not by reason of any truth in the things. For this is truth concerning not-being; and not-being has not truth of itself, but only so far as our intellect apprehends it. Hence it is true to say that truth did not exist, in so far as we apprehend its not-being as preceding its being. Article 8. Whether truth is immutable? Objection 1. It seems that truth is immutable. For Augustine says (De Lib. Arbit. ii, 12), that "Truth and mind do not rank as equals, otherwise truth would be mutable, as the mind is." Objection 2. Further, what remains after every change is immutable; as primary matter is unbegotten and incorruptible, since it remains after all generation and corruption. But truth remains after all change; for after every change it is true to say that a thing is, or is not. Therefore truth is immutable. Objection 3. Further, if the truth of an enunciation changes, it changes mostly with the changing of the thing. But it does not thus change. For truth, according to Anselm (De Verit. viii), "is a certain rightness" in so far as a thing answers to that which is in the divine mind concerning it. But this proposition that "Socrates sits", receives from the divine mind the signification that Socrates does sit; and it has the same signification even though he does not sit. Therefore the truth of the proposition in no way changes. Objection 4. Further, where there is the same cause, there is the same effect. But the same thing is the cause of the truth of the three propositions, "Socrates sits, will sit, sat." Therefore the truth of each is the same. But one or other of these must be the true one. Therefore the truth of these propositions remains immutable; and for the same reason that of any other. On the contrary, It is written (Psalm 11:2),"Truths are decayed from among the children of men." I answer that, Truth, properly speaking, resides only in the intellect, as said before (Article 1); but things are called true in virtue of the truth residing in an intellect. Hence the mutability of truth must be regarded from the point of view of the intellect, the truth of which consists in its conformity to the thing understood. Now this conformity may vary in two ways, even as any other likeness, through change in one of the two extremes. Hence in one way truth varies on the part of the intellect, from the fact that a change of opinion occurs about a thing which in itself has not changed, and in another way, when the thing is changed, but not the opinion; and in either way there can be a change from true to false. If, then, there is an intellect wherein there can be no alternation of opinions, and the knowledge of which nothing can escape, in this is immutable truth. Now such is the divine intellect, as is clear from what has been said before (I:14:15). Hence the truth of the divine intellect is immutable. But the truth of our intellect is mutable; not because it is itself the subject of change, but in so far as our intellect changes from truth to falsity, for thus forms may be called mutable. Whereas the truth of the divine intellect is that according to which natural things are said to be true, and this is altogether immutable. Reply to Objection 1. Augustine is speaking of divine truth. Reply to Objection 2. The true and being are convertible terms. Hence just as being is not generated nor corrupted of itself, but accidentally, in so far as this being or that is corrupted or generated, as is said in Phys. i, so does truth change, not so as that no truth remains, but because that truth does not remain which was before. Reply to Objection 3. A proposition not only has truth, as other things are said to have it, in so far, that is, as they correspond to that which is the design of the divine intellect concerning them; but it said to have truth in a special way, in so far as it indicates the truth of the intellect, which consists in the conformity of the intellect with a thing. When this disappears, the truth of an opinion changes, and consequently the truth of the proposition. So therefore this proposition, "Socrates sits," is true, as long as he is sitting, both with the truth of the thing, in so far as the expression is significative, and with the truth of signification, in so far as it signifies a true opinion. When Socrates rises, the first truth remains, but the second is changed. Reply to Objection 4. The sitting of Socrates, which is the cause of the truth of the proposition, "Socrates sits," has not the same meaning when Socrates sits, after he sits, and before he sits. Hence the truth which results, varies, and is variously signified by these propositions concerning present, past, or future. Thus it does not follow, though one of the three propositions is true, that the same truth remains invariable. The Summa Theologiæ of St. Thomas Aquinas Second and Revised Edition, 1920 Literally translated by Fathers of the English Dominican Province Online Edition Copyright © 2017 by Kevin Knight Nihil Obstat. F. Innocentius Apap, O.P., S.T.M., Censor. Theol. Imprimatur. Edus. Canonicus Surmont, Vicarius Generalis. Westmonasterii. APPROBATIO ORDINIS Nihil Obstat. F. Raphael Moss, O.P., S.T.L. and F. Leo Moore, O.P., S.T.L. Imprimatur. F. Beda Jarrett, O.P., S.T.L., A.M., Prior Provincialis Angliæ MARIÆ IMMACULATÆ - SEDI SAPIENTIÆ \| \| \| Copyright © 2023 by New Advent LLC. Dedicated to the Immaculate Heart of Mary. \| CONTACT US \| ADVERTISE WITH NEW ADVENT
3876	https://artofproblemsolving.com/wiki/index.php/2015_AMC_12A_Problems/Problem_13?srsltid=AfmBOopo5IAF3_CGKv9bXrzWXpxjwwTnfWy5kbxwX7bJJ7Bq23brf4kZ	Art of Problem Solving 2015 AMC 12A Problems/Problem 13 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2015 AMC 12A Problems/Problem 13 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2015 AMC 12A Problems/Problem 13 Contents [hide] 1 Problem 2 Solution 1 3 Solution 2 4 Solution 3 5 See Also Problem A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores? Solution 1 We can eliminate answer choices and because there are an even number of scores, so if one is false, the other must be false too. Answer choice must be true since every team plays every other team, so it is impossible for two teams to lose every game. Answer choice must be true since each game gives out a total of two points, and there are games, for a total of points. Answer choice is false. If everyone draws each of their 11 games, then every team will tie for first place with 11 points each. Quick question, why does the answer key say 3? Solution 2 We will proceed by process of elimination: : We know that this must be true, since any tied results in a 1 point (which is odd) for both teams. Hence, there must be 0 or a positive even number of odd scores. : This is true too, because each non-tie generates 2 points for the winner, and 0 points for the loser, which are both even scores. Hence, there must be 0 or a positive even number of even scores as well. : This must be true since every team plays every other team, so it is impossible for two teams to lose every game. : This is true as well. Since any game gives out a net total of two points, and there are games, there are a total of points for any round-robin tournament. Therefore, answer choice is false. If everyone ties, every team will be tied for the first place with 11 points each. ~xHypotenuse Solution 3 This one tries not to use the process of elimination, even though that would be far easier. Statement assumes one team wins at least 6 games (they need 12 points, ignoring tie possibilities). Can we prevent the number one team from winning 6 games? Yes, all we need to do is ensure EVERY team wins 5 games out of 11. This gets us 60 wins, and thus we will have 60 corresponding losses. The remaining 12 points are distributed through ties since there's only one game left per team. Each team rounds off with 10+1 = 11 points. We just found an exception to statement . Thus, it is our "not" answer. ~panikd See Also 2015 AMC 12A (Problems • Answer Key • Resources) Preceded by Problem 12Followed by Problem 14 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 12 Problems and Solutions Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
3877	https://dokumen.pub/jawetz-melnick-amp-adelbergs-medical-microbiology-26thnbsped-0071790314-9780071790314-print.html	Jawetz, Melnick & Adelberg’s Medical Microbiology [26th ed.] 0071790314, 9780071790314 (print) - DOKUMEN.PUB Anmelden Registrierung Deutsch English Español Português Français Dom Najlepsze kategorie CAREER & MONEY PERSONAL GROWTH POLITICS & CURRENT AFFAIRS SCIENCE & TECH HEALTH & FITNESS LIFESTYLE ENTERTAINMENT BIOGRAPHIES & HISTORY FICTION Najlepsze historie Najlepsze historie Dodaj historię Moje historie Home Jawetz, Melnick & Adelberg’s Medical Microbiology [26th ed.] 0071790314, 9780071790314 (print) Jawetz, Melnick & Adelberg’s Medical Microbiology [26th ed.] 0071790314, 9780071790314 (print) The twenty-sixth edition of Jawetz, Melnick & Adelberg’s Medical Microbiology delivers a concise, up-to-date overvie 14,800 1,397 26MB English Pages 879 Year 2013 Report DMCA / Copyright DOWNLOAD FILE Polecaj historie ###### Jawetz, Melnick & Adelberg’s Medical Microbiology [28th Edition] 9781260012026 Since 1954, Jawetz, Melnick & Adelberg’s Medical Microbiology has been hailed by students, instructors, and clinicia 19,829 5,931 93MB Read more ###### Medical Microbiology 9783110517736, 9783110517644 Medical Microbiology is an excellent and easy-to-use textbook which explains the roles of microorganisms in human health 1,126 176 3MB Read more ###### Medical Microbiology [9th Edition] 9780323674508 9,835 2,409 86MB Read more ###### Medical Microbiology [9 ed.] 9780323673228, 9780323674508 Cleaned version with corrected bookmarks and removed watermarks. 24,195 4,815 87MB Read more ###### Medical Microbiology Il(75064415X) 0750601876, 9780750601870 Medical microbiology is essential in the management of patients with infection. A good knowledge of micro-organisms, the 467 88 5MB Read more ###### Medical Microbiology E-Book 9780323359528, 2015030867 4,073 927 39MB Read more ###### Medical Microbiology [9 ed.] 0323673228, 9780323673228 The foremost text in this complex and fast-changing field, Medical Microbiology, 9th Edition, provides concise, up-to-da 933 91 34MB Read more ###### Essentials of Medical Microbiology 9789352704798, 9352704797 The new edition of this comprehensive guide provides students with the latest information and advances in medical microb 17,765 3,167 146MB Read more ###### Essentials of Medical Microbiology, Fourth Edition 4 127 11 12MB Read more ###### Basic Medical Microbiology [1 ed.] 0323476767, 9780323476768 Authored by the lead author of the bestselling Medical Microbiology and written in the same tradition, Basic Medical Mic 1,629 232 5MB Read more Author / Uploaded Geo. F. Brooks Karen C. Carroll Janet S. Butel Stephen A. Morse Timothy A. Mietzner Categories Biology Microbiology _Table of contents : SECTION I: FUNDAMENTALS OF MICROBIOLOGY The Science of Microbiology Cell Structure Classification of Bacteria The Growth, Survival, and Death of Microorganisms Cultivation of Microorganisms Microbial Metabolism Microbial Genetics SECTION II: IMMUNOLOGY I mmunology SECTION III: BACTERIOLOGY Pathogenesis of Bacterial Infection Normal Human Microbiota Spore-Forming Gram-Positive Bacilli: Bacillus and Clostridium Species Aerobic Non-Spore-Forming Gram-Positive Bacilli: Corynebacterium, Listeria, Erysipelothrix, Actinomycetes, and Related Pathogens The Staphylococci The Streptococci, Enterococci, and Related Genera Enteric Gram-Negative Rods (Enterobacteriaceae) Pseudomonads, Acinetobacters, and Uncommon Gram-Negative Bacteria Vibrios, Campylobacters, Helicobacter, and Associated Bacteria Haemophilus, Bordetella, Brucella, and Francisella Yersinia and Pasteurella The Neisseriae Infections Caused by Anaerobic Bacteria Legionellae, Bartonella, and Unusual Bacterial Pathogens Mycobacteria Spirochetes and Other Spiral Microorganisms Mycoplasmas and Cell Wall-Defective Bacteria Rickettsia and Related Genera Chlamydia Spp. Antimicrobial Chemotherapy SECTION IV: VIROLOGY General Properties of Viruses Pathogenesis and Control of Viral Diseases Parvoviruses Adenoviruses Herpesviruses Poxviruses Hepatitis Viruses Picornaviruses (Enterovirus and Rhinovirus Groups) Reoviruses, Rotaviruses, and Caliciviruses Arthropod-Borne and Rodent-Borne Viral Diseases Orthomyxoviruses (Influenza Viruses) Paramyxoviruses and Rubella Virus Coronaviruses Rabies, Slow Virus Infections, and Prion Diseases Human Cancer Viruses AIDS and Lentiviruses SECTION V: MYCOLOGY Medical Mycology SECTION VI: PARASITOLOGY Medical Parasitology SECTION VII: DIAGNOSTIC MEDICAL MICROBIOLOGY and CLINICAL CORRELATION Principles of Diagnostic Medical Microbiology Cases and Clinical Correlations_ Citation preview a LANGE medical book Jawetz, Melnick, & Adelberg’s Medical Microbiology Twenty-Sixth Edition Geo. F. Brooks, MD Professor of Laboratory Medicine and Microbiology and Immunology Chief, Microbiology Section Clinical Laboratories University of California San Francisc, California Karen C. Carroll, MD Professor of Pathology The Johns Hopkins University School of Medicine Director, Division Medical Microbiology The Johns Hopkins Hospital Baltimore, Maryland Janet S. Butel, PhD Distinguished Service Professor Chair, Department of Molecular Virology and Microbiology Baylor College of Medicine Houston, Texas Stephen A. Morse, PhD Associate Director for Environmental Microbiology Division of Foodborne, Waterborne, and Environmental Diseases National Center for Emerging and Zoonotic Infectious Diseases Centers for Disease Control and Prevention Atlanta, Georgia Timothy A. Mietzner, PhD Associate Professor Department of Microbiology and Molecular Genetics University of Pittsburgh School of Medicine Pittsburgh Adjunct Associate Professor of Microbiology Arizona School of Dentistry and Oral Health Mesa, Arizona New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-181578-9 MHID: 0-07-181578-3 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-179031-4, MHID: 0-07-179031-4. All trademarks are trademarks of their respective owners. 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Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. Contents iii Contents Preface xi S E C T I O N I Fundamentals of Microbiology 1 Stephen A. Morse, PhD, and Timothy A. Meitzner, PhD The Science of Microbiology 1 Introduction 1 Biologic Principles Illustrated by Microbiology 1 Viruses 2 Prions 2 Prokaryotes 3 Protists 6 Chapter Summary 8 Review Questions 8 Cell Structure 11 Optical Methods 11 Eukaryotic Cell Structure 13 Prokaryotic Cell Structure 15 Staining 39 Morphologic Changes During Growth 40 Chapter Summary 40 Review Questions 41 Classification of Bacteria 43 Taxonomy—The Vocabulary of Medical Microbiology 43 Criteria for Classification of Bacteria 44 Classification Systems 45 Description of the Major Categories and Groups of Bacteria 48 Subtyping and Its Application 50 Nucleic Acid–Based Taxonomy 51 Nonculture Methods for the Identification of Pathogenic Microorganisms 53 Objectives 53 Review Questions 53 The Growth, Survival, and Death of Microorganisms 55 Survival of Microorganisms in the Natural Environment 55 The Meaning of Growth 55 Exponential Growth 55 The Growth Curve 57 Maintenance of Cells in the Exponential Phase 58 Definition and Measurement of Death 58 Antimicrobial Agents 60 Objectives 65 Review Questions 65 Cultivation of Microorganisms 67 Requirements for Growth 67 Sources of Metabolic Energy 67 Nutrition 68 Environmental Factors Affecting Growth 69 Cultivation Methods 72 Chapter Summary 75 Review Questions 76 Microbial Metabolism 77 Role of Metabolism in Biosynthesis and Growth 77 Focal Metabolites and Their Interconversion 77 Assimilatory Pathways 80 Biosynthetic Pathways 88 Patterns of Microbial Energy-Yielding Metabolism 91 Regulation of Metabolic Pathways 96 Chapter Summary 98 Review Questions 99 Microbial Genetics 101 Organization of Genes 101 Replication 106 Transfer of DNA 107 Mutation and Gene Rearrangement 111 Gene Expression 111 Genetic Engineering 115 Characterization of Cloned DNA 118 Site-Directed Mutagenesis 119 Analysis With Cloned DNA: Hybridization Probes 119 iii iv Contents Manipulation of Cloned DNA 120 Objectives 121 Objectives 121 S E C T I O N II IMMUNOLOGY 123 Barbara Detrick, PhD Aerobic Non–Spore-Forming Gram-Positive Immunology 123 Overview 123 Innate Immunity 123 Adaptive Immunity 127 Complement 138 Cytokines 140 Hypersensitivity 141 Deficiencies of the Immune Response 142 Clinical Immunology Laboratory (Diagnostic Testing) 143 Chapter Summary 145 Review Questions 147 S E C T I O N III BACTERIOLOGY 149 Karen C. Carroll, MD Pathogenesis of Bacterial Infection 149 Identifying Bacteria That Cause Disease 150 Transmission of Infection 151 The Infectious Process 152 Genomics and Bacterial Pathogenicity 152 Regulation of Bacterial Virulence Factors 153 Bacterial Virulence Factors 154 Chapter Summary 161 Review Questions 162 Normal Human Microbiota 165 Human Microbiome Project 165 Role of the Resident Microbiota 165 Normal Microbiota of the Skin 167 Normal Microbiota of the Mouth and Upper Respiratory Tract 167 Normal Microbiota of the Urethra 172 Normal Microbiota of the Vagina 172 Normal Microbiota of the Conjunctiva 172 Chapter Summary 172 Review Questions 173 Spore-Forming Gram-Positive Bacilli: Bacillus and Clostridium Species 175 Bacillus Species 175 Bacillus anthracis 175 Bacillus cereus 178 Clostridium Species 178 Clostridium botulinum 179 Clostridium tetani 180 Clostridia That Produce Invasive Infections 181 Clostridium difficile and Diarrheal Disease 183 Review Questions 183 Bacilli: Corynebacterium, Listeria, Erysipelothrix, Actinomycetes, and Related Pathogens 187 Corynebacterium diphtheriae 188 Other Coryneform Bacteria 191 Listeria monocytogenes 192 Erysipelothrix rhusiopathiae 193 Actinomycetes 194 Nocardiosis 194 Actinomycetoma 195 Review Questions 195 The Staphylococci 199 Chapter Summary 205 Review Questions 206 The Streptococci, Enterococci, and Related Genera 209 Classification of Streptococci 209 Streptococci of Particular Medical Interest 211 Streptococcus pyogenes 211 Streptococcus agalactiae 216 Groups C and G 217 Group D Streptococci 217 Streptococcus anginosus Group 217 Group N Streptococci 217 Groups E, F, G, H, and K–U Streptococci 217 Viridans Streptococci 218 Nutritionally Variant Streptococci 218 Peptostreptococcus and Related Genera 218 Streptococcus pneumoniae 218 Enterococci 222 Other Catalase-Negative Gram-Positive Cocci 224 Review Questions 225 Enteric Gram-Negative Rods (Enterobacteriaceae) 229 Classification 229 Diseases Caused by Enterobacteriaceae Other Than Salmonella and Shigella 233 The Shigellae 236 The Salmonella-Arizona Group 238 Chapter Summary 241 Review Questions 241 Contents v Pseudomonads, Acinetobacters, and Uncommon Gram-Negative Bacteria 245 The Pseudomonad Group 245 Pseudomonas aeruginosa 245 Burkholderia pseudomallei 248 Burkholderia mallei 248 Burkholderia cepacia Complex and Burkholderia Gladioli 248 Stenotrophomonas maltophilia 249 Acinetobacter 249 Other Pseudomonads 249 Uncommon Gram-Negative Bacteria 250 Aggregatibacter 250 Achromobacter and Alcaligenes 250 Ochrobactrum 250 Capnocytophaga 250 Cardiobacterium 250 Chromobacteria 250 Eikenella corrodens 251 Chryseobacterium 251 Kingella 251 Moraxella 251 Chapter Summary 251 Review Questions 251 Vibrios, Campylobacters, Helicobacter, and Associated Bacteria 255 The Vibrios 255 Vibrio Cholerae 255 Vibrio Parahaemolyticus and Other Vibrios 258 Aeromonas 259 Plesiomonas 259 Campylobacter 259 Campylobacter Jejuni and Campylobacter Coli 259 Campylobacter fetus 261 Other Campylobacters 261 Helicobacter Pylori 261 Review Questions 263 Haemophilus, Bordetella, Brucella, and Francisella 265 The Haemophilus Species 265 Haemophilus influenzae 265 Haemophilus aegyptius 267 Aggregatibacter aphrophilus 268 Haemophilus ducreyi 268 Other Haemophilus Species 268 The Bordetellae 268 Bordetella pertussis 268 Bordetella parapertussis 270 Bordetella bronchiseptica 270 The Brucellae 271 Francisella Tularensis and Tularemia 273 Review Questions 275 Yersinia and Pasteurella 279 Yersinia pestis and Plague 279 Yersinia enterocolitica and Yersinia pseudotuberculosis 281 Pasteurella 282 Review Questions 282 The Neisseriae 285 Neisseria gonorrhoeae 285 Neisseria meningitidis 291 Other Neisseriae 292 Chapter Summary 293 Review Questions 293 Infections Caused by Anaerobic Bacteria 295 Physiology and Growth Conditions for Anaerobes 295 Anaerobic Bacteria Found in Human Infections 296 Bacteria That Cause Vaginosis 297 Gardnerella vaginalis 297 Mobiluncus Species 297 Pathogenesis of Anaerobic Infections 300 Immunity in Anaerobic Infections 300 The Polymicrobial Nature of Anaerobic Infections 300 Diagnosis of Anaerobic Infections 301 Treatment of Anaerobic Infections 301 Chapter Summary 301 Review Questions 302 Legionellae, Bartonella, and Unusual Bacterial Pathogens 305 Legionella pneumophila and Other Legionellae 305 Bartonella 308 Streptobacillus moniliformis 310 Whipple Disease 310 Review Questions 310 Mycobacteria 313 Mycobacterium tuberculosis 313 Other Mycobacteria 321 Mycobacterium leprae 323 Review Questions 324 Spirochetes and Other Spiral Microorganisms 327 Treponema 327 Treponema pallidum and Syphilis 327 Diseases Related To Syphilis 331 Borrelia 331 Borrelia Species and Relapsing Fever 331 Borrelia burgdorferi and Lyme Disease 333 Leptospira and Leptospirosis 335 Other Spirochetal Diseases 337 Spirillum minor (Spirillum morsus muris) 337 vi Contents Spirochetes of the Normal Mouth and Mucous Membranes 337 Review Questions 338 Mycoplasmas and Cell Wall–Defective Bacteria 341 Mycoplasmas 341 Mycoplasma pneumoniae and Atypical Pneumonias 343 Mycoplasma hominis 344 Ureaplasma urealyticum 345 Mycoplasma genitalium 345 Cell Wall–Defective Bacteria 345 Chapter Summary 345 Review Questions 345 Rickettsia and Related Genera 349 General 349 Rickettsia and Orientia 349 Ehrlichia and Anaplasma 353 Coxiella Burnetii 354 Review Questions 356 Chlamydia Spp. 359 Chlamydia Trachomatis Ocular, Genital, and Respiratory Infections 362 Trachoma 362 Chlamydia trachomatis Genital Infections and Inclusion Conjunctivitis 363 Chlamydia Trachomatis And Neonatal Pneumonia 364 Lymphogranuloma Venereum 364 Chlamydia pneumoniae and Respiratory Infections 365 Chlamydia psittaci and Psittacosis 366 Chapter Summary 368 Review Questions 368 Antimicrobial Chemotherapy 371 Mechanisms Of Action Of Antimicrobial Drugs 371 Selective Toxicity 371 Inhibition of Cell Wall Synthesis 371 Inhibition of Cell Membrane Function 373 Inhibition of Protein Synthesis 373 Inhibition of Nucleic Acid Synthesis 375 Resistance To Antimicrobial Drugs 375 Origin of Drug Resistance 376 Cross-Resistance 376 Limitation of Drug Resistance 376 Clinical Implications of Drug Resistance 377 Factors Affecting Antimicrobial Activity 378 Antimicrobial Activity In Vitro 378 Measurement of Antimicrobial Activity 379 Drug–Pathogen Relationships 379 Antimicrobial Activity In Vivo 379 Host–Pathogen Relationships 380 Clinical Use Of Antibiotics 381 Selection of Antibiotics 381 Dangers of Indiscriminate Use 381 Antimicrobial Drugs Used in Combination 382 Antimicrobial Chemoprophylaxis 383 Antimicrobial Drugs For Systemic Administration 384 Penicillins 384 Cephalosporins 390 Other b-Lactam Drugs 393 Tetracyclines 394 Glycylcyclines 394 Chloramphenicol 395 Erythromycins 395 Clindamycin and Lincomycin 396 Glycopeptides and Lipopeptides 396 Streptogramins 397 Oxazolidinones 397 Bacitracin 397 Polymyxins 397 Aminoglycosides 398 Quinolones 399 Sulfonamides and Trimethoprim 401 Other Drugs With Specialized Uses 401 Drugs Used Primarily To Treat Mycobacterial Infections 402 Review Questions 403 S E C T I O N IV VIROLOGY 407 Jane Butel, PhD General Properties of Viruses 407 Terms and Definitions in Virology 407 Evolutionary Origin of Viruses 408 Classification of Viruses 408 Principles of Virus Structure 414 Chemical Composition of Viruses 415 Cultivation and Assay of Viruses 416 Purification and Identification of Viruses 418 Laboratory Safety 419 Reaction To Physical and Chemical Agents 419 Replication of Viruses: An Overview 420 Genetics of Animal Viruses 425 Natural History (Ecology) and Modes of Transmission of Viruses 427 Chapter Summary 428 Review Questions 429 Contents vii Pathogenesis and Control of Viral Diseases 431 Principles of Viral Diseases 431 Pathogenesis of Viral Diseases 431 Prevention and Treatment of Viral Infections 441 Chapter Summary 449 Review Questions 449 Parvoviruses 451 Properties of Parvoviruses 451 Parvovirus Infections in Humans 452 Chapter Summary 455 Review Questions 455 Adenoviruses 457 Properties of Adenoviruses 457 Adenovirus Infections in Humans 461 Chapter Summary 464 Review Questions 464 Herpesviruses 467 Properties of Herpesviruses 467 Herpesvirus Infections in Humans 471 Herpes Simplex Viruses 471 Varicella-Zoster Virus 476 Cytomegalovirus 480 Epstein-Barr Virus 484 Human Herpesvirus 6 487 Human Herpesvirus 7 487 Human Herpesvirus 8 488 B Virus 488 Chapter Summary 489 Review Questions 489 Poxviruses 493 Properties of Poxviruses 493 Poxvirus Infections in Humans: Vaccinia and Variola 496 Monkeypox Infections 501 Cowpox Infections 501 Buffalopox Infections 501 Orf Virus Infections 501 Molluscum Contagiosum 501 Tanapox and Yaba Monkey Tumor Poxvirus Infections 503 Chapter Summary 504 Review Questions 504 Hepatitis Viruses 507 Properties of Hepatitis Viruses 507 Hepatitis Virus Infections in Humans 512 Chapter Summary 524 Review Questions 524 Picornaviruses (Enterovirus and Rhinovirus Groups) 527 Properties of Picornaviruses 527 Enterovirus Group 531 Polioviruses 531 Coxsackieviruses 533 Other Enteroviruses 536 Enteroviruses in the Environment 537 Rhinoviruses 538 Parechovirus Group 539 Foot-And-Mouth Disease (Aphthovirus Of Cattle) 539 Chapter Summary 540 Review Questions 540 Reoviruses, Rotaviruses, and Caliciviruses 543 Reoviruses and Rotaviruses 543 Rotaviruses 544 Reoviruses 548 Caliciviruses 548 Orbiviruses and Coltiviruses 548 Astroviruses 551 Chapter Summary 551 Review Questions 551 Arthropod-Borne and Rodent-Borne Viral Diseases 553 Human Arbovirus Infections 553 Togavirus and Flavivirus Encephalitis 555 Yellow Fever 562 Dengue 564 Bunyavirus Encephalitis 566 Sandfly Fever 566 Rift Valley Fever 566 Colorado Tick Fever 567 Rodent-Borne Hemorrhagic Fevers 567 Bunyavirus Diseases 567 Arenavirus Diseases 569 Filovirus Diseases 571 Chapter Summary 573 Review Questions 573 Orthomyxoviruses (Influenza Viruses) 577 Properties of Orthomyxoviruses 577 Influenza Virus Infections in Humans 583 Chapter Summary 588 Review Questions 589 Paramyxoviruses and Rubella Virus 591 Properties of Paramyxoviruses 591 Parainfluenza Virus Infections 594 Respiratory Syncytial Virus Infections 598 Human Metapneumovirus Infections 600 Mumps Virus Infections 601 viii Contents Measles (Rubeola) Virus Infections 603 Hendra Virus and Nipah Virus Infections 606 Rubella (German Measles) Virus Infections 607 Postnatal Rubella 607 Congenital Rubella Syndrome 609 Chapter Summary 609 Review Questions 510 Coronaviruses 613 Properties of Coronaviruses 613 Coronavirus Infections in Humans 615 Chapter Summary 617 Review Questions 617 Rabies, Slow Virus Infections, and Prion Diseases 619 Rabies 619 Borna Disease 626 Slow Virus Infections and Prion Diseases 626 Chapter Summary 629 Review Questions 629 Human Cancer Viruses 633 General Features of Viral Carcinogenesis 633 Retroviruses 635 Cellular Oncogenes 641 Tumor Suppressor Genes 642 DNA Tumor Viruses 642 Polyomaviruses 642 Papillomaviruses 644 Adenoviruses 647 Herpesviruses 648 Poxviruses 648 Hepatitis B Virus and Hepatitis C Virus 648 How to Prove That a Virus Causes Human Cancer 649 Chapter Summary 649 Review Questions 649 AIDS and Lentiviruses 653 Properties of Lentiviruses 653 HIV Infections in Humans 657 Chapter Summary 667 Review Questions 667 S E C T I O N V MYCOLOGY 671 Thomas G. Mitchell, PhD Medical Mycology 671 General Properties and Classification of Fungi 672 Growth and Isolation of Fungi 676 Superficial Mycoses 676 Cutaneous Mycoses 677 Key Concepts: Superficial and Cutaneous Mycoses 681 Subcutaneous Mycoses 681 Sporotrichosis 681 Chromoblastomycosis 682 Phaeohyphomycosis 684 Mycetoma 684 Key Concepts: Subcutaneous Mycoses 685 Endemic Mycoses 685 Coccidioidomycosis 686 Histoplasmosis 689 Blastomycosis 692 Paracoccidioidomycosis 693 Key Concepts: Endemic Mycoses 694 Opportunistic Mycoses 694 Candidiasis 694 Cryptococcosis 697 Aspergillosis 699 Mucormycosis 701 Pneumocystis Pneumonia 702 Penicilliosis 702 Other Opportunistic Mycoses 702 Key Concepts: Opportunistic Mycoses 703 Antifungal Prophylaxis 703 Hypersensitivity to Fungi 703 Mycotoxins 704 Antifungal Chemotherapy 704 Topical Antifungal Agents 709 Key Concepts: Antifungal Chemotherapy 710 Review Questions 710 S E C T I O N VI PARASITOLOGY 715 Judy A. Sakanari, PhD, and James H. McKerrow, MD, PhD Medical Parasitology 715 Classification of Parasites 715 Intestinal Protozoan Infections 719 Giardia lamblia (Intestinal Flagellate) 719 Key Concepts: Parasitic Protozoa 719 Entamoeba histolytica (Intestinal and Tissue Ameba) 720 Other Intestinal Amebae 722 Cryptosporidium (Intestinal Sporozoa) 722 Cyclospora (Intestinal Sporozoa) 723 Sexually Transmitted Protozoan Infection 723 Contents ix Trichomonas vaginalis (Genitourinary Flagellate) 723 Blood and Tissue Protozoan Infections 723 Blood Flagellates 723 Trypanosoma brucei rhodesiense and T b gambiense (Blood Flagellates) 724 Trypanosoma cruzi (Blood Flagellate) 725 Leishmania Species (Blood Flagellates) 725 Entamoeba histolytica (Tissue Ameba)—See Intestinal Protozoan Infections Section 727 Naegleria fowleri, Acanthamoeba castellanii, and Balamuthia mandrillaris (Free-Living Amebae) 727 Plasmodium Species (Blood Sporozoa) 727 Babesia microti (Blood Sporozoa) 731 Toxoplasma gondii (Tissue Sporozoa) 732 Microsporidia 733 Intestinal Helminthic Infections 733 Key Concepts: Parasitic Helminths 733 Enterobius vermicularis (Pinworm—Intestinal Nematode) 734 Trichuris trichiura (Whipworm—Intestinal Nematode) 734 Ascaris lumbricoides (Human Roundworm— Intestinal Nematode) 738 Ancylostoma duodenale and Necator americanus (Human Hookworms—Intestinal Nematode) 739 Strongyloides stercoralis (Human Threadworm— Intestinal and Tissue Nematode) 740 Trichinella spiralis (Intestinal And Tissue Nematode) 741 Fasciolopsis buski (Giant Intestinal Fluke— Intestinal Trematode) 741 Taenia saginata (Beef Tapeworm—Intestinal Cestode) and Taenia Solium (Pork Tapeworm— Intestinal and Tissue Cestode) 741 Diphyllobothrium latum (Broad Fish Tapeworm— Intestinal Cestode) 742 Hymenolepis nana (Dwarf Tapeworm—Intestinal Cestode) 742 Dipylidium caninum (Dog Tapeworm—Intestinal Cestode) 743 Wuchereria bancrofti and Brugia Malayi (Lymphatic Filariasis—Tissue Nematodes) 743 Blood and Tissue Helminthic Infections 743 Onchocerca volvulus (River Blindness—Tissue Nematode) 743 Dracunculus medinensis (Guinea Worm—Tissue Nematode) 744 Larva Migrans (Zoonotic Larval Nematode Infections) 745 Clonorchis sinensis (Chinese Liver Fluke), Fasciola hepatica (Sheep Liver Fluke), and Paragonimus westermani (Lung Fluke)—Tissue Trematodes 745 Schistosoma mansoni, S japonicum, and S haematobium (Blood Flukes) 746 Tissue Cestode Infections (Caused by the Larval Stages) 746 Taenia solium—Cysticercosis/ Neurocysticercosis 746 Echinococcus granulosus (Hydatid Cyst) 746 Review Questions 748 S E C T I O N VII DIAGNOSTIC MEDICAL MICROBIOLOGY and CLINICAL CORRELATION 753 Karen C. Carroll, MD Principles of Diagnostic Medical Microbiology 753 Communication Between Physician and Laboratory 753 Diagnosis of Bacterial and Fungal Infections 754 The Importance of Normal Bacterial and Fungal Microbiota 765 Laboratory Aids in the Selection of Antimicrobial Therapy 766 Diagnosis of Infection by Anatomic Site 767 Anaerobic Infections 773 Diagnosis of Chlamydial Infections 773 Diagnosis of Viral Infections 775 Review Questions 783 Cases and Clinical Correlations 785 Central Nervous System 785 Respiratory 789 Heart 793 Abdomen 795 Urinary Tract 800 Bone and Soft Tissue 802 Sexually Transmitted Diseases 803 Mycobacterium Tuberculosis Infections 806 HIV-1 and Aids 809 Infections in Transplant Patients 813 Biologic Warfare and Bioterrorism 817 Index 823 This page intentionally left blank Contents xi Preface The twenty-sixth edition of Jawetz, Melnick, & Adelberg’s Medical Microbiology remains true to the goals of the first edition published in 1954 “to provide a brief, accurate and up-to-date presentation of those aspects of medical micro biology that are of particular significance to the fields of clinical infections and chemotherapy.” The 26th edition has included the following new features! • Addition of concept checks after major sections within chapters. • Chapter Summaries at the end of each chapter. • Increased number of new and revised review questions. • Full color photographs and photomicrographs of the previous edition. • All chapters have been revised extensively consistent with the tremendous expansion of medical knowledge afforded by molecular mechanisms, advances in our understanding of microbial pathogenesis and the discovery of novel pathogens. New also to this edition is Barbara Detrick, PhD, Professor in the Division of Clinical Immunology in the Department of Pathology at the Johns Hopkins University School of Medicine. Dr. Detrick’s extensive expertise in clinical immunology, and in particular the role of cytokines in health and disease, will add significantly to the current and future editions and we welcome her participation. The authors hope that the changes to this edition will be helpful to the student of microbiology. Geo. F. Brooks San Francisco, California Karen C. Carroll Baltimore, Maryland Janet S. Butel Houston, Texas Stephen A. Morse Atlanta, Georgia Timothy A. Meitzner Mesa, Arizona November 2012 xi This page intentionally left blank SECTION I FUNDAMENTALS OF MICROBIOLOGY C The Science of Microbiology INTRODUCTION Microbiology is the study of microorganisms, a large and diverse group of microscopic organisms that exist as single cells or cell clusters; it also includes viruses, which are microscopic but not cellular. Microorganisms have a tremendous impact on all life and the physical and chemical makeup of our planet. They are responsible for cycling the chemical elements essential for life, including carbon, nitrogen, sulfur, hydrogen, and oxygen; more photosynthesis is carried out by microorganisms than by green plants. Furthermore, there are 100 million times as many bacteria in the oceans (13 × 1028) as there are stars in the known universe. The rate of viral infections in the oceans is about 1 × 1023 infections per second, and these infections remove 20–40% of all bacterial cells each day. It has been estimated that 5 × 1030 microbial cells exist on earth; excluding cellulose, these cells constitute about 90% of the biomass of the entire biosphere. Humans also have an intimate relationship with microorganisms; more than 90% of the cells in our bodies are microbes. The bacteria present in the average human gut weigh about 1 kg, and a human adult will excrete his or her own weight in fecal bacteria each year. The number of genes contained within this gut flora outnumber that contained within our genome 150-fold, and even in our own genome, 8% of the DNA is derived from remnants of viral genomes. BIOLOGIC PRINCIPLES ILLUSTRATED BY MICROBIOLOGY Nowhere is biologic diversity demonstrated more dramatically than by microorganisms, creatures that are not directly 1 H A P T E R visible to the unaided eye. In form and function, be it biochemical property or genetic mechanism, analysis of microorganisms takes us to the limits of biologic understanding. Thus, the need for originality—one test of the merit of a scientific hypothesis—can be fully met in microbiology. A useful hypothesis should provide a basis for generalization, and microbial diversity provides an arena in which this challenge is ever present. Prediction, the practical outgrowth of science, is a product created by a blend of technique and theory. Biochemistry, molecular biology, and genetics provide the tools required for analysis of microorganisms. Microbiology, in turn, extends the horizons of these scientific disciplines. A biologist might describe such an exchange as mutualism, that is, one that benefits all of the contributing parties. Lichens are an example of microbial mutualism. Lichens consist of a fungus and phototropic partner, either an alga (a eukaryote) or a cyanobacterium (a prokaryote). The phototropic component is the primary producer, and the fungus provides the phototroph with an anchor and protection from the elements. In biology, mutualism is called symbiosis, a continuing association of different organisms. If the exchange operates primarily to the benefit of one party, the association is described as parasitism, a relationship in which a host provides the primary benefit to the parasite. Isolation and characterization of a parasite—such as a pathogenic bacterium or virus—often require effective mimicry in the laboratory of the growth environment provided by host cells. This demand sometimes represents a major challenge to investigators. The terms mutualism, symbiosis, and parasitism relate to the science of ecology, and the principles of environmental biology are implicit in microbiology. Microorganisms are 1 2 SECTION I Fundamentals of Microbiology the products of evolution, the biologic consequence of natural selection operating on a vast array of genetically diverse organisms. It is useful to keep the complexity of natural history in mind before generalizing about microorganisms, the most heterogeneous subset of all living creatures. A major biologic division separates the eukaryotes, organisms containing a membrane-bound nucleus, from prokaryotes, organisms in which DNA is not physically separated from the cytoplasm. As described in this chapter and in Chapter 2, further major distinctions can be made between eukaryotes and prokaryotes. Eukaryotes, for example, are distinguished by their relatively large size and by the presence of specialized membrane-bound organelles such as mitochondria. As described more fully later in this chapter, eukaryotic microorganisms—or, phylogenetically speaking, the Eukarya—are unified by their distinct cell structure and phylogenetic history. Among the groups of eukaryotic microorganisms are the algae, the protozoa, the fungi, and the slime molds. VIRUSES The unique properties of viruses set them apart from living creatures. Viruses lack many of the attributes of cells, including the ability to replicate. Only when it infects a cell does a virus acquire the key attribute of a living system— reproduction. Viruses are known to infect all cells, including microbial cells. Recently, viruses called virophages have been discovered that infect other viruses. Host–virus interactions tend to be highly specific, and the biologic range of viruses mirrors the diversity of potential host cells. Further diversity of viruses is exhibited by their broad array of strategies for replication and survival. Viral particles are generally small (eg, adenovirus is 90 nm) and consist of a nucleic acid molecule, either DNA or RNA, enclosed in a protein coat, or capsid (sometimes itself enclosed by an envelope of lipids, proteins, and carbohydrates). Proteins—frequently glycoproteins—in the capsid determine the specificity of interaction of a virus with its host cell. The capsid protects the nucleic acid and facilitates attachment and penetration of the host cell by the virus. Inside the cell, viral nucleic acid redirects the host’s enzymatic machinery to functions associated with replication of the virus. In some cases, genetic information from the virus can be incorporated as DNA into a host chromosome. In other instances, the viral genetic information can serve as a basis for cellular manufacture and release of copies of the virus. This process calls for replication of the viral nucleic acid and production of specific viral proteins. Maturation consists of assembling newly synthesized nucleic acid and protein subunits into mature viral particles, which are then liberated into the extracellular environment. Some very small viruses require the assistance of another virus in the host cell for their duplication. The delta agent, also known as hepatitis D virus, is too small to code for even a single capsid protein and needs help from hepatitis B virus for transmission. Viruses are known to infect a wide variety of plant and animal hosts as well as protists, fungi, and bacteria. However, most viruses are able to infect specific types of cells of only one host species. Some viruses are large and complex. For example, Mimivirus, a DNA virus infecting Acanthamoeba, a freeliving soil ameba, has a diameter of 400–500 nm and a genome that encodes 979 proteins, including the first four aminoacyl tRNA synthetases ever found outside of cellular organisms and enzymes for polysaccharide biosynthesis. An even larger marine virus has recently been discovered (Megavirus); its genome (1,259,197-bp) encodes 1120 putative proteins and is larger than that of some bacteria (Table 7-1). Because of their large size, these viruses resemble bacteria when observed in stained preparations by light microscopy; however, they do not undergo cell division or contain ribosomes. A number of transmissible plant diseases are caused by viroids—small, single-stranded, covalently closed circular RNA molecules existing as highly base-paired rodlike structures. They range in size from 246 to 375 nucleotides in length. The extracellular form of the viroid is naked RNA— there is no capsid of any kind. The RNA molecule contains no protein-encoding genes, and the viroid is therefore totally dependent on host functions for its replication. Viroid RNA is replicated by the DNA-dependent RNA polymerase of the plant host; preemption of this enzyme may contribute to viroid pathogenicity. The RNAs of viroids have been shown to contain inverted repeated base sequences at their 3′ and 5′ ends, a characteristic of transposable elements (see Chapter 7) and retroviruses. Thus, it is likely that they have evolved from transposable elements or retroviruses by the deletion of internal sequences. The general properties of animal viruses pathogenic for humans are described in Chapter 29. Bacterial viruses are described in Chapter 7. PRIONS A number of remarkable discoveries in the past 3 decades have led to the molecular and genetic characterization of the transmissible agent causing scrapie, a degenerative central nervous system disease of sheep. Studies have identified a scrapie-specific protein in preparations from scrapie-infected brains of sheep that is capable of reproducing the symptoms of scrapie in previously uninfected sheep (Figure 1-1). Attempts to identify additional components, such as nucleic acid, have been unsuccessful. To distinguish this agent from viruses and viroids, the term prion was introduced to emphasize its proteinaceous and infectious nature. The cellular form of the prion protein (PrPc) is encoded by the host’s chromosomal DNA. PrPc is a sialoglycoprotein with a molecular mass of 33,000–35,000 daltons and a high content of α-helical secondary structure that is sensitive to proteases and soluble in detergent. PrPc is expressed on the surface of neurons via a glycosylphosphatidyl inositol anchor in both infected and CHAPTER 1 The Science of Microbiology 3 Both normal prion protein (NP) and abnormal prion protein (PP) are present. PP NP Step 1 Abnormal prion protein interacts with the normal prion protein. PP Step 2 The normal prion protein is converted to the abnormal prion protein. Neuron NP 50 µm Converted NPs Figure 1-1 Prion. Prions isolated from the brain of a scrapieinfected hamster. This neurodegenerative disease is caused by a prion. (Reproduced with permission from Stanley B. Prusiner.) Original PP Steps 3 and 4 The abnormal prion proteins continue to interact with normal prion proteins until they convert all of the normal prion proteins to abnormal prion proteins. Converted NP uninfected brains. A conformational change occurs in the prion protein, changing it from its normal or cellular form PrPc to the disease-causing conformation, PrPSc (Figure 1-2). When PrPSc is present in an individual (owing to spontaneous conformational conversion or to infection), it is capable of recruiting PrPc and converting it to the disease form. Thus, prions replicate using the PrPc substrate that is present in the host. There are additional prion diseases of importance (Table 1-1 and Chapter 42). Kuru, Creutzfeldt-Jakob disease (CJD), Gerstmann-Sträussler-Scheinker disease, and fatal familial insomnia affect humans. Bovine spongiform encephalopathy, which is thought to result from the ingestion of feeds and bone meal prepared from rendered sheep offal, has been responsible for the deaths of more than 184,000 cattle in Great Britain since its discovery in 1985. A new variant of CJD (vCJD) has been associated with human ingestion of prion-infected beef in the United Kingdom and France. A common feature of all of these diseases is the conversion of a host-encoded sialoglycoprotein to a protease-resistant form as a consequence of infection. Human prion diseases are unique in that they manifest as sporadic, genetic, and infectious diseases. The study of prion biology is an important emerging area of biomedical investigation, and much remains to be learned. The distinguishing features of the nonliving members of the microbial world are given in Table 1-2. Abnormal prion proteins Figure 1-2 Proposed mechanism by which prions replicate. The normal and abnormal prion proteins differ in their tertiary structure. (Reproduced with permission from Nester EW, Anderson DG, Roberts CE, Nester MT (editors): Microbiology: A Human Perspective, 6th ed. McGraw-Hill, 2009, p. 342.) PROKARYOTES The primary distinguishing characteristics of the prokaryotes are their relatively small size, usually on the order of 1 μm in diameter, and the absence of a nuclear membrane. The DNA of almost all bacteria is a circle with a length of about 1 mm; this is the prokaryotic chromosome. Most prokaryotes have only a single chromosome. The chromosomal DNA must be folded more than 1000-fold just to fit within the prokaryotic cell membrane. Substantial evidence suggests that the folding may be orderly and may bring specified regions of the DNA 4 SECTION I Fundamentals of Microbiology TABLE 1-1 Common Human and Animal Prion Diseases Type Name Etiology Human prion diseases Acquired Variant Creutzfeldt-Jakob diseasea Associated with ingestion or inoculation of prion-infected material Kuru Iatrogenic Creutzfeldt-Jakob diseaseb Sporadic Creutzfeldt-Jakob disease Source of infection unknown Familial Gerstmann-Sträussler-Scheinker Associated with specific mutations within the gene encoding PrP Fatal familial insomnia Creutzfeldt-Jakob disease Animal prion diseases a Cattle Bovine spongiform encephalopathy Exposure to prion-contaminated meat and bone meal Sheep Scrapie Ingestion of scrapie-contaminated material Deer, elk Chronic wasting disease Ingestion of prion-contaminated material Mink Transmissible mink encephalopathy Source of infection unknown Cats Feline spongiform encephalopathy Exposure to prion-contaminated meat and bone meal a Associated with exposure to bovine spongiform encephalopathy–contaminated materials. Associated with prion-contaminated biologic materials, such as dura mater grafts, corneal transplants, and cadaver-derived human growth hormone, or prion-contaminated surgical instruments. b PrP, prion protein. Reproduced with permission from the American Society for Microbiology. Priola SA: How animal prions cause disease in humans. Microbe 2008;3(12):568. Table 1-2 Distinguishing Characteristics of Viruses, Viroids, and Prions Viruses Viroids Prions Obligate intracellular agents Obligate intracellular agents Abnormal form of a cellular protein Consist of either DNA or RNA surrounded by a protein coat Consist only of RNA; no protein coat Consist only of protein; no DNA or RNA Reproduced with permission from Nester EW, Anderson DG, Roberts CE, Nester MT (editors): Microbiology: A Human Perspective, 6th ed. McGraw-Hill, 2009, p. 13. into proximity. The specialized region of the cell containing DNA is termed the nucleoid and can be visualized by electron microscopy as well as by light microscopy after treatment of the cell to make the nucleoid visible. Thus, it would be a mistake to conclude that subcellular differentiation, clearly demarcated by membranes in eukaryotes, is lacking in prokaryotes. Indeed, some prokaryotes form membrane-bound subcellular structures with specialized function such as the chromatophores of photosynthetic bacteria (see Chapter 2). Prokaryotic Diversity The small size of the prokaryotic chromosome limits the amount of genetic information it can contain. Recent data based on genome sequencing indicate that the number of genes within a prokaryote may vary from 468 in Mycoplasma genitalium to 7825 in Streptomyces coelicolor, and many of these genes must be dedicated to essential functions such as energy generation, macromolecular synthesis, and cellular replication. Any one prokaryote carries relatively few genes that allow physiologic accommodation of the organism to its environment. The range of potential prokaryotic environments is unimaginably broad, and it follows that the prokaryotic group encompasses a heterogeneous range of specialists, each adapted to a rather narrowly circumscribed niche. The range of prokaryotic niches is illustrated by consideration of strategies used for generation of metabolic energy. Light from the sun is the chief source of energy for life. Some prokaryotes such as the purple bacteria convert light energy to metabolic energy in the absence of oxygen production. Other prokaryotes, exemplified by the blue-green bacteria (Cyanobacteria), produce oxygen that can provide energy through respiration in the absence of light. Aerobic organisms depend on respiration with oxygen for their energy. CHAPTER 1 The Science of Microbiology 5 Some anaerobic organisms can use electron acceptors other than oxygen in respiration. Many anaerobes carry out fermentations in which energy is derived by metabolic rearrangement of chemical growth substrates. The tremendous chemical range of potential growth substrates for aerobic or anaerobic growth is mirrored in the diversity of prokaryotes that have adapted to their utilization. Prokaryotic Communities A useful survival strategy for specialists is to enter into consortia, arrangements in which the physiologic characteristics of different organisms contribute to survival of the group as a whole. If the organisms within a physically interconnected community are directly derived from a single cell, the community is a clone that may contain up to 108 cells. The biology of such a community differs substantially from that of a single cell. For example, the high cell number virtually ensures the presence within the clone of at least one cell carrying a variant of any gene on the chromosome. Thus, genetic variability—the wellspring of the evolutionary process called natural selection—is ensured within a clone. The high number of cells within clones also is likely to provide physiologic protection to at least some members of the group. Extracellular polysaccharides, for example, may afford protection against potentially lethal agents such as antibiotics or heavy metal ions. Large amounts of polysaccharides produced by the high number of cells within a clone may allow cells within the interior to survive exposure to a lethal agent at a concentration that might kill single cells. Many bacteria exploit a cell–cell communication mechanism called quorum sensing to regulate the transcription of genes involved in diverse physiologic processes, including bioluminescence, plasmid conjugal transfer, and the production of virulence determinants. Quorum sensing depends on the production of one or more diffusible signal molecules termed autoinducers or pheromones that enable a bacterium to monitor its own cell population density. It is an example of multicellular behavior in prokaryotes. A distinguishing characteristic of prokaryotes is their capacity to exchange small packets of genetic information. This information may be carried on plasmids, small and specialized genetic elements that are capable of replication within at least one prokaryotic cell line. In some cases, plasmids may be transferred from one cell to another and thus may carry sets of specialized genetic information through a population. Some plasmids exhibit a broad host range that allows them to convey sets of genes to diverse organisms. Of particular concern are drug resistance plasmids that may render diverse bacteria resistant to antibiotic treatment. The survival strategy of a single prokaryotic cell line may lead to a range of interactions with other organisms. These may include symbiotic relationships illustrated by complex nutritional exchanges among organisms within the human gut. These exchanges benefit both the microorganisms and their human host. Parasitic interactions can be quite deleterious to the host. Advanced symbiosis or parasitism can lead to loss of functions that may not allow growth of the symbiont or parasite independent of its host. The mycoplasmas, for example, are parasitic prokaryotes that have lost the ability to form a cell wall. Adaptation of these organisms to their parasitic environment has resulted in incorporation of a substantial quantity of cholesterol into their cell membranes. Cholesterol, not found in other prokaryotes, is assimilated from the metabolic environment provided by the host. Loss of function is exemplified also by obligate intracellular parasites, the chlamydiae and rickettsiae. These bacteria are extremely small (0.2–0.5 μm in diameter) and depend on the host cell for many essential metabolites and coenzymes. This loss of function is reflected by the presence of a smaller genome with fewer genes (see Table 7-1). The most widely distributed examples of bacterial symbionts appear to be chloroplasts and mitochondria, the energy-yielding organelles of eukaryotes. A substantial body of evidence points to the conclusion that ancestors of these organelles were endosymbionts, prokaryotes that established symbiosis within the cell membrane of the ancestral eukaryotic host. The presence of multiple copies of the organelles may have contributed to the relatively large size of eukaryotic cells and to their capacity for specialization, a trait ultimately reflected in the evolution of differentiated multicellular organisms. Classification of the Prokaryotes An understanding of any group of organisms requires their classification. An appropriate classification system allows a scientist to choose characteristics that allow swift and accurate categorization of a newly encountered organism. The categorization allows prediction of many additional traits shared by other members of the category. In a hospital setting, successful classification of a pathogenic organism may provide the most direct route to its elimination. Classification may also provide a broad understanding of relationships among different organisms, and such information may have great practical value. For example, elimination of a pathogenic organism will be relatively long-lasting if its habitat is occupied by a nonpathogenic variant. The principles of prokaryotic classification are discussed in Chapter 3. At the outset, it should be recognized that any prokaryotic characteristic might serve as a potential criterion for classification. However, not all criteria are equally effective in grouping organisms. Possession of DNA, for example, is a useless criterion for distinguishing organisms because all cells contain DNA. The presence of a broad host range plasmid is not a useful criterion because such plasmids may be found in diverse hosts and need not be present all of the time. Useful criteria may be structural, physiologic, biochemical, or genetic. Spores—specialized cell structures that may allow survival in extreme environments—are useful structural criteria for classification because well-characterized subsets of bacteria form spores. Some bacterial groups can 6 SECTION I Fundamentals of Microbiology be effectively subdivided on the basis of their ability to ferment specified carbohydrates. Such criteria may be ineffective when applied to other bacterial groups that may lack any fermentative capability. A biochemical test, the Gram stain, is an effective criterion for classification because response to the stain reflects fundamental and complex differences in the bacterial cell surface that divide most bacteria into two major groups. Genetic criteria are increasingly used in bacterial classification, and many of these advances are made possible by the development of DNA-based technologies. It is now possible to design DNA probe or DNA amplification assays (eg, polymerase chain reaction [PCR] assays) that swiftly identify organisms carrying specified genetic regions with common ancestry. Comparison of DNA sequences for some genes led to the elucidation of phylogenetic relationships among prokaryotes. Ancestral cell lines can be traced, and organisms can be grouped on the basis of their evolutionary affinities. These investigations have led to some striking conclusions. For example, comparison of cytochrome c sequences suggests that all eukaryotes, including humans, arose from one of three different groups of purple photosynthetic bacteria. This conclusion in part explains the evolutionary origin of eukaryotes, but it does not fully take into account the generally accepted view that the eukaryotic cell was derived from the evolutionary merger of different prokaryotic cell lines. Bacteria and Archaebacteria: The Major Subdivisions Within the Prokaryotes A major success in molecular phylogeny has been the demonstration that prokaryotes fall into two major groups. Most investigations have been directed to one group, the bacteria. The other group, the archaebacteria, has received relatively little attention until recently, partly because many of its representatives are difficult to study in the laboratory. Some archaebacteria, for example, are killed by contact with oxygen, and others grow at temperatures exceeding that of boiling water. Before molecular evidence became available, the major subgroupings of archaebacteria seemed disparate. The methanogens carry out an anaerobic respiration that gives rise to methane, the halophiles demand extremely high salt concentrations for growth, and the thermoacidophiles require high temperature and acidity. It has now been established that these prokaryotes share biochemical traits such as cell wall or membrane components that set the group entirely apart from all other living organisms. An intriguing trait shared by archaebacteria and eukaryotes is the presence of introns within genes. The function of introns—segments of DNA that interrupts informational DNA within genes—is not established. What is known is that introns represent a fundamental characteristic shared by the DNA of archaebacteria and eukaryotes. This common trait has led to the suggestion that—just as mitochondria and chloroplasts appear to be evolutionary derivatives of the bacteria—the eukaryotic nucleus may have arisen from an archaebacterial ancestor. PROTISTS The “true nucleus” of eukaryotes (from Gr karyon, “nucleus”) is only one of their distinguishing features. The membranebound organelles, the microtubules, and the microfilaments of eukaryotes form a complex intracellular structure unlike that found in prokaryotes. The agents of motility for eukaryotic cells are flagella or cilia—complex multistranded structures that do not resemble the flagella of prokaryotes. Gene expression in eukaryotes takes place through a series of events achieving physiologic integration of the nucleus with the endoplasmic reticulum, a structure that has no counterpart in prokaryotes. Eukaryotes are set apart by the organization of their cellular DNA in chromosomes separated by a distinctive mitotic apparatus during cell division. In general, genetic transfer among eukaryotes depends on fusion of haploid gametes to form a diploid cell containing a full set of genes derived from each gamete. The life cycle of many eukaryotes is almost entirely in the diploid state, a form not encountered in prokaryotes. Fusion of gametes to form reproductive progeny is a highly specific event and establishes the basis for eukaryotic species. This term can be applied only metaphorically to the prokaryotes, which exchange fragments of DNA through recombination. Taxonomic groupings of eukaryotes frequently are based on shared morphologic properties, and it is noteworthy that many taxonomically useful determinants are those associated with reproduction. Almost all successful eukaryotic species are those in which closely related cells, members of the same species, can recombine to form viable offspring. Structures that contribute directly or indirectly to the reproductive event tend to be highly developed and—with minor modifications among closely related species—extensively conserved. Microbial eukaryotes—protists—are members of the four following major groups: algae, protozoa, fungi, and slime molds. It should be noted that these groupings are not necessarily phylogenetic: Closely related organisms may have been categorized separately because underlying biochemical and genetic similarities may not have been recognized. Algae The term algae has long been used to denote all organisms that produce O2 as a product of photosynthesis. One major subgroup of these organisms—the blue-green bacteria, or cyanobacteria—are prokaryotic and no longer are termed algae. This classification is reserved exclusively for photosynthetic eukaryotic organisms. All algae contain chlorophyll in the photosynthetic membrane of their subcellular chloroplast. Many algal species are unicellular microorganisms. Other algae may form extremely large multicellular structures. Kelps of brown algae sometimes are several hundred meters in length. A number of algae produce toxins that are poisonous to humans and other animals. Dinoflagellates, a unicellular alga, cause algal blooms, or red tides, in the ocean CHAPTER 1 The Science of Microbiology 7 Fungi FIGURE 1-3 The dinoflagellate Gymnodinium scanning electron micrograph (4000×). (Reproduced with permission from David M. Phillips/Visuals Unlimited.) (Figure 1-3). Red tides caused by the dinoflagellate Gonyaulax species are serious because this organism produces neurotoxins such as saxitoxin and gonyautoxins, which accumulate in shellfish (eg, clams, mussels, scallops, oysters) that feed on this organism. Ingestion of these shellfish by humans results in symptoms of paralytic shellfish poisoning and can lead to death. Protozoa Protozoa are unicellular nonphotosynthetic protists. The most primitive protozoa appear to be flagellated forms that in many respects resemble representatives of the algae. It seems likely that the ancestors of these protozoa were algae that became heterotrophs—the nutritional requirements of such organisms are met by organic compounds. Adaptation to a heterotrophic mode of life was sometimes accompanied by loss of chloroplasts, and algae thus gave rise to the closely related protozoa. Similar events have been observed in the laboratory to be the result of either mutation or physiologic adaptation. From flagellated protozoa appear to have evolved the ameboid and the ciliated types; intermediate forms are known that have flagella at one stage in the life cycle and pseudopodia (characteristic of the ameba) at another stage. A fourth major group of protozoa, the sporozoa, are strict parasites that are usually immobile; most of these reproduce sexually and asexually in alternate generations by means of spores. Protozoan parasites of humans are discussed in Chapter 46. The fungi are nonphotosynthetic protists growing as a mass of branching, interlacing filaments (“hyphae”) known as a mycelium. The largest known contiguous fungal mycelium covered an area of 2400 acres (9.7 km2) at a site in eastern Oregon. Although the hyphae exhibit cross walls, the cross walls are perforated and allow free passage of nuclei and cytoplasm. The entire organism is thus a coenocyte (a multinucleated mass of continuous cytoplasm) confined within a series of branching tubes. These tubes, made of polysaccharides such as chitin, are homologous with cell walls. The mycelial forms are called molds; a few types, yeasts, do not form a mycelium but are easily recognized as fungi by the nature of their sexual reproductive processes and by the presence of transitional forms. The fungi probably represent an evolutionary offshoot of the protozoa; they are unrelated to the actinomycetes, mycelial bacteria that they superficially resemble. The major subdivisions (phyla) of fungi are Chytridiomycota, Zygomycota (the zygomycetes), Ascomycota (the ascomycetes), Basidiomycota (the basidiomycetes), and the “deuteromycetes” (or imperfect fungi). The evolution of the ascomycetes from the phycomycetes is seen in a transitional group, members of which forms a zygote but then transform this directly into an ascus. The basidiomycetes are believed to have evolved in turn from the ascomycetes. The classification of fungi and their medical significance are discussed further in Chapter 45. Slime Molds These organisms are characterized by the presence, as a stage in their life cycle, of an ameboid multinucleate mass of cytoplasm called a plasmodium. The plasmodium of a slime mold is analogous to the mycelium of a true fungus. Both are coenocytic. Whereas in the latter, cytoplasmic flow is confined to the branching network of chitinous tubes, in the former, the cytoplasm can flow in all directions. This flow causes the plasmodium to migrate in the direction of its food source, frequently bacteria. In response to a chemical signal, 3′, 5′-cyclic AMP (see Chapter 7), the plasmodium, which reaches macroscopic size, differentiates into a stalked body that can produce individual motile cells. These cells, flagellated or ameboid, initiate a new round in the life cycle of the slime mold (Figure 1-4). The cycle frequently is initiated by sexual fusion of single cells. The life cycle of the slime molds illustrates a central theme of this chapter—the interdependency of living forms. The growth of slime molds depends on nutrients provided by bacterial or, in some cases, plant cells. Reproduction of the slime molds via plasmodia can depend on intercellular recognition and fusion of cells from the same species. Full understanding of a microorganism requires both knowledge of the other organisms with which it coevolved and an appreciation of the range of physiologic responses that may contribute to survival. 8 SECTION I Fundamentals of Microbiology Spores Fruiting bodies release spores Germination Myxamoebae Fruiting body Plasmodium A B Figure 1-4 Slime molds. A: Life cycle of an acellular slime mold. B: Fruiting body of a cellular slime mold. (Reproduced with permission from Carolina Biological Supply/Phototake, Inc.) CHAPTER SUMMARY • • • • • • • Microorganisms are a large and diverse group of microorganisms existing as single cells or clusters; they also include viruses, which are microscopic but not cellular. A virus consists of a nucleic acid molecule, either DNA or RNA, enclosed in a protein coat, or capsid, sometimes enclosed by an envelope composed of lipids, proteins, and carbohydrates. A prion is an infectious protein, which is capable of causing chronic neurologic diseases. Prokaryotes consist of bacteria and archaebacteria. Prokaryotes are haploid. Microbial eukaryotes, or protists, are members of four major groups: algae, protozoa, fungi, and slime molds. Eukaryotes have a true nucleus and are diploid. REVIEW QUESTIONS 1. Which one of the following terms characterizes the interaction between a fungus and algae in a lichen? (A) Parasitism (B) Symbiosis (C) Endosymbiosis (D) Endoparasitism (E) Consortia 2. Which one of the following agents lacks nucleic acid? (A) Bacteria (B) Viruses (C) Viroids (D) Prions (E) Protozoa 3. Which one of the following is not a protist? (A) Bacteria (B) Algae (C) Protozoa (D) Fungi (E) Slime molds 4. Which one of the following agents simultaneously contains both DNA and RNA? (A) Bacteria (B) Viruses (C) Viroids (D) Prions (E) Plasmids 5. A 65-year-old man develops dementia, progressive over several months, along with ataxia and somnolence. An electroencephalographic pattern shows paroxysms with high voltages and slow waves, suggestive of Creutzfeldt-Jakob disease. This disease is caused by which of the following agents? (A) Bacterium (B) Virus (C) Viroid (D) Prion (E) Plasmid 6. Which of the following cannot be infected by viruses? (A) Bacteria (B) Protozoa (C) Human cells (D) Viruses (E) None of the above CHAPTER 1 The Science of Microbiology 9 7. Viruses, bacteria, and protists are uniquely characterized by their respective size. True or false? (A) True (B) False 8. Which of the following are prokaryotes? (A) Archaebacteria (B) Protozoa (C) Viruses (D) Prions (E) Fungi 9. Quorum sensing in prokaryotes involves (A) Cell–cell communication (B) Production of pheromones (C) An example of multicellular behavior (D) Regulation of genes involved in diverse physiologic processes (E) All of the above 10. Twenty minutes after ingesting a raw clam, a 35-year-old man experiences paresthesias of the mouth and extremities, headache, and ataxia. These symptoms are the result of a neurotoxin produced by algae called (A) Amoeba (B) Blue-green algae (C) Dinoflagellates (D) Kelp (E) None of the above Answers 1. B A B D D A A E E C REFERENCES Arslan D, Legendre M, Seltzer V, et al: Distant Mimivirus relative with a larger genome highlights the fundamental features of Megaviridae. Proc Natl Acad Sci U S A 2011;108:17486. Belay ED: Transmissible spongiform encephalopathies in humans. Annu Rev Microbiol 1999;53:283. Colby DW, Prusiner SB: De novo generation of prion strains. Nature Rev Microbiol 2011;9:771. Diener TO: Viroids and the nature of viroid diseases. Arch Virol 1999;15(Suppl):203. Fournier PE, Raoult D: Prospects for the future using genomics and proteomics in clinical microbiology. Annu Rev Microbiol 2011;65:169. Lederberg J (editor): Encyclopedia of Microbiology, 4 vols. Academic Press, 1992. Olsen GJ, Woese CR: The winds of (evolutionary) change: Breathing new life into microbiology. J Bacteriol 1994;176:1. Pelczar MJ Jr, Chan ECS, Krieg NR: Microbiology: Concepts and Applications. McGraw-Hill, 1993. Priola SA: How animal prions cause disease in humans. Microbe 2008;3:568. Prusiner SB: Biology and genetics of prion diseases. Annu Rev Microbiol 1994;48:655. Reisser W (editor): Algae and Symbiosis: Plants, Animals, Fungi, Viruses, Interactions Explored. Biopress, 1992. Schloss PD, Handlesman J: Status of the microbial census. Microbiol Mol Biol Rev 2004;68:686. Sleigh MA: Protozoa and Other Protists. Chapman & Hall, 1990. Whitman WB, Coleman DC, Wiebe WJ: Prokaryotes: The unseen majority. Proc Natl Acad Sci U S A 1998;95:6578. This page intentionally left blank C Cell Structure This chapter discusses the basic structure and function of the components that make up eukaryotic and prokaryotic cells. The chapter begins with a discussion of the microscope. Historically, the microscope first revealed the presence of bacteria and later the secrets of cell structure. Today it remains a powerful tool in cell biology. OPTICAL METHODS The Light Microscope The resolving power of the light microscope under ideal conditions is about half the wavelength of the light being used. (Resolving power is the distance that must separate two point sources of light if they are to be seen as two distinct images.) With yellow light of a wavelength of 0.4 mm, the smallest separable diameters are thus about 0.2 mm (ie, onethird the width of a typical prokaryotic cell). The useful magnification of a microscope is the magnification that makes visible the smallest resolvable particles. Several types of light microscopes are commonly used in microbiology: A. Bright‑Field Microscope The bright-field microscope is most commonly used in microbiology courses and consists of two series of lenses (objective and ocular lens), which function together to resolve the image. These microscopes generally employ a 100-power objective lens with a 10-power ocular lens, thus magnifying the specimen 1000 times. Particles 0.2 mm in diameter are therefore magnified to about 0.2 mm and so become clearly visible. Further magnification would give no greater resolution of detail and would reduce the visible area (field). With this microscope, specimens are rendered visible because of the differences in contrast between them and the surrounding medium. Many bacteria are difficult to see well because of their lack of contrast with the surrounding medium. Dyes (stains) can be used to stain cells or their organelles and increase their contrast so they can be more easily seen in the bright-field microscope. 2 H A P T E R B. Phase Contrast Microscope The phase contrast microscope was developed to improve contrast differences between cells and the surrounding medium, making it possible to see living cells without staining them; with bright-field microscopes, killed and stained preparations must be used. The phase contrast microscope takes advantage of the fact that light waves passing through transparent objects, such as cells, emerge in different phases depending on the properties of the materials through which they pass. This effect is amplified by a special ring in the objective lens of a phase contrast microscope, leading to the formation of a dark image on a light background. C. Dark‑Field Microscope The dark-field microscope is a light microscope in which the lighting system has been modified to reach the specimen from the sides only. This is accomplished through the use of a special condenser that both blocks direct light rays and deflects light off a mirror on the side of the condenser at an oblique angle. This creates a “dark field” that contrasts against the highlighted edge of the specimens and results when the oblique rays are reflected from the edge of the specimen upward into the objective of the microscope. Resolution by dark-field microscopy is quite high. Thus, this technique has been particularly useful for observing organisms such as Treponema pallidum, a spirochete that is smaller than 0.2 mm in diameter and therefore cannot be observed with a brightfield or phase contrast microscope (Figure 2-1A). D. Fluorescence Microscope The fluorescence microscope is used to visualize specimens that fluoresce, which is the ability to absorb short wavelengths of light (ultraviolet) and give off light at a longer wavelength (visible). Some organisms fluoresce naturally because of the presence within the cells of naturally fluorescent substances such as chlorophyll. Those that do not naturally fluoresce may be stained with a group of fluorescent dyes called fluorochromes. Fluorescence microscopy is widely used in clinical diagnostic microbiology. For example, the 11 12 SECTION I Fundamentals of Microbiology fluorochrome auramine O, which glows yellow when exposed to ultraviolet light, is strongly absorbed by Mycobacterium tuberculosis, the bacterium that causes tuberculosis. When the dye is applied to a specimen suspected of containing M tuberculosis and exposed to ultraviolet light, the bacterium can be detected by the appearance of bright yellow organisms against a dark background. The principal use of fluorescence microscopy is a diagnostic technique called the fluorescent-antibody (FA) technique or immunofluorescence. In this technique, specific antibodies (eg, antibodies to Legionella pneumophila) are chemically labeled with a fluorochrome such as fluorescein isothiocyanate (FITC). These fluorescent antibodies are then added to a microscope slide containing a clinical specimen. If the specimen contains L pneumophila, the fluorescent antibodies will bind to antigens on the surface of the bacterium, causing it to fluoresce when exposed to ultraviolet light (Figure 2-1B). E. Differential Interference Contrast Microscope A B Differential interference contrast (DIC) microscopes employ a polarizer to produce polarized light. The polarized light beam passes through a prism that generates two distinct beams; these beams pass through the specimen and enter the objective lens, where they are recombined into a single beam. Because of slight differences in refractive index of the substances each beam passed through, the combined beams are not totally in phase but instead create an interference effect, which intensifies subtle differences in cell structure. Structures such as spores, vacuoles, and granules appear three-dimensional. DIC microscopy is particularly useful for observing unstained cells because of its ability to generate images that reveal internal cell structures that are less apparent by bright-field techniques. 10 µm The Electron Microscope C Figure 2-1 A: Positive dark-field examination. Treponemes are recognizable by their characteristic corkscrew shape and deliberate forward and backward movement with rotation about the longitudinal axis. (Reproduced with permission from Charles Stratton/Visuals Unlimited.) B: Fluorescence photomicrograph. A rod-shaped bacterium tagged with a fluorescent marker. (© Evans Roberts.) C: Scanning electron microscope of bacteria— Staphylococcus aureus (32,000×). (Reproduced with permission from David M. Phillips/Photo Researchers, Inc.) The high resolving power of electron microscopes has enabled scientists to observe the detailed structures of prokaryotic and eukaryotic cells. The superior resolution of the electron microscope is due to the fact that electrons have a much shorter wavelength than the photons of white light. There are two types of electron microscopes in general use: the transmission electron microscope (TEM), which has many features in common with the light microscope, and the scanning electron microscope (SEM). The TEM was the first to be developed and uses a beam of electrons projected from an electron gun and directed or focused by an electromagnetic condenser lens onto a thin specimen. As the electrons strike the specimen, they are differentially scattered by the number and mass of atoms in the specimen; some electrons pass through the specimen and are gathered and focused by an electromagnetic objective lens, which presents an image of the specimen to the projector lens system for further enlargement. The image is visualized by allowing it to impinge on a screen that fluoresces when struck with the electrons. The image can be recorded on photographic film. TEM can resolve particles 0.001 mm CHAPTER 2 Cell Structure 13 apart. Viruses with diameters of 0.01–0.2 mm can be easily resolved. The SEM generally has a lower resolving power than the TEM; however, it is particularly useful for providing threedimensional images of the surface of microscopic objects. Electrons are focused by means of lenses into a very fine point. The interaction of electrons with the specimen results in the release of different forms of radiation (eg, secondary electrons) from the surface of the material, which can be captured by an appropriate detector, amplified, and then imaged on a television screen (Figure 2-1C). An important technique in electron microscopy is the use of “shadowing.” This involves depositing a thin layer of heavy metal (eg, platinum) on the specimen by placing it in the path of a beam of metal ions in a vacuum. The beam is directed at a low angle to the specimen so that it acquires a “shadow” in the form of an uncoated area on the other side. When an electron beam is then passed through the coated preparation in the electron microscope and a positive print is made from the “negative” image, a three-dimensional effect is achieved (eg, see Figure 2-22). Other important techniques in electron microscopy include the use of ultrathin sections of embedded material, a method of freeze-drying specimens that prevents the distortion caused by conventional drying procedures, and the use of negative staining with an electron-dense material such as phosphotungstic acid or uranyl salts (eg, see Figure 42-1). Without these heavy metal salts, there would not be enough contrast to detect the details of the specimen. Confocal Scanning Laser Microscope The confocal scanning laser microscope (CSLM) couples a laser light source to a light microscope. In confocal scanning laser microscopy, a laser beam is bounced off a mirror that directs the beam through a scanning device. Then the laser beam is directed through a pinhole that precisely adjusts the plane of focus of the beam to a given vertical layer within the specimen. By precisely illuminating only a single plane of the specimen, illumination intensity drops off rapidly above and below the plane of focus, and stray light from other planes of focus are minimized. Thus, in a relatively thick specimen, various layers can be observed by adjusting the plane of focus of the laser beam. Cells are often stained with fluorescent dyes to make them more visible. Alternatively, false color images can be generated by adjusting the microscope in such a way as to make different layers take on different colors. The CSLM is equipped with computer software to assemble digital images for subsequent image processing. Thus, images obtained from different layers can be stored and then digitally overlaid to reconstruct a three-dimensional image of the entire specimen. Scanning Probe Microscopes A new class of microscopes, called scanning probe microscopes, measure surface features by moving a sharp probe over the object’s surface. The scanning tunneling microscope and the atomic force microscope are examples of this new class of microscopes, which enable scientists to view atoms or molecules on the surface of a specimen. For example, interactions between proteins of the bacterium Escherichia coli can be studied with the atomic force microscope. EUKARYOTIC CELL STRUCTURE The Nucleus The nucleus contains the cell’s genome. It is bounded by a membrane that consists of a pair of unit membranes separated by a space of variable thickness. The inner membrane is usually a simple sac, but the outermost membrane is, in many places, continuous with the endoplasmic reticulum (ER). The nuclear membrane exhibits selective permeability because of pores, which consist of a complex of several proteins whose function is to import substances into and export substances out of the nucleus. The chromosomes of eukaryotic cells contain linear DNA macromolecules arranged as a double helix. They are only visible with a light microscope when the cell is undergoing division and the DNA is in a highly condensed form; at other times, the chromosomes are not condensed and appear as in Figure 2-2. Eukaryotic DNA macromolecules are associated with basic proteins called histones that bind to the DNA by ionic interactions. A structure often visible within the nucleus is the nucleolus, an area rich in RNA that is the site of ribosomal RNA synthesis (see Figure 2-2). Ribosomal proteins synthesized in the cytoplasm are transported into the nucleolus and combine with ribosomal RNA to form the small and large subunits of the eukaryotic ribosome. These are then exported to the cytoplasm, where they associate to form an intact ribosome that can function in protein synthesis. Cytoplasmic Structures The cytoplasm of eukaryotic cells is characterized by the presence of an ER, vacuoles, self-reproducing plastids, and an elaborate cytoskeleton composed of microtubules, microfilaments, and intermediate filaments. The endoplasmic reticulum (ER) is a network of membrane-bound channels continuous with the nuclear membrane. Two types of ER are recognized: rough, which contains attached 80S ribosomes, and smooth, which does not (see Figure 2-2). Rough ER is a major producer of glycoproteins and produces new membrane material that is transported throughout the cell; smooth ER participates in the synthesis of lipids and in some aspects of carbohydrate metabolism. The Golgi apparatus consists of a stack of membranes that function in concert with the ER to chemically modify and sort products of the ER into those destined to be secreted and those that function in other membranous structures of the cell. The plastids include mitochondria and chloroplasts. Several lines of evidence suggest that mitochondria and chloroplasts were descendents of ancient prokaryotic 14 SECTION I Fundamentals of Microbiology Pores Chromatin Nucleolus Figure 2-2 Electron micrograph of a thin section of a typical eukaryotic nucleus showing a prominent nucleolus and large aggregations of heterochromatin against the nuclear membrane, which is traversed by pores (arrows). Inset upper left: Two nuclear pores and their pore diaphragms. Inset lower right: The fibrous lamina present in the inner aspect of the nuclear envelope. The endoplasmic reticulum and several mitochondria are visible in the cytoplasm. (Reproduced with permission from Fawcett DW: Bloom and Fawcett, A Textbook of Histology, 12th ed. Copyright © 1994. By permission of Chapman & Hall, New York, NY.) organisms and arose from the engulfment of a prokaryotic cell by a larger cell (endosymbiosis). Mitochondria are of prokaryotic size, and its membrane, which lacks sterols, is much less rigid than the eukaryotic cell’s cytoplasmic membrane, which does contain sterols. Mitochondria contain two sets of membranes. The outermost membrane is rather permeable, having numerous minute channels that allow passage of ions and small molecules (eg, adenosine triphosphate [ATP]). Invagination of the outer membrane forms a system of inner folded membranes called cristae. The cristae are the sites of enzymes involved in respiration and ATP production. Cristae also contain specific transport proteins that regulate passage of metabolites into and out of the mitochondrial matrix. The matrix contains a number of enzymes, particularly those of the citric acid cycle. Chloroplasts are photosynthetic cell organelles that are capable of converting the energy of sunlight into chemical energy through photosynthesis. Chlorophyll and all other components needed for photosynthesis are located in a series of flattened membrane discs called thylakoids. The size, shape, and number of chloroplasts per cell vary markedly; in contrast to mitochondria, chloroplasts are generally much larger than prokaryotes. Mitochondria and chloroplasts contain their own DNA, which exists in a covalently closed circular form and codes for some (not all) of their constituent proteins and transfer RNAs. Mitochondria and chloroplasts also contain 70S ribosomes, the same as those of prokaryotes. Some eukaryotic microorganisms (eg, Trichomonas vaginalis) lack mitochondria and contain instead a membraneenclosed respiratory organelle called the hydrogenosome. Hydrogenosomes may have arisen by endosymbiosis, and some have been identified that contain DNA and ribosomes. The hydrogenosome, although similar in size to mitochondria, lacks cristae and the enzymes of the tricarboxylic acid cycle. Pyruvate is taken up by the hydrogenosome, and H2, CO2, acetate, and ATP are produced. Lysosomes are membrane-enclosed sacs that contain various digestive enzymes that the cell uses to digest macromolecules such as proteins, fats, and polysaccharides. The lysosome allows these enzymes to be partitioned away from the cytoplasm proper, where they could destroy key cellular macromolecules if not contained. After the hydrolysis of macromolecules in the lysosome, the resulting monomers pass from the lysosome into the cytoplasm, where they serve as nutrients. The peroxisome is a membrane-enclosed structure whose function is to produce H2O2 from the reduction of O2 by various hydrogen donors. The H2O2 produced in the peroxisome is subsequently degraded to H2O and O2 by the enzyme catalase. The cytoskeleton is a three-dimensional structure that fills the cytoplasm. The primary types of fibers comprising the cytoskeleton are microfilaments, intermediate filaments, and microtubules. Microfilaments are about 3–6 nm in diameter and are polymers composed of subunits of the protein actin. These fibers form scaffolds throughout the cell, defining and maintaining the shape of the cell. Microfilaments can also carry out cellular movements, including gliding, contraction, and cytokinesis. Microtubules are cylindrical tubes 20–25 nm in diameter and are composed of subunits of the protein tubulin. Microtubules assist microfilaments in maintaining cell structure, form the spindle fibers for separating chromosomes during mitosis, and play an important role in cell motility. Intermediate filaments are about 10 nm in diameter and provide tensile strength for the cell. Surface Layers The cytoplasm is enclosed within a plasma membrane composed of protein and phospholipid similar to the prokaryotic cell membrane illustrated later (see Figure 2-10). Most animal cells have no other surface layers; however, plant cells have an outer cell wall composed of cellulose. Many eukaryotic CHAPTER 2 Cell Structure 15 of the cell, and cilia, which are shorter than flagella, surround the cell (Figure 2-3). Both the flagella and the cilia of eukaryotic cells have the same basic structure and biochemical composition. Both consist of a series of microtubules, hollow protein cylinders composed of a protein called tubulin surrounded by a membrane. The arrangement of the microtubules is called the “9 + 2 system” because it consists of nine peripheral pairs of microtubules surrounding two single central microtubules (Figure 2-4). PROKARYOTIC CELL STRUCTURE 20 µm Figure 2-3 A paramecium moves with the aid of cilia on the cell surface. (© Manfred Kage). microorganisms also have an outer cell wall, which may be composed of a polysaccharide such as cellulose or chitin or may be inorganic (eg, the silica wall of diatoms). Motility Organelles Many eukaryotic microorganisms have organelles called flagella (eg, T vaginalis) or cilia (eg, Paramecium) that move with a wavelike motion to propel the cell through water. Eukaryotic flagella emanate from the polar region The prokaryotic cell is simpler than the eukaryotic cell at every level, with one exception: The cell envelope is more complex. The Nucleoid Prokaryotes have no true nuclei; instead they package their DNA in a structure known as the nucleoid. The nucleoid can be seen with the light microscope in stained material (Figure 2-5). It is Feulgen positive, indicating the presence of DNA. The negatively charged DNA is at least partially neutralized by small polyamines and magnesium ions, but histone-like proteins exist in bacteria and presumably play a role similar to that of histones in eukaryotic chromatin. Electron micrographs of a typical prokaryotic cell such as Figure 2-5 reveal the absence of a nuclear membrane and a mitotic apparatus. The exception to this rule is the Outer dynein arm Inner dynein arm Central microtubule Spoke head Radial spoke Doublet microtubule Nexin link Central sheath Subtubule A Subtubule B A B Figure 2-4 Cilia and flagella structure. A: An electron micrograph of a cilium cross section. Note the two central microtubles surrounded by nine microtubule doublets (160,000×). (Reproduced with permission from KG Murti/Visuals Unlimited.) B: A diagram of cilia and flagella structure. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ [editors]: Prescott, Harley, and Klein’s Microbiology, 7th ed. New York: McGraw-Hill; 2008. © The McGraw-Hill Companies, Inc.) 16 SECTION I Fundamentals of Microbiology A 0.5 µm Figure 2-5 Nucleoids of Bacillus cereus (2500×). (Reproduced with permission from Robinow C: The chromatin bodies of bacteria. Bacteriol Rev 1956;20:207.) DNA fibers planctomycetes, a divergent group of aquatic bacteria, which have a nucleoid surrounded by a nuclear envelope consisting of two membranes. The distinction between prokaryotes and eukaryotes that still holds is that prokaryotes have no eukaryotic-type mitotic apparatus. The nuclear region (Figure 2-6) is filled with DNA fibrils. The nucleoid of most bacterial cells consists of a single continuous circular molecule ranging in size from 0.58 to almost 10 million base pairs. However, a few bacteria have been shown to have two, three, or even four dissimilar chromosomes. For example, Vibrio cholerae and Brucella melitensis have two dissimilar chromosomes. There are exceptions to this rule of circularity because some prokaryotes (eg, Borrelia burgdorferi and Streptomyces coelicolor) have been shown to have a linear chromosome. In bacteria, the number of nucleoids, and therefore the number of chromosomes, depend on the growth conditions (Figure 2-5). Rapidly growing bacteria have more nucleoids per cell than slowly growing ones; however, when multiple copies are present, they are all the same (ie, prokaryotic cells are haploid). Cytoplasmic Structures Prokaryotic cells lack autonomous plastids, such as mitochondria and chloroplasts; the electron transport enzymes are localized instead in the cytoplasmic membrane. The photosynthetic pigments (carotenoids, bacteriochlorophyll) of photosynthetic bacteria are contained in intracytoplasmic membrane systems of various morphologies. Membrane vesicles (chromatophores) or lamellae are commonly observed membrane types. Some photosynthetic bacteria have specialized nonunit membrane-enclosed structures called chlorosomes. Membrane Ruptured cell B Figure 2-6 The nucleoid. A: Color-enhanced transmission electron micrograph of Escherichia coli with the DNA shown in red. (© CNRI/SPL/Photo Researchers, Inc.) B: Chromosome released from a gently lysed cell of E coli. Note how tightly packaged the DNA must be inside the bacterium. (© Dr. Gopal Murti/SPL/Photo Researchers.) In some Cyanobacteria (formerly known as blue-green algae), the photosynthetic membranes often form multilayered structures known as thylakoids (Figure 2-7). The major accessory pigments used for light harvesting are the phycobilins found on the outer surface of the thylakoid membranes. Bacteria often store reserve materials in the form of insoluble granules, which appear as refractile bodies in the cytoplasm when viewed by phase contrast microscopy. These so-called inclusion bodies almost always function in the storage of energy or as a reservoir of structural building blocks. Most cellular inclusions are bounded by a thin nonunit membrane consisting of lipid, which serves to separate the inclusion from the cytoplasm proper. One of the most common inclusion bodies consists of poly-ahydroxybutyric acid (PHB), a lipid-like compound consisting of chains of β-hydroxybutyric acid units connected through ester linkages. PHB is produced when the source CHAPTER 2 Cell Structure 17 t A Plasma membrane Cell wall Phycobilisomes Thylakoids 1µm Carboxysome 70S ribosome B Figure 2-7 A: Thin section of Synechococcus lividus showing an extensive thylakoid system. The phycobillisomes lining these thylakoids are clearly visible as granules at location t (85,000×). (Reproduced with permission from Elizabeth Gentt/Visuals Unlimited.) B: Thin section of Synechocystis during division. Many structures are visible. (Reproduced with permission from Carlsberg Research Communications 42:77-98, 1977, With kind permission of Springer Science+Business Media.) of nitrogen, sulfur, or phosphorous is limited and there is excess carbon in the medium (Figure 2-8A). Another storage product formed by prokaryotes when carbon is in excess is glycogen, which is a polymer of glucose. PHB and glycogen are used as carbon sources when protein and nucleic acid synthesis are resumed. A variety of prokaryotes are capable of oxidizing reduced sulfur compounds such as hydrogen sulfide and thiosulfate, producing intracellular granules of elemental sulfur (Figure 2-8C). As the reduced sulfur source becomes limiting, the sulfur in the granules is oxidized, usually to sulfate, and the granules slowly disappear. Many bacteria accumulate large reserves of inorganic phosphate in the form of granules of polyphosphate (Figure 2-8B). These granules can be degraded and used as sources of phosphate for nucleic acid and phospholipid synthesis to support growth. These granules are sometimes termed volutin granules or metachromatic granules because they stain red with a blue dye. They are characteristic features of the corynebacteria (see Chapter 13). Certain groups of autotrophic bacteria that fix carbon dioxide to make their biochemical building blocks contain polyhedral bodies surrounded by a protein shell (carboxysomes) containing the key enzyme of CO2 fixation, ribulosebisphosphate carboxylase (Figure 2-7B). Magnetosomes are intracellular crystal particles of the iron mineral magnetite (Fe3O4) that allow certain aquatic bacteria to exhibit magnetotaxis (ie, migration or orientation of the cell with respect to the earth’s magnetic field). Magnetosomes are surrounded by a nonunit membrane containing phospholipids, proteins, and glycoproteins. Gas vesicles are found almost exclusively in microorganisms from aquatic habitats, where they provide buoyancy. The gas vesicle membrane is a 2-nm-thick layer of protein, impermeable to water and solutes but permeable to gases; thus, gas vesicles exist as gas-filled structures surrounded by the constituents of the cytoplasm (Figure 2-9). Bacteria contain proteins resembling both the actin and nonactin cytoskeletal proteins of eukaryotic cells as additional proteins that play cytoskeletal roles (Figure 2-10). Actin homologs (eg, MreB, Mbl) perform a variety of functions, helping to determine cell shape, segregate chromosomes, and localize proteins with the cell. Nonactin homologs (eg, FtsZ) and unique bacterial cytoskeletal proteins (eg, SecY, MinD) are involved in determining cell shape and in regulation of cell division and chromosome segregation. The Cell Envelope Prokaryotic cells are surrounded by complex envelope layers that differ in composition among the major groups. These structures protect the organisms from hostile environments, such as extreme osmolarity, harsh chemicals, and even antibiotics. 18 SECTION I Fundamentals of Microbiology PM PHB R LI pm L II L III N pp M pb CW A c pp Thylakoids L IV 0.1µm C B Figure 2-8 Inclusion Bodies in Bacteria. A: Electron micrograph of Bacillus megaterium (30,500×) showing poly-β-hydroxybutyric acid inclusion body, PHB; cell wall, CW; nucleoid, N; plasma membrane, PM; “mesosome,” M; and ribosomes, R. (Reproduced with permission from Ralph A. Slepecky/Visuals Unlimited.) B: Ultrastructure of the cyanobacterium Anacystis nidulans. The bacterium is dividing, and a septum is partially formed, LI and LII. Several structural features can be seen, including cell wall layers, LIII and LIV; polyphosphate granules, pp; a polyhedral body, pb; cyanophycin material, c; and plasma membrane, pm. (Reproduced with permission from National Research Council of Canada.) C: Cromatium vinosum, a purple sulfur bacterium, with intracellular sulfur granules, bright field microscopy (2,000×). (Reproduced with permission from John Holt (editor): The Shorter Bergey’s Manual of Determinative Bacteriology, 8th ed, 1977. Copyright Bergey’s Manual Trust. Published by Williams & Wilkins.) The Cell Membrane A. Structure The bacterial cell membrane, also called the cytoplasmic membrane, is visible in electron micrographs of thin sections (see Figure 2-15). It is a typical “unit membrane” composed of phospholipids and upward of 200 different kinds of proteins. Proteins account for approximately 70% of the mass of the membrane, which is a considerably higher proportion than that of mammalian cell membranes. Figure 2-11 illustrates a model of membrane organization. The membranes of prokaryotes are distinguished from those of eukaryotic cells by the absence of sterols, the only exception being mycoplasmas that incorporate sterols, such as cholesterol, into their membranes when growing in sterol-containing media. The cell membranes of the Archaea (see Chapter 1) differ from those of the Bacteria. Some Archaeal cell membranes contain unique lipids, isoprenoids, rather than fatty acids, linked to glycerol by ether rather than an ester linkage. Some of these lipids have no phosphate groups, and therefore, they are not phospholipids. In other species, the cell membrane is made up of a lipid monolayer consisting of long lipids (about twice as long as a phospholipid) with glycerol ethers at both ends (diglycerol tetraethers). The molecules orient themselves with the polar glycerol groups on the surfaces and the nonpolar hydrocarbon chain in the interior. These unusual lipids contribute to the ability of many Archaea to grow under environmental conditions such as high salt, low pH, or very high temperature. CHAPTER 2 Cell Structure 19 Figure 2-9 Transverse section of a dividing cell of the cyanobacterium Microcystis species showing hexagonal stacking of the cylindric gas vesicles (31,500×). (Micrograph by HS Pankratz. Reproduced with permission from Walsby AE: Gas vesicles. Microbiol Rev 1994;58:94.) B. Function A B Figure 2-10 The prokaryotic cytoskeleton. Visualization of the MreB-like cytoskeletal protein (Mbl) of Bacillus subtilis. The Mbl protein has been fused with green fluorescent protein, and live cells have been examined by fluorescence microscopy. A: Arrows point to the helical cytoskeleton cables that extend the length of the cells. B: Three of the cells from A are shown at a higher magnification. (Courtesy of Rut Carballido-Lopez and Jeff Errington.) The major functions of the cytoplasmic membrane are (1) selective permeability and transport of solutes; (2) electron transport and oxidative phosphorylation in aerobic species; (3) excretion of hydrolytic exoenzymes; (4) bearing the enzymes and carrier molecules that function in the biosynthesis of DNA, cell wall polymers, and membrane lipids; and (5) bearing the receptors and other proteins of the chemotactic and other sensory transduction systems. At least 50% of the cytoplasmic membrane must be in the semifluid state for cell growth to occur. At low temperatures, this is achieved by greatly increased synthesis and incorporation of unsaturated fatty acids into the phospholipids of the cell membrane. Permeability and transport—The cytoplasmic mem- brane forms a hydrophobic barrier impermeable to most hydrophilic molecules. However, several mechanisms (transport systems) exist that enable the cell to transport nutrients into and waste products out of the cell. These transport systems work against a concentration gradient to increase the concentration 20 SECTION I Fundamentals of Microbiology Oligosaccharide Hydrophobic α helix Integral protein Glycolipid Hopanoid Peripheral Phospholipid protein Figure 2-11 Bacterial Plasma Membrane Structure. This diagram of the fluid mosaic model of bacterial membrane structure shown the integral proteins (green and red) floating in a lipid bilayer. Peripheral proteins (yellow) are associated loosely with the inner membrane surface. Small spheres represent the hydrophilic ends of membrane phospholipids and wiggly tails, the hydrophobic fatty acid chains. Other membrane lipids such as hopanoids (purple) may be present. For the sake of clarity, phospholipids are shown proportionately much larger size than in real membranes. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ [editors]: Prescott, Harley, and Klein’s Microbiology, 7th ed. New York: McGraw-Hill; 2008. © The McGraw-Hill Companies, Inc.) of nutrients inside the cell, a function that requires energy in some form. There are three general transport mechanisms involved in membrane transport: passive transport, active transport, and group translocation. a. Passive transport—This mechanism relies on diffusion, uses no energy, and operates only when the solute is at higher concentration outside than inside the cell. Simple diffusion accounts for the entry of very few nutrients, including dissolved oxygen, carbon dioxide, and water itself. Simple diffusion provides neither speed nor selectivity. Facilitated diffusion also uses no energy so the solute never achieves an internal concentration greater than what exists outside the cell. However, facilitated diffusion is selective. Channel proteins form selective channels that facilitate the passage of specific molecules. Facilitated diffusion is common in eukaryotic microorganisms (eg, yeast) but is rare in prokaryotes. Glycerol is one of the few compounds that enters prokaryotic cells by facilitated diffusion. b. Active transport—Many nutrients are concentrated more than a thousand-fold as a result of active transport. There are two types of active transport mechanisms depending on the source of energy used: ion-coupled transport and ATPbinding cassette (ABC) transport. 1) Ion-coupled transport—These systems move a molecule across the cell membrane at the expense of a previously established ion gradient such as protonmotive or sodiummotive force. There are three basic types: uniport, symport, and antiport (Figure 2-12). Ion-coupled transport is particularly common in aerobic organisms, which have an easier time generating an ion-motive force than do anaerobes. Uniporters catalyze the transport of a substrate independent of any coupled ion. Symporters catalyze the simultaneous transport of two substrates in the same direction by a single carrier; for example, an H+ gradient can permit symport of an oppositely charged ion (eg, glycine) or a neutral molecule (eg, galactose). Antiporters catalyze the simultaneous transport of two likecharged compounds in opposite directions by a common carrier (eg, H+:Na+). Approximately 40% of the substrates transported by E coli use this mechanism. 2) ABC transport—This mechanism uses ATP directly to transport solutes into the cell. In gram-negative bacteria, the transport of many nutrients is facilitated by specific binding proteins located in the periplasmic space; in gram-positive cells, the binding proteins are attached to the outer surface of the cell membrane. These proteins function by transferring the bound substrate to a membrane-bound protein complex. Hydrolysis of ATP is then triggered, and the energy is used to open the membrane pore and allow the unidirectional movement of the substrate into the cell. Approximately 40% of the substrates transported by E coli use this mechanism. c. Group translocation—In addition to true transport, in which a solute is moved across the membrane without change in structure, bacteria use a process called group translocation (vectorial metabolism) to effect the net uptake of certain sugars (eg, glucose and mannose), the substrate becoming phosphorylated during the transport process. In a strict sense, group translocation is not active transport because CHAPTER 2 Cell Structure 21 Uniport Outside Inside Symport H+ H+ H+ H+ H+ H+ + H H+ H+ H+ Antiport d. Special transport processes—Iron (Fe) is an essential nutrient for the growth of almost all bacteria. Under anaerobic conditions, Fe is generally in the +2 oxidation state and soluble. However, under aerobic conditions, Fe is generally in the +3 oxidation state and insoluble. The internal compartments of animals contain virtually no free Fe; it is sequestered in complexes with such proteins as transferrin and lactoferrin. Some bacteria solve this problem by secreting siderophores— compounds that chelate Fe and promote its transport as a soluble complex. One major group of siderophores consists of derivatives of hydroxamic acid (–CONH2OH), which chelate Fe3+ very strongly. The iron–hydroxamate complex is actively transported into the cell by the cooperative action of a group of proteins that span the outer membrane, periplasm, and inner membrane. The iron is released, and the hydroxamate can exit the cell and be used again for iron transport. Some pathogenic bacteria use a fundamentally different mechanism involving specific receptors that bind host transferrin and lactoferrin (as well as other iron-containing host proteins). The Fe is removed and transported into the cell by an energy-dependent process. Electron transport and oxidative phosphoryla‑ tion—The cytochromes and other enzymes and components Figure 2-12 Three types of porters: A: uniporters, B: symporters, and C: antiporters. Uniporters catalyze the transport of a single species independently of any other, symporters catalyze the cotransport of two dissimilar species (usually a solute and a positively charged ion, H +) in the same direction, and antiporters catalyze the exchange transport of two similar solutes in opposite directions. A single transport protein may catalyze just one of these processes, two of these processes, or even all three of these processes, depending on conditions. Uniporters, symporters, and antiporters have been found to be structurally similar and evolutionarily related, and they function by similar mechanisms. (Reproduced with permission from Saier MH Jr: Peter Mitchell and his chemiosmotic theories. ASM News 1997;63:13.) no concentration gradient is involved. This process allows bacteria to use their energy resources efficiently by coupling transport with metabolism. In this process, a membrane carrier protein is first phosphorylated in the cytoplasm at the expense of phosphoenolpyruvate; the phosphorylated carrier protein then binds the free sugar at the exterior membrane face and transports it into the cytoplasm, releasing it as sugar phosphate. Such systems of sugar transport are called phosphotransferase systems. Phosphotransferase systems are also involved in movement toward these carbon sources (chemotaxis) and in the regulation of several other metabolic pathways (catabolite repression). of the respiratory chain, including certain dehydrogenases, are located in the cell membrane. The bacterial cell membrane is thus a functional analog of the mitochondrial membrane— a relationship which has been taken by many biologists to support the theory that mitochondria have evolved from symbiotic bacteria. The mechanism by which ATP generation is coupled to electron transport is discussed in Chapter 6. Excretion of hydrolytic exoenzymes and pathoge‑ nicity proteins—All organisms that rely on macromo- lecular organic polymers as a source of nutrients (eg, proteins, polysaccharides, lipids) excrete hydrolytic enzymes that degrade the polymers to subunits small enough to penetrate the cell membrane. Higher animals secrete such enzymes into the lumen of the digestive tract; bacteria (both gram positive and gram negative) secrete them directly into the external medium or into the periplasmic space between the peptidoglycan layer and the outer membrane of the cell wall in the case of gram-negative bacteria (see The Cell Wall, later). In gram-positive bacteria, proteins are secreted directly, but proteins secreted by gram-negative bacteria must traverse the outer membrane as well. Six pathways of protein secretion have been described in bacteria: the type I, type II, type III, type IV, type V, and type VI secretion systems. A schematic overview of the type to V systems is presented in Figure 2-12. The type I and IV secretion systems have been described in both gram-negative and gram-positive bacteria, but the type II, III, V, and VI secretion systems have been found only in gram-negative bacteria. Proteins secreted by the type I and type III pathways traverse the inner membrane (IM) and outer membrane (OM) in one step, but proteins secreted by 22 SECTION I Fundamentals of Microbiology the type II and type V pathways cross the IM and OM in separate steps. Proteins secreted by the type II and type V pathways are synthesized on cytoplasmic ribosomes as preproteins containing an extra leader or signal sequence of 15–40 amino acids—most commonly about 30 amino acids—at the amino terminal and require the sec system for transport across the IM. In E coli, the sec pathway comprises a number of IM proteins (SecD to SecF, SecY), a cell membrane–associated ATPase (SecA) that provides energy for export, a chaperone (SecB) that binds to the preprotein, and the periplasmic signal peptidase. After translocation, the leader sequence is cleaved off by the membrane-bound signal peptidase, and the mature protein is released into Type I Type III the periplasmic space. In contrast, proteins secreted by the type I and type III systems do not have a leader sequence and are exported intact. In gram-negative and gram-positive bacteria, another plasma membrane translocation system, called the tat pathway, can move proteins across the plasma membrane. In gram-negative bacteria, these proteins are then delivered to the type II system (Figure 2-13). The tat pathway is distinct from the sec system in that it translocates already folded proteins. Although proteins secreted by the type II and type V systems are similar in the mechanism by which they cross the IM, differences exist in how they traverse the OM. Proteins Type II Type V Type IV Cell exterior TolC Outer membrane Yop PulS YscJ Periplasmic space Tat Plasma membrane ADP + Pi SecD EFGY Sec ATP ATP Chaperone ADP + Pi ATP ADP + Pi Cytoplasm ATP ADP + Pi ATP ATP ADP + Pi Chaperone Protein Figure 2-13 The protein secretion systems of gram-negative bacteria. Five secretion systems of gram-negative bacteria are shown. The Sec-dependent and Tat pathways deliver proteins from the cytoplasm to the periplasmic space. The type II, type V, and sometimes type IV systems complete the secretion process begun by the Sec-dependent pathway. The Tat system appears to deliver proteins only to the type II pathway. The type I and type III systems bypass the Sec-dependent and Tat pathways, moving proteins directly from the cytoplasm, through the outer membrane, to the extracellular space. The type IV system can work either with the Sec-dependent pathway or can work alone to transport proteins to the extracellular space. Proteins translocated by the Sec-dependent pathway and the type III pathway are delivered to those systems by chaperone proteins. ADP, adenosine diphosphate; ATP, adenosine triphosphate; EFGY, ; PuIS, ; SecD, TolC, ; Yop. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ [editors]: Prescott, Harley, and Klein’s Microbiology, 7th ed. New York: McGraw-Hill; 2008. © The McGraw-Hill Companies, Inc.) CHAPTER 2 Cell Structure 23 secreted by the type II system are transported across the OM by a multiprotein complex (Figure 2-13). This is the primary pathway for the secretion of extracellular degradative enzymes by gram-negative bacteria. Elastase, phospholipase C, and exotoxin A are secreted by this system in Pseudomonas aeruginosa. However, proteins secreted by the type V system autotransport across the outer membrane by virtue of a carboxyl terminal sequence, which is enzymatically removed upon release of the protein from the OM. Some extracellular proteins—eg, the IgA protease of Neisseria gonorrhoeae and the vacuolating cytotoxin of Helicobacter pylori—are secreted by this system. The type I and type III secretion pathways are sec independent and thus do not involve amino terminal processing of the secreted proteins. Protein secretion by these pathways occurs in a continuous process without the presence of a cytoplasmic intermediate. Type I secretion is exemplified by the a-hemolysin of E coli and the adenylyl cyclase of Bordetella pertussis. Type I secretion requires three secretory proteins: an IM ATP-binding cassette (ABC transporter), which provides energy for protein secretion; an OM protein; and a membrane fusion protein, which is anchored in the inner membrane and spans the periplasmic space (see Figure 2-13). Instead of a signal peptide, the information is located within the carboxyl terminal 60 amino acids of the secreted protein. The type III secretion pathway is a contact-dependent system. It is activated by contact with a host cell, and then injects a toxin protein into the host cell directly. The type III secretion apparatus is composed of approximately 20 proteins, most of which are located in the IM. Most of these IM components are homologous to the flagellar biosynthesis apparatus of both gram-negative and gram-positive bacteria. As in type I secretion, the proteins secreted via the type III pathway are not subject to amino terminal processing during secretion. Type IV pathways secrete either polypeptide toxins (directed against eukaryotic cells) or protein–DNA complexes either between two bacterial cells or between a bacterial and a eukaryotic cell. Type IV secretion is exemplified by the protein–DNA complex delivered by Agrobacterium tumefaciens into a plant cell. Additionally, B pertussis and H pylori possess type IV secretion systems that mediate secretion of pertussis toxin and interleukin-8–inducing factor, respectively. The sec-independent type VI secretion was recently described in P aeruginosa, where it contributes to pathogenicity in patients with cystic fibrosis. This secretion system is composed of 15–20 proteins whose biochemical functions are not well understood. However, recent studies suggest that some of these proteins share homology with bacteriophage tail proteins. The characteristics of the protein secretion systems of bacteria are summarized in Table 9-6. Biosynthetic functions—The cell membrane is the site of the carrier lipids on which the subunits of the cell wall are assembled (see the discussion of synthesis of cell wall substances in Chapter 6) as well as of the enzymes of cell wall biosynthesis. The enzymes of phospholipid synthesis are also localized in the cell membrane. Chemotactic systems—Attractants and repellents bind to specific receptors in the bacterial membrane (see Flagella, later). There are at least 20 different chemoreceptors in the membrane of E coli, some of which also function as a first step in the transport process. The Cell Wall The internal osmotic pressure of most bacteria ranges from 5 to 20 atm as a result of solute concentration via active transport. In most environments, this pressure would be sufficient to burst the cell were it not for the presence of a high-tensilestrength cell wall (Figure 2-14). The bacterial cell wall owes its strength to a layer composed of a substance variously referred to as murein, mucopeptide, or peptidoglycan (all are synonyms). The structure of peptidoglycan is discussed below. Most bacteria are classified as gram-positive or gramnegative according to their response to the Gram-staining procedure. This procedure was named for the histologist Hans Christian Gram, who developed this differential staining procedure in an attempt to stain bacteria in infected tissues. The Gram stain depends on the ability of certain bacteria (the gram-positive bacteria) to retain a complex of crystal violet (a purple dye) and iodine after a brief wash with alcohol or acetone. Gram-negative bacteria do not retain Figure 2-14 Isolated gram-positive cell wall. The peptidoglycan wall from Bacillus megaterium, a gram-positive bacterium. The latex spheres have a diameter of 0.25 mm. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ [editors]: Prescott, Harley, and Klein’s Microbiology, 7th ed. New York: McGraw-Hill; 2008. © The McGraw-Hill Companies, Inc.) 24 SECTION I Fundamentals of Microbiology P The gram-negative cell wall The gram-positive cell wall PM OM Cell wall M PM W Peptidoglycan Plasma membrane Cell wall Outer membrane Peptidoglycan Plasma membrane P Periplasmic space Figure 2-15 Gram-positive and gram-negative cell walls. The gram-positive envelope is from Bacillus licheniiformis (left), and the gramnegative micrograph is of Aquaspirillum serpens (right). IM, plasma membrane; M; peptidoglycan or murein layer; OM, outer membrane; P, periplasmic space; W, gram-positive peptidoglycan wall. (Reproduced with permission from T. J. Beveridge/Biological Photo Service.) the dye–iodine complex and become translucent, but they can then be counterstained with safranin (a red dye). Thus, gram-positive bacteria look purple under the microscope, and gram-negative bacteria look red. The distinction between these two groups turns out to reflect fundamental differences in their cell envelopes (Figure 2-15). In addition to giving osmotic protection, the cell wall plays an essential role in cell division as well as serving as a primer for its own biosynthesis. Various layers of the wall are the sites of major antigenic determinants of the cell surface, and one component—the lipopolysaccharide of gram-negative cell walls—is responsible for the nonspecific endotoxin activity of gram-negative bacteria. The cell wall is, in general, nonselectively permeable; one layer of the gram-negative wall, however—the outer membrane—hinders the passage of relatively large molecules (see below). The biosynthesis of the cell wall and the antibiotics that interfere with this process are discussed in Chapter 6. A. The Peptidoglycan Layer Peptidoglycan is a complex polymer consisting, for the purposes of description, of three parts: a backbone, composed of alternating N-acetylglucosamine and N-acetylmuramic acid connected by β1→4 linkages; a set of identical tetrapeptide side chains attached to N-acetylmuramic acid; and a set of identical peptide cross-bridges (Figure 2-16). The backbone is the same in all bacterial species; the tetrapeptide side chains and the peptide cross-bridges vary from species to species, those of Staphylococcus aureus being illustrated in Figure 2-16. In many gram-negative cell walls, the cross-bridge consists of a direct peptide linkage between the diaminopimelic acid (DAP) amino group of one side chain and the carboxyl group of the terminal d-alanine of a second side chain. The tetrapeptide side chains of all species, however, have certain important features in common. Most have l-alanine at position 1 (attached to N-acetylmuramic acid), d-glutamate or substituted d-glutamate at position 2, and d-alanine at position 4. Position 3 is the most variable one: Most gramnegative bacteria have diaminopimelic acid at this position, to which is linked the lipoprotein cell wall component discussed below. Gram-positive bacteria usually have l-lysine at position 3; however, some may have diaminopimelic acid or another amino acid at this position. Diaminopimelic acid is a unique element of bacterial cell walls. It is never found in the cell walls of Archaea or eukaryotes. Diaminopimelic acid is the immediate precursor of lysine in the bacterial biosynthesis of that amino acid (see Figure 6-18). Bacterial mutants that are blocked before diaminopimelic acid in the biosynthetic pathway grow normally when provided with diaminopimelic acid in the medium; when given l-lysine alone, however, they lyse, because they continue to grow but are specifically unable to make new cell wall peptidoglycan. The fact that all peptidoglycan chains are cross-linked means that each peptidoglycan layer is a single giant molecule. In gram-positive bacteria, there are as many as 40 sheets of peptidoglycan, comprising up to 50% of the cell wall material; in gram-negative bacteria, there appears to be only one or two sheets, comprising 5–10% of the wall material. Bacteria owe their shapes, which are characteristic of particular species, to their cell wall structure. CHAPTER 2 Cell Structure 25 A β-1,4 linkage cleaved by lysozyme (N-Acetylglucosamine) 6 CH OH 2 O 5 H (N-Acetylmuramic acid peptide) CH2OH O H H O H 4 CH H OH 3 H H NH COCH3 NH (α-NH) H OH H H NH H H COCH3 H NH (α-NH) L-Alanine D-Isoglutamine D-Isoglutamine L-Lysine L-Lysine (α-COOH) D-Alanine MurNAc COCH3 (α-COOH) [Gly]5 GlcNAc MurNAc L-Ala L-Ala D-I-Glu-N D-I-Glu-N L-Lys L-Lys D-Ala GlcNAc D-Ala [Gly]5 GlcNAc O CO O [Gly]5 GlcNAc CH L-Alanine D-Alanine B COCH3 CH3 O H H H H O CO O 2 (N-Acetylmuramic acid peptide) CH2OH O H H CH3 O 1 (N-Acetylglucosamine) CH2OH O H MurNAc [Gly]5 GlcNAc MurNAc L-Ala L-Ala D-I-Glu-N D-I-Glu-N L-Lys L-Lys D-Ala GlcNAc D-Ala [Gly]5 [Gly]5 Figure 2-16 A: A segment of the peptidoglycan of Staphylococcus aureus. The backbone of the polymer consists of alternating subunits of N-acetylglucosamine and N-acetylmuramic acid connected by β1→4 linkages. The muramic acid residues are linked to short peptides, the composition of which varies from one bacterial species to another. In some species, the l-lysine residues are replaced by diaminopimelic acid, an amino acid that is found in nature only in prokaryotic cell walls. Note the d-amino acids, which are also characteristic constituents of prokaryotic cell walls. The peptide chains of the peptidoglycan are cross-linked between parallel polysaccharide backbones, as shown in B. B: Schematic representation of the peptidoglycan lattice that is formed by cross-linking. Bridges composed of pentaglycine peptide chains connect the a-carboxyl of the terminal d-alanine residue of one chain with the ε-amino group of the l-lysine residue of the next chain. The nature of the cross-linking bridge varies among different species. 26 SECTION I Fundamentals of Microbiology B. Special Components of Gram-Positive Cell Walls Most gram-positive cell walls contain considerable amounts of teichoic and teichuronic acids, which may account for up to 50% of the dry weight of the wall and 10% of the dry weight of the total cell. In addition, some gram-positive walls may contain polysaccharide molecules. Teichoic and teichuronic acids—The term teichoic acids encompasses all wall, membrane, or capsular polymers containing glycerophosphate or ribitol phosphate residues. These polyalcohols are connected by phosphodiester linkages and usually have other sugars and d-alanine attached (Figure 2-17A). Because they are negatively charged, teichoic acids are partially responsible for the negative charge of the cell surface as a whole. There are two types of teichoic acids: wall teichoic acid (WTA), covalently linked to peptidoglycan, and membrane teichoic acid, covalently linked to membrane glycolipid. Because the latter are intimately associated with lipids, they have been called lipoteichoic acids (LTA). Together with peptidoglycan, WTA and LTA make up a polyanionic network or matrix that provides functions relating to the elasticity, porosity, tensile strength, and electrostatic properties of the envelope. Although not all gram-positive bacteria have conventional LTA and WTA, those that lack these polymers generally have functionally similar ones. Most teichoic acids contain large amounts of d-alanine, usually attached to position 2 or 3 of glycerol or position 3 or 4 of ribitol. In some of the more complex teichoic acids, however, d-alanine is attached to one of the sugar residues. In addition to d-alanine, other substituents may be attached to the free hydroxyl groups of glycerol and ribitol (eg, glucose, galactose, N-acetylglucosamine, N-acetylgalactosamine, or succinate). A given species may have more than one type of sugar substituent in addition to d-alanine; in such cases, it is not certain whether the different sugars occur on the same or on separate teichoic acid molecules. The composition of the teichoic acid formed by a given bacterial species can vary with the composition of the growth medium. The teichoic acids constitute major surface antigens of those gram-positive species that possess them, and their accessibility to antibodies has been taken as evidence that they lie on the outside surface of the peptidoglycan. Their activity is often increased, however, by partial digestion of the peptidoglycan; thus, much of the teichoic acid may lie between the cytoplasmic membrane and the peptidoglycan layer, possibly extending upward through pores in the latter (Figure 2-17B). In the pneumococcus (Streptococcus pneumoniae), the teichoic acids bear the antigenic determinants called Forssman antigen. In Streptococcus pyogenes, LTA is associated with the M protein that protrudes from the cell membrane through the peptidoglycan layer. The long M protein molecules together with the LTA form microfibrils that facilitate the attachment of S pyogenes to animal cells. The teichuronic acids are similar polymers, but the repeat units include sugar acids (eg, N-acetylmannosuronic or d-glucosuronic acid) instead of phosphoric acids. They Teichoic acid Lipoteichoic acid O O P O– O CH2 O R Peptidoglycan H C CH2 O O P O– O CH2 H C O R Periplasmic space Plasma membrane CH2 O O P O O– B A Figure 2-17 A: Teichoic acid structure. The segment of a teichoic acid made of phosphate, glycerol, and a side chain, R. R may represent glucose, or other molecules. B: Teichoic and lipoteichoic acids of the gram-positive envelope. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ [editors]: Prescott, Harley, and Klein’s Microbiology, 7th ed. New York: McGraw-Hill; 2008.) d-alanine, CHAPTER 2 Cell Structure 27 are synthesized in place of teichoic acids when phosphate is limiting. Polysaccharides—The hydrolysis of gram-positive walls has yielded, from certain species, neutral sugars such as mannose, arabinose, rhamnose, and glucosamine and acidic sugars such as glucuronic acid and mannuronic acid. It has been proposed that these sugars exist as subunits of polysaccharides in the cell wall; the discovery, however, that teichoic and teichuronic acids may contain a variety of sugars (Figure 2-17A) leaves the true origin of these sugars uncertain. C. Special Components of Gram-Negative Cell Walls Gram-negative cell walls contain three components that lie outside of the peptidoglycan layer: lipoprotein, outer membrane, and lipopolysaccharide (Figure 2-18). Outer membrane—The outer membrane is chemically distinct from all other biological membranes. It is a bilayered structure; its inner leaflet resembles in composition that of the cell membrane, and its outer leaflet contains a distinctive component, a lipopolysaccharide (LPS) (see below). As a result, the leaflets of this membrane are asymmetrical, and the properties of this bilayer differ considerably from those of a symmetrical biologic membrane such as the cell membrane. The ability of the outer membrane to exclude hydrophobic molecules is an unusual feature among biologic membranes and serves to protect the cell (in the case of enteric bacteria) from deleterious substances such as bile salts. Because of its lipid nature, the outer membrane would be expected to exclude hydrophilic molecules as well. However, the outer membrane has special channels, consisting of protein molecules called porins that permit the passive diffusion of low-molecular-weight hydrophilic compounds such as sugars, amino acids, and certain ions. Large antibiotic molecules penetrate the outer membrane relatively slowly, which accounts for the relatively high antibiotic resistance of gramnegative bacteria. The permeability of the outer membrane O-antigen repeat GlcNAc Lipopoly- Glucose saccharide Galactose Heptose Porin KDO Outer core Inner core Lipid A Outer membrane Lipoprotein Peptidoglycan Periplasm MDO Phospholipids Inner membrane Cytoplasm Proteins Figure 2-18 Molecular representation of the envelope of a gram-negative bacterium. Ovals and rectangles represent sugar residues, and circles depict the polar head groups of the glycerophospholipids (phosphatidylethanolamine and phosphatidylglycerol). The core region shown is that of Escherichia coli K-12, a strain that does not normally contain an O-antigen repeat unless transformed with an appropriate plasmid. MDO, membrane-derived oligosaccharides. (Reproduced with permission from Raetz CRH: Bacterial endotoxins: Extraordinary lipids that activate eucaryotic signal transduction. J Bacteriol 1993;175:5745.) 28 SECTION I Fundamentals of Microbiology varies widely from one gram-negative species to another; in P aeruginosa, for example, which is extremely resistant to antibacterial agents, the outer membrane is 100 times less permeable than that of E coli. The major proteins of the outer membrane, named according to the genes that code for them, have been placed into several functional categories on the basis of mutants in which they are lacking and on the basis of experiments in which purified proteins have been reconstituted into artificial membranes. Porins, exemplified by OmpC, D, and F and PhoE of E coli and Salmonella typhimurium, are trimeric proteins that penetrate both faces of the outer membrane (Figure 2-19). They form relatively nonspecific pores that permit the free diffusion of small hydrophilic solutes across the membrane. The porins of different species have different exclusion limits, ranging from molecular weights of about 600 in E coli and S typhimurium to more than 3000 in P aeruginosa. Members of a second group of outer membrane proteins, which resemble porins in many ways, are exemplified by LamB and Tsx. LamB, an inducible porin that is also the receptor for lambda bacteriophage, is responsible for most of the transmembrane diffusion of maltose and maltodextrins; Tsx, the receptor for T6 bacteriophage, is responsible for the transmembrane diffusion of nucleosides and some amino acids. LamB allows some passage of other solutes; however, its relative specificity may reflect weak interactions of solutes with configuration-specific sites within the channel. The OmpA protein is an abundant protein in the outer membrane. The OmpA protein participates in the anchoring of the outer membrane to the peptidoglycan layer; it is also the sex pilus receptor in F-mediated bacterial conjugation (see Chapter 7). The outer membrane also contains a set of less abundant proteins that are involved in the transport of specific molecules such as vitamin B12 and iron-siderophore complexes. They show high affinity for their substrates and probably function like the classic carrier transport systems of the cytoplasmic membrane. The proper function of these proteins requires energy coupled through a protein called TonB. Additional minor proteins include a limited number of enzymes, among them phospholipases and proteases. The topology of the major proteins of the outer membrane, based on cross-linking studies and analyses of functional relationships, is shown in Figure 2-18. The outer membrane is connected to both the peptidoglycan layer and the cytoplasmic membrane. The connection with the peptidoglycan layer is primarily mediated by the outer membrane lipoprotein (see below). About one-third of the lipoprotein molecules are covalently linked to peptidoglycan and help hold the two structures together. A noncovalent association of some of the porins with the peptidoglycan layer plays a lesser role C N A B Figure 2-19 A: General fold of a porin monomer (OmpF porin from Escherichia coli). The large hollow β-barrel structure is formed by antiparallel arrangement of 16 β-strands. The strands are connected by short loops or regular turns on the periplasmic rim (bottom), and long irregular loops face the cell exterior (top). The internal loop, which connects β-strands 5 and 6 and extends inside the barrel, is highlighted in dark. The chain terminals are marked. The surface closest to the viewer is involved in subunit contacts. B: Schematic representation of the OmpF trimer. The view is from the extracellular space along the molecular threefold symmetry axis. (Reproduced with permission from Schirmer T: General and specific porins from bacterial outer membranes. J Struct Biol 1998;121:101.) CHAPTER 2 Cell Structure 29 in connecting the outer membrane with this structure. Outer membrane proteins are synthesized on ribosomes bound to the cytoplasmic surface of the cell membrane; how they are transferred to the outer membrane is still uncertain, but one hypothesis suggests that transfer occurs at zones of adhesion between the cytoplasmic and outer membranes, which are visible in the electron microscope. Unfortunately, firm evidence for such areas of adhesion has proven hard to come by. Lipopolysaccharide (LPS)—The LPS of gram-negative cell walls consists of a complex glycolipid, called lipid A, to which is attached a polysaccharide made up of a core and a terminal series of repeat units (Figure 2-20A). The lipid A component is embedded in the outer leaflet of the membrane anchoring the LPS. LPS is synthesized on the cytoplasmic membrane and transported to its final exterior position. The presence of LPS is required for the function of many outer membrane proteins. Lipid A consists of phosphorylated glucosamine disaccharide units to which are attached a number of long-chain fatty acids (Figure 2-20). β-Hydroxymyristic acid, a C14 fatty acid, is always present and is unique to this lipid; the other Man fatty acids, along with substituent groups on the phosphates, vary according to the bacterial species. The polysaccharide core, shown in Figure 2-20A and B is similar in all gram-negative species that have LPS and includes two characteristic sugars, ketodeoxyoctanoic acid (KDO) and a heptose. Each species, however, contains a unique repeat unit, that of Salmonella being shown in Figure 2-20A. The repeat units are usually linear trisaccharides or branched tetra- or pentasaccharides. The repeat unit is referred to as the O antigen. The hydrophilic carbohydrate chains of the O antigen cover the bacterial surface and exclude hydrophobic compounds. The negatively charged LPS molecules are noncovalently cross-bridged by divalent cations (ie, Ca 2+ and Mg2+); this stabilizes the membrane and provides a barrier to hydrophobic molecules. Removal of the divalent cations with chelating agents or their displacement by polycationic antibiotics such as polymyxins and aminoglycosides renders the outer membrane permeable to large hydrophobic molecules. Lipopolysaccharide, which is extremely toxic to animals, has been called the endotoxin of gram-negative bacteria Abe Rha Gal Man n O side chain Abe Rha Gal Glc NAG Gal Glc Gal Core polysaccharide Hep Hep P ethanolamine P KDO KDO P GlcN KDO GlcN P ethanolamine P Fatty acid A Lipid A B Figure 2-20 Lipopolysaccharide structure. A: The lipopolysaccharide from Salmonella. This slightly simplified diagram illustrates one form of the LPS. Abe, abequose; Gal, galactose; GlcN, glucosamine; Hep, heptulose; KDO, 2-keto-3-deoxyoctonate; Man, mannose; NAG, N-acetylglucosamine; P, phosphate; Rha, l-rhamnose. Lipid A is buried in the outer membrane. B: Molecular model of an Escherichia coli lipopolysaccharide. The lipid A and core polysaccharide are straight; the O side chain is bent at an angle in this model. (Reproduced with permission from Willey VM, Sherwood LM, Woolverton CJ: Prescott, Harley, & Klein’s Microbiology, 7th ed. McGraw-Hill, 2008. © The McGraw-Hill Companies, Inc.) 30 SECTION I Fundamentals of Microbiology because it is firmly bound to the cell surface and is released only when the cells are lysed. When LPS is split into lipid A and polysaccharide, all of the toxicity is associated with the former. The O antigen is highly immunogenic in a vertebrate animal. Antigenic specificity is conferred by the O antigen because this antigen is highly variable among species and even in strains within a species. The number of possible antigenic types is very great: Over 1000 have been recognized in Salmonella alone. Not all gram-negative bacteria have outer membrane LPS composed of a variable number of repeated oligosaccharide units (see Figure 2-20); the outer membrane glycolipids of bacteria that colonize mucosal surfaces (eg, Neisseria meningitidis, N gonorrhoeae, Haemophilus influenzae, and Haemophilus ducreyi) have relatively short, multiantennary (ie, branched) glycans. These smaller glycolipids have been compared with the “R-type” truncated LPS structures, which lack O antigens and are produced by rough mutants of enteric bacteria such as E coli. However, their structures more closely resemble those of the glycosphingolipids of mammalian cell membranes, and they are more properly termed lipooligosaccharides (LOS). These molecules exhibit extensive antigenic and structural diversity even within a single strain. LOS is an important virulence factor. Epitopes have been identified on LOS that mimic host structures and may enable these organisms to evade the immune response of the host. Some LOS (eg, those from N gonorrhoeae, N meningitidis, and H ducreyi) have a terminal N-acetyllactosamine (Galβ-1→4-GlcNAc) residue that is immunochemically similar to the precursor of the human erythrocyte i antigen. In the presence of a bacterial enzyme called sialyltransferase and a host or bacterial substrate (cytidine monophosphoN-acetylneuraminic acid, CMP-NANA), the N-acetyllactosamine residue is sialylated. This sialylation, which occurs in vivo, provides the organism with the environmental advantages of molecular mimicry of a host antigen and the biologic masking thought to be provided by sialic acids. Lipoprotein—Molecules of an unusual lipoprotein cross-link the outer membrane and peptidoglycan layers (Figure 2-18). The lipoprotein contains 57 amino acids, representing repeats of a 15-amino-acid sequence; it is peptide-linked to DAP residues of the peptidoglycan tetrapeptide side chains. The lipid component, consisting of a diglyceride thioether linked to a terminal cysteine, is noncovalently inserted in the outer membrane. Lipoprotein is numerically the most abundant protein of gram-negative cells (ca 700,000 molecules per cell). Its function (inferred from the behavior of mutants that lack it) is to stabilize the outer membrane and anchor it to the peptidoglycan layer. The periplasmic space—The space between the inner and outer membranes, called the periplasmic space, contains the peptidoglycan layer and a gel-like solution of proteins. The periplasmic space is approximately 20–40% of the cell volume, which is far from insignificant. The periplasmic proteins include binding proteins for specific substrates (eg, amino acids, sugars, vitamins, and ions), hydrolytic enzymes (eg, alkaline phosphatase and 5′-nucleotidase) that break down nontransportable substrates into transportable ones, and detoxifying enzymes (eg, β-lactamase and aminoglycosidephosphorylase) that inactivate certain antibiotics. The periplasm also contains high concentrations of highly branched polymers of d-glucose, eight to 10 residues long, which are variously substituted with glycerol phosphate and phosphatidylethanolamine residues; some contain O-succinyl esters. These so-called membrane-derived oligosaccharides appear to play a role in osmoregulation because cells grown in media of low osmolarity increase their synthesis of these compounds 16-fold. D. The Acid-Fast Cell Wall Some bacteria, notably the tubercle bacillus (M tuberculosis) and its relatives have cell walls that contain large amounts of waxes, complex branched hydrocarbons (70–90 carbons long) known as mycolic acids. The cell wall is composed of peptidoglycan and an external asymmetric lipid bilayer; the inner leaflet contains mycolic acids linked to an arabinoglycan, and the outer leaflet contains other extractable lipids. This is a highly ordered lipid bilayer in which proteins are embedded, forming water-filled pores through which nutrients and certain drugs can pass slowly. Some compounds can also penetrate the lipid domains of the cell wall albeit slowly. This hydrophobic structure renders these bacteria resistant to many harsh chemicals, including detergents and strong acids. If a dye is introduced into these cells by brief heating or treatment with detergents, it cannot be removed by dilute hydrochloric acid, as in other bacteria. These organisms are therefore called acid fast. The permeability of the cell wall to hydrophilic molecules is 100- to 1000-fold lower than for E coli and may be responsible for the slow growth rate of mycobacteria. E. Cell Walls of the Archaea The Archaea do not have cell walls like the Bacteria. Some have a simple S-layer (see below) often composed of glycoproteins. Some Archaea have a rigid cell wall composed of polysaccharides or a peptidoglycan called pseudomurein. The pseudomurein differs from the peptidoglycan of bacteria by having l-amino acids rather than d-amino acids and disaccharide units with an a-1→3 rather than a β-1→4 linkage. Archaea that have a pseudomurein cell wall are gram-positive. F. Crystalline Surface Layers Many bacteria, both gram-positive and gram-negative bacteria as well as Archaebacteria, possess a two-dimensional crystalline, subunit-type layer lattice of protein or glycoprotein molecules (S-layer) as the outermost component of the cell envelope. In both gram-positive and gram-negative bacteria, this structure is sometimes several molecules thick. In some Archaea, they are the only layer external to the cell membrane. CHAPTER 2 Cell Structure 31 S-layers are generally composed of a single kind of protein molecule, sometimes with carbohydrates attached. The isolated molecules are capable of self-assembly (ie, they make sheets similar or identical to those present on the cells). S-layer proteins are resistant to proteolytic enzymes and proteindenaturing agents. The function of the S-layer is uncertain but is probably protective. In some cases, it has been shown to protect the cell from wall-degrading enzymes, from invasion by Bdellovibrio bacteriovorous (a predatory bacterium), and from bacteriophages. It also plays a role in the maintenance of cell shape in some species of Archaebacteria, and it may be involved in cell adhesion to host epidermal surfaces. G. Enzymes That Attack Cell Walls The β1→4 linkage of the peptidoglycan backbone is hydrolyzed by the enzyme lysozyme (Figure 2-16), which is found in animal secretions (tears, saliva, nasal secretions) as well as in egg white. Gram-positive bacteria treated with lysozyme in low-osmotic-strength media lyse; if the osmotic strength of the medium is raised to balance the internal osmotic pressure of the cell, free spherical bodies called protoplasts are liberated. The outer membrane of the gram-negative cell wall prevents access of lysozyme unless disrupted by an agent such as ethylene-diaminetetraacetic acid (EDTA), a compound that chelates divalent cations; in osmotically protected media, cells treated with EDTA-lysozyme form spheroplasts that still possess remnants of the complex gram-negative wall, including the outer membrane. Bacteria themselves possess a number of autolysins, hydrolytic enzymes that attack peptidoglycan, including muramidases, glucosaminidases, endopeptidases, and carboxypeptidases. These enzymes catalyze the turnover or degradation of peptidoglycan in bacteria. These enzymes presumably participate in cell wall growth and turnover and in cell separation, but their activity is most apparent during the dissolution of dead cells (autolysis). Enzymes that degrade bacterial cell walls are also found in cells that digest whole bacteria (eg, protozoa and the phagocytic cells of higher animals). H. Cell Wall Growth Cell wall synthesis is necessary for cell division; however, the incorporation of new cell wall material varies with the shape of the bacterium. Rod-shaped bacteria (eg, E coli, Bacillus subtilis) have two modes of cell wall synthesis; new peptidoglycan is inserted along a helical path leading to elongation of the cell and is inserted in a closing ring around the future division site, leading to the formation of the division septum. Coccoid cells such as S aureus do not seem to have an elongation mode of cell wall synthesis. Instead, new peptidoglycan is inserted only at the division site. A third form of cell wall growth is exemplified by S pneumoniae, which are not true cocci, because their shape is not totally round but instead have the shape of a rugby ball. S pneumoniae synthesize cell wall not only at the septum but also at the so-called equatorial rings (Figure 2-21). A B Bacillus subtilis or Escherichia coli Streptococcus pneumoniae Staphylococcus aureus Figure 2-21 Incorporation of new cell wall in differently shaped bacteria. Rod-shaped bacteria such as Bacillus subtilis or Escherichia coli have two modes of cell wall synthesis: new peptidoglycan is inserted along a helical path (A), leading to elongation of the lateral wall and is inserted in a closing ring around the future division site, leading to the formation of the division septum (B). Streptococcus pneumoniae cells have the shape of a rugby ball and elongate by inserting new cell wall material at the so-called equatorial rings (A), which correspond to an outgrowth of the cell wall that encircles the cell. An initial ring is duplicated, and the two resultant rings are progressively separated, marking the future division sites of the daughter cells. The division septum is then synthesized in the middle of the cell (B). Round cells such as Staphylococcus aureus do not seem to have an elongation mode of cell wall synthesis. Instead, new peptidoglycan is inserted only at the division septum (B). (Reproduced with permission from Scheffers DJ and Pinho MG: Bacterial cell wall synthesis: new insights from localization studies. Microbiol Mol Biol Rev 2005;69:585.) 32 SECTION I Fundamentals of Microbiology I. Protoplasts, Spheroplasts, and L Forms Removal of the bacterial wall may be accomplished by hydrolysis with lysozyme or by blocking peptidoglycan synthesis with an antibiotic such as penicillin. In osmotically protective media, such treatments liberate protoplasts from grampositive cells and spheroplasts (which retain outer membrane and entrapped peptidoglycan) from gram-negative cells. If such cells are able to grow and divide, they are called L forms. L forms are difficult to cultivate and usually require a medium that is solidified with agar as well as having the right osmotic strength. L forms are produced more readily with penicillin than with lysozyme, suggesting the need for residual peptidoglycan. Some L forms can revert to the normal bacillary form upon removal of the inducing stimulus. Thus, they are able to resume normal cell wall synthesis. Others are stable and never revert. The factor that determines their capacity to revert may again be the presence of residual peptidoglycan, which normally acts as a primer in its own biosynthesis. Some bacterial species produce L forms spontaneously. The spontaneous or antibiotic-induced formation of L forms in the host may produce chronic infections, the organisms persisting by becoming sequestered in protective regions of the body. Because L-form infections are relatively resistant to antibiotic treatment, they present special problems in chemotherapy. Their reversion to the bacillary form can produce relapses of the overt infection. J. The Mycoplasmas The mycoplasmas are cell wall-lacking bacteria containing no peptidoglycan (Figure 2-22). There are also wall-less Archaea, but they have been less well studied. Genomic analysis places the mycoplasmas close to the gram-positive bacteria from which they may have been derived. Mycoplasmas lack a target for cell wall-inhibiting antimicrobial agents (eg, penicillins and cephalosporins) and are therefore resistant to these drugs. Some, such as Mycoplasma pneumoniae, an agent of pneumonia, contain sterols in their membranes. The difference between L forms and mycoplasmas is that when the murein is allowed to reform, L forms revert to their original bacteria shape, but mycoplasmas never do. Capsule and Glycocalyx Many bacteria synthesize large amounts of extracellular polymer when growing in their natural environments. With one known exception (the poly-D-glutamic acid capsules of Bacillus anthracis and Bacillus licheniformis), the extracellular material is polysaccharide (Table 2-1). The terms capsule and slime layer are frequently used to describe polysaccharide layers; the more inclusive term glycocalyx is also used. Glycocalyx is defined as the polysaccharide-containing material lying outside the cell. A condensed, well-defined layer closely surrounding the cell that excludes particles, such as India ink, is referred to as a capsule (Figure 2-23). If the glycocalyx is loosely associated with the cell and does not exclude particles, it is referred to as a slime layer. Extracellular polymer is synthesized by enzymes located at the surface of the bacterial cell. Streptococcus mutans, for example, uses two enzymes—glucosyl transferase and fructosyl transferase—to synthesize long-chain dextrans (poly-D-glucose) and levans (poly-D-fructose) from sucrose. These polymers are called homopolymers. Polymers containing more than one kind of monosaccharide are called heteropolymers. The capsule contributes to the invasiveness of pathogenic bacteria—encapsulated cells are protected from phagocytosis unless they are coated with anticapsular antibody. The glycocalyx plays a role in the adherence of bacteria to surfaces in their environment, including the cells of plant and animal hosts. S mutans, for example, owes its capacity to adhere tightly to tooth enamel to its glycocalyx. Bacterial cells of the same or different species become entrapped in the glycocalyx, which forms the layer known as plaque on the tooth surface; acidic products excreted by these bacteria cause dental caries (see Chapter 11). The essential role of the glycocalyx in this process—and its formation from sucrose—explains the correlation of dental caries with sucrose consumption by the human population. Because outer polysaccharide layers bind a significant amount of water, the glycocalyx layer may also play a role in resistance to desiccation. Flagella A. Structure 2 µm Figure 2-22 Mycoplasma pneumoniae. These cells vary in shape because they lack a cell wall. (Courtesy of Dr. Edwin S. Boatman.) Bacterial flagella are thread-like appendages composed entirely of protein, 12–30 nm in diameter. They are the organs of locomotion for the forms that possess them. Three types of arrangement are known: monotrichous (single polar flagellum), lophotrichous (multiple polar flagella), and peritrichous (flagella distributed over the entire cell). The three types are illustrated in Figure 2-24. CHAPTER 2 Cell Structure 33 TABLE 2-1 Chemical Composition of the Extracellular Polymer in Selected Bacteria Organism Polymer Chemical Subunits Bacillus anthracis Polypeptide d-Glutamic Enterobacter aerogenes Complex polysaccharide Glucose, fucose, glucuronic acid Haemophilus influenzae Serogroup b Ribose, ribitol, phosphate Neisseria meningitidis Homopolymers and heteropolymers, eg, acid Serogroup A Partially O-acetylated N-acetylmannosaminephosphate Serogroup B N-Acetylneuraminic acid (sialic acid) Serogroup C Acetylated sialic acid Serogroup 135 Galactose, sialic acid Pseudomonas aeruginosa Alginate d-Manuronic Streptococcus pneumoniae Complex polysaccharide (many types), eg, (pneumococcus) Type II Rhamnose, glucose, glucuronic acid Type III Glucose, glucuronic acid Type VI Galactose, glucose, rhamnose Type XIV Galactose, glucose, N-acetylglucosamine Type XVIII Rhamnose, glucose Streptococcus pyogenes (group A) Hyaluronic acid N-Acetylglucosamine, glucuronic acid Streptococcus salivarius Levan Fructose A acid, l-glucuronic acid B Figure 2-23 Bacterial capsules. A: Bacillus anthracis M’Faydean capsule stain, grown at 35°C, in defibrinated horse blood. B: Demonstration of the presence of a capsule in B. anthracis by negative staining with India ink. This method is useful for improving visualization of encapsulated bacteria in clinical samples such as blood, blood culture bottles, or cerebrospinal fluid. (CDC, courtesy of Larry Stauffer, Oregon State Public Health Laboratory.) A bacterial flagellum is made up of several thousand molecules of a protein subunit called flagellin. In a few organisms (eg, Caulobacter), flagella are composed of two types of flagellin, but in most, only a single type is found. The flagellum is formed by the aggregation of subunits to form a helical structure. If flagella are removed by mechanically agitating a suspension of bacteria, new flagella are rapidly formed by the synthesis, aggregation, and extrusion of flagellin subunits; motility is restored within 3–6 minutes. The flagellins of different bacterial species presumably differ from one another in primary structure. They 34 SECTION I Fundamentals of Microbiology A C B Figure 2-24 Bacterial flagellation. A: Vibrio metchnikovii, a monotrichous bacterium (7500×). (Reproduced with permission from van Iterson W: Biochim Biophys Acta 1947;1:527.) B: Electron micrograph of Spirillum serpens, showing lophotrichous flagellation (9000×). (Reproduced with permission from van Iterson W: Biochim Biophys Acta 1947;1:527.) C: Electron micrograph of Proteus vulgaris, showing peritrichous flagellation (9000×). Note basal granules. (Reproduced with permission from Houwink A, van Iterson W: Electron microscopical observations on bacterial cytology; a study on flagellation. Biochim Biophys Acta 1950;5:10.) are highly antigenic (H antigens), and some of the immune responses to infection are directed against these proteins. The flagellum is attached to the bacterial cell body by a complex structure consisting of a hook and a basal body. The hook is a short curved structure that appears to act as the universal joint between the motor in the basal structure and the flagellum. The basal body bears a set of rings, one pair in gram-positive bacteria and two pairs in gram-negative bacteria. An interpretative diagram of the gram-negative structure is shown in Figure 2-25; the rings labeled L and P are absent A B Filament cap (FliD) Filament (FliC) Filament Hook-filament junction (FlgK FlgL) 20 nm 10 µm Hook (FlgE) Hook Propellor L ring (FlgH) Bushing Outer membrane Periplasmic space Basal body Cell membrane Motor Switch Export apparatus Switch (FliG, FliM, FliN) Export (FlhA, FliH, Flil) ? P ring (Flgl) Distal rod (Flgl) Proximal rod (FliE, FlgB, FlgC, FlgF) MS ring (FliF) Transmission shaft Mounting plate Motor (MotA, MotB) Figure 2-25 A: General structure of the flagellum of a gram-negative bacterium, such as Escherichia coli or Salmonella typhimurium. The filament-hook-basal body complex has been isolated and extensively characterized. The location of the export apparatus has not been demonstrated. B: An exploded diagram of the flagellum showing the substructures and the proteins from which they are constructed. The FliF protein is responsible for the M-ring feature, S-ring feature, and collar feature of the substructure shown, which is collectively termed the MS ring. The location of FliE with respect to the MS ring and the rod—and the order of the FlgB, FlgC, and FlgF proteins within the proximal rod— is not known. (From Macnab RM: Genetics and biogenesis of bacterial flagella. Annu Rev Genet 1992;26:131. Reproduced with permission from Annual Review of Genetics, Volume 26, © 1992 by Annual Reviews.) CHAPTER 2 Cell Structure 35 in gram-positive cells. The complexity of the bacterial flagellum is revealed by genetic studies, which show that over 40 gene products are involved in its assembly and function. Flagella are made stepwise (see Figure 2-25). First, the basal body is assembled and inserted into the cell envelope. Then the hook is added, and finally, the filament is assembled progressively by the addition of flagellin subunits to its growing tip. The flagellin subunits are extruded through a hollow central channel in the flagella; when it reaches the tip, it condenses with its predecessors, and thus the filament elongates. B. Motility Bacterial flagella are semirigid helical rotors to which the cell imparts a spinning movement. Rotation is driven by the flow of protons into the cell down the gradient produced by the primary proton pump (see earlier discussion); in the absence of a metabolic energy source, it can be driven by a proton motive force generated by ionophores. Bacteria living in alkaline environments (alkalophiles) use the energy of the sodium ion gradient—rather than the proton gradient—to drive the flagellar motor (Figure 2-26). All of the components of the flagellar motor are located in the cell envelope. Flagella attached to isolated, sealed cell envelopes rotate normally when the medium contains a suitable substrate for respiration or when a proton gradient is artificially established. When a peritrichous bacterium swims, its flagella associate to form a posterior bundle that drives the cell forward in a straight line by counterclockwise rotation. At intervals, the flagella reverse their direction of rotation and momentarily dissociate, causing the cell to tumble until swimming resumes in a new, randomly determined direction. This behavior makes possible the property of chemotaxis: A cell that is moving away from the source of a chemical attractant tumbles and reorients itself more frequently than one that is moving toward the attractant, the result being the net movement of the cell toward the source. The presence of a chemical attractant (eg, a sugar or an amino acid) is sensed by specific receptors located in the cell membrane (in many cases, the same receptor also participates in membrane transport of that molecule). The bacterial cell is too small to be able to detect the existence of a spatial chemical gradient (ie, a gradient between its two poles); rather, experiments show that it detects temporal gradients, that is, concentrations that Filament Hook Outer membrane Murein + H H+ + – + – H+ + – H+ + + – + – Proton motive force Basal body H+ Periplasmic space + Cell membrane – Motor Switch H+ Figure 2-26 Structural components within the basal body of the flagellum allow the inner portion of this structure, the rods of the basal body, and the attached hook–filament complex to rotate. The outer rings remain statically in contact with the inner and outer cell membranes and cell wall (murein), anchoring the flagellum complex to the bacterial cell envelope. Rotation is driven by the flow of protons through the motor from the periplasmic space, outside the cell membrane, into the cytoplasm in response to the electric field and proton gradient across the membrane, which together constitute the proton motive force. A switch determines the direction of rotation, which in turn determines whether the bacteria swim forward (by counterclockwise rotation of the flagellum) or tumble (caused by clockwise rotation of the flagellum). (Reproduced with permission from Saier MH Jr: Peter Mitchell and his chemiosmotic theories. ASM News 1997;63:13.) 36 SECTION I Fundamentals of Microbiology decrease with time during which the cell is moving away from the attractant source and increase with time during which the cell is moving toward it. Some compounds act as repellants rather than attractants. One mechanism by which cells respond to attractants and repellents involves a cGMP-mediated methylation and demethylation of specific proteins in the membrane. Whereas attractants cause a transient inhibition of demethylation of these proteins, repellents stimulate their demethylation. The mechanism by which a change in cell behavior is brought about in response to a change in the environment is called sensory transduction. Sensory transduction is responsible not only for chemotaxis but also for aerotaxis (movement toward the optimal oxygen concentration), phototaxis (movement of photosynthetic bacteria toward the light), and electron acceptor taxis (movement of respiratory bacteria toward alternative electron acceptors, such as nitrate and fumarate). In these three responses, as in chemotaxis, net movement is determined by regulation of the tumbling response. Pili (Fimbriae) Many gram-negative bacteria possess rigid surface appendages called pili (L “hairs”) or fimbriae (L “fringes”). They are shorter and finer than flagella; similar to flagella, they are composed of structural protein subunits termed pilins. Some pili contain a single type of pilin, others more than one. Minor proteins termed adhesins are located at the tips of pili and are responsible for the attachment properties. Two classes can be distinguished: ordinary pili, which play a role in the adherence of symbiotic and pathogenic bacteria to host cells, and sex pili, which are responsible for the attachment of donor and recipient cells in bacterial conjugation (see Chapter 7). Pili are illustrated in Figure 2-27, in which the sex pili have been coated with phage particles for which they serve as specific receptors. Sex pilus Flagellum Other pili Motility via pili is completely different from flagellar motion. Pilin molecules are arranged helically to form a straight cylinder that does not rotate and lacks a complete basal body. Their tips strongly adhere to surfaces at a distance from the cells. Pili then depolymerize from the inner end, thus retracting inside the cell. The result is that the bacterium moves in the direction of the adhering tip. This kind of surface motility is called twitching and is widespread among piliated bacteria. Unlike flagella, pili grow from the inside of the cell outward. The virulence of certain pathogenic bacteria depends on the production not only of toxins but also of “colonization antigens,” which are ordinary pili that provide the cells with adherent properties. In enteropathogenic E coli strains, both the enterotoxins and the colonization antigens (pili) are genetically determined by transmissible plasmids, as discussed in Chapter 7. In one group of gram-positive cocci, the streptococci, fimbriae are the site of the main surface antigen, the M protein. Lipoteichoic acid, associated with these fimbriae, is responsible for the adherence of group A streptococci to epithelial cells of their hosts. Pili of different bacteria are antigenically distinct and elicit the formation of antibodies by the host. Antibodies against the pili of one bacterial species will not prevent the attachment of another species. Some bacteria (see Chapter 21), such as N gonorrhoeae, are able to make pili of different antigenic types (antigenic variation) and thus can still adhere to cells in the presence of antibodies to their original type of pili. Similar to capsules, pili inhibit the phagocytic ability of leukocytes. Endospores Members of several bacterial genera are capable of forming endospores (Figure 2-28). The two most common are gram-positive rods: the obligately aerobic genus Bacillus and the obligately anaerobic genus Clostridium. The other bacteria known to form endospores are Thermoactinomyces, Sporolactobacillus, Sporosarcina, Sporotomaculum, Sporomusa, and Sporohalobacter spp. These organisms undergo a cycle of differentiation in response to environmental conditions: The process, sporulation, is triggered by near depletion of any of several nutrients (carbon, nitrogen, or phosphorous). Each cell forms a single internal spore that is liberated when the mother cell undergoes autolysis. The spore is a resting cell, highly resistant to desiccation, heat, and chemical agents; when returned to favorable nutritional conditions and activated (see below), the spore germinates to produce a single vegetative cell. A. Sporulation 1 µm Figure 2-27 Pili. Pili on an Escherichia coli cell. The short pili (fimbriae) mediate adherence; the sex pilus is involved in DNA transfer. (Courtesy of Dr. Charles Brinton, Jr.) The sporulation process begins when nutritional conditions become unfavorable, near depletion of the nitrogen or carbon source (or both) being the most significant factor. Sporulation occurs massively in cultures that have terminated exponential growth as a result of this near depletion. CHAPTER 2 Cell Structure 37 A B C Figure 2-28 Sporulating cells of bacillus species. A: Unidentified bacillus from soil. B: Bacillus cereus. C: Bacillus megaterium. (Reproduced with permission from Robinow CF: Structure. In Gunsalus IC, Stanier RY [editors]. The Bacteria: A Treatise on Structure and Function, Vol 1. Academic Press, 1960.) Sporulation involves the production of many new structures, enzymes, and metabolites along with the disappearance of many vegetative cell components. These changes represent a true process of differentiation: A series of genes whose products determine the formation and final composition of the spore are activated. These changes involve alterations in the transcriptional specificity of RNA polymerase, which is determined by the association of the polymerase core protein with one or another promoterspecific protein called a sigma factor. During vegetative growth, a sigma factor designated σA predominates. Then, during sporulation, five other sigma factors are formed that cause various spore genes to be expressed at various times in specific locations. The sequence of events in sporulation is highly complex: Differentiation of a vegetative cell of B subtilis into an endospore takes about 7 hours under laboratory conditions. Different morphologic and chemical events occur at sequential stages of the process. Seven different stages have been identified. Morphologically, sporulation begins with the formation of an axial filament (Figure 2-29). The process continues with an infolding of the membrane so as to produce a doublemembrane structure whose facing surfaces correspond to the cell wall–synthesizing surface of the cell envelope. The growing points move progressively toward the pole of the cell so as to engulf the developing spore. The two spore membranes now engage in the active synthesis of special layers that will form the cell envelope: the spore wall and the cortex, lying outside the facing membranes. In the newly isolated cytoplasm, or core, many vegetative cell enzymes are degraded and are replaced by a set of unique spore constituents. B. Properties of Endospores 1. Core—The core is the spore protoplast. It contains a com- plete nucleus (chromosome), all of the components of the protein-synthesizing apparatus, and an energy-generating system based on glycolysis. Cytochromes are lacking even in aerobic species, the spores of which rely on a shortened electron transport pathway involving flavoproteins. A number of vegetative cell enzymes are increased in amount (eg, alanine racemase), and a number of unique enzymes are formed (eg, dipicolinic acid synthetase). Spores contain no reduced pyridine nucleotides or ATP. The energy for germination is stored as 3-phosphoglycerate rather than as ATP. The heat resistance of spores is partly attributable to their dehydrated state and in part to the presence in the core of large amounts (5–15% of the spore dry weight) of calcium dipicolinate, which is formed from an intermediate of the lysine biosynthetic pathway (see Figure 6-18). In some way not yet understood, these properties result in the stabilization of the spore enzymes, most of which exhibit normal heat lability when isolated in soluble form. Spore wall—The innermost layer surrounding the inner spore membrane is called the spore wall. It contains normal peptidoglycan and becomes the cell wall of the germinating vegetative cell. Cortex—The cortex is the thickest layer of the spore envelope. It contains an unusual type of peptidoglycan, with many fewer cross-links than are found in cell wall peptidoglycan. Cortex peptidoglycan is extremely sensitive to lysozyme, and its autolysis plays a role in spore germination. 38 SECTION I Fundamentals of Microbiology Cell wall 0 1 Axial filament formation DNA 2 3 Forespore septum formation Engulfment of forespore Spore mother cell 4 Cortex synthesis 5 Coat deposition 6 Cortex Germ cell wall Maturation Spore coats 7 Lysis of mother cell Spore Figure 2-29 The stages of endospore formation. (Reproduced with permission from Merrick MJ: Streptomyces. In: Parish JH [editor]. Developmental Biology of Procaryotes. Univ California Press, 1979.) Coat—The coat is composed of a keratin-like protein con- taining many intramolecular disulfide bonds. The impermeability of this layer confers on spores their relative resistance to antibacterial chemical agents. Exosporium—The exosporium is composed of proteins, lipids, and carbohydrates. It consists of a paracrystalline basal layer and a hairlike outer region. The function of the exosporium is unclear. Spores of some Bacillus species (eg, B anthracis and B cereus) possess an exosporium, but other species (eg, B atrophaeus) have spores that lack this structure. C. Germination The germination process occurs in three stages: activation, initiation, and outgrowth. Activation—Most endospores cannot germinate immediately after they have formed. But they can germinate after they have rested for several days or are first activated in a nutritionally rich medium by one or another agent that damages the spore coat. Among the agents that can overcome spore dormancy are heat, abrasion, acidity, and compounds containing free sulfhydryl groups. CHAPTER 2 Cell Structure 39 Initiation—After activation, a spore will initiate germination if the environmental conditions are favorable. Different species have evolved receptors that recognize different effectors as signaling a rich medium: Thus, initiation is triggered by l-alanine in one species and by adenosine in another. Binding of the effector activates an autolysin that rapidly degrades the cortex peptidoglycan. Water is taken up, calcium dipicolinate is released, and a variety of spore constituents are degraded by hydrolytic enzymes. 3. Outgrowth—Degradation of the cortex and outer layers results in the emergence of a new vegetative cell consisting of the spore protoplast with its surrounding wall. A period of active biosynthesis follows; this period, which terminates in cell division, is called outgrowth. Outgrowth requires a supply of all nutrients essential for cell growth. The Acid-Fast Stain Acid-fast bacteria are those that retain carbolfuchsin (basic fuchsin dissolved in a phenol–alcohol–water mixture) even when decolorized with hydrochloric acid in alcohol. A smear of cells on a slide is flooded with carbolfuchsin and heated on a steam bath. After this, the discolorization with acid-alcohol is carried out, and finally a contrasting (blue or green) counterstain is applied (see Chapter 47). Acid-fast bacteria (mycobacteria and some of the related actinomycetes) appear red; others take on the color of the counterstain. Negative Staining This procedure involves staining the background with an acidic dye, leaving the cells contrastingly colorless. The black dye nigrosin is commonly used. This method is used for cells or structures that are difficult to stain directly (see Figure 2-23B). STAINING Stains combine chemically with the bacterial protoplasm; if the cell is not already dead, the staining process itself will kill it. The process is thus a drastic one and may produce artifacts. The commonly used stains are salts. Basic stains consist of a colored cation with a colorless anion (eg, methylene blue+ chloride -); acidic stains are the reverse (eg, sodium+ eosinate -). Bacterial cells are rich in nucleic acid, bearing negative charges as phosphate groups. These combine with the positively charged basic dyes. Acidic dyes do not stain bacterial cells and hence can be used to stain background material a contrasting color (see Negative Staining). The basic dyes stain bacterial cells uniformly unless the cytoplasmic RNA is destroyed first. Special staining techniques can be used, however, to differentiate flagella, capsules, cell walls, cell membranes, granules, nucleoids, and spores. The Flagella Stain Flagella are too fine (12–30 nm in diameter) to be visible in the light microscope. However, their presence and arrangement can be demonstrated by treating the cells with an unstable colloidal suspension of tannic acid salts, causing a heavy precipitate to form on the cell walls and flagella. In this manner, the apparent diameter of the flagella is increased to such an extent that subsequent staining with basic fuchsin makes the flagella visible in the light microscope. Figure 2-30 shows cells stained by this method. The Gram Stain An important taxonomic characteristic of bacteria is their response to Gram stain. The Gram-staining property appears to be a fundamental one because the Gram reaction is correlated with many other morphologic properties in phylogenetically related forms (see Chapter 3). An organism that is potentially gram positive may appear so only under a particular set of environmental conditions and in a young culture. The Gram-staining procedure (see Chapter 47 for details) begins with the application of a basic dye, crystal violet. A solution of iodine is then applied; all bacteria will be stained blue at this point in the procedure. The cells are then treated with alcohol. Gram-positive cells retain the crystal violet– iodine complex, remaining blue; gram-negative cells are completely decolorized by alcohol. As a last step, a counterstain (eg, the red dye safranin) is applied so that the decolorized gram-negative cells will take on a contrasting color; the gram-positive cells now appear purple. The basis of the differential Gram reaction is the structure of the cell wall, as discussed earlier in this chapter. Figure 2-30 Flagella stain of Pseudomonas species. (Reproduced with permission from Leifson E: Staining, shape and arrangement of bacterial flagella. J Bacteriol 1951;62:377.) 40 SECTION I Fundamentals of Microbiology In peritrichous bacteria, the flagella form into bundles during movement, and such bundles may be thick enough to be observed on living cells by dark-field or phase contrast microscopy. The Capsule Stain Capsules are usually demonstrated by the negative staining procedure or a modification of it (see Figure 2-23). One such “capsule stain” (Welch method) involves treatment with hot crystal violet solution followed by a rinsing with copper sulfate solution. The latter is used to remove excess stain because the conventional washing with water would dissolve the capsule. The copper salt also gives color to the background, with the result that the cell and background appear dark blue and the capsule a much paler blue. Staining of Nucleoids Nucleoids are stainable with the Feulgen stain, which is specific for DNA (see Figure 2-5). The Spore Stain Spores are most simply observed as intracellular refractile bodies (see Figure 2-28) in unstained cell suspensions or as colorless areas in cells stained by conventional methods . The spore wall is relatively impermeable, but dyes can be made to penetrate it by heating the preparation. The same impermeability then serves to prevent decolorization of the spore by a period of alcohol treatment sufficient to decolorize vegetative cells. The latter can finally be counterstained. Spores are commonly stained with malachite green or carbolfuchsin (Figure 2-31). MORPHOLOGIC CHANGES DURING GROWTH Cell Division Most bacteria divide by binary fission into two equal progeny cells. In a growing culture of a rod-shaped bacterium such as E coli, cells elongate and then form a partition that eventually separates the cell into two daughter cells. The partition is referred to as a septum and is a result of the inward growth of the cytoplasmic membrane and cell wall from opposing directions until the two daughter cells are pinched off. The chromosomes, which have doubled in number preceding the division, are distributed equally to the two daughter cells. Although bacteria lack a mitotic spindle, the septum is formed in such a way as to separate the two sister chromosomes formed by chromosomal replication. This is accomplished by the attachment of the chromosome to the cell membrane. According to one model, completion of a cycle of DNA replication triggers active membrane synthesis between the sites of attachment of the two sister chromosomes. The chromosomes are then pushed apart by the inward growth of the septum, one copy going to each daughter cell. Cell Groupings If the cells remain temporarily attached after division, certain characteristic groupings result. Depending on the plane of division and the number of divisions through which the cells remain attached, the following may occur in the coccal forms: chains (streptococci), pairs (diplococci), cubical bundles (sarcinae), or flat plates. Rods may form pairs or chains. After fission of some bacteria, characteristic postdivision movements occur. For example, a “whipping” motion can bring the cells into parallel positions; repeated division and whipping result in the “palisade” arrangement characteristic of diphtheria bacilli. CHAPTER SUMMARY • • • 10 µm Figure 2-31 Endospore stain. Endospores retain the green primary stain, malachite green. Counterstaining with safranin imparts a red color to other cells. (© Jack M. Bostrack/Visuals Unlimited.) icroscopy has played an important role in our underM standing of cell structure. Eukaryotic cells are characterized by a membrane-bound nucleus, an endoplasmic reticulum, 80S ribosomes, and plastids (mitochondria and chloroplasts). The plasma membrane is characterized by the presence of sterols (cholesterol). Prokaryotic cells lack a true nucleus and are haploid. The cytoplasm contains 70S ribosomes, and they do not have mitochondria and chloroplasts. The major functions of the cell membrane of prokaryotic cells are (1) selective permeability and transport of solutes; (2) electron transport and oxidative phosphorylation, in aerobic species; (3) excretion of hydrolytic enzymes and other proteins; (4) bearing the enzymes and carrier molecules that function in the biosynthesis of DNA, cell wall polymers, and membrane lipids; and CHAPTER 2 Cell Structure 41 • • • • • (5) bearing the receptors and proteins of the chemotactic and other sensory transduction systems. Most bacteria are classified as gram-positive or gram– negative according to their response to the Gram-staining procedure. The differences between these two groups are reflected by fundamental differences in their cell envelopes. Gram-positive cell wall consists of a plasma membrane and thick peptidoglycan layer; the gram-negative cell wall consists of a plasma membrane, a thin peptidoglycan layer, and an outer membrane containing lipopolysaccharide (endotoxin). The space between the plasma membrane and outer membrane is referred to as the periplasmic space. Many bacteria synthesize large amounts of extracellular polymers. When this polymer forms a condensed, welldefined layer surrounding the cell that excludes particles such as India ink, it is referred to as a capsule. Capsules are an important virulence factor and protect the cell from phagocytosis. Cell surface structures such as pili and flagella are important for attachment and motility, respectively. The formation of endospores is a characteristic of the genera Bacillus and Clostridium and is triggered by near depletion of nutrients in the environment. Endospores (spores) are resting cells, highly resistant to desiccation, heat, and chemical agents; when returned to favorable nutritional conditions and activated, the spore germinates to produce a vegetative cell. REVIEW QUESTIONS 1. A 22-year-old man presents with a painless 1-cm ulcer on the shaft of his penis. Inguinal lymphadenopathy is present. The patient admits trading drugs for sex and has several sexual partners. An RPR test result is positive, and syphilis is suspected; however, a Gram stain of a swab specimen from the ulcer shows no bacteria. Treponema pallidum, the causative agent of syphilis, cannot be visualized by light microscopy because (A) It is transparent. (B) It cannot be stained by ordinary stains. (C) It has a diameter of less than 0.2 mm. (D) The wavelength of white light is too long. (E) Rapid movement of the organism prevents visualization. 2. Chloramphenicol, an antibiotic that inhibits bacterial protein synthesis, will also affect which of the following eukaryotic organelles? (A) Mitochondria (B) Golgi apparatus (C) Microtubules (D) Endoplasmic reticulum (E) Nuclear membrane 3. Which of the following structures is not part of the bacterial cell envelope? (A) Peptidoglycan (B) Lipopolysaccharide (C) Capsule (D) Gas vacuole (E) S-layer Which of the following transport mechanisms functions without the requirement for energy? (A) Binding protein dependent (B) Group translocation (C) Symport (D) Uniport (E) Facilitated diffusion 5. Which of the following components is present in gram-negative bacteria but not in gram-positive bacteria? (A) Peptidoglycan (B) Lipid A (C) Capsule (D) Flagella (E) Pili 6. Which of the following components is present in gram-positive bacteria but not in gram-negative bacteria? (A) Peptidoglycan (B) Capsule (C) Flagella (D) Teichoic acid (E) Diaminopimelic acid 7. In the fall of 2001, a series of letters containing spores of Bacillus anthracis were mailed to members of the media and to U.S. Senate offices. The result was 22 cases of anthrax, with five deaths. The heat resistance of bacterial spores, such as those of Bacillus anthracis, is partly attributable to their dehydrated state and partly to the presence of large amounts of (A) Diaminopimelic acid (B) d-Glutamic acid (C) Calcium dipicolinate (D) Sulfhydryl-containing proteins (E) Lipid A 8. Which of the following terms does NOT describe the bacterial chromosome? (A) Haploid (B) Diploid (C) Circular (D) Nucleoid (E) Feulgen positive 9. Lysozyme cleaves the β1→4 linkage between (A) d-Alanine and the pentaglycine bridge (B) N-Acetylmuramic acid and d-alanine (C) Lipid A and KDO (D) N-Acetylmuramic acid and N-acetylglucosamine (E) d-Alanine and d-alanine 10. Mycoplasma species lack which of the following components? (A) Ribosomes (B) Plasma membrane (C) Both DNA and RNA (D) Lipids (E) Peptidoglycan Answers 1. C E C A B B D D D E 42 SECTION I Fundamentals of Microbiology REFERENCES Balows A et al (editors): The Prokaryotes, A Handbook on the Biology of Bacteria: Ecophysiology, Isolation, Identification, Applications, 2nd ed, 4 vols. Springer, 1992. Barreteau H, Kovac A, Boniface A, Sova M, Gobec S, Blanot D: Cytoplasmic steps of peptidoglycan biosynthesis. FEMS Microbiol Rev 2008;32:168. Barton LL: Structural and Functional Relationships in Prokaryotes. Springer, 2005. Bermudes D, Hinkle G, Margulis L: Do prokaryotes contain microtubules? Microbiol Rev 1994;58:387. Blair DF: How bacteria sense and swim. Annu Rev Microbiol 1995;49:489. Craig L, Pique ME, Tainer JA: Type IV pilus structure and bacterial pathogenicity. Nat Rev Microbiol 2004;2:363. Dautin N, Bernstein HD: Protein secretion in gram-negative bacteria via the autotransporter pathway. Annu Rev Microbiol 2007;61:89. Drlica K, Riley M (editors): The Bacterial Chromosome. American Society for Microbiology, 1990. Economou A, Christie PJ, Fernandez RC, Palmer T, Plano GV, Pugsley AP: Secretion by the numbers: Protein traffic in prokaryotes. Mol Microbiol 2006;62:308. Henriques AO, Moran CP Jr: Structure, assembly, and function of the spore surface layers. Annu Rev Microbiol 2007;61:555. Hinnebusch J, Tilly K: Linear plasmids and chromosomes in bacteria. Mol Microbiol 1993;10:917. Hueck CJ: Type III protein secretion systems in bacterial pathogens of animals and plants. Microbiol Mol Biol Rev 1998;62:379. Leiman PG et al: Type VI secretion apparatus and phage tailassociated protein complexes share a common evolutionary origin. Proc Natl Acad Sci U S A 2009;106:4154. Liu J, Barry CE III, Besra GS, Nikaido H: Mycolic acid structure determines the fluidity of the mycobacterial cell wall. J Biol Chem 1996;271:29545. Messner P et al: III. Biochemistry of S-layers. FEMS Microbiol Rev 1997;20:25. Moat AG, Foster JW: Microbial Physiology, 3rd ed. Wiley-Liss, 1995. Naroninga N: Morphogenesis of Escherichia coli. Microbiol Mol Biol Rev 1998;62:110. Neuhaus FC, Baddiley J: A continuum of anionic charge: Structures and functions of D-alanyl-teichoic acids in gram-positive bacteria. Microbiol Mol Biol Rev 2003;67:686. Nikaido H: Molecular basis of bacterial outer membrane permeability revisited. Microbiol Mol Biol Rev 2003;67:593. Rachel R et al: Fine structure of S-layers. FEMS Microbiol Rev 1997;20:13. Sauvage E, Kerff F, Terrak M, Ayala JA, Charlier P: The penicillinbinding proteins: Structure and role in peptidoglycan biosynthesis. FEMS Microbiol Rev 2008;32:234. Schaechter M, Ingraham JL, Neidhardt FC: Microbe. American Society for Microbiology, 2006. Scheffers DJ, Pinho MG: Bacterial cell wall synthesis: New insights from localization studies. Microbiol Mol Biol Rev 2005;69:585. Schirmer T: General and specific porins from bacterial outer membranes. J Struct Biol 1998;121:101. [PMID: 9615433] Scott JR, Barnett TC: Surface proteins of gram-positive bacteria and how they get there. Annu Rev Microbiol 2006;60:397. Sonenshein AL, Hoch JA, Losick R: Bacillus subtilis and Its Closest Relatives. American Society for Microbiology, 2002. Vaara M: Agents that increase the permeability of the outer membrane. Microbiol Rev 1992;56:395. Vollmer W, Blanot D, de Pedro MA: Peptidoglycan structure and architecture. FEMS Microbiol Rev 2008;32:149. Walsby AE: Gas vesicles. Microbiol Rev 1994;58:94. Whittaker CJ, Klier CM, Kolenbrander PE: Mechanisms of adhesion by oral bacteria. Annu Rev Microbiol 1996;50:513. C Classification of Bacteria TAXONOMY—THE VOCABULARY OF MEDICAL MICROBIOLOGY One has only to peruse the table of contents of this book to appreciate the diversity of medical pathogens that are associated with infectious diseases. It has been estimated that we currently have the capacity to identify fewer than 10% of the pathogens responsible for causing human disease because of our inability to culture or target these organisms using molecular probes. Yet the diversity of even these identifiable pathogens is so great that it is important to understand the subtle differences associated with each infectious agent. The reason for understanding these subtleties is significant because each infectious agent has specifically adapted to a particular mode(s) of transmission, a mechanism(s) to grow in human hosts (colonization), and a mechanism(s) to cause disease (pathology). As such, a vocabulary that consistently communicates the unique characteristics of infectious organisms to students, microbiologists, and health care workers is critical to avoid the chaos that would ensue without the organizational restraints of bacterial taxonomy (Gk. taxon = arrangement; eg, the classification of organisms in an ordered system that indicates a natural relationship). Classification, nomenclature, and identification are three separate but interrelated areas of bacterial taxonomy. Classification is the categorization of organisms into taxonomic groups. Classification of bacteria requires experimental and observational techniques; this is because biochemical, physiologic, genetic, and morphologic properties are often necessary for an adequate description of a taxon. Nomenclature refers to the naming of an organism by international rules (established by a recognized group of medical professionals) according to its characteristics. Identification is practical use of a classification scheme to (1) isolate and distinguish desirable organisms from undesirable ones, (2) verify the authenticity or special properties of a culture in a clinical setting, and (3) isolate and identify the causative agent of a disease. The latter may lead to the selection of specific pharmacologic treatments directed toward their eradication, a vaccine mitigating their pathology, or a public health measure (eg, handwashing or use of a condom) that prevents their transmission. 3 H A P T E R Identification schemes are not classification schemes, although there may be some superficial similarity. An identification scheme for a group of organisms can be devised only after that group has first been classified (ie, recognized as being different from other organisms). For example, the popular literature has reported Escherichia coli as being a cause of hemolytic uremic syndrome (HUS) in infants. There are hundreds of different strains that are classified as E coli but only a few that are associated with HUS. These strains can be distinguished from the many other E coli strains by antibody reactivity with their O- and H-antigens, as described in Chapter 2 (eg, E coli O157:H7). Taxonomy, and the nomenclature that accompanies it, is an imprecise and evolving science. Just as our societal vocabulary evolves, so does the vocabulary of medical microbiology. Any professional associated with infectious disease should be aware of the evolving taxonomy of infectious microorganisms. Taxonomic ranks form the basis for the organization of bacteria. Linnaean taxonomy is the system most familiar to biologists. It uses the formal taxonomic ranks of kingdom, phylum, class, order, family, genus, and species. The lower ranks are approved by a consensus of experts in the scientific community (Table 3-1). Of these ranks, the family, genus, and species are the most useful. TABle 3-1 Taxonomic ranks Formal rank example Kingdom Prokaryotae Division Gracilicutes Class Scotobacteria Order Eubacteriales Family Enterobacteriaceae Genus Escherichia Species coli Subtype Escherichia coli O157: H7 43 44 SECTION I Fundamentals of Microbiology CRITERIA FOR CLASSIFICATION OF BACTERIA Growth on Media Suitable criteria for purposes of bacterial classification include many of the properties that were described in the preceding chapter. One criterion is growth on bacteriologic media. In contrast to viruses and most parasites, many bacterial pathogens can be isolated on solid agar-containing media. The general cultivation of most bacteria requires media rich in metabolic nutrients. These media generally include agar, a carbon source, and an acid hydrolysate or enzymatically degraded source of biologic material (eg, casein). Because of the undefined composition of the latter, these types of media are referred to as complex media. Clinical samples from normally nonsterile sites (eg, the throat or the colon) contain multiple species of organisms, including potential pathogens and resident microbial flora. Media can be nonselective or selective; the latter are used to distinguish among the various bacteria in a clinical sample containing many different organisms. A. Nonselective Media Blood agar and chocolate agar are examples of complex, nonselective media, which support the growth of many different bacteria. These media are intended to cultivate as many species as possible, thus giving rise to numerous types of bacterial colonies. B. Selective Media Because of the diversity of microorganisms that typically reside at some sampling sites (eg, the skin, respiratory tract, intestines, vagina), selective media are used to eliminate (or reduce) the large numbers of irrelevant bacteria in these specimens. The basis for selective media is the incorporation of an inhibitory agent that specifically selects against the growth of irrelevant bacteria. Examples of such agents are: • • • Sodium azide—selects for gram-positive bacteria over gram-negative bacteria Bile salts (sodium deoxycholate)—select for gram- negative enteric bacteria and inhibit gram-negative mucosal and most gram-positive bacteria Colistin and nalidixic acid—inhibit the growth of many gram-negative bacteria Examples of selective media are MacConkey agar (contains bile) that selects for the Enterobacteriaceae and CNA blood agar (contains colistin and nalidixic acid) that selects for Staphylococci and Streptococci. C. Differential Media Upon culture, some bacteria produce characteristic pigments, and others can be differentiated on the basis of their complement of extracellular enzymes; the activity of these enzymes often can be detected as zones of clearing surrounding colonies grown in the presence of insoluble substrates (eg, zones of hemolysis in agar medium containing red blood cells). Many of the members of the Enterobacteriaceae can be differentiated on the basis of their ability to metabolize lactose. For example, whereas pathogenic salmonellae and shigellae do not ferment lactose on a MacConkey plate form white colonies, lactose-fermenting members of the Enterobacteriaceae (eg, E coli) form red or pink colonies. The number of differential media used in today’s clinical laboratories is far beyond the scope of this chapter. However, it should be noted that biochemical identification is an important means to classify microbial pathogens. Bacterial Microscopy Historically, the Gram stain, together with visualization by light microscopy, has been among the most informative methods for classifying the eubacteria. This staining technique broadly divides bacteria on the basis of fundamental differences in the structure of their cell walls (see Chapter 2). This typically represents the first step in identifying individual microbial specimens (eg, are they gram negative or gram positive?) grown in culture or even directly from patient specimens (eg, urine specimens). Biochemical Tests Tests such as the oxidase test, which uses an artificial electron acceptor, can be used to distinguish organisms on the basis of the presence or absence of a respiratory enzyme, cytochrome C, the lack of which differentiates the Enterobacteriaceae from other gram-negative rods. Similarly, catalase activity can be used, for example, to differentiate between the gram-positive cocci; the species staphylococci are catalase positive whereas the species streptococci are catalase negative. If the organism is demonstrated to be catalase positive (Staphylococcus spp.), the species can be subdivided by a coagulase test into Staphylococcus aureus (coagulase positive) or Staphylococcus epidermitidis (coagulase negative) as demonstrated in Figure 3-1. Ultimately, there are many examples of biochemical tests that can ascertain the presence of characteristic metabolic functions and be used to group bacteria into a specific taxon. A nonexhaustive list of common biochemical tests is given in Table 3-2. Immunologic Tests—Serotypes, Serogroups, and Serovars The designation “sero” simply indicates the use of antibodies (polyclonal or monoclonal) that react with specific bacterial cell surface structures such as lipopolysaccharide (LPS), flagella, or capsular antigens. The terms “serotype,” “serogroups,” and “serovars” are, for all practical purposes, identical—they all use the specificity of these antibodies to subdivide strains of a particular bacterial species. This has been described earlier in this chapter as it relates to the relationship E coli O157:H7 and HUS. CHAPTER 3 Classification of Bacteria 45 Table 3-2 Common Microbial Biochemical Tests Gram + cocci catalase test Catalase + staphylococci Used to Differentiate Among Bacteria Catalase – streptococci Coagulase test S. aureus – S. epidermitis Figure 3-1 Algorithm for differentiating the gram-positive cocci. Genetic Instability The value of a taxonomic criterion depends upon the biologic group being compared. Traits shared by all or none of the members of a group cannot be used to distinguish its members, but they may define a group (eg, all staphylococci produce the enzyme catalase). Developments in molecular biology now make it possible to investigate the relatedness of genes or genomes by comparing sequences among different bacteria (see Chapter 7). For these cases, genetic instability can cause some traits to be highly variable within a biologic group or even within a specific taxonomic group. For example, antibiotic resistance genes or genes encoding enzymes (eg, lactose utilization) may be carried on plasmids or bacteriophages (see Chapter 7), extrachromosomal genetic elements that may be transferred among unrelated bacteria or that may be lost from a subset of bacterial strains identical in all other respects. Many organisms are difficult to cultivate, and in these instances, techniques that reveal relatedness by measurement of nucleic acid hybridization or by DNA sequence analysis may be of particular value. CLASSIFICATION SYSTEMS Keys Keys organize bacterial traits in a manner that permits efficient identification of organisms. The ideal system should contain the minimum number of features required for a correct categorization. Groups are split into smaller subgroups based on the presence (+) or absence (–) of a diagnostic character. Continuation of the process with different characteristics guides the investigator to the smallest defined subgroup containing the analyzed organism. In the early stages of this process, organisms may be assigned to subgroups on the basis of characteristics that do not reflect genetic relatedness. It would be perfectly reasonable, for example, for a bacterial key to include a group such as “bacteria forming red pigments when propagated on a defined medium” even though this would include such unrelated forms as Serratia Carbohydrate breakdown. The ability to produce acidic metabolic products, fermentatively or oxidatively, from a range of carbohydrates (eg, glucose, sucrose, and lactose) has been applied to the identification of most groups of bacteria. Such tests are crude and imperfect in defining mechanisms, but have proved useful for taxonomic purposes. More recently, gas chromatographic identification of specific short-chain fatty acids produced by fermentation of glucose has proved useful in classifying many anaerobic bacteria. 2. Catalase production. The enzyme catalase catalyzes the conversion of hydrogen peroxide to water and oxygen. When a colony is placed in hydrogen peroxide, liberation of oxygen as gas bubbles can be seen. The test is particularly useful in differentiation of staphylococci (positive) from streptococci (negative), but also has taxonomic application to Gram-negative bacteria. 3. Citrate utilization. An agar medium that contains sodium citrate as the sole carbon source may be used to determine ability to use citrate. Bacteria that grow on this medium are termed citrate-positive. 4. Coagulase. The enzyme coagulase acts with a plasma factor to convert fibrinogen to a fibrin clot. It is used to differentiate Staphylococcus aureus from other, less pathogenic staphylococci. 5. Decarboxylases and deaminases. The decarboxylation or deamination of the amino acids lysine, ornithine, and arginine is detected by the effect of the amino products on the pH of the reaction mixture or by the formation of colored products. These tests are used primarily with Gram-negative rods. 6. Hydrogen sulfide. The ability of some bacteria to produce H2S from amino acids or other sulfur-containing compounds is helpful in taxonomic classification. The black color of the sulfide salts formed with heavy metals such as iron is the usual means of detection. 7. Indole. The indole reaction tests the ability of the organism to produce indole, a benzopyrrole, from tryptophan. Indole is detected by the formation of a red dye after addition of a benzaldehyde reagent. A spot test can be done in seconds using isolated colonies. 8. Nitrate reduction. Bacteria may reduce nitrates by several mechanisms. This ability is demonstrated by detection of the nitrites and/or nitrogen gas formed in the process. 9. O-Nitrophenyl-β-d-galactoside (ONPG) breakdown. The ONPG test is related to lactose fermentation. Organisms that possess the β-galactoside necessary for lactose fermentation but lack a permease necessary for lactose to enter the cell are ONPG-positive and lactose-negative. 10. Oxidase production. The oxidase tests detect the c component of the cytochrome–oxidase complex. The reagents used change from clear to colored when converted from the reduced to the oxidized state. The oxidase reaction is commonly demonstrated in a spot test, which can be done quickly from isolated colonies. 11. Proteinase production. Proteolytic activity is detected by growing the organism in the presence of substrates such as gelatin or coagulated egg. 12. Urease production. Urease hydrolyzes urea to yield two molecules of ammonia and one of CO2. This reaction can be detected by the increase in medium pH caused by ammonia production. Urease-positive species vary in the amount of enzyme produced; bacteria can thus be designated as positive, weakly positive, or negative. 13. Voges–Proskauer test. The Voges–Proskauer test detects acetylmethylcarbinol (acetoin), an intermediate product in the butene glycol pathway of glucose fermentation. Copyright © The McGraw-Hill Companies. (Reproduced with permission from Ryan KJ, Ray CG (editors): Sherris Medical Microbiology, 5th ed. McGraw-Hill, 2010.) 46 SECTION I Fundamentals of Microbiology marcescens (see Chapter 15) and purple photosynthetic bacteria (see Chapter 6). These two disparate bacterial assemblages occupy distinct niches and depend on entirely different forms of energy metabolism. Nevertheless, preliminary grouping of the assemblages would be useful because it would immediately make it possible for an investigator having to identify a red-pigmented culture to narrow the range of possibilities to relatively few groups. Numerical Taxonomy Numerical taxonomy (also referred to as phenetics or taxometrics) became widely used in the 1970s. These classification schemes use a large number (frequently greater than 100) of unweighted taxonomically useful characteristics. The Analytical Profile Index (APITM) is a method commonly used to identify a wide range of microorganisms. APIs consist of a number of plastic strips, each of which has about 20 miniature compartments containing biochemical reagents. Almost all cultivatable bacterial groups and more than 550 different species can be identified using the results of these API tests. These identification systems have extensive databases of microbial biochemical reactions. The numerical clusters derived from these tests identify different strains at selected levels of overall similarity (usually >80% at the species level) on the basis of the frequency with which they share traits. In addition, numerical classification provides percentage frequencies of positive character states for all strains within each cluster. The limitation of this approach is that it is a static system, which does not allow for the evolution of bacteria and routine discovery of new bacterial pathogens (Figure 3-2). Phylogenetic Classification: Toward an Understanding of Evolutionary Relationships Among Bacteria Phylogenetic classifications are measures between two organisms and imply that they share a common ancestor. The fossil record has made such inferences relatively easy to draw for many representatives of plants and animals. However, no such record exists for bacteria, and in the absence of molecular evidence, the distinction between convergent and divergent evolution for bacterial traits can be difficult to establish. The genetic properties of bacteria allow genes to be exchanged among distantly related organisms. Furthermore, multiplication of bacteria is almost entirely vegetative, and their mechanisms of genetic exchange rarely involve recombination among large portions of their genomes (see Chapter 7). Therefore, the concept of a species—the fundamental unit of eukaryotic phylogeny—has an entirely different meaning when applied to bacteria. A eukaryotic species is a biologic group capable of interbreeding, with the ultimate intent to produce variable offspring. Because bacteria replicate clonally by binary fission, they do not require a complementary set of chromosomes in order to reproduce. Consequently, the definition of a bacterial species is necessarily pragmatic and is operationally defined. For the purposes of categorizing bacteria, a species is a genomically coherent group of individual isolates or strains sharing a high degree of similarity in many independent features when comparably tested under highly standardized conditions. The decision to circumscribe clusters of organisms within a bacterial species is made by the taxonomist, who may choose to subdivide the group into biotypes and to cluster species within genera. Broader groupings such as families may be proposed. The formal ranks used in the taxonomy of bacteria are listed in Table 3-1. For practical purposes, only the ranks of the family, genus, and species are commonly used. There is considerable genetic diversity among bacteria. Chemical characterization of bacterial genomic DNA reveals a wide range of nucleotide base compositions among different bacterial strains. The guanine + cytosine (G + C) content of closely related bacteria is similar, indicating that genetic relatedness of DNA from similar organisms can be used as a measure of taxonomic relatedness. The parameter of DNA–DNA similarity based on the difference in thermal denaturation midpoint has been a gross method for species delineation. A more precise method is DNA sequencing. This method has become a routine procedure, and comparison of the DNA sequences of divergent genes can give a measure of their relatedness. Genes for different functions, such as those encoding surface antigens to escape immune surveillance, diverge at different rates relative to “housekeeping” genes such as those that encode cytochromes. Thus, DNA sequence differences among rapidly diverging genes can be used to ascertain the genetic distance of closely related groups of bacteria, and sequence differences among Figure 3-2 APITM test demonstrating how bacteria can be differentiated using a series of biochemical tests. Each small compartment contains a dehydrated powder that can be inoculated from a bacterial culture. After incubation, the colorimetric changes can be scored numerically to produce a number that matches to a specific bacterial species and genus. (Courtesy of bioMerieux, Inc.) TABLE 3-3 Major Categories and Groups of Bacteria That Cause Disease in Humans as Part of an Identification Scheme Described in Bergey’s Manual of Determinative Bacteriology, 9th Ed. Bergey’s Manual of Systematic Bacteriology I. Gram-negative eubacteria that have cell walls Group 1: The spirochetes Group 2: Aerobic/microaerophilic, motile helical/vibroid gram-negative bacteria Group 3: Nonmotile (or rarely motile) curved bacteria Group 4: Gram-negative aerobic/microaerophilic rods and cocci Group 5: Facultatively anaerobic gram-negative rods Group 6: Gram-negative, anaerobic, straight, curved, and helical rods Group 7: Dissimilatory sulfate- or sulfur-reducing bacteria Group 8: Anaerobic gram-negative cocci Group 9: The rickettsiae and chlamydiae Group 10: Anoxygenic phototrophic bacteria Group 11: Oxygenic phototrophic bacteria Group 12: Aerobic chemolithotrophic bacteria and assorted organisms Group 13: Budding or appendaged bacteria Group 14: Sheathed bacteria Group 15: Nonphotosynthetic, nonfruiting gliding bacteria Group 16: Fruiting gliding bacteria: the myxobacteria II. Gram-positive bacteria that have cell walls Group 17: Gram-positive cocci Group 18: Endospore-forming gram-positive rods and cocci Group 19: Regular, nonsporing gram-positive rods Group 20: Irregular, nonsporing gram-positive rods Group 21: The mycobacteria Groups 22–29: Actinomycetes III. Cell wall-less eubacteria: The mycoplasmas or mollicutes Group 30: Mycoplasmas IV. Archaebacteria Group 31: The methanogens Group 32: Archaeal sulfate reducers Group 33: Extremely halophilic archaebacteria Group 34: Cell wall-less archaebacteria Group 35: Extremely thermophilic and hyperthermophilic sulfur metabolizers Treponema Borrelia Leptospira Campylobacter Helicobacter Spirillum None Alcaligenes Bordetella Brucella Francisella Legionella Moraxella Neisseria Pseudomonas Rochalimaea Bacteroides (some species) Escherichia (and related coliform bacteria) Klebsiella Proteus Providencia Salmonella Shigella Yersinia Vibrio Haemophilus Pasteurella Bacteroides Fusobacterium Prevotella None None Rickettsia Coxiella Chlamydia None None None None None Capnocytophaga None Enterococcus Peptostreptococcus Staphylococcus Streptococcus Bacillus Clostridium Erysipelothrix Listeria Actinomyces Corynebacterium Mobiluncus Mycobacterium Nocardia Streptomyces Rhodococcus Mycoplasma Ureaplasma None None None None None 47 48 SECTION I Fundamentals of Microbiology Bacteria Archaea Spirochetes Green Filamentous bacteria Eucarya Entamoebae Slime molds Fungi Methanosarcina Proteobacteria Gram positives Cyanobacteria Planctomyces Bacteroides Cytophaga Methanobacterium Animals Halophiles Methanococcus T. celer Plants Ciliates Flagellates Thermoproteus Pyrodicticum Trichomonads Microsporidia Thermotoga Diplomonads Aquifex Figure 3-3 A phylogenetic tree based on rRNA data, showing the separation of bacteria, archaea, and eukaryotes families. The groups of the major known pathogenic bacteria are designated in grey. The only group of pathogenic bacteria that does not cluster in this shaded area is the Bacteroides group. housekeeping genes can be used to measure the relatedness of widely divergent groups of bacteria. Ribosomal RNA Ribosomes have an essential role in protein synthesis for all organisms. Genetic sequence encodings both ribosomal RNAs (rRNA) and proteins (both of which are required to comprise a functional ribosome) have been highly conserved throughout evolution and have diverged more slowly than other chromosomal genes. Comparison of the nucleotide sequence of 16S ribosomal RNA from a range of prokaryotic sources revealed evolutionary relationships among widely divergent organisms and has led to the elucidation of a new kingdom, the Archaebacteria. The phylogenetic tree based on ribosomal RNA (rRNA) data, showing the separation of bacteria, archaea, and eukaryote families, is depicted in Figure 3-3, which shows the three major domains of biological life as they are currently understood. From this diagram, two kingdoms, the Eubacteria (true bacteria) and the Archaebacteria, are distinct from the Eukaryotic branch. classifies, in the form of a key, known bacteria that have or have not been cultured or well-described. A companion volume, Bergey’s Manual of Determinative Bacteriology, serves as an aid in the identification of bacteria that have been described and cultured. The major bacteria that cause infectious diseases, as categorized in Bergey’s Manual, are listed in Table 3-3. Because it is likely that emerging information concerning phylogenetic relationships will lead to further modifications in the organization of bacterial groups within Bergey’s Manual, its designations must be regarded as a work in progress. As discussed in Chapter 2, there are two different groups of prokaryotic organisms, eubacteria and archaebacteria. Both are small unicellular organisms that replicate asexually. Eubacteria refer to classic bacteria as science has historically understood them. They lack a true nucleus, have characteristic lipids that make up their membranes, possess a peptidoglycan cell wall, and have a protein and nucleic acid synthesis machinery that can be selectively inhibited by antimicrobial agents. In contrast, archaebacteria do not have a classic peptidoglycan cell wall and have many characteristics (eg, protein synthesis and nucleic acid replication machinery) that are similar to those of eukaryotic cells (Table 3-4). DESCRIPTION OF THE MAJOR CATEGORIES AND GROUPS OF BACTERIA The Eubacteria Bergey’s Manual of Systematic Bacteriology This is a heterogeneous group of bacteria that have a complex (gram-negative type) cell envelope consisting of an outer membrane, a periplasmic space containing a thin peptidoglycan layer, and a cytoplasmic membrane. The cell shape (Figure 3-4) may be spherical, oval, straight or curved rods, helical, or filamentous; some of these forms may be sheathed The definitive work on the taxonomic organization of bacteria is the latest edition of Bergey’s Manual of Systematic Bacteriology. First published in 1923, this publication taxonomically A. Gram-Negative Eubacteria CHAPTER 3 Classification of Bacteria 49 TABLE 3-4 Key Characteristics Shared by Archaebacteria and Eukaryotic Cells That are Absent from Eubacteria Eubacteria Archaebacteria, Eukaryotes Elongation factor-2 (EF-2) contains the amino acid diphthamide and is therefore ADP-ribosylable by diphtheria toxin No Yes The methionyl initiator tRNA is not formylated No Yes Some tRNA genes contain introns No Yes in eukaryotes Protein synthesis is inhibited by anisomycin but not by chloramphenicol No Yes DNA-dependent RNA polymerases are multicomponent enzymes insensitive to the antibiotics rifampin and streptomycin No Yes DNA-dependent RNA polymerases are multicomponent enzymes and are insensitive to the antibiotics rifampin and streptolydigin No Yes Characteristic or encapsulated. Reproduction is by binary fission, but some groups reproduce by budding. Fruiting bodies and myxospores may be formed by the myxobacteria. Motility, if present, occurs by means of flagella or by gliding motility. Members of this category may be phototrophic or nonphototrophic (see Chapter 5) bacteria and include aerobic, anaerobic, facultatively anaerobic, and microaerophilic species. B. Gram-Positive Eubacteria These bacteria have a cell wall profile of the gram-positive type; cells generally, but not always, stain gram positive. The cell envelope of gram-positive organisms consists of a thick cell wall that determines cellular shape and a cytoplasmic membrane. These cells may be encapsulated and can exhibit flagellar-mediated motility. Cells may be spherical, rods, or filaments; the rods and filaments may be nonbranching or may show true branching. Reproduction is generally by binary fission. Some bacteria in this category produce spores (eg, Bacillus and Clostridium spp.) as resting forms that are highly resistant to disinfection. The gram-positive eubacteria are generally chemosynthetic heterotrophs (see Chapter 5) A B and include aerobic, anaerobic, and facultatively anaerobic species. The groups within this category include simple asporogenous and sporogenous bacteria as well as the structurally complex actinomycetes and their relatives. C. Eubacteria Lacking Cell Walls These are microorganisms that lack cell walls (commonly called mycoplasmas and making up the class Mollicutes) and do not synthesize the precursors of peptidoglycan. They are enclosed by a unit membrane, the plasma membrane (Figure 3-5). They resemble the L-forms that can be generated from many species of bacteria (notably gram-positive eubacteria); unlike L-forms, however, mycoplasmas never revert to the walled state, and there are no antigenic relationships between mycoplasmas and eubacterial L-forms. Six genera have been designated as mycoplasmas on the basis of their habitat; however, only two genera contain animal pathogens. Mycoplasmas are highly pleomorphic organisms and range in size from vesicle-like forms to very small (0.2 μm), filterable forms (meaning that they are too small to be captured on filters that routinely trap most bacteria). C Figure 3-4 The cell shapes that occur among unicellular true bacteria. A: Coccus. B: Rod. C: Spiral. (Phase contrast, 1500×.) (Reproduced with permission from Stanier RY, Doudoroff M, Adelberg EA: The Microbial World, 3rd ed. Copyright © 1970. By permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.) 50 SECTION I Fundamentals of Microbiology some molecular features with eubacteria (Table 3-4). Cells may have a diversity of shapes, including spherical, spiral, and plate or rod shaped; unicellular and multicellular forms in filaments or aggregates also occur. Multiplication occurs by binary fission, budding, constriction, fragmentation, or other unknown mechanisms. SUBTYPING AND ITS APPLICATION Figure 3-5 Electron micrograph of cells of a member of the mycoplasma group, the agent of bronchopneumonia in the rat (1960×). (Reproduced with permission from Klieneberger-Nobel E, Cuckow FW: A study of organisms of the pleuropneumonia group by electron microscopy. J Gen Microbiol 1955;12:99.) Reproduction may be by budding, fragmentation, or binary fission, singly or in combination. Most species require a complex medium for growth and tend to form characteristic “fried egg” colonies on a solid medium. A unique characteristic of the Mollicutes is that some genera require cholesterol for growth; unesterified cholesterol is a unique component of the membranes of both sterol-requiring and non–sterolrequiring species if present in the medium. The Archaebacteria These organisms are predominantly inhabitants of extreme terrestrial and aquatic environments (high salt, high temperature, anaerobic) and are often referred to as “extremeophiles”; some are symbionts in the digestive tract of humans and animals. The archaebacteria consist of aerobic, anaerobic, and facultatively anaerobic organisms that are chemolithotrophs, heterotrophs, or facultative heterotrophs. Some species are mesophiles, but others are capable of growing at temperatures above 100°C. These hyperthermophilic archaebacteria are uniquely adapted for growth at high temperatures. With few exceptions, enzymes isolated from these organisms are intrinsically more thermostable than their counterparts from mesophilic organisms. Some of these thermostable enzymes, such as the DNA polymerase from Thermus aquaticus (Taq polymerase), are important components of DNA amplification methods such as the polymerase chain reaction (PCR). Archaebacteria can be distinguished from eubacteria in part by their lack of a peptidoglycan cell wall, possession of isoprenoid diether or diglycerol tetraether lipids, and characteristic ribosomal RNA sequences. Archaebacteria also share Under certain circumstances (eg, an epidemic), it is important to distinguish among strains of a given species or to identify a particular strain. This is called subtyping and is accomplished by examining bacterial isolates for characteristics that allow discrimination below the species level. Classically, subtyping has been accomplished by biotyping, serotyping, antimicrobial susceptibility testing, and bacteriophage typing. For example, more than 130 serogroups of Vibrio cholerae have been identified on the basis of antigenic differences in the O-polysaccharide of their LPS; however, only the O1 and O139 serogroups are associated with epidemic and pandemic cholera. Within these serogroups, only strains that produce a particular bundle-forming pilus and cholera toxin are virulent and cause the disease cholera. By contrast, nontoxigenic V cholerae strains, which are not associated with epidemic cholera, have been isolated from environmental specimens, from food, and from patients with sporadic diarrhea. Serologic Typing Clonality with respect to isolates of microorganisms from a common source outbreak (point source spread) is an important concept in the epidemiology of infectious diseases. Etiologic agents associated with these outbreaks are generally clonal; in other words, they are the progeny of a single cell and thus, for all practical purposes, are genetically identical. Thus, subtyping plays an important role in discriminating among these particular microorganisms. Recent advances in biotechnology have dramatically improved our ability to subtype microorganisms. Hybridoma technology has resulted in the development of monoclonal antibodies against cell surface antigens, which have been used to create highly standardized antibody-based subtyping systems that describe bacterial serotypes. This is an important tool for defining the epidemiologic spread of a bacterial infection. Other organisms cannot be identified as unique serotypes. For example, some pathogens (eg, Neisseria gonorrhoeae) are transmitted as an inoculum composed of quasispecies (meaning that there is extensive antigenic variation among the bacteria present in the inoculum). In these cases, groups of hybridomas that recognize variants of the original organisms are used to categorize serovariants or serovars. Genotyping Genotyping multilocus enzyme electrophoresis (MLEE), which has been a standard method for investigating eukaryotic population genetics, has also been used to study the genetic diversity and clonal structure of pathogenic CHAPTER 3 Classification of Bacteria 51 microorganisms. MLEE involves the determination of the mobilities of a set of soluble enzymes (usually 15–25 enzymes) by starch gel electrophoresis. Because the rate of migration of a protein during electrophoresis and its net electrostatic charge are determined by its amino acid sequence, mobility variants (referred to as electromorphs or allozymes) of an enzyme reflect amino acid substitutions in the protein sequence, which reflect changes in the DNA sequence encoding the protein. The enzyme-encoding structural genes of E coli exhibit extensive genetic diversity; however, by using MLEE, investigators at the Centers for Disease Control and Prevention were able to ascertain that the HUS pathogen E coli serotype O157:H7 are descended from a clone that is widely distributed in North America. Chemical Fingerprinting The characterization or identification of isolates has been improved by applying physical methods to prokaryotic cells, such as Fourier transform infrared spectroscopy (FTIR), pyrolysis/mass spectrometry, and matrix-assisted laser desorption/ionization with time-of-flight (Maldi/Tof) or spray ionization mass spectrometry. The equipment required for these powerful techniques is expensive and is not routinely available to clinical laboratories. NUCLEIC ACID–BASED TAXONOMY Since 1975, developments in nucleic acid isolation, amplification, and sequencing spurred the evolution of nucleic acid–based subtyping systems. These include plasmid profile analysis; restriction endonuclease analysis; ribotyping; pulsed field gel electrophoresis; PCR amplification and restriction endonuclease digestion of specific genes; arbitrarily primed PCR; and nucleic acid sequence analysis. Plasmid Analysis Plasmid profile analysis was the first and the simplest nucleic acid–based technique applied to epidemiologic studies. Plasmids, which are extrachromosomal genetic elements (see Chapter 7), are isolated from each bacterium and then separated by agarose gel electrophoresis to determine their number and size. However, plasmids of identical size with different sequences can exist in many bacteria. Thus, digesting the plasmids with restriction endonucleases and then comparing the number and size of the resulting restriction fragments often provides additional useful information. Plasmid analysis has been shown to be most useful for examining outbreaks that are restricted in time and place (eg, an outbreak in a hospital), particularly when they are combined with other identification methods. Restriction Endonucleases Analysis The use of restriction enzymes to cleave DNA into discrete fragments is one of the most basic procedures in molecular biology. Restriction endonucleases recognize short DNA sequences (restriction sequence), and they cleave doublestranded DNA within or adjacent to this sequence. Restriction sequences range from four to more than 12 bases in length and occur throughout the bacterial chromosome. Restriction enzymes that recognize short sequences (eg, four base pairs) occur more frequently than those that recognize longer sequences (eg, 12 base pairs). Thus, enzymes that recognize short DNA sequences produce more fragments than enzymes that recognize infrequently occurring long DNA sequences. Several subtyping methods use restriction endonucleasedigested DNA. The basic method involves digesting DNA with an enzyme that recognizes a frequently occurring restriction site and separating the fragments, which generally range from 0.5 to 50 kb in length, by agarose gel electrophoresis followed by visualization under ultraviolet light after staining with ethidium bromide. One of the major limitations of this technique is the difficulty in interpreting the complex profiles consisting of hundreds of bands that may be unresolved and overlapping. The use of restriction endonucleases that cut at infrequently occurring restriction sites has circumvented this problem. Digestion of DNA with these enzymes generally results in five to 20 fragments ranging from approximately 10–800 kb in length. Separation of these large DNA fragments is accomplished by a technique called pulsed field gel electrophoresis (PFGE), which requires specialized equipment. Theoretically, all bacterial isolates can be typed by this method. Its advantage is that the restriction profile consists of a finite number of well-resolved bands representing the entire bacterial chromosome in a single gel. Southern Blot Analysis This analysis was named after its inventor, Edwin Mellor Southern, and has been used as a subtyping method to identify isolates associated with outbreaks. For this analysis, DNA preparations from bacterial isolates are subjected to restriction endonuclease digestion. After agarose gel electrophoresis, the separated restriction fragments are transferred to a nitrocellulose or nylon membrane. These double-stranded DNA fragments are first converted into single-stranded linear sequences. Using a labeled fragment of DNA as a probe, it is possible to identify the restriction fragments containing sequences (loci) that are homologous to the probe by complementation to the bound single-stranded fragments (Figure 3-6). Restriction fragment length polymorphisms (RFLPs) refer to variations in both the number of loci that are homologous to the probe and the location of restriction sites that are within or flanking those loci. Ribotyping This method uses Southern blot analysis to detect polymorphisms of rRNA genes, which are present in all bacteria. Because ribosomal sequences are highly conserved, they can be detected with a common probe prepared from the 16S and 23S rRNA of a eubacterium E coli. Many organisms have 52 SECTION I Fundamentals of Microbiology Restriction Enzymes 1) Digest DNA using restriction enzymes 2) Separate fragments by agarosegel electrophoresis Agarose gel Nylon membrane 3) Transfer separated fragments to a nylon membrane Labeled DNA probe Detection film 4) Hybridize labeled DNA probe with DNA bound to nylon membrane 5) Detect labeled fragment Figure 3-6 Southern blot procedure showing how specific loci on separated DNA fragments can be detected with a labeled DNA probe. This procedure in essence allows for the discrimination of DNA at three levels: (1) at the level of restriction enzyme recognition, (2) by the size of the DNA fragment, and (3) by the hybridization of a DNA probe to a specific locus defined by a specific band at a specific position of the membrane. multiple copies (five to seven) of these genes, resulting in patterns with a sufficient number of bands to provide good discriminatory power; however, ribotyping are of limited value for some microorganisms, such as mycobacteria, which have only a single copy of these genes. Repetitive Sequences In the current genomic era of molecular medicine, hundreds of microbial genomes have now been sequenced. With this era have come bioinformatical tools to mine this wealth of DNA sequence information to identify novel targets for pathogen subtyping, such as the repetitive sequences that have been found in different species (see Chapter 7). These repetitive sequences have been termed satellite DNA and have repeating units that range from 10 to 100 bp. They are commonly referred to as variable number tandem repeats (VNTRs). VNTRs have been found in regions controlling gene expression and within open reading frames. The repeat unit and the number of copies repeated side by side define each VNTR CHAPTER 3 Classification of Bacteria 53 locus. A genotyping approach using PCR, referred to as multiple-locus VNTR analysis (MLVA), takes advantage of the levels of diversity generated by both repeat unit size variation and copy number among a number of characterized loci. It has proved especially useful in subtyping monomorphic species such as Bacillus anthracis, Yersinia pestis, and Francisella tularensis. Microbial Forensics Genotyping methods are progressing toward the identification of single nucleotide polymorphisms (SNPs) in both open reading frames and intergenic regions to address a diverse range of epidemiologic and evolutionary questions. The field of microbial forensics has developed in the wake of bioterrorist attacks with spores of Bacillus anthracis (anthrax) in the fall of 2001. Microbial forensics was part of the criminal investigation used to identify the precise strain and substrain of the microorganism used in this biocrime. NONCULTURE METHODS FOR THE IDENTIFICATION OF PATHOGENIC MICROORGANISMS OBJECTIVES Attempts to estimate total numbers of eubacteria, archaebacteria, and viruses are problematic because of difficulties such as detection in and recovery from the environment, our incomplete knowledge of obligate microbial associations, and the problem of the species concept in these groups. Nevertheless, estimates indicate that the numbers of uncultured microbial taxa greatly exceed those of the cultured organisms (Table 3-5). However, more recent estimates suggest that the number of bacterial species in the world is between 107 and 109. Until very recently, microbial identification required the isolation of pure cultures followed by testing for multiple physiologic and biochemical traits. Clinicians have long been aware of human diseases that are associated with visible but nonculturable microorganisms. Scientists are now using a PCR-assisted approach using rRNA to identify TABLE 3-5 Known and Estimated Numbers of Biologic Speciesa Group Known Species Estimated Total Species Percentage of Known Species Viruses 5000 130,000 4 Bacteria 4760 40,000 12 Fungi 69,000 1,500,000 Algae 40,000 60,000 67 Protozoa 30,800 100,000 31 5 Modified with permission of Annual Reviews, Inc., from Bull AT et al: Biodiversity as a source of innovation in biotechnology. Ann Rev Microbiol 1992;46:219. Permission conveyed through Copyright Clearance Center, Inc. a pathogenic microorganisms in situ. The first phase of this approach involves the extraction of DNA from a suitable specimen, the use of standard molecular techniques to obtain a clone library, the retrieval of rRNA sequence information, and a comparative analysis of the retrieved sequences. This yields information on the identity or relatedness of the sequences in comparison with the available data base. In the second phase, proof that the sequences are from cells in the original specimen is obtained by in situ hybridization using sequence-specific probes. This approach has been used in the identification of pathogenic microorganisms. For example, a previously uncharacterized pathogen has been identified as the Whipple-disease–associated rodshaped bacterium now designated Tropheryma whipplei. This rRNA approach has also been used to identify the etiologic agent of bacillary angiomatosis as Bartonella henselae and to show that the opportunistic pathogen Pneumocystis jiroveci is a member of the fungi. Undoubtedly, these and other techniques will identify additional etiologic agents in the future. U nderstand how the vocabulary of taxonomy is critical to communicating the science of infectious diseases. 2. K now the taxonomic ranks. 3. Appreciate the growth, biochemical, and genetic characteristics that are used in differentiating bacteria. 4. Understand the differences between the Eubacteria, Archaebacteria, and Eukaryotes. 5. Be conversant in the different tools for nucleic acid–based taxonomy. REVIEW QUESTIONS 1. Eubacteria that lack cell walls and do not synthesize the precursors of peptidoglycan are called (A) Gram-negative bacteria (B) Viruses (C) Mycoplasmas (D) Serovar variant (E) Bacilli 2. Archaebacteria can be distinguished from eubacteria by their lack of (A) DNA (B) RNA (C) Ribosomes (D) Peptidoglycan (E) Nucleus 3. A 16-year-old cystic fibrosis patient is admitted to the hospital. A sputum culture yields Burkholderia cepacia. Subsequently, there are two other patients with B. cepacia bacteremia, and the organism is cultured from the sputum of four additional patients. During this nosocomial outbreak of B. cepacia, 50 environmental and seven patient isolates are being subtyped to identify the 54 SECTION I Fundamentals of Microbiology source of the outbreak. Which of the following techniques would be most useful in this endeavor? (A) Culture (B) Ribotyping (C) 16S rRNA sequencing (D) Antimicrobial susceptibility testing (E) Nucleic acid sequencing 4. An unculturable gram-positive microorganism has been visualized in tissue specimens obtained from patients with a previously undescribed disease. Which of the following techniques would be most useful in identifying this organism? (A) Serology (B) PCR amplification and sequencing of rRNA genes (C) Multilocus enzyme electrophoresis (D) SDS-polyacrylamide gel electrophoresis (E) Pulsed field gel electrophoresis 5. The DNA polymerase from Thermus aquaticus is an important component of DNA amplification methods such as the polymerase chain reaction. This organism is capable of growing at temperatures above 100°C. Organisms that are capable of growth at these temperatures are referred to as (A) Mesophiles (B) Psychrophiles (C) Halophiles (D) Thermophiles (E) Chemolithotrophs Answers 1. C E D B D REFERENCES Achtman M, Wagner M: Microbial diversity and the genetic nature of microbial species. Nat Rev Microbiol 2008;6:431. Amann RI, Ludwig W, Schleiffer K-H: Phylogenetic identification and in situ detection of individual microbial cells without culture. Microbiol Rev 1995;59:143. Boone DR, Castenholz RW (editors): Bergey’s Manual of Systematic Bacteriology: The Archaea and the Deeply Branching and Phototrophic Bacteria, vol. 1, 2nd ed. Springer, 2001. Breeze RG, Budowle B, Schutzer SE (editors): Microbial Forensics. Elsevier, 2005. Brenner DJ, Krieg NR, Staley JT (editors): Part A. Introductory essays. Bergey’s Manual of Systematic Bacteriology: The Proteobacteria, vol 2. Springer, 2005. Brenner DJ, Krieg NR, Staley JT (editors): Part B. The gammaproteobacteria. Bergey’s Manual of Systematic Bacteriology: The Proteobacteria, vol 2. Springer, 2005. Brenner DJ, Krieg NR, Staley JT (editors): Part C. The alpha-, beta-, delta-, and epsilonproteobacteria. Bergey’s Manual of Systematic Bacteriology: The Proteobacteria, vol 3. Springer, 2005. Colwell RR, Grimes DJ (editors): Nonculturable Microorganisms in the Environment. ASM Press, 2000. Curtis TP, Sloan WT, Scannell JW: Estimating prokaryotic diversity and its limits. Proc Natl Acad Sci U S A 2002;99:10494. Edman JC et al: Ribosomal RNA sequence shows Pneumocystis carinii to be a member of the fungi. Nature (London) 1988;334:519. Fernandez LA: Exploring prokaryotic diversity: There are other molecular worlds. Molec Microbiol 2005;55:5–15. Fredericks DN, Relman DA: Sequence-based identification of microbial pathogens: A reconsideration of Koch’s postulates. Clin Microbiol Rev 1996;9:18. Holt JG et al (editors): Bergey’s Manual of Determinative Bacteriology, 9th ed. Williams & Wilkins, 1994. Medini D et al: Microbiology in the post-genomic era. Nat Rev Microbiol 2008;6:429. Persing DH et al (editors): Molecular Microbiology. Diagnostic Principles and Practice. ASM Press, 2004. Riley LW: Molecular Epidemiology of Infectious Diseases. Principles and Practices. ASM Press, 2004. Rosello-Mora R, Amann R: The species concept for prokaryotes. FEMS Microbiol Rev 2001;25:39. Schloss PD, Handelsman J: Status of the microbial census. Microbiol Molec Biol Rev 2004;68:686. Stringer JR et al: A new name (Pneumocystis jiroveci) for Pneumocystis from humans. Emerg Infect Dis 2002;8:891. Whitman WB, Coleman DC, Wiebe WJ: Prokaryotes: The unseen majority. Proc Natl Acad Sci U S A 1998;95:6578. C The Growth, Survival, and Death of Microorganisms SURVIVAL OF MICROORGANISMS IN THE NATURAL ENVIRONMENT The population of microorganisms in the biosphere remains roughly constant because the growth of microorganisms is in turn balanced by the death of these organisms. The survival of any microbial group, within a specific niche, is ultimately influenced by successful competition for nutrients and by maintenance of a pool of living cells, often composed of host cells and a consortia of different microorganisms. Consequently, understanding competition for nutritional resources within a given microenvironment is essential to understanding the growth, survival, and evolution of bacterial species (also known as physiology). Much of our understanding of microbial physiology has come from the study of isolated cultures grown under optimal conditions in laboratories (nutrient excess); these observations form the basis for this section. To the contrary, most microorganisms compete in the natural environment under nutritional stress. Furthermore, it should be recognized that a vacant microbial niche in the environment will soon be fi lled with another consortiae of bacteria. It is somewhat of a conundrum that public health procedures stress the elimination of microorganisms, yet by clearing their niche, these spaces can be fi lled with other bacterial species. In the end, understanding the complex interactions that ensure the survival of a specific bacterium in a microbially diverse biosphere is a matter of physiologic efficiency. THE MEANING OF GROWTH Growth is the orderly increase in the sum of all the components of an organism. The increase in size that results when a cell takes up water or deposits lipid or polysaccharide is not true growth. Cell multiplication is a consequence of cell division of unicellular organisms, growth leads to an increase in the number of single bacteria making up a population, referred to as a culture. The Measurement of Microbial Concentrations Microbial concentrations can be measured in terms of cell concentration (the number of viable cells per unit volume of 4 H A P T E R culture) or of biomass concentration (dry weight of cells per unit volume of culture). These two parameters are not always equivalent because the average dry weight of the cell varies at different stages in the history of a culture. Nor are they of equal significance: In studies of microbial genetics and the inactivation of microbes, cell concentration is the significant quantity; in studies on microbial biochemistry or nutrition, biomass concentration is the significant quantity. A. Cell Concentration The viable cell count (Table 4-1) is typically considered the measure of cell concentration. For most purposes, the turbidity of a culture, measured by photoelectric means, is related to the viable count in the form of a standard curve. As an alternative a rough visual estimate is sometimes possible: For example, a barely turbid suspension of Escherichia coli contains about 107 cells per milliliter, and a fairly turbid suspension contains about 108 cells per milliliter. In using turbidimetric measurements, the correlation between turbidity and viable count can vary during the growth and death of a culture; cells may lose viability without producing a loss in turbidity of the culture. B. Biomass Density In principle, biomass can be measured directly by determining the dry weight of a microbial culture after it has been washed with distilled water. In practice, this procedure is cumbersome, and the investigator customarily prepares a standard curve that correlates dry weight with turbidity. Alternatively, the concentration of biomass can be estimated indirectly by measuring an important cellular component such as protein or by determining the volume occupied by cells that have settled out of suspension. EXPONENTIAL GROWTH The Growth Rate Constant The growth rate of cells unlimited by nutrient is first order: The rate of growth (measured in grams of biomass produced 55 56 SECTION I Fundamentals of Microbiology TABLE 4-1 Example of a Viable Count Dilution a 16 Plate Count a Too many to count 10-1 Too many to count 10-2 510 10-3 72 10-4 6 10-5 1 8 Biomass (B) Undiluted Each count is the average of three replicate plates. 2 per hour) is the product of time (t), the growth rate constant (k), and the biomass concentration B: dB kB dt 1 td (1) Rearrangement of equation (1) demonstrates that the growth rate constant is the rate at which cells are producing more cells: Bdt (2) k dB A growth rate constant of 4.3 h−1, one of the highest recorded, means that each gram of cells produces 4.3 g of cells per hour during this period of growth. Slowly growing organisms may have growth rate constants as low as 0.02 h−1. With this growth rate constant, each gram of cells in the culture produces 0.02 g of cells per hour. Integration of equation (1) yields 4 (3) The natural logarithm of the ratio of B1 (the biomass at time 1 [t1]) to B0 (the biomass at time zero [t0]) is equal to the product of the growth rate constant (k) and the difference in time (t1 – t0). Growth obeying equation (3) is termed exponential because biomass increases exponentially with respect to time. Linear correlations of exponential growth are produced by plotting the logarithm of biomass concentration (B) as a function of time (t). 2td 3td 4td Doubling time (td) Figure 4-1 Exponential growth. The biomass (B) doubles with each doubling time (td). required for doubling the biomass is td (Figure 4-1). The growth rate constant can be calculated from the doubling time by substituting the value 2 for B1/B0 and td for t1 – t0 in equation (3), which yields In2 kt d k In2 td (4) A rapid doubling time corresponds to a high growth rate constant. For example, a doubling time of 10 minutes (0.17 hour) corresponds to a growth rate constant of 4.1 h−1. The relatively long doubling time of 35 hours indicates a growth rate constant of 0.02 h−1. The calculated growth rate constant can be used either to determine the amount of growth that will occur in a specified period of time or to calculate the amount of time required for a specified amount of growth. The amount of growth within a specified period of time can be predicted on the basis of the following rearrangement of equation (3): B1 k(t1 − t 0) B0 2. 3 Calculation of the Growth Rate Constant and Prediction of the Amount of Growth (5) It is possible to determine the amount of growth that would occur if a culture with a growth rate constant of 4.1 h−1 grew exponentially for 5 hours: Bacteria reproduce by binary fission, and the average time required for the population, or the biomass, to double is known as the generation time or doubling time (td). Usually the td is determined by plotting the amount of growth on a semi-logarithmic scale as a function of time; the time (6) In this example, the increase in biomass is 10−9; a single bacterial cell with a dry weight of 2 × 10−13 g would give rise to log10 CHAPTER 4 The Growth, Survival, and Death of Microorganisms 57 0.2 mg of biomass, a quantity that would densely populate a 5-mL culture. Clearly, this rate of growth cannot be sustained for a long period of time. Another 5 hours of growth at this rate would produce 200 kg dry weight of biomass, or roughly 1 ton of cells. Another rearrangement of equation (3) allows calculation of the amount of time required for a specified amount of growth to take place. In equation (7), shown below, N, cell concentration, is substituted for B, biomass concentration, to permit calculation of the time required for a specified increase in cell number. (7) Using equation (7), it is possible, for example, to determine the time required for a slowly growing organism with a growth rate constant of 0.02 h−1 to grow from a single cell into a barely turbid cell suspension with a concentration of 107 cells per milliliter. 2. 3 × 7 (8) t1 − t 0 0.02h −1 Solution of equation (8) reveals that about 800 hours— slightly more than a month—would be required for this amount of growth to occur. The survival of slowly growing organisms implies that the race for biologic survival is not always to the swift—these species flourish that compete successfully for nutrients and avoid annihilation by predators and other environmental hazards. THE GROWTH CURVE If a fixed volume of liquid medium is inoculated with microbial cells taken from a culture that has previously been grown to saturation and the number of viable cells per milliliter is determined periodically and plotted, a curve of the type shown in Figure 4-2 is usually obtained. The phases Log cell concentration Stationary phase Death or logarithmic decline phase Log or exponential growth phase Lag phase Time Figure 4-2 A bacterial growth curve. TABLE 4-2 Phases of the Microbial Growth Curve Phase Growth Rate Lag Zero Exponential Constant Maximum stationary Zero Decline Negative (death) of the bacterial growth curve shown in Figure 4-2 are reflections of the events in a population of cells, not in individual cells. This type of culture is referred to as a batch culture. The typical growth curve may be discussed in terms of four phases (Table 4-2). Batch culture is a closed system with finite resources; this is very different from the environment of the human host. The Lag Phase The lag phase represents a period during which cells, depleted of metabolites and enzymes as the result of the unfavorable conditions that existed at the end of their previous culture history, adapt to their new environment. Enzymes and intermediates are formed and accumulate until they are present in concentrations that permit growth to resume. If the cells are taken from an entirely different medium, it often happens that they are genetically incapable of growth in the new medium. In such cases, a long lag in growth may occur, representing the period necessary for a few mutants in the inoculum to multiply sufficiently for a net increase in cell number to be apparent. The Exponential Phase During the exponential phase, the cells are in a steady state. New cell material is being synthesized at a constant rate, but the new material is itself catalytic, and the mass increases in an exponential manner. This continues until one of two things happens: either one or more nutrients in the medium become exhausted or toxic metabolic products accumulate and inhibit growth. For aerobic organisms, the nutrient that becomes limiting is usually oxygen. When the cell concentration exceeds about 1 × 107/mL (in the case of bacteria), the growth rate decreases unless oxygen is forced into the medium by agitation or by bubbling in air. When the bacterial concentration reaches 4-5 × 109/mL, the rate of oxygen diffusion cannot meet the demand even in an aerated medium, and growth is progressively slowed. The Maximum Stationary Phase Eventually, the exhaustion of nutrients or the accumulation of toxic products causes growth to cease completely. In most cases, 58 SECTION I Fundamentals of Microbiology however, cell turnover takes place in the stationary phase: There is a slow loss of cells through death, which is balanced by the formation of new cells through growth and division. When this occurs, the total cell count slowly increases, although the viable count stays constant. The Phase of Decline: The Death Phase After a period of time in the stationary phase, which varies with the organism and with the culture conditions, the death rate increases until it reaches a steady level. The mathematics of steady-state death is discussed below. In most cases, the rate of cell death is much slower than that of exponential growth. Frequently, after the majority of cells have died, the death rate decreases drastically, so that a small number of survivors may persist for months or even years. This persistence may in some cases reflect cell turnover, a few cells growing at the expense of nutrients released from cells that die and lyse. A phenomenon, in which cells are called viable but not culturable (VBNC), is thought to be the result of a genetic response triggered in starving, stationary phase cells. Just as some bacteria form spores as a survival mechanism, others are able to become dormant without changes in morphology. When the appropriate conditions are available (eg, passage through an animal), VNBC microbes resume growth. MAINTENANCE OF CELLS IN THE EXPONENTIAL PHASE Cells can be maintained in the exponential phase by transferring them repeatedly into fresh medium of identical composition while they are still growing exponentially. This is referred to as continuous culture; the most common type of continuous culture device used is a chemostat. Continuous culture is more similar to conditions that organisms encounter in the real world (eg, the human body), where nutrients are constantly being replaced. The Chemostat This device consists of a culture vessel equipped with an overflow siphon and a mechanism for dripping in fresh medium from a reservoir at a regulated rate. The medium in the culture vessel is stirred by a stream of sterile air; each drop of fresh medium that enters causes a drop of culture to siphon out. The medium is prepared so that one nutrient limits growth yield. The vessel is inoculated, and the cells grow until the limiting nutrient is exhausted; fresh medium from the reservoir is then allowed to flow in at such a rate that the cells use up the limiting nutrient as fast as it is supplied. Under these conditions, the cell concentration remains constant, and the growth rate is directly proportionate to the flow rate of the medium. Biofilms It has been increasingly recognized that many infections are caused by bacteria that do not grow individually (or planktonically); rather, they exist in intimate and complex communities communicating among themselves. It is routine to debride our teeth every day to remove the film of bacteria that grow up while we sleep. Similarly, biofilms are associated with Pseudomonas aeruginosa lung infections and Legionella pneumophila colonization of hospital water systems, among many others. This growth pattern begins by a single bacterium nucleating a surface followed by replicating themselves or recruiting other bacteria into a forming colony. The recognition of this growth pattern has been increasingly appreciated. The conceptual strategy of biofilm formation is logical: (1) By forming layer upon layer of bacterial growth, the initial bacterium are less susceptible to immune clearance, and (2) penetration by antitherapeutics is shielded by external growth. The ability to study biofilms in the laboratory and in vivo is an evolving science. DEFINITION AND MEASUREMENT OF DEATH The Meaning of Death For a microbial cell, death means the irreversible loss of the ability to reproduce (grow and divide). The empirical test of death is the culture of cells on solid media: A cell is considered dead if it fails to give rise to a colony on any medium. Obviously, then, the reliability of the test depends on the choice of medium and conditions: A culture in which 99% of the cells appear “dead” in terms of ability to form colonies on one medium may prove to be 100% viable if tested on another medium. Furthermore, the detection of a few viable cells in a large clinical specimen may not be possible by directly plating a sample because the sample fluid itself may be inhibitory to microbial growth. In such cases, the sample may have to be diluted first into liquid medium, permitting the outgrowth of viable cells before plating. The conditions of incubation in the first hour after treatment are also critical in the determination of “killing.” For example, if bacterial cells are irradiated with ultraviolet light and plated immediately on any medium, it may appear that 99.99% of the cells have been killed. If such irradiated cells are first incubated in a suitable buffer for 20 minutes, however, plating will indicate only 10% killing. In other words, irradiation determines that a cell will “die” if plated immediately but will live if allowed to repair radiation damage before plating. A microbial cell that is not physically disrupted is thus “dead” only in terms of the conditions used to test viability. The Measurement of Death When dealing with microorganisms, one does not customarily measure the death of an individual cell but the death of CHAPTER 4 The Growth, Survival, and Death of Microorganisms 59 S S0 e − kt (9) where S0 is the number of survivors at time zero and S is the number of survivors at any later time t. As in the case of exponential growth, −k represents the rate of exponential death when the fraction ln (S/S0) is plotted against time. The one-hit curve shown in Figure 4-3A is typical of the kinetics of inactivation observed with many antimicrobial agents. The fact that it is a straight line from time zero (dose zero)—rather than exhibiting an initial shoulder—means that a single “hit” by the inactivating agent is sufficient to kill the cell (ie, only a single target must be damaged for the entire cell to be inactivated). Such a target might be the chromosome of a haploid bacterium or the cell membrane; conversely, it could not be an enzyme or other cell constituent that is present in multiple copies. A cell that contains several copies of the target to be inactivated exhibits a multihit curve of the type shown in Figure 4-3B. Extrapolation of the straight-line portion of the curve to the ordinate permits an estimate of the number of targets (eg, 4 in Figure 4-3B). Sterilization In practice, we speak of “sterilization” as the process of killing all of the organisms in a preparation. From the above considerations, however, we see that no set of conditions is guaranteed to sterilize a preparation. Consider Figure 4-3, for example. At 60 minutes, there is one organism (10 0) left per milliliter. At 70 minutes, there would be 10−1; at 80 minutes 10−2; and so on. By 10−2 organisms per milliliter, we mean that in a total volume of 100 mL, one organism would survive. How long, then, does it take to “sterilize” the culture? All we can say is that after any given time of treatment, the probability of having any surviving organisms in 1 mL is that given by the curve in Figure 4-3. After 2 hours, in the above example, the probability is 1 × 10−6. This would usually be considered a safe sterilization time, but a 1000-L lot may still contain a single viable organism. Note that such calculations depend on the curve’s remaining unchanged in slope over the entire time range. 6 5 4 3 2 1 0 10 20 30 40 50 60 40 50 60 Minutes B 7 6 Log10 number surviving cells/mL A Log10 number surviving cells/mL a population. This is a statistical problem: Under any condition that may lead to cell death, the probability of a given cell’s dying is constant per unit time. For example, if a condition is used that causes 90% of the cells to die in the first 10 minutes, the probability of any one cell dying in a 10-minute interval is 0.9. Thus, it may be expected that 90% of the surviving cells will die in each succeeding 10-minute interval, and a death curve similar to those shown in Figure 4-3 will be obtained. The number of cells dying in each time interval is thus a function of the number of survivors present, so that death of a population proceeds as an exponential process according to the general formula: 5 4 3 2 1 0 10 20 30 Minutes Figure 4-3 Death curve of a suspension of 106 viable microorganisms per milliliter. A: Single-hit curve. B: Multihit curve. The straight-line portion extrapolates to 6.5, corresponding to 4 × 106 cells. The number of targets is thus 4 × 106, or four per cell. Unfortunately, it is very common for the curve to bend upward after a certain period as a result of the population being heterogeneous with respect to sensitivity to the inactivation agent. Extrapolations are dangerous and can lead to errors such as those encountered in early preparations of sterile polio vaccine. 60 SECTION I Fundamentals of Microbiology The Effect of Drug Concentration 2.4 When antimicrobial substances (drugs) are used to inactivate microbial cells, it is commonly observed that the concentration of drug used is related to the time required to kill a given fraction of the population by the following expression: 2.0 In this equation, C is the drug concentration, t is the time required to kill a given fraction of the cells, and n and K are constants. This expression says that, for example, if n = 6 (as it is for phenol), then doubling the concentration of the drug will reduce the time required to achieve the same extent of inactivation 64-fold. That the effectiveness of a drug varies with the sixth power of the concentration suggests that six molecules of the drug are required to inactivate a cell, although there is no direct chemical evidence for this conclusion. To determine the value of n for any drug, inactivation curves are obtained for each of several concentrations, and the time required at each concentration to inactivate a fixed fraction of the population is determined. For example, let the first concentration used be C1 and the time required to inactivate 99% of the cells be t1. Similarly, let C2 and t2 be the second concentration and time required to inactivate 99% of the cells. From equation (10), we see that: n 1 n 2 C t1 C t 2 (11) Solving for n gives: n log t 2 − log t1 log C1 − log C2 (12) Thus, n can be determined by measuring the slope of the line that results when log t is plotted against log C (Figure 4-4). If n is experimentally determined in this manner, K can be determined by substituting observed values for C, t, and n in equation (10). ANTIMICROBIAL AGENTS Definitions The following terms are commonly used in connection with antimicrobial agents and their uses. A. Biocide A chemical or physical agent, usually broad spectrum, that inactivates microorganisms (Table 4-3). Chemical biocides include hydrogen peroxide, alcohols, bleach, cycloheximide, and phenols, and physical biocides include heat and radiation. Biocides are generally broad spectrum, in contrast to anti-infectives, which have a narrower range of antimicrobial activity. Log10 t (in minutes) (10) C nt K Slope = n 1.6 1.2 0.8 0.4 0 1.00 1.10 1.20 1.30 Log10 C (in parts per 1000) Figure 4-4 Relationship between drug concentration and time required to kill a given fraction of a cell population. B. Bacteriostatic A specific term referring to the property by which a biocide is able to inhibit bacterial multiplication; upon removal of the agent, multiplication resumes. (The terms “fungistatic” and “sporostatic” refer to biocides that inhibit the growth of fungi and spores, respectively.) C. Bactericidal A specific term referring to the property by which a biocide is able to kill bacteria. Bactericidal action differs from bacteriostasis only in being irreversible (ie, the “killed” organism can no longer reproduce even after being removed from contact with the agent). In some cases, the agent causes lysis (dissolution) of the cells; in other cases, the cells remain intact and may even continue to be metabolically active. (The terms “fungicidal,” “sporicidal,” and “virucidal” refer to the property whereby biocides are able to kill fungi, spores, and viruses, respectively.) D. Sterilization A defined process used to render a surface or product free from viable organisms, including bacterial spores. E. Disinfectants Products or biocides used to reduce the number of viable microorganisms, or bioburden, on or in a product or surface to a level previously specified as appropriate for its intended further handling or use. Disinfectants are not necessarily sporicidal but are sporostatic, inhibiting germination or outgrowth. TABLE 4-3 Some Common Biocides Used for Antisepsis, Disinfection, Preservation, and Other Purposes Agent Formula Alcohols Ethanol Antisepsis, disinfection, preservation CH32CHOH Isopropanol CH3 CHOH CH3 Aldehydes Glutaraldehyde H O Disinfection, sterilization, preservation H CCH2CH2CH2C Formaldehyde Biguanides Chlorhexidine Uses O H C5O H N(HCN)2H(CH2)6N(HCN)2H Cl NH Antisepsis, antiplaque activity, preservation, disinfection NH Bisphenols Triclosan Cl Antisepsis, antiplaque activity OH Cl Hexachlorophene Cl O Cl Deodorant, preservation OH OH Cl Cl CH2 Cl Cl Cl Halogen-releasing agents Chlorine compounds Cl Disinfection, antisepsis → OCI-, HOCl, Cl2 Iodine compounds → I2 Heavy metal derivatives Silver compounds Ag Preservation, antisepsis Mercury compounds Hg Disinfection Organic acids Benzoic acid Propionic acid Preservation COOH Sodium or calcium salt used for preservation CH32CH22COOH Peroxygens Hydrogen peroxide Disinfection, sterilization H2O2 Ozone O3 Peracetic acid CH3COOOH Phenols and cresols Phenol Disinfection, preservation OH Cresol OH CH3 Quaternary ammonium compounds R1 R3 N 2 R Disinfection, antisepsis, preservation + X– 4 R (continued) 62 SECTION I Fundamentals of Microbiology TABLE 4-3 Some Common Biocides Used for Antisepsis, Disinfection, Preservation, and Other Purposes (continued) Agent Formula Cetrimide H3C Uses Br – N H3C C0H2n+1 Benzalkonium chloride CH2 CH3 H3C C0H2n+1 Vapor phase Ethylene oxide Sterilization, disinfection O H2C CH2 H H Hydrogen peroxide F. Septic Characterized by the presence of pathogenic microbes in living tissues or associated fluids. G. Antiseptic A biocide or product that destroys or inhibits the growth of microorganisms in or on living tissue (eg, skin) or biologic fluids (eg, mucosal secretions). H. Aseptic Free of, or using methods to keep free of, microorganisms. I. Preservation The prevention of multiplication of microorganisms in formulated products, including pharmaceuticals and foods. J. Antibiotics Naturally occurring and synthetically derived organic compounds that inhibit or destroy selective bacteria, generally at low concentrations. Modes of Action A. Damage to DNA A number of physical and chemical agents act by damaging DNA; these include ionizing radiations, ultraviolet light, and DNA-reactive chemicals. Among the last category are Brooks_CH04_p055-p066.indd 62 Cl – N Formaldehyde Disinfection, antisepsis, preservation + CH3 C O H2O2 alkylating agents and other compounds that react covalently with purine and pyrimidine bases to form DNA adducts or interstrand cross-links. Radiation damage DNA in several ways: Ultraviolet light, for example, induces cross-linking between adjacent pyrimidines on one or the other of the two polynucleotide strands, forming pyrimidine dimers; ionizing radiations produce breaks in single and double strands. Radiation-induced and chemically-induced DNA lesions kill the cell mainly by interfering with DNA replication. See Chapter 7 for a discussion of DNA repair systems. B. Protein Denaturation Proteins exist in a folded, three-dimensional state determined primarily by intramolecular noncovalent interactions such as ionic, hydrophobic, and hydrogen bonds or covalent disulfide linkages. This state is called the tertiary structure of the protein; it is readily disrupted by a number of physical (eg, heat) or chemical (eg, alcohol) agents, causing the protein to become nonfunctional. The disruption of the tertiary structure of a protein is called protein denaturation. C. Disruption of the Cell Membrane or Wall The cell membrane acts as a selective barrier, allowing some solutes to pass through and excluding others. Many compounds are actively transported through the membrane, becoming concentrated within the cell. The membrane is also the site of enzymes involved in the biosynthesis of components of the cell envelope. Substances that concentrate at the cell surface may alter the physical and chemical properties of 16/08/12 11:45 AM CHAPTER 4 The Growth, Survival, and Death of Microorganisms 63 the membrane, preventing its normal functions and therefore killing or inhibiting the cell. The cell wall acts as a corseting structure (best characterized as a fishing net), protecting the cell against osmotic lysis. Thus, agents that destroy the wall (eg, lysozyme, which cleaves the sugar linkages) or prevent its normal synthesis (eg, penicillin, which interrupts peptidyl crosslinkages) may bring about lysis of the cell. D. Disruption of Free Sulfhydryl Groups Enzymes containing cysteine have side chains terminating in sulfhydryl groups. In addition to these, coenzymes such as coenzyme A and dihydrolipoate contain free sulfhydryl groups. Such enzymes and coenzymes cannot function unless the sulfhydryl groups remain free and reduced. Oxidizing agents thus interfere with metabolism by forming disulfide linkages between neighboring sulfhydryl groups: R SH HS 2H R − →R S S R Many metals such as mercuric ion likewise interfere by combining with sulfhydryls. There are sulfhydryl-containing enzymes in the cell, so oxidizing agents and heavy metals do widespread damage. E. Chemical Antagonism The interference by a chemical agent with the normal reaction between a specific enzyme and its substrate is known as chemical antagonism. The antagonist acts by combining with some part of the holoenzyme (the protein apoenzyme, the mineral activator, or the coenzyme), thereby preventing attachment of the normal substrate. (Substrate is used here in the broad sense to include cases in which the inhibitor combines with the apoenzyme, thereby preventing attachment to it of coenzyme.) An antagonist combines with an enzyme because of its chemical affinity for an essential site on that enzyme. Enzymes perform their catalytic function by virtue of their affinity for their natural substrates; hence any compound structurally resembling a substrate in essential aspects may also have an affinity for the enzyme. If this affinity is great enough, the “analog” will displace the normal substrate and prevent the proper reaction from taking place. Many holoenzymes include a mineral ion as a bridge either between enzyme and coenzyme or between enzyme and substrate. Chemicals that combine readily with these minerals will again prevent attachment of coenzyme or substrate (eg, carbon monoxide and cyanide combine with the iron atom in heme-containing enzymes and prevent their function in respiration). Chemical antagonists can be conveniently discussed under two headings: (a) antagonists of energy-yielding processes and (b) antagonists of biosynthetic processes. The former include poisons of respiratory enzymes (carbon monoxide, cyanide) and of oxidative phosphorylation (dinitrophenol); the latter include analogs of the building blocks of proteins (amino acids) and of nucleic acids (nucleotides). In some cases, the analog simply prevents incorporation of the normal metabolite (eg, 5-methyltryptophan prevents incorporation of tryptophan into protein), and in other cases, the analog replaces the normal metabolite in the macromolecule, causing it to be nonfunctional. The incorporation of p-fluorophenylalanine in place of phenylalanine in proteins is an example of the latter type of antagonism. Reversal of Antibacterial Action In the section on definitions, the point was made that bacteriostatic action is, by definition, reversible. Reversal can be brought about in several ways. A. Removal of Agent When cells that are inhibited by the presence of a bacteriostatic agent are removed by flushing the surface or centrifugation that removes bacteria from the bacteriostatic substance they will resume normal multiplication. B. Reversal by Substrate When a chemical antagonist of the analog type binds reversibly with the enzyme, it is possible to displace it by adding a high concentration of the normal substrate. Such cases are termed “competitive inhibition.” The ratio of inhibitor concentration to concentration of substrate reversing the inhibition is called the antimicrobial index; it is usually very high (100–10,000), indicating a much greater affinity of enzyme for the analog over its normal substrate. C. Inactivation of Agent An agent can often be inactivated by adding to the medium a substance that combines with it, preventing its combination with cellular constituents. For example, mercuric ion can be inactivated by addition to the medium of sulfhydryl compounds such as thioglycolic acid. D. Protection Against Lysis Osmotic lysis can be prevented by making the medium isotonic for naked bacterial protoplasts. Concentrations of 10–20% sucrose are required. Under such conditions, penicillin-induced protoplasts (ie, the living material of a bacterial cell, including the protoplasm and plasma membrane after the cell wall has been removed) remain viable and continue to grow as L-forms. Resistance to Antibacterial Agents The ability of bacteria to become resistant to antibacterial agents is an important factor in their control. The mechanisms by which resistance is acquired are discussed in Chapters 7: Microbial Genetics and 28: Antimicrobial Chemotherapy. 64 SECTION I Fundamentals of Microbiology Physical Agents A. Heat Application of heat is the simplest means of sterilizing materials, provided the material is itself resistant to heat damage. A temperature of 100°C will kill all but spore forms of eubacteria within 2–3 minutes in laboratory-scale cultures; a temperature of 121°C for 15 minutes is used to kill spores. Steam is generally used, both because bacteria are more quickly killed when moist and because steam provides a means for distributing heat to all parts of the sterilizing vessel. At sea level, steam must be kept at a pressure of 15 lb/sq inches (psi) in excess of atmospheric pressure to obtain a temperature of 121°C; autoclaves or pressure cookers are used for this purpose. At higher altitudes, the pressure would need to be higher than 15 psi to reach 121°C. For sterilizing materials that must remain dry, circulating hot air electric ovens are available; because heat is less effective on dry material, it is customary to apply a temperature of 160–170°C for 1 hour or more. Under these conditions (ie, excessive temperatures applied for long periods of time), heat acts by denaturing cell proteins and nucleic acids and by disrupting cell membranes. B. Radiation Ultraviolet light and ionizing radiations have various appli cations as sterilizing agents. Their modes of action have been discussed above. Chemical Agents The chemical structures and uses of biocides are shown in Table 4-3. A. Alcohols Ethyl alcohol, isopropyl alcohol, and n-propanol exhibit rapid, broad-spectrum antimicrobial activity against vegetative bacteria, viruses, and fungi but are not sporicidal. Activity is optimal when they are diluted to a concentration of 60% to 90% with water. B. Aldehydes Glutaraldehyde is used for low-temperature disinfection and sterilization of endoscopes and surgical equipment. It is normally used as a 2% solution to achieve sporicidal activity. Formaldehyde is bactericidal, sporicidal, and virucidal. C. Biguanides Chlorhexidine is widely used in handwashing and oral products and as a disinfectant and preservative. The Mycobacteria, because of their unique waxy cell envelope, are generally highly resistant to these compounds. D. Bisphenols The bisphenols are widely used in antiseptic soaps and hand rinses. In general, they are broad spectrum but have little activity against Pseudomonas aeruginosa and molds. Triclosan and hexachlorophene are bactericidal and sporostatic. E. Halogen-Releasing Agents The most important types of chlorine-releasing agents are sodium hypochlorite, chlorine dioxide, and sodium dichloroisocyanurate, which are oxidizing agents that destroy the cellular activity of proteins. Hypochlorous acid is the active compound responsible for the bactericidal and virucidal effect of these compounds. At higher concentrations, this group is sporicidal. Iodine is rapidly bactericidal, fungicidal, tuberculocidal, virucidal, and sporicidal. Iodophors (eg, povidone-iodine) are complexes of iodine and a solubilizing agent or carrier, which acts as a reservoir of the active I2. F. Heavy Metal Derivatives Silver (Ag+) sulfadiazine, a combination of two antibacterial agents, Ag+ and sulfadiazine, has a broad spectrum of activity. Binding to cell components such as DNA may be responsible for its inhibitory properties. G. Organic Acids Organic acids are used as preservatives in the pharmaceutical and food industries. Benzoic acid is fungistatic; propionic acid is both bacteriostatic and fungistatic. H. Peroxygens Hydrogen peroxide has broad-spectrum activity against viruses, bacteria, yeasts, and bacterial spores. Sporicidal activity requires higher concentrations (10–30%) of H2O2 and longer contact times. I. Phenols Phenol and many phenolic compounds have antiseptic, disinfectant, or preservative properties. J. Quaternary Ammonium Compounds These compounds have two regions in their molecular structures, one a water-repelling (hydrophobic) group and the other a water-attracting (hydrophilic) group. Cationic detergents, as exemplified by quaternary ammonium compounds (QACs), are useful antiseptics and disinfectants. QACs have been used for a variety of clinical purposes (eg, preoperative disinfection of unbroken skin) as well as for cleaning hard surfaces. They are sporostatic; they inhibit the outgrowth of spores but not the actual germination process. QACs are also mycobacteriostatic and have an effect on lipid-enveloped but not lipid-nonenveloped viruses. K. Vapor-Phase Sterilants Heat-sensitive medical devices and surgical supplies can be effectively sterilized by vapor-phase systems using ethylene oxide, formaldehyde, hydrogen peroxide, or peracetic acid. CHAPTER 4 The Growth, Survival, and Death of Microorganisms 65 OBJECTIVES 1. U nderstand the differences between growth in a closed system (liquid culture), growth in continuous culture, and growth within a biofilm. 2. Appreciate the differences between bacteriostatic and bacteriocidal concepts. 3. Know the conditions required for antimicrobial sterilization. 4. Be aware of the mechanism of action of the common disinfective agents. REVIEW QUESTIONS 1. A 23-year-old woman has 10 Escherichia coli inoculated into her bladder while having sex. These E coli have a generation time of 20 minutes. After a lag of 20 minutes, the E coli enter the logarithmic phase of growth. After 3 hours of logarithmic growth, the total number of cells is (A) 2560 (B) 5012 (C) 90 (D) 1028 (E) 1,000,000 2. A 73-year-old woman is admitted to the hospital for intravenous treatment of an abscess caused by Staphylococcus aureus. Subsequent to her treatment and discharge from the hospital, it is necessary to disinfect the hospital room. One thousand of the S aureus cells are exposed to a disinfectant. After 10 minutes, 90% of the cells are killed. How many cells remain viable after 20 minutes? (A) 500 (B) 100 (C) 10 (D) 1 (E) 0 3. The action of which of the following agents or processes on bacteria can be reversed? (A) A disinfectant (B) A bactericidal agent (C) A bacteriostatic agent (D) Autoclaving at 121°C for 15 minutes (E) Dry heat at 160–170°C for 1 hour 4. The growth rate of bacteria during the exponential phase of growth is (A) Zero (B) Increasing (C) Constant (D) Decreasing (E) Negative 5. The growth rate of bacteria during the maximum stationary phase of growth is (A) Zero (B) Increasing (C) Constant (D) Decreasing (E) Negative Answers 1. A C C C A REFERENCES Block SS (editor): Disinfection, Sterilization, and Preservation, 5th ed. Lippincott Williams & Wilkins, 2001. Colwell RR, Grimes DJ (editors): Nonculturable Microorganisms in the Environment. ASM Press, 2000. Donohue WD: The cell cycle of Escherichia coli. Annu Rev Microbiol 1993;47:199. Gerhardt P et al (editors): Manual of Methods for General Bacteriology. American Society for Microbiology, 1981. Hans-Curt Flemming, Jost Wingender: The biofilm matrix. Nat Rev Microbiol 2010;8:623-633. Kjelleberg S (editor): Starvation in Bacteria. Plenum Press, 1993. Kjelleberg S, Hermansson M, Mårdén P, Jones GW: The transient phase between growth and nongrowth of heterotrophic bacteria. Annu Rev Microbiol 1987;41:25. Kolter R, Siegels DA, Tormo A: The stationary phase of the bacterial life cycle. J Bacteriol 1992;174:345. McDonnell GE: Antisepsis, Disinfection, and Sterilization: Types, Action, and Resistance. ASM Press, 2007. McDonnell G, Russell AD: Antiseptics and disinfectants: Activity, action, and resistance. Clin Microbiol Rev 1999; 12:147. Olmstad RN (editor): APIC Infection Control and Applied Epidemiology: Principles and Practices. Mosby Year Book, 1996. Russell AD, Hugo WB, Ayliffe GAJ (editors): Principles and Practice of Disinfection, Preservation and Sterilization, 3rd ed. Blackwell Scientific Publications, 1999. Sancar A, Sancar GB: DNA repair enzymes. Annu Rev Biochem 1988;57:29. Siegels DA, Kolter R: Life after log. J Bacteriol 1992;174:345. This page intentionally left blank C Cultivation of Microorganisms Cultivation is the process of propagating organisms by providing the proper environmental conditions. Growing microorganisms are making replicas of themselves, and they require the elements present in their chemical composition. Nutrients must provide these elements in metabolically accessible form. In addition, organisms require metabolic energy to synthesize macromolecules and maintain essential chemical gradients across their membranes. Factors that must be controlled during growth include the nutrients, pH, temperature, aeration, salt concentration, and ionic strength of the medium. H 5 A P T E R ionic or molecular gradients across the membrane, to synthesize anhydride bonds in ATP, or for a combination of these purposes. Alternatively, cells given a source of ATP may use its anhydride bond energy to create a proton motive force that in turn may be used to move the cell and to maintain chemical gradients. To grow, an organism requires all of the elements in its organic matter and the full complement of ions required for energetics and catalysis. In addition, there must be a source of energy to establish the proton motive force and to allow macromolecular synthesis. Microorganisms vary widely in their nutritional demands and their sources of metabolic energy. REQUIREMENTS FOR GROWTH Most of the dry weight of microorganisms is organic matter containing the elements carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur. In addition, inorganic ions such as potassium, sodium, iron, magnesium, calcium, and chloride are required to facilitate enzymatic catalysis and to maintain chemical gradients across the cell membrane. For the most part, the organic matter is in macromolecules formed by anhydride bonds between building blocks. Synthesis of the anhydride bonds requires chemical energy, which is provided by the two phosphodiester bonds in ATP (adenosine triphosphate; see Chapter 6). Additional energy required to maintain a relatively constant cytoplasmic composition during growth in a range of extracellular chemical environments is derived from the proton motive force. The proton motive force is the potential energy that can be derived by passage of a proton across a membrane. In eukaryotes, the membrane may be part of the mitochondrion or the chloroplast. In prokaryotes, the membrane is the cytoplasmic membrane of the cell. The proton motive force is an electrochemical gradient with two components, a difference in pH (hydrogen ion concentration) and a difference in ionic charge. The charge on the outside of the bacterial membrane is more positive than the charge on the inside, and the difference in charge contributes to the free energy released when a proton enters the cytoplasm from outside the membrane. Metabolic processes that generate the proton motive force are discussed in Chapter 6. The free energy may be used to move the cell, to maintain SOURCES OF METABOLIC ENERGY The three major mechanisms for generating metabolic energy are fermentation, respiration, and photosynthesis. At least one of these mechanisms must be used if an organism is to grow. Fermentation The formation of ATP in fermentation is not coupled to the transfer of electrons. Fermentation is characterized by substrate phosphorylation, an enzymatic process in which a pyrophosphate bond is donated directly to ADP (adenosine diphosphate) by a phosphorylated metabolic intermediate. The phosphorylated intermediates are formed by metabolic rearrangement of a fermentable substrate such as glucose, lactose, or arginine. Because fermentations are not accompanied by a change in the overall oxidation-reduction state of the fermentable substrate, the elemental composition of the products of fermentation must be identical to those of the substrates. For example, fermentation of a molecule of glucose (C6H12O6) by the Embden-Meyerhof pathway (see Chapter 6) yields a net gain of two pyrophosphate bonds in ATP and produces two molecules of lactic acid (C3H6O3). respiration Respiration is analogous to the coupling of an energydependent process to the discharge of a battery. Chemical 67 68 SECTION I Fundamentals of Microbiology reduction of an oxidant (electron acceptor) through a specific series of electron carriers in the membrane establishes the proton motive force across the bacterial membrane. The reductant (electron donor) may be organic or inorganic (eg, lactic acid serves as a reductant for some organisms, and hydrogen gas is a reductant for other organisms). Gaseous oxygen (O2) often is used as an oxidant, but alternative oxidants that are used by some organisms include carbon dioxide (CO2), sulfate (SO42-), and nitrate (NO3-). Photosynthesis Photosynthesis is similar to respiration in that the reduction of an oxidant via a specific series of electron carriers establishes the proton motive force. The difference in the two processes is that in photosynthesis, the reductant and oxidant are created photochemically by light energy absorbed by pigments in the membrane; thus, photosynthesis can continue only as long as there is a source of light energy. Plants and some bacteria are able to invest a substantial amount of light energy in making water a reductant for carbon dioxide. Oxygen is evolved in this process, and organic matter is produced. Respiration, the energetically favorable oxidation of organic matter by an electron acceptor such as oxygen, can provide photosynthetic organisms with energy in the absence of light. NUTRITION Nutrients in growth media must contain all the elements necessary for the biologic synthesis of new organisms. In the following discussion, nutrients are classified according to the elements they supply. Carbon Source As already mentioned, plants and some bacteria are able to use photosynthetic energy to reduce carbon dioxide at the expense of water. These organisms belong to the group of autotrophs, creatures that do not require organic nutrients for growth. Other autotrophs are the chemolithotrophs, organisms that use an inorganic substrate such as hydrogen or thiosulfate as a reductant and carbon dioxide as a carbon source. Heterotrophs require organic carbon for growth, and the organic carbon must be in a form that can be assimilated. Naphthalene, for example, can provide all of the carbon and energy required for respiratory heterotrophic growth, but very few organisms possess the metabolic pathway necessary for naphthalene assimilation. Glucose, on the other hand, can support the fermentative or respiratory growth of many organisms. It is important that growth substrates be supplied at levels appropriate for the microbial strain that is being grown: Levels that will support the growth of one organism may inhibit the growth of another organism. Carbon dioxide is required for a number of biosynthetic reactions. Many respiratory organisms produce more than enough carbon dioxide to meet this requirement, but others require a source of carbon dioxide in their growth medium. Nitrogen Source Nitrogen is a major component of proteins, nucleic acids, and other compounds, accounting for approximately 5% of the dry weight of a typical bacterial cell. Inorganic dinitrogen (N2) is very prevalent, comprising 80% of the earth’s atmosphere. It is also a very stable compound, primarily because of the high activation energy required to break the nitrogen–nitrogen triple bond. However, nitrogen may be supplied in a number of different forms, and microorganisms vary in their abilities to assimilate nitrogen (Table 5-1). The end product of all pathways for nitrogen assimilation is the most reduced form of the element, ammonia (NH3). When NH3 is available, it diffuses into most bacteria through transmembrane channels as dissolved gaseous NH3 rather than ionic ammonium ion (NH4+). The ability to assimilate N2 reductively via NH3, which is called nitrogen fixation, is a property unique to prokaryotes, and relatively few bacteria are capable of breaking the nitrogen–nitrogen triple bond. This process (see Chapter 6) requires a large amount of metabolic energy and is readily inactivated by oxygen. The capacity for nitrogen fixation is found in widely divergent bacteria that have evolved quite different biochemical strategies to protect their nitrogen-fixing enzymes from oxygen. Most microorganisms can use NH3 as a sole nitrogen source, and many organisms possess the ability to produce NH3 from amines (R—NH2) or from amino acids (RCHNH2COOH), generally intracellularly. Production of NH3 from the deamination of amino acids is called ammonification. Ammonia is introduced into organic matter by biochemical pathways involving glutamate and glutamine. These pathways are discussed in Chapter 6. Many microorganisms possess the ability to assimilate nitrate (NO3 -) and nitrite (NO2 -) reductively by conversion of these ions into NH3. These processes are termed assimilatory nitrate reduction and assimilatory nitrite reduction, respectively. These pathways for assimilation differ from pathways used for dissimilation of nitrate and nitrite. The dissimilatory pathways are used by organisms that use these TABLE 5-1 Sources of Nitrogen in Microbial Nutrition Compound NO3- +5 NO2- +3 N2 a Valence of N 0 NH4+ -3 R-NH2a -3 R, organic radical. CHAPTER 5 Cultivation of Microorganisms 69 ions as terminal electron acceptors in respiration. Some autotrophic bacteria (eg, Nitrosomonas, Nitrobacter spp.) are able to convert NH3 to gaseous N2 under anaerobic conditions; this process is known as denitrification. Our understanding of the nitrogen cycle continues to evolve. In the mid-1990s, the anammox reaction was discovered. The reaction NH4+ + NO2– → N2 + 2H2O in which ammonia is oxidized by nitrite is a microbial process that occurs in anoxic waters of the ocean and is a major pathway by which nitrogen is returned to the atmosphere. Sulfur Source Similar to nitrogen, sulfur is a component of many organic cell substances. It forms part of the structure of several coenzymes and is found in the cysteinyl and methionyl side chains of proteins. Sulfur in its elemental form cannot be used by plants or animals. However, some autotrophic bacteria can oxidize it to sulfate (SO42–). Most microorganisms can use sulfate as a sulfur source, reducing the sulfate to the level of hydrogen sulfide (H2S). Some microorganisms can assimilate H2S directly from the growth medium, but this compound can be toxic to many organisms. Phosphorus Source Phosphate (PO43–) is required as a component of ATP; nucleic acids; and such coenzymes as NAD, NADP, and flavins. In addition, many metabolites, lipids (phospholipids, lipid A), cell wall components (teichoic acid), some capsular polysaccharides, and some proteins are phosphorylated. Phosphate is always assimilated as free inorganic phosphate (Pi). Mineral Sources Numerous minerals are required for enzyme function. Magnesium ion (Mg2+) and ferrous ion (Fe2+) are also found in porphyrin derivatives: magnesium in the chlorophyll molecule, and iron as part of the coenzymes of the cytochromes and peroxidases. Mg2+ and K+ are both essential for the function and integrity of ribosomes. Ca 2+ is required as a constituent of gram-positive cell walls, although it is dispensable for gram-negative bacteria. Many marine organisms require Na+ for growth. In formulating a medium for the cultivation of most microorganisms, it is necessary to provide sources of potassium, magnesium, calcium, and iron, usually as their ions (K+, Mg2+, Ca2+, and Fe2+). Many other minerals (eg, Mn2+, Mo2+, Co2+, Cu2+, and Zn2+) are required; these frequently can be provided in tap water or as contaminants of other medium ingredients. The uptake of iron, which forms insoluble hydroxides at neutral pH, is facilitated in many bacteria and fungi by their production of siderophores—compounds that chelate iron and promote its transport as a soluble complex. These include hydroxamates (–CONH2OH) called sideramines, and derivatives of catechol (eg, 2,3-dihydroxy-benzoylserine). Plasmiddetermined siderophores play a major role in the invasiveness of some bacterial pathogens (see Chapter 7). Siderophoreand nonsiderophore-dependent mechanisms of iron uptake by bacteria are discussed in Chapter 9. Growth Factors A growth factor is an organic compound that a cell must contain to grow but that it is unable to synthesize. Many microorganisms, when provided with the nutrients listed above, are able to synthesize all of the building blocks for macromolecules (Figure 5-1), which are amino acids; purines, pyrimidines, and pentoses (the metabolic precursors of nucleic acids); additional carbohydrates (precursors of polysaccharides); and fatty acids and isoprenoid compounds. In addition, free-living organisms must be able to synthesize the complex vitamins that serve as precursors of coenzymes. Each of these essential compounds is synthesized by a discrete sequence of enzymatic reactions; each enzyme is produced under the control of a specific gene. When an organism undergoes a gene mutation resulting in failure of one of these enzymes to function, the chain is broken, and the end product is no longer produced. The organism must then obtain that compound from the environment: The compound has become a growth factor for the organism. This type of mutation can be readily induced in the laboratory. Different microbial species vary widely in their growth factor requirements. The compounds involved are found in and are essential to all organisms; the differences in requirements reflect differences in synthetic abilities. Some species require no growth factors, but others—such as some of the lactobacilli—have lost, during evolution, the ability to synthesize as many as 30–40 essential compounds and hence require them in the medium. ENVIRONMENTAL FACTORS AFFECTING GROWTH A suitable growth medium must contain all the nutrients required by the organism to be cultivated, and such factors as pH, temperature, and aeration must be carefully controlled. A liquid medium is used; the medium can be gelled for special purposes by adding agar or silica gel. Agar, a polysaccharide extract of a marine alga, is uniquely suitable for microbial cultivation because it is resistant to microbial action and because it dissolves at 100°C but does not gel until cooled below 45°C; cells can be suspended in the medium at 45°C and the medium quickly cooled to a gel without harming them. Nutrients On the previous pages, the function of each type of nutrient is described, and a list of suitable substances presented. In general, 70 SECTION I Fundamentals of Microbiology H2O Macromolecules Building blocks Percent Dry Weight in Typical Cell H2O Amino acids Proteins 50 Nucleic acids 20 Polysaccharides 10 H2O Mononucleotides H2O Monosaccharides H2O Isoprenoid precursors Acetate D H2 Lipids D H2O Acceptors Fatty acids 10 H2O Figure 5-1 Macromolecular synthesis. Polymerization of building blocks into macromolecules is achieved largely by the introduction of anhydride bonds. Formation of fatty acids from acetate requires several steps of biochemical reduction using organic hydrogen donors (D • H2). the following must be provided: (1) hydrogen donors and acceptors, about 2 g/L; (2) carbon source, about 1 g/L; (3) nitrogen source, about 1 g/L; (4) minerals: sulfur and phosphorus, about 50 mg/L of each, and trace elements, 0.1–1 mg/L of each; (5) growth factors: amino acids, purines, and pyrimidines, about 50 mg/L of each, and vitamins, 0.1–1 mg/L of each. For studies of microbial metabolism, it is usually necessary to prepare a completely synthetic medium in which the exact characteristics and concentration of every ingredient are known. Otherwise, it is much cheaper and simpler to use natural materials such as yeast extract, protein digest, or similar substances. Most free-living microbes grow well on yeast extract; parasitic forms may require special substances found only in blood or in extracts of animal tissues. Nevertheless, some parasitic microbes (eg, Treponema pallidum) cannot be grown in vitro or grow inside eukaryotic cells (eg, Chlamydia trachomatis). For many organisms, a single compound (eg, an amino acid) may serve as energy source, carbon source, and nitrogen source; others require a separate compound for each. If natural materials for nonsynthetic media are deficient in any particular nutrient, they must be supplemented. Hydrogen Ion Concentration (pH) Most organisms have a fairly narrow optimal pH range. The optimal pH must be empirically determined for each species. Most organisms (neutralophiles) grow best at a pH of 6.0– 8.0, although some forms (acidophiles) have optima as low as pH 3.0, and others (alkaliphiles) have optima as high as pH 10.5. Microorganisms regulate their internal pH over a wide range of external pH values by pumping protons in or out of their cells. Acidophiles maintain an internal pH of about 6.5 over an external range of 1.0–5.0, neutralophiles maintain an internal pH of about 7.5 over an external range of 5.5–8.5, and alkaliphiles maintain an internal pH of about 9.5 over an external range of 9.0–11.0. Internal pH is regulated by a set of proton transport systems in the cytoplasmic membrane, including a primary, ATP-driven proton pump and an Na+/H+ exchanger. A K+/H+ exchange system has also been proposed to contribute to internal pH regulation in neutralophiles. Temperature Different microbial species vary widely in their optimal temperature ranges for growth (Figure 5-2): Psychrophilic forms grow best at low temperatures (-5–15°C) and are usually found in such environments as the Arctic and Antarctic regions; psychrotrophs have a temperature optimum between 20°C and 30°C but grow well at lower temperatures. They are an important cause of food spoilage. Mesophilic forms grow best at 30–37°C, and most thermophilic forms grow best at 50–60°C. Some organisms are hyperthermophilic and can grow at well above the temperature of boiling water, which exists under high pressure in the depths of the ocean. Most organisms are mesophilic; 30°C is optimal for many free-living forms, and the body temperature of the host is optimal for symbionts of warm-blooded animals. The upper end of the temperature range tolerated by any given species correlates well with the general thermal stability of that species’ proteins as measured in cell extracts. CHAPTER 5 Cultivation of Microorganisms 71 Growth rate Optimum Thermophile Mesophile Hyperthermophile Psychrotroph 10 20 30 40 50 60 70 80 90 100 110 120 Minimum 0 Maximum -10 Log k Psychrophile Temperature (°C) Figure 5-2 Temperature requirements for growth. Prokaryotes are commonly divided into five groups based on their optimum growth temperatures. Note that the optimum temperature, the point at which the growth rate is highest, is near the upper limit of the range. (Reproduced with permission from Nester EW, Anderson DG, Roberts CE, Nester MT [editors]: Microbiology: A Human Perspective, 6th ed. McGraw-Hill, 2009, p. 91. © The McGraw-Hill Companies, Inc.) Microorganisms share with plants and animals the heat-shock response, a transient synthesis of a set of “heat-shock proteins,” when exposed to a sudden rise in temperature above the growth optimum. These proteins appear to be unusually heat resistant and to stabilize the heat-sensitive proteins of the cell. The relationship of growth rate to temperature for any given microorganism is seen in a typical Arrhenius plot (Figure 5-3). Arrhenius showed that the logarithm of the velocity of any chemical reaction (log k) is a linear function of the reciprocal of the temperature (1/T); because cell growth is the result of a set of chemical reactions, it might be expected to show this relationship. Figure 5-3 shows this to be the case over the normal range of temperatures for a given species; log k decreases linearly with 1/T. Above and below the normal range, however, log k drops rapidly, so that maximum and minimum temperature values are defined. Beyond their effects on growth rate, extremes of temperature kill microorganisms. Extreme heat is used to sterilize preparations (see Chapter 4); extreme cold also kills microbial cells, although it cannot be used safely for sterilization. Bacteria also exhibit a phenomenon called cold shock, which is the killing of cells by rapid—as opposed to slow—cooling. For example, the rapid cooling of Escherichia coli from 37°C to 5°C can kill 90% of the cells. A number of compounds protect cells from either freezing or cold shock; glycerol and dimethyl sulfoxide are most commonly used. Aeration The role of oxygen as hydrogen acceptor is discussed in Chapter 6. Many organisms are obligate aerobes, specifically High temperature Normal temperature Low temperature 1/T (K) Figure 5-3 General form of an Arrhenius plot of bacterial growth. (Reproduced with permission from Ingraham JL: Growth of psychrophilic bacteria. J Bacteriol 1958;76(1):75-80.) requiring oxygen as hydrogen acceptor; some are facultative anaerobes, able to live aerobically or anaerobically; some are obligate anaerobes requiring a substance other than oxygen as hydrogen acceptor and are sensitive to oxygen inhibition; some are microaerophiles, which require small amounts of oxygen (2%–10%) for aerobic respiration (higher concentrations are inhibitory); and others are aerotolerant anaerobes, which are indifferent to oxygen. They can grow in its presence, but they do not use it as a hydrogen acceptor (Figure 5-4). The natural by-products of aerobic metabolism are the reactive compounds hydrogen peroxide (H2O2) and superoxide (O2–). In the presence of iron, these two species can generate hydroxyl radicals (•OH), which can damage any biologic macromolecule: Fe /Fe O2- + H2O2 −−−−−−−→ O2 + OH- + •OH 3+ 2+ Many aerobes and aerotolerant anaerobes are protected from these products by the presence of superoxide dismutase, an enzyme that catalyzes the reaction 2O2- + 2H+ → O2 + H2O2 and by the presence of catalase, an enzyme that catalyzes the reaction 2H2O2 → 2H2O + O2 72 SECTION I Fundamentals of Microbiology Obligate aerobe Facultative anaerobe Obligate anaerobe Microaerophile Aerotolerant Bacteria Bacteria Enzymes in Cells for O2 Detoxification Catalase: 2H2O2 2H2O + O2 Superoxide dismutase: O +H O 2O - + 2H+ 2 2 Catalase, superoxide dismutase Neither catalase nor superoxide dismutase in most Small amounts of catalase and superoxide dismutase Superoxide dismutase 2 2 Figure 5-4 Oxygen (O2) requirements of prokaryotes. (Reproduced with permission from Nester EW, Anderson DG, Roberts CE, Nester MT [editors]: Microbiology: A Human Perspective, 6th ed. McGraw-Hill, 2009, p. 92. © The McGraw-Hill Companies, Inc.) Some fermentative organisms (eg, Lactobacillus plantarum) are aerotolerant but do not contain catalase or superoxide dismutase. Oxygen is not reduced, and therefore H2O2 and O2– are not produced. All strict anaerobes lack both superoxide dismutase and catalase. Some anaerobic organisms (eg, Peptococcus anaerobius) have considerable tolerance to oxygen as a result of their ability to produce high levels of an enzyme (NADH oxidase) that reduces oxygen to water according to the reaction NADH + H+ + 1/2O2 −−→ NAD+ + H2O Hydrogen peroxide owes much of its toxicity to the damage it causes to DNA. DNA repair-deficient mutants are exceptionally sensitive to hydrogen peroxide; the recA gene product, which functions in both genetic recombination and repair, has been shown to be more important than either catalase or superoxide dismutase in protecting E coli cells against hydrogen peroxide toxicity. The supply of air to cultures of aerobes is a major technical problem. Vessels are usually shaken mechanically to introduce oxygen into the medium or air is forced through the medium by pressure. The diffusion of oxygen often becomes the limiting factor in growing aerobic bacteria; when a cell concentration of 4–5 × 109/mL is reached, the rate of diffusion of oxygen to the cells sharply limits the rate of further growth. Obligate anaerobes, on the other hand, present the problem of oxygen exclusion. Many methods are available for this: Reducing agents such as sodium thioglycolate can be added to liquid cultures, tubes of agar can be sealed with a layer of petrolatum and paraffin, the culture vessel can be placed in a container from which the oxygen is removed by evacuation or by chemical means, or the organism can be handled within an anaerobic glove box. Ionic Strength and Osmotic Pressure To a lesser extent, such factors as osmotic pressure and salt concentration may have to be controlled. For most organisms, the properties of ordinary media are satisfactory; however, for marine forms and organisms adapted to growth in strong sugar solutions, for example, these factors must be considered. Organisms requiring high salt concentrations are called halophilic; those requiring high osmotic pressures are called osmophilic. Most bacteria are able to tolerate a wide range of external osmotic pressures and ionic strengths because of their ability to regulate internal osmolality and ion concentration. Osmolality is regulated by the active transport of K+ ions into the cell; internal ionic strength is kept constant by a compensating excretion of the positively charged organic polyamine putrescine. Because putrescine carries several positive charges per molecule, a large drop in ionic strength is effected at only a small cost in osmotic strength. CULTIVATION METHODS Two problems will be considered: the choice of a suitable medium and the isolation of a bacterial organism in pure culture. The Medium The technique used and the type of medium selected depend on the nature of the investigation. In general, three situations may be encountered: (1) One may need to raise a crop of cells of a particular species that is on hand, (2) one may need to determine the numbers and types of organisms present in a given material, or (3) one may wish to isolate a particular type of microorganism from a natural source. A. Growing Cells of a Given Species Microorganisms observed microscopically to be growing in a natural environment may prove exceedingly difficult to grow in pure culture in an artificial medium. Certain parasitic forms have never been cultivated outside the host. In general, however, a suitable medium can be devised by carefully CHAPTER 5 Cultivation of Microorganisms 73 reproducing the conditions found in the organism’s natural environment. The pH, temperature, and aeration are easy to duplicate; the nutrients present the major problem. The contribution made by the living environment is important and difficult to analyze; a parasite may require an extract of the host tissue, and a free-living form may require a substance excreted by a microorganism with which it is associated in nature. Considerable experimentation may be necessary to determine the requirements of the organism, and success depends on providing a suitable source of each category of nutrient listed at the beginning of this chapter. The cultivation of obligate parasites such as chlamydiae is discussed in Chapter 27. B. Microbiologic Examination of Natural Materials A given natural material may contain many different microenvironments, each providing a niche for a different species. Plating a sample of the material under one set of conditions will allow a selected group of forms to produce colonies but will cause many other types to be overlooked. For this reason, it is customary to plate out samples of the material using as many different media and conditions of incubation as is practicable. Six to eight different culture conditions are not an unreasonable number if most of the forms present are to be discovered. Because every type of organism present must have a chance to grow, solid media are used, and crowding of colonies is avoided. Otherwise, competition will prevent some types from forming colonies. C. Isolation of a Particular Type of Microorganism A small sample of soil, if handled properly, will yield a different type of organism for every microenvironment present. For fertile soil (moist, aerated, rich in minerals and organic matter), this means that hundreds or even thousands of types can be isolated. This is done by selecting for the desired type. One gram of soil, for example, is inoculated into a flask of liquid medium that has been made up for the purpose of favoring one type of organism, such as aerobic nitrogen fixers (Azotobacter). In this case, the medium contains no combined nitrogen and is incubated aerobically. If cells of Azotobacter are present in the soil, they will grow well in this medium; forms unable to fix nitrogen will grow only to the extent that the soil has introduced contaminating fixed nitrogen into the medium. When the culture is fully grown, therefore, the percentage of Azotobacter in the total population will have increased greatly; the method is thus called enrichment culture. Transfer of a sample of this culture to fresh medium will result in further enrichment of Azotobacter; after several serial transfers, the culture can be plated out on a solidified enrichment medium and colonies of Azotobacter isolated. Liquid medium is used to permit competition and hence optimal selection even when the desired type is represented in the soil as only a few cells in a population of millions. Advantage can be taken of “natural enrichment.” For example, in looking for kerosene oxidizers, oil-laden soil is chosen because it is already an enrichment environment for such forms. Enrichment culture, then, is a procedure whereby the medium is prepared so as to duplicate the natural environment (“niche”) of the desired microorganism, thereby selecting for it. An important principle involved in such selection is the following: The organism selected for will be the type whose nutritional requirements are barely satisfied. Azotobacter, for example, grows best in a medium containing organic nitrogen, but its minimum requirement is the presence of N2; hence, it is selected for in a medium containing N2 as the sole nitrogen source. If organic nitrogen is added to the medium, the conditions no longer select for Azotobacter but rather for a form for which organic nitrogen is the minimum requirement. When searching for a particular type of organism that is part of a mixed population, selective or differential media are used. Selective media inhibit the growth of organisms other than the one being sought. For example, Thayer-Martin agar is used to isolate Neisseria gonorrhoeae, the cause of gonorrhea, from clinical specimens. Differential media contain a substance(s) that certain bacteria change in a recognizable way. For example, colonies of E coli have a characteristic iridescent sheen on agar containing the dyes eosin and methylene blue (EMB agar). EMB agar containing a high concentration of one sugar will also cause organisms that ferment that sugar to form reddish colonies. Differential media are used for such purposes as recognizing the presence of enteric bacteria in water or milk and the presence of certain pathogens in clinical specimens. Table 5-2 presents examples of enrichment culture conditions and the types of bacteria they will select. However, despite our best efforts, many environments contain numerous uncultured bacteria. Isolation of Microorganisms in Pure Culture To study the properties of a given organism, it is necessary to handle it in pure culture free of all other types of organisms. To do this, a single cell must be isolated from all other cells and cultivated in such a manner that its collective progeny also remain isolated. Several methods are available. A. Plating Unlike cells in a liquid medium, cells in or on a gelled medium are immobilized. Therefore, if few enough cells are placed in or on a gelled medium, each cell will grow into an isolated colony. The ideal gelling agent for most microbiologic media is agar, an acidic polysaccharide extracted from certain red algae. A 1.5–2% suspension in water dissolves at 100°C, forming a clear solution that gels at 45°C. Thus, a sterile agar solution can be cooled to 50°C, bacteria or other 74 SECTION I Fundamentals of Microbiology TABLE 5-2 Some Enrichment Cultures Nitrogen Source N2 Carbon Source CO2 Alcohol, fatty acids, etc Glucose NaNO3 CO2 Alcohol, fatty acids, etc Glucose NH4Cl CO2 Alcohol, fatty acids, etc Glucose Predominant Organism Initially Enriched Atmosphere Illumination Aerobic or anaerobic Dark None Light Cyanobacteria Anaerobic Dark None Air Dark Azotobacter Anaerobic Dark Clostridium pasteurianum Air Dark Azotobacter Aerobic or anaerobic Dark None Light Green algae and cyanobacteria Anaerobic Dark Denitrifiers Air Dark Aerobes Anaerobic Dark Fermenters Air Dark Aerobes Anaerobic Dark None Aerobic Dark Nitrosomonas Aerobic or anaerobic Light Green algae and cyanobacteria Anaerobic Dark Sulfate or carbonate reducers Aerobic Dark Aerobes Anaerobic Dark Fermenters Aerobic Dark Aerobes Note: Constituents of all media: MgSO4, K 2HPO4, FeCl3, CaCl2, CaCO3, and trace elements. microbial cells added, and then the solution quickly cooled below 45°C to form a gel. (Although most microbial cells are killed at 50°C, the time course of the killing process is sufficiently slow at this temperature to permit this procedure; see Figure 4-3.) Once gelled, agar will not again liquefy until it is heated above 80°C, so that any temperature suitable for the incubation of a microbial culture can subsequently be used. In the pour-plate method, a suspension of cells is mixed with melted agar at 50°C and poured into a Petri dish. When the agar solidifies, the cells are immobilized in the agar and grow into colonies. If the cell suspension is sufficiently dilute, the colonies will be well separated, so that each has a high probability of being derived from a single cell (Figure 5-5). To make certain of this, however, it is necessary to pick a colony of the desired type, suspend it in water, and replate. Repeating this procedure several times ensures that a pure culture will be obtained. Alternatively, the original suspension can be streaked on an agar plate with a wire loop (streak-plate technique). As the streaking continues, fewer and fewer cells are left on the loop, and finally the loop may deposit single cells on the agar (Figure 5-6). The plate is incubated, and any well-isolated colony is then removed, resuspended in water, and again streaked on agar. If a suspension (and not just a bit of growth from a colony or slant) is streaked, this method is just as reliable as and much faster than the pour-plate method. In the spread plate technique, a small volume of dilute microbial suspension containing ca 30–300 cells is transferred to the center of an agar plate and spread evenly over the surface with a sterile bent-glass rod. The dispersed cells develop into isolated colonies. Because the number of colonies should equal the number of viable organisms in a sample, spread plates can be used to count the microbial population. B. Dilution A much less reliable method is that of extinction dilution. The suspension is serially diluted, and samples of each dilution are plated. If only a few samples of a particular dilution exhibit growth, it is presumed that some of the colonies started from single cells. This method is not used unless plating is for some reason impossible. An undesirable feature of this method is that it can only be used to isolate the predominant type of organism in a mixed population. CHAPTER 5 Cultivation of Microorganisms 75 1.0 ml Original sample 1.0 ml 9 ml H2O (10–1 dilution) 1.0 ml 9 ml H2O (10–2 dilution) Mix with warm agar and pour. 1.0 ml 9 ml H2O (10–3 dilution) 1.0 ml 9 ml H2O (10–4 dilution) 1.0 ml Figure 5-5 The pour-plate technique. The original sample is diluted several times to thin out the population sufficiently. The most diluted samples are then mixed with warm agar and poured into Petri dishes. Isolated cells grow into colonies and are used to establish pure cultures. The surface colonies are circular; subsurface colonies are lenticular (lens shaped). (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ: Prescott, Harley, & Klein’s Microbiology, 7th ed, McGraw-Hill, 2008. © The McGraw-Hill Companies, Inc.) Note: This method only works if the spreading tool (usually an inoculating loop) is resterilized after each of steps 1– 4. 1 2 3 4 A Steps in a Streak Plate 5 B Figure 5-6 Streak-plate technique. A: A typical streaking pattern. (Reproduced with permission from Willey JM, Sherwood CJ, Woolverton CJ: Prescott, Harley, & Klein’s Microbiology, 7th ed, McCgraw-Hill, 2008. © The McGraw-Hill Companies, Inc.) B: An example of a streak plate. (Reproduced with permission from Kathy Park Talaro.) CHAPTER SUMMARY • An organism requires all of the elements in its organic matter and the full complement of ions required for energetics in order to grow. Nutrients are classified according to the elements they provide, including carbon source, nitrogen source, sulfur source, phosphorous source, and mineral sources. • Growth factors are organic compounds that a cell must have to grow but that it is unable to synthesize. • There must be a source of energy present to establish a proton motive force and to allow macromolecular 76 SECTION I Fundamentals of Microbiology synthesis. The three major mechanisms for generating metabolic energy are fermentation, respiration, and photosynthesis. • Environmental factors such as pH, temperature, and aeration are important for growth. Most human pathogens are neutralophiles (grow best at pH of 6.0–8.0) and mesophilic (grow best at 30–37oC). • Organisms vary widely in their ability to use oxygen as a hydrogen acceptor and to their ability to inactivate toxic by-products of aerobic metabolism. They may be grouped as obligate aerobes, facultative anaerobes, obligate anaerobes, microaerophiles, and aerotolerant anaerobes. • Microbiologic media can be formulated to permit the growth of a particular type of microorganism present in low numbers (enrichment culture), identify specific types of microorganisms (differential medium), or isolate a specific organism from a mixed population (selective medium). REVIEW QUESTIONS 1. Most microorganisms pathogenic for humans grow best in the laboratory when cultures are incubated at (A) 15–20°C (B) 20–30°C (C) 30–37°C (D) 38–50°C (E) 50–55°C 2. The process by which microorganisms form ATP during the fermentation of glucose is characterized by (A) Coupling of ATP production with the transfer of electrons (B) Denitrification (C) The reduction of oxygen (D) Substrate phosphorylation (E) Anaerobic respiration 3. Which of the following culture techniques and media would enumerate the greatest number of microbial species in a soil sample? (A) Enrichment culture (B) A plate of selective medium (C) A plate of differential medium (D) A tube of nutrient broth (E) A number of different media and conditions of incubation 4. Polymerization of building blocks (eg, amino acids) into macromolecules (eg, proteins) is achieved largely by (A) Dehydration (B) Reduction (C) Oxidation (D) Assimilation (E) Hydrolysis 5. A strain of E coli undergoes a mutation such that it can no longer grow in a defined medium consisting of glucose, mineral salts, and ammonium chloride. However, it is capable of growth in this medium if methionine is added. The methionine is referred to as (A) An inorganic salt (B) A sulfur source (C) A growth factor (D) An energy source (E) A nitrogen source 6. Which of the following is NOT a mechanism for generating metabolic energy by microorganisms? (A) Fermentation (B) Protein synthesis (C) Respiration (D) Photosynthesis (E) C and D 7. Which of the following terms best describes a microorganisms that grows at 20 oC? (A) Neutralophile (B) Psychrotroph (C) Mesophile (D) Osmophile (E) Thermophile 8. The ability to assimilate N2 reductively via NH3 is called (A) Ammonification (B) Anammox (C) Assimilatory nitrate reduction (D) Deamination (E) Nitrogen fixation 9. Which of the following is NOT assimilated by eukaryotic cells? (A) Glucose (B) Lactate (C) Sulfate (SO42-) (D) Nitrogen (N2) (E) Phosphate (PO43-) 10. Bacteria that are obligate intracellular pathogens of humans (eg, Chlamydia trachomatis) are considered to be (A) Autotrophs (B) Photosynthetic (C) Chemolithotrophs (D) Hyperthermophiles (E) Heterotrophs Answers 1. C A B D C E E B D E REFERENCES Adams MW: Enzymes and proteins from organisms that grow near or above 100°C. Annu Rev Med 1993;47:627. Koch AL: Microbial physiology and ecology of slow growth. Microbiol Molec Biol Rev 1997;61:305. Maier RM, Pepper IL, Gerba CP: Environmental Microbiology. Academic Press, 1992. Marzlut GA: Regulation of sulfur and nitrogen metabolism in filamentary fungi. Annu Rev Microbiol 1993;42:89. Pelczar MJ Jr, Chan ECS, Krieg NR: Microbiology: Concepts and Applications. McGraw-Hill, 1993. Schloss PD, Handelsman J: Status of the microbial census. Microbiol Molec Biol Rev 2004;68:686. Wood JM: Bacterial osmoregulation: A paradigm for the study of cellular homeostasis. Annu Rev Microbiol 2011;65:215. C Microbial Metabolism ROLE OF METABOLISM IN BIOSYNTHESIS AND GROWTH Microbial growth requires the polymerization of biochemical building blocks into proteins, nucleic acids, polysaccharides, and lipids. The building blocks must come preformed in the growth medium or must be synthesized by the growing cells. Additional biosynthetic demands are placed by the requirement for coenzymes that participate in enzymatic catalysis. Biosynthetic polymerization reactions demand the transfer of anhydride bonds from adenosine triphosphate (ATP). Growth demands a source of metabolic energy for the synthesis of anhydride bonds and for the maintenance of transmembrane gradients of ions and metabolites. Metabolism has two components, catabolism and anabolism (Figure 6-1). Catabolism encompasses processes that harvest energy released from the breakdown of compounds (eg, glucose), and using that energy to synthesize ATP. In contrast, anabolism, or biosynthesis, includes processes that utilize the energy stored in ATP to synthesize and assemble the subunits, or building blocks, of macromolecules that make up the cell. The sequence of building blocks within a macromolecule is determined in one of two ways. In nucleic acids and proteins, it is template-directed: DNA serves as the template for its own synthesis and for the synthesis of the various types of RNA; messenger RNA serves as the template for the synthesis of proteins. In carbohydrates and lipids, on the other hand, the arrangement of building blocks is determined entirely by enzyme specificities. Once the macromolecules have been synthesized, they self-assemble to form the supramolecular structures of the cell, eg, ribosomes, membranes, cell wall, flagella, and pili. The rate of macromolecular synthesis and the activity of metabolic pathways must be regulated so that biosynthesis is balanced. All of the components required for macromolecular synthesis must be present for orderly growth, and control must be exerted so that the resources of the cell are not expended on products that do not contribute to growth or survival. This chapter contains a review of microbial metabolism and its regulation. Microorganisms represent extremes of evolutionary divergence, and a vast array of metabolic pathways 6 H A P T E R is found within the group. For example, any of more than half a dozen different metabolic pathways may be used for assimilation of a relatively simple compound, benzoate, and a single pathway for benzoate assimilation may be regulated by any of more than half a dozen control mechanisms. Our goal is to illustrate the principles that underlie metabolic pathways and their regulation. The primary principle that determines metabolic pathways is that they are achieved by organizing relatively few biochemical type reactions in a specific order. Many biosynthetic pathways can be deduced by examining the chemical structures of the starting material, the end product, and perhaps one or two metabolic intermediates. The primary principle underlying metabolic regulation is that enzymes tend to be called into play only when their catalytic activity is demanded. The activity of an enzyme may be changed by varying either the amount of enzyme or the amount of substrate. In some cases, the activity of enzymes may be altered by the binding of specific effectors, metabolites that modulate enzyme activity. FOCAL METABOLITES AND THEIR INTERCONVERSION glucose 6-Phosphate and Carbohydrate interconversions The biosynthetic origins of building blocks and coenzymes can be traced to relatively few precursors, called focal metabolites. Figures 6-2–6-5 illustrate how the respective focal metabolites glucose 6-phosphate (G6PD), phosphoenolpyruvate, oxaloacetate, and a-ketoglutarate give rise to most biosynthetic end products. Figure 6-2 illustrates how G6PD is converted to a range of biosynthetic end products via phosphate esters of carbohydrates with different chain lengths. Carbohydrates possess the empirical formula (CH2O)n, and the primary objective of carbohydrate metabolism is to change n, the length of the carbon chain. Mechanisms by which the chain lengths of carbohydrate phosphates are interconverted are summarized in Figure 6-6. In one case, oxidative reactions are used to remove a single carbon from G6PD, producing the pentose 77 78 SECTION I Fundamentals of Microbiology CATABOLISM ANABOLISM Energy source (glucose) Cell structures (cell wall, membrane, ribosomes, surface structures) Energy Macromolecules (proteins, nucleic acids) Energy Subunits (amino acids, nucleotides) Energy Precursors Waste products (acids, carbon dioxide) Nutrients (source of nitrogen, sulfur, etc.) Figure 6-1 The relationship between catabolism and anabolism. Catabolism encompasses processes that harvest energy released during disassembly of compounds, using it to synthesize adenosine triphosphate (ATP); it also provides precursor metabolites used in biosynthesis. Anabolism, or biosynthesis, includes processes that utilize ATP and precursor metabolites to synthesize and assemble subunits of macromolecules that make up the cell. (Reproduced with permission from Nester EW, Anderson DG, Roberts CE, Nester MT [editors]: Microbiology: A Human Perspective, 6th ed. McGraw-Hill, 2009, p. 127. © The McGraw-Hill Companies, Inc.) derivative ribulose 5-phosphate. Isomerase and epimerase reactions interconvert the most common biochemical forms of the pentoses: ribulose 5-phosphate, ribose 5-phosphate, and xylulose 5-phosphate. Transketolases transfer a twocarbon fragment from a donor to an acceptor molecule. These reactions allow pentoses to form or to be formed from carbohydrates of varying chain lengths. As shown in Figure 6-6, two pentose 5-phosphates (n = 5) are interconvertible with triose 3-phosphate (n = 3) and heptose 7-phosphate (n = 7); pentose 5-phosphate (n = 5) and tetrose 4-phosphate (n = 4) are interconvertible with triose 3-phosphate (n = 3) and hexose 6-phosphate (n = 6). The six-carbon hexose chain of fructose 6-phosphate can be converted to two three-carbon triose derivatives by the consecutive action of a kinase and an aldolase on fructose 6-phosphate. Alternatively, aldolases, acting in conjunction with phosphatases, can be used to lengthen carbohydrate molecules: Triose phosphates give rise to fructose 6-phosphate; a triose phosphate and tetrose 4-phosphate form heptose 7-phosphate. The final form of carbohydrate chain length interconversion is the transaldolase reaction, which interconverts heptose 7-phosphate and triose 3-phosphate with tetrose 4-phosphate and hexose 6-phosphate. The coordination of different carbohydrate rearrangement reactions to achieve an overall metabolic goal is illustrated by the hexose monophosphate shunt (Figure 6-7). This metabolic cycle is used by Cyanobacteria for the reduction of NAD+ (nicotinamide adenine dinucleotide) to NADH (reduced nicotinamide adenine dinucleotide), which serves as a reductant for respiration in the dark. Many organisms use the hexose monophosphate shunt to reduce NADP+ (nicotinamide adenine dinucleotide phosphate) to NADPH (reduced nicotinamide adenine dinucleotide phosphate), which is used for biosynthetic reduction reactions. The first steps in the hexose monophosphate shunt are the oxidative reactions that shorten six hexose 6-phosphates (abbreviated as six C6 in Figure 6-7) to six pentose 5-phosphates (abbreviated six C5). Carbohydrate rearrangement reactions convert the six C5 molecules to five C6 molecules so that the oxidative cycle may continue. Clearly, all reactions for interconversion of carbohydrate chain lengths are not called into play at the same time. Selection of specific sets of enzymes, essentially the determination of the metabolic pathway taken, is dictated by the source of carbon and the biosynthetic demands of the cell. For example, a cell given triose phosphate as a source of carbohydrate will use the aldolase-phosphatase combination to form fructose 6-phosphate; the kinase that acts on fructose 6-phosphate in its conversion to triose phosphate would not be expected to be active under these circumstances. If demands for pentose 5-phosphate are high, as in the case of photosynthetic carbon dioxide assimilation, transketolases that can give rise to pentose 5-phosphates are very active. In sum, G6PD can be regarded as a focal metabolite because it serves both as a direct precursor for metabolic building blocks and as a source of carbohydrates of varying length that are used for biosynthetic purposes. G6PD itself may be generated from other phosphorylated carbohydrates by selection of pathways from a set of reactions for chain length interconversion. The reactions chosen are determined by the genetic potential of the cell, the primary carbon source, and the biosynthetic demands of the organism. Metabolic regulation is required to ensure that reactions that meet the requirements of the organism are selected. Formation and Utilization of Phosphoenolpyruvate Triose phosphates, formed by the interconversion of carbohydrate phosphoesters, are converted to phosphoenolpyruvate by the series of reactions shown in Figure 6-8. Oxidation of glyceraldehyde 3-phosphate by NAD+ is accompanied by the formation of the acid anhydride bond on the one carbon of 1,3-diphosphoglycerate. This phosphate anhydride is transferred in a substrate phosphorylation to adenosine diphosphate (ADP), yielding an energy-rich bond in ATP. Another energy-rich phosphate bond is formed by dehydration of CHAPTER 6 Microbial Metabolism 79 Focal metabolite Intermediates End products Hexose phosphates Polysaccharides Nucleic acids Pentose phosphates Histidine Tryptophan Glucose 6-phosphate Tetrose phosphate Chorismate Phenylalanine Tyrosine Triose phosphates Lipids Glycine 3-Phosphoglycerate Serine Cysteine Tryptophan Figure 6-2 Biosynthetic end products formed from glucose 6-phosphate. Carbohydrate phosphate esters of varying chain length serve as intermediates in the biosynthetic pathways. Focal metabolite Intermediates End products Triose phosphates Glycine 3-Phosphoglycerate Serine Cysteine Tryptophan Chorismate Phenylalanine Tyrosine Phosphoenolpyruvate Polysaccharides Alanine Pyruvate Valine Isoleucine Acetyl-CoA Lipids Figure 6-3 Biosynthetic end products formed from phosphoenolpyruvate. 2-phosphoglycerate to phosphoenolpyruvate; via another substrate phosphorylation, phosphoenolpyruvate can donate the energy-rich bond to ADP, yielding ATP and pyruvate. Thus, two energy-rich bonds in ATP can be obtained by the metabolic conversion of triose phosphate to pyruvate. This is an oxidative process, and in the absence of an exogenous electron acceptor, the NADH generated by oxidation of glyceraldehyde 3-phosphate must be oxidized to NAD+ by pyruvate or by metabolites derived from pyruvate. The products formed as a result of this process vary and, as described later 80 SECTION I Fundamentals of Microbiology Focal metabolite End products Asparagine Oxaloacetate Aspartate Threonine Isoleucine Methionine Coenzymes Pyrimidines Nucleic acids Figure 6-4 Biosynthetic end products formed from oxaloacetate. The end products aspartate, threonine, and pyrimidines serve as intermediates in the synthesis of additional compounds. Focal metabolite Intermediates End products Lysine α-Ketoglutarate Glutamate Glutamic semialdehyde Glutamine Arginine Proline Figure 6-5 Biosynthetic end products formed from a-ketoglutarate. in this chapter, can be used in the identification of clinically significant bacteria. Formation of phosphoenolpyruvate from pyruvate requires a substantial amount of metabolic energy, and two anhydride ATP bonds invariably are invested in the process. Some organisms—Escherichia coli, for example—directly phosphorylate pyruvate with ATP, yielding adenosine monophosphate (AMP) and inorganic phosphate (Pi). Other organisms use two metabolic steps: One ATP pyrophosphate bond is invested in the carboxylation of pyruvate to oxaloacetate, and a second pyrophosphate bond (often carried by guanosine triphosphate [GTP] rather than ATP) is used to generate phosphoenolpyruvate from oxaloacetate. Formation and Utilization of Oxaloacetate As already described, many organisms form oxaloacetate by the ATP-dependent carboxylation of pyruvate. Other organisms, such as E coli, which form phosphoenolpyruvate directly from pyruvate, synthesize oxaloacetate by carboxylation of phosphoenolpyruvate. Succinyl-CoA is a required biosynthetic precursor for the synthesis of porphyrins and other essential compounds. Some organisms form succinyl-CoA by reduction of oxaloacetate via malate and fumarate. These reactions represent a reversal of the metabolic flow observed in the conventional tricarboxylic acid cycle (see Figure 6-11). Formation of `-Ketoglutarate From Pyruvate Conversion of pyruvate to a-ketoglutarate requires a metabolic pathway that diverges and then converges (Figure 6-9). In one branch, oxaloacetate is formed by carboxylation of pyruvate or phosphoenolpyruvate. In the other branch, pyruvate is oxidized to acetyl-CoA. It is noteworthy that, regardless of the enzymatic mechanism used for the formation of oxaloacetate, acetyl-CoA is required as a positive metabolic effector for this process. Thus, the synthesis of oxaloacetate is balanced with the production of acetyl-CoA. Condensation of oxaloacetate with acetyl-CoA yields citrate. Isomerization of the citrate molecule produces isocitrate, which is oxidatively decarboxylated to a-ketoglutarate. ASSIMILATORY PATHWAYS Growth With Acetate Acetate is metabolized via acetyl-CoA, and many organisms possess the ability to form acetyl-CoA (Figure 6-10). AcetylCoA is used in the biosynthesis of a-ketoglutarate, and in most respiratory organisms, the acetyl fragment in acetyl-CoA is oxidized completely to carbon dioxide via the tricarboxylic acid cycle (Figure 6-11). The ability to use acetate as a net source of carbon, however, is limited to relatively few microorganisms CHAPTER 6 Microbial Metabolism 81 Dehydrogenases + NAD NADH+H+ NAD+ NADH+H+ Glucose 6-phosphate (C6) CO2 Ribulose 5-phosphate (C5) Transketolases Glyceraldehyde 3-phosphate (C3) Xylulose 5-phosphate (C5) Sedoheptulose 7-phosphate (C7) Ribose 5-phosphate (C5) Glyceraldehyde 3-phosphate (C3) Xylulose 5-phosphate (C5) Fructose 6-phosphate (C6) Erythrose 4-phosphate (C4) Kinase, Aldolase ADP Fructose 6-phosphate (C6) Dihydroxyacetone phosphate (C3) ATP Fructose 1,6-diphosphate Glyceraldehyde 3-phosphate (C3) Aldolase, Phosphatase Dihydroxyacetone phosphate (C3) H2O Phosphate Fructose 1,6-diphosphate Fructose 6-phosphate (C6) Glyceraldehyde 3-phosphate (C3) Dihydroxyacetone phosphate (C3) H2O Phosphate Sedoheptulose 1,7-diphosphate Sedoheptulose 7-phosphate (C7) Erythrose 4-phosphate (C4) Transaldolase Sedoheptulose 7-phosphate (C7) Erythrose 4-phosphate (C4) Glyceraldehyde 3-phosphate (C3) Fructose 6-phosphate (C6) Figure 6-6 Biochemical mechanisms for changing the length of carbohydrate molecules. The general empirical formula for carbohydrate phosphate esters, (CnH2nOn)-N-phosphate, is abbreviated (Cn) to emphasize changes in chain length. and plants. Net synthesis of biosynthetic precursors from acetate is achieved by coupling reactions of the tricarboxylic acid cycle with two additional reactions catalyzed by isocitrate lyase and malate synthase. As shown in Figure 6-12, these reactions allow the net oxidative conversion of two acetyl moieties from acetyl-CoA to one molecule of succinate. Succinate may be used for biosynthetic purposes after its conversion to oxaloacetate, a-ketoglutarate, phosphoenolpyruvate, or G6PD. 82 SECTION I Fundamentals of Microbiology Net reaction + H2O Glucose 6-phosphate + 12NAD+ 2C5 6NAD+ 6NAD+ 6NADH Transketolase 6CO2 + 12NADH + Phosphate 2C3 Transaldolase 2C6 6NADH 6C6 6C5 2C5 2C4 2C7 6CO2 Transketolase 2C6 H2O Phosphate 2C5 2C3 Aldolase, phosphatase C6 Figure 6-7 The hexose monophosphate shunt. Oxidative reactions (see Figure 6-6) reduce NAD+ (nicotinamide adenine dinucleotide phosphate) and produce CO2, resulting in the shortening of the six hexose phosphates (abbreviated C6) to six pentose phosphates (abbreviated C5). Carbohydrate rearrangements (see Figure 6-6) convert the pentose phosphates to hexose phosphates so that the oxidative cycle may continue. OXIDATION CH2OH C CHO O O NAD+ NADH+H+ 2– CH2OPO3 ADP COPO32– HCOH HCOH 2– SUBSTRATE PHOSPHORYLATION CH2OPO3 Triose phosphates CO2– HCOH 2– CH2OPO32– CH2OPO3 Pi ATP 1,3-Diphosphoglycerate 3-Phosphoglycerate SUBSTRATE PHOSPHORYLATION CO2– C ATP ADP COPO32– O CH3 Pyruvate CO2– CH2 Phosphoenolpyruvate H2O CO2– HCOPO32– CH2OH 2-Phosphoglycerate Figure 6-8 Formation of phosphoenolpyruvate and pyruvate from triose phosphate. The figure draws attention to two sites of substrate phosphorylation and to the oxidative step that results in the reduction of NAD+ (nicotinamide adenine dinucleotide phosphate) to NADH (nicotinamide adenine dinucleotide hydride). Repetition of this energy-yielding pathway demands a mechanism for oxidizing NADH to NAD+. Fermentative organisms achieve this goal by using pyruvate or metabolites derived from pyruvate as oxidants. Growth With Carbon Dioxide: The Calvin Cycle Similar to plants and algae, a number of microbial species can use carbon dioxide as a sole source of carbon. In almost all of these organisms, the primary route of carbon assimilation is via the Calvin cycle, in which carbon dioxide and ribulose diphosphate combine to form two molecules of 3-phosphoglycerate (Figure 6-13A). 3-Phosphoglycerate is phosphorylated to 1,3-diphosphoglycerate, and this compound is reduced to the triose derivative, glyceraldehyde 3-phosphate. Carbohydrate rearrangement reactions (see Figure 6-6) allow triose phosphate to be converted to the pentose derivative ribulose 5-phosphate, which is phosphorylated to regenerate the acceptor molecule, ribulose 1,5-diphosphate (Figure 6-13B). Additional reduced carbon, formed by the reductive assimilation of carbon dioxide, is converted to focal metabolites for biosynthetic pathways. CHAPTER 6 Microbial Metabolism 83 CO2 NAD+ HSCoA NADH+H+ CO2– O C O CH3 CH3CSCoA Pyruvate Acetyl-CoA CO2 H2O ATP HSCoA ADP HOCCO2– CO2– C O O CH3 Pyruvate H2O CH2CO2– Acetyl-CoA required for activity CCO2– CH2CO2– CCO2– CH2CO2– Aconitate Citrate CH2CO2– CHCO2– Oxaloacetate H2O Pi ATP H2O CO2– COPO32– AMP CO2 CH2 Phosphoenolpyruvate O CCO2– CO2 NADH+H+ CCO2– O NAD+ HOCHCO – 2 CH2 CHCO2– CHCO2– CH2CO2– CH2CO2– CHCO2– α-Ketoglutarate Isocitrate Oxalosuccinate Figure 6-9 Conversion of pyruvate to a-ketoglutarate. Pyruvate is converted to a-ketoglutarate by a branched biosynthetic pathway. In one branch, pyruvate is oxidized to acetyl-CoA; in the other, pyruvate is carboxylated to oxaloacetate. CO2– C + NAD O NADH+H+ CH3 Pyruvate HSCoA CO2 HSCoA PPi O CH3CO2– CH3CSCoA Acetate ATP Acetyl-CoA AMP β-OXIDATION HSCoA PPi H3(CH2CH2)nCSCoA Fatty acyl-CoA CH3(CH2CH2)nCO2– Fatty acids O ATP AMP Figure 6-10 Biochemical sources of acetyl-CoA. AMP, adenosine monophosphate; ATP, adenosine triphosphate. 84 SECTION I Fundamentals of Microbiology O CH3CSCoA CH2CO2– Acetyl-CoA H2O – HOCCO2 O CCO2– HSCoA CH2CO2– H2O CHCO2– CH2CO2– CCO2– Citrate CH2CO2– Aconitate Oxaloacetate NADH+H+ H2O HOCHCO2– NAD+ CHCO2– – HOCHCO2 CH2CO2– – CH2CO2 Isocitrate L-Malate NAD+ Net reaction Acetyl-CoA + 3NAD+ + Enz(FAD) + GDP + Pi + 2H2O → HSCoA + 2CO2 + 3NADH + 3H+ + Enz(FADH2) + GTP H2O NADH+H+ O CHCO2– CHCO2– CHCO2– CH2CO2– Fumarate Oxalosuccinate O Enz(FADH2) Enz(FAD) CCO2– CH2 CH2CO2– CH2CO2– CCO2– CO2 O Succinate CH2CSCoA GTP GDP CH2CO2– HSCoA HSCoA CH2CO2– CO2 α-Ketoglutarate + NAD NADH ++ Succinyl-CoA H Figure 6-11 The tricarboxylic acid cycle. There are four oxidative steps, three giving rise to NADH (nicotinamide adenine dinucleotide hydride) and one giving rise to a reduced flavoprotein, Enz(FADH2). The cycle can continue only if electron acceptors are available to oxidize the NADH and reduced flavoprotein. GDP, guanosine diphosphate; GTP, guanosine triphosphate. Cells that can use carbon dioxide as a sole source of carbon are termed autotrophic, and the demands for this pattern of carbon assimilation can be summarized briefly as follows: In addition to the primary assimilatory reaction giving rise to 3-phosphoglycerate, there must be a mechanism for regenerating the acceptor molecule, ribulose 1,5-diphosphate. This process demands the energy-dependent reduction of 3phosphoglycerate to the level of carbohydrate. Thus, autotrophy requires carbon dioxide, ATP, NADPH, and a specific set of enzymes. Depolymerases Many potential growth substrates occur as building blocks within the structure of biologic polymers. These large molecules are not readily transported across the cell membrane and often are affixed to even larger cellular structures. Many microorganisms elaborate extracellular depolymerases that hydrolyze proteins, nucleic acids, polysaccharides, and lipids. The pattern of depolymerase activities can be useful in the identification of microorganisms. Oxygenases Many compounds in the environment are relatively resistant to enzymatic modification, and utilization of these compounds as growth substrates demands a special class of enzymes, oxygenases. These enzymes directly use the potent oxidant molecular oxygen as a substrate in reactions that convert a relatively intractable compound to a form in which it CHAPTER 6 Microbial Metabolism 85 O CH2CO2– CH3CSCoA HOCCO2– Acetyl-CoA H2O CH2CO2– Citrate HSCoA CHCO2– H2O CCO2– CCO2– CH2CO2– CH2CO2– Oxaloacetate Aconitate O NADH+H+ H2O NAD+ O HOCHCO2– CHCO2– CH3CSCoA CH2CO2– L-Malate HOCHCO2– HSCoA CH2CO2– Acetyl-CoA Isocitrate H2O MALATE SYNTHASE ISOCITRATE LYASE O CHCO2– Glyoxylate CH2CO2– CH2CO2– Succinate Net reaction 2Acetyl-CoA + NAD+ + 2H2O → Succinate + 2HSCoA + NADH + H+ Figure 6-12 The glyoxylate cycle. Note that the reactions that convert malate to isocitrate are shared with the tricarboxylic acid cycle (see Figure 6-11). Metabolic divergence at the level of isocitrate and the action of two enzymes, isocitrate lyase and malate synthase, modify the tricarboxylic acid cycle so that it reductively converts two molecules of acetyl-CoA to succinate. can be assimilated by thermodynamically favored reactions. The action of oxygenases is illustrated in Figure 6-14, which shows the role of two different oxygenases in the utilization of benzoate. Reductive Pathways Some microorganisms live in extremely reducing environments that favor chemical reactions that would not occur in organisms using oxygen as an electron acceptor. In these organisms, powerful reductants can be used to drive reactions that allow the assimilation of relatively intractable compounds. An example is the reductive assimilation of benzoate, a process in which the aromatic ring is reduced and opened to form the dicarboxylic acid pimelate. Further metabolic reactions convert pimelate to focal metabolites. Nitrogen Assimilation The reductive assimilation of molecular nitrogen, also referred to as nitrogen fixation, is required for continuation of life on our planet. Nitrogen fixation is accomplished by a variety of bacteria and Cyanobacteria using a multicomponent nitrogenase enzyme complex. Despite the variety of organisms capable of fixing nitrogen, the nitrogenase complex is similar in most of them (Figure 6-15). Nitrogenase is a complex of two enzymes—one enzyme (dinitrogenase reductase) contains iron and the other (dinitrogenase) contains iron and molybdenum. Together, these enzymes catalyze the following reaction: N2 + 6H+ + 6e - + 12ATP −−−→ 2NH3 + 12ADP + 12Pi Because of the high activation energy of breaking the very strong triple bond that joins two nitrogen atoms, this 86 SECTION I Fundamentals of Microbiology A CH2OH C O ATP CH2OPO32– CO2 C O ADP HCOH HCOH HCOH O 2ADP CO2– 2NADPH 2NADP+ COPO32– 2HCOH CHO 2HCOH 2HCOH 2– HCOH CH2OPO32– 2ATP CH2OPO32– 2– CH2OPO3 CH2OPO3 CH2OPO32– Ribulose 5-phosphate (C5) Ribose 1,5-diphosphate 2 3-Phosphoglycerate 2 Glyceraldehyde 3-phosphate (2 C3) 2 1,3-Diphosphoglycerate B Focal metabolites and biosynthesis 2C3 Aldolase, phosphatase 4C3 12C3 2C6 Transketolase 2C3 2C5 2C4 Aldolase, phosphatase 2C7 Transketolase 2C5 2C3 2C3 2C5 Reductive assimilation of CO2 6C5 + 12NADP 12NADPH 12ADP 12ATP 6CO2 6ADP 6ATP Net reaction + 6CO2 + 12NADPH + 18ATP → 2 Triose phosphate (C3) + 12NADP + 18ADP + 18Pi Figure 6-13 The Calvin cycle. A: Reductive assimilation of CO2. Adenosine triphosphate (ATP) and NADPH (nicotinamide adenine dinucleotide phosphate) are used to reductively convert pentose 5-phosphate (C5) to two molecules of triose phosphate (C3). B: The Calvin cycle is completed by carbohydrate rearrangement reactions (Figure 6-6) that allow the net synthesis of carbohydrate and the regeneration of pentose phosphate so that the cycle may continue. ADP, adenosine diphosphate. reductive assimilation of nitrogen demands a substantial amount of metabolic energy. Somewhere between 20 and 24 molecules of ATP are hydrolyzed as a single N2 molecule is reduced to two molecules of NH3. Additional physiologic demands are placed by the fact that nitrogenase is readily inactivated by oxygen. Aerobic organisms that use nitrogenase have developed elaborate mechanisms to protect the enzyme against inactivation. Some form specialized cells in which nitrogen fixation takes place, and others have developed elaborate electron transport chains to protect nitrogenase against inactivation by oxygen. The most significant of these bacteria in agriculture are the CHAPTER 6 Microbial Metabolism 87 1 CO2– 2 O2 CO2 CO2– OH O2 CO2– OH Benzoate NADH ++ H NAD OH + NAD + NADH ++ H CO2– OH Catechol Succinyl-CoA + Acetyl-CoA 5 steps Figure 6-14 The role of oxygenases in aerobic utilization of benzoate as a carbon source. Molecular oxygen participates directly in the reactions that disrupt the aromaticity of benzoate and catechol. O2 Leghemoglobin Terminal oxidase system Carbohydrate (from glycolysis or photosynthesis) 16MgATP 16MgADP + Pi 8NAD+ 8NADH + H+ Fd-8e– 8Fd Fe protein 2H+ + 2e– Fe – Mo + – Pro tein 2H + 2e H2 Uptake hydrogenase + 6H + 6e– N2 2NH3 Figure 6-15 Reduction of N2 to two molecules of NH3. In addition to reductant, the nitrogenase reaction requires a substantial amount of metabolic energy. The number of adenosine triphosphate (ATP) molecules required for reduction of a single nitrogen molecule to ammonia is uncertain; the value appears to lie between 12 and 16. The overall reaction requires 8NADH+ (nicotinamide adenine dinucleotide phosphate) H+. Six of these are used to reduce N2 to 2NH3, and two are used to form H2. The uptake hydrogenase returns H2 to the system, thus conserving energy. (Redrawn and reproduced, with permission, from Moat AG, Foster JW: Microbial Physiology, 4th ed. Wiley-Liss, 2002. Reprinted by permission of John Wiley & Sons, Inc.) Rhizobiaceae, organisms that fix nitrogen symbiotically in the root nodules of leguminous plants. The capacity to use ammonia as a nitrogen source is widely distributed among organisms. The primary portal of entry of nitrogen into carbon metabolism is glutamate, which is formed by reductive amination of a-ketoglutarate. As shown in Figure 6-16, there are two biochemical mechanisms by which this can be achieved. One, the single-step reduction catalyzed by glutamate dehydrogenase (Figure 6-16A) is effective in environments in which there is an ample supply of ammonia. The other, a two-step process in which glutamine is an intermediate (Figure 6-16B), is used in environments in which ammonia is in short supply. The latter mechanism allows cells to invest the free energy formed by hydrolysis of a pyrophosphate bond in ATP into the assimilation of ammonia from the environment. The amide nitrogen of glutamine, an intermediate in the two-step assimilation of ammonia into glutamate (see Figure 6-16B), is also transferred directly into organic nitrogen appearing in the structures of purines, pyrimidines, arginine, tryptophan, and glucosamine. The activity and synthesis of glutamine synthase are regulated by the ammonia supply and by the availability of metabolites containing nitrogen derived directly from the amide nitrogen of glutamine. 88 SECTION I Fundamentals of Microbiology A. High concentrations of ammonia. CO2– C O NH3 + CH2 + CO2– + H3NCH CH2 NADPH + NADP+ + ADP CH2 CH2 CO2– – CO2 Glutamate α-Ketoglutarate B. Low concentrations of ammonia. CO2– + H3NCH CO2– + H3NCH ATP + CH2 + NH3 CH2 CH2 + Pi CH2 – C CO2 O NH2 Glutamate Glutamine CO2– H3NCH C + CH2 CH2 C CO2– CO2– + NADPH + CH2 CH2 CH2 CH2 O H3NCH H3NCH O CH2 CO2– – NADP+ CH2 – CO2 + CO2– CO2 NH2 Glutamine α-Ketoglutarate 2 Glutamates Figure 6-16 Mechanisms for the assimilation of NH3. A: When the NH3 concentration is high, cells are able to assimilate the compound via the glutamate dehydrogenase reaction. B: When, as most often is the case, the NH3 concentration is low, cells couple the glutamine synthase and glutamate synthase reactions to invest the energy produced by hydrolysis of a pyrophosphate bond into ammonia assimilation. Most of the organic nitrogen in cells is derived from the a-amino group of glutamate, and the primary mechanism by which the nitrogen is transferred is transamination. The usual acceptor in these reactions is an a-keto acid, which is transformed to the corresponding a-amino acid. aKetoglutarate, the other product of the transamination reaction, may be converted to glutamate by reductive amination (see Figure 6-16). BIOSYNTHETIC PATHWAYS Tracing the Structures of Biosynthetic Precursors: Glutamate and Aspartate In many cases, the carbon skeleton of a metabolic end product may be traced to its biosynthetic origins. Glutamine, an obvious example, clearly is derived from glutamate (Figure 6-17). The glutamate skeleton in the structures of arginine and proline (see Figure 6-17) is less obvious but readily discernible. Similarly, the carbon skeleton of aspartate, directly derived from the focal metabolite oxaloacetate, is evident in the structures of asparagine, threonine, methionine, and pyrimidines (Figure 6-18). CO2– + H3NCH CO2– + H3NCH CO2– CH2 CH2 CH2 HN CH2 CH2 H2C C O NH2 NH C CH C H2 CH2 NH NH2 Glutamine Arginine Proline Figure 6-17 Amino acids formed from glutamate. CHAPTER 6 Microbial Metabolism 89 CO2– + H3NCH CO2– + H3NCH CH2 C CO2– + H3NCH O HN CH2 CHOH CH2 CH3 S NH2 form in Figure 6-20A. The synthesis of peptidoglycan begins with the stepwise synthesis in the cytoplasm of UDPN-acetylmuramic acid-pentapeptide. N-Acetylglucosamine is first attached to uridine diphosphate (UDP) and then converted to UDP-N-acetylmuramic acid by condensation with phosphoenolpyruvate and reduction. The amino acids of the pentapeptide are sequentially added, each addition catalyzed by a different enzyme and each involving the split of ATP to ADP + Pi. The UDP–N-acetylmuramic acid–pentapeptide is attached to bactoprenol (a lipid of the cell membrane) and receives a molecule of N-acetylglucosamine from UDP. Some bacteria (eg, Staphylococcus aureus) form a pentaglycine derivative in a series of reactions using glycyl-tRNA as the donor; the completed disaccharide is polymerized to an oligomeric intermediate before being transferred to the growing end of a glycopeptide polymer in the cell wall. Final cross-linking (Figure 6-20B) is accomplished by a transpeptidation reaction in which the free amino group of a pentaglycine residue displaces the terminal D-alanine residue of a neighboring pentapeptide. Transpeptidation is catalyzed by one of a set of enzymes called penicillin-binding proteins (PBPs). PBPs bind penicillin and other β-lactam antibiotics covalently, partly because of a structural similarity between these antibiotics and the pentapeptide precursor. Some PBPs O O C C CH2 CH2 N H CH3 Asparagine Threonine Uracil Methionine Figure 6-18 Biosynthetic end products formed from aspartate. In some cases, different carbon skeletons combine in a biosynthetic pathway. For example, aspartate semialdehyde and pyruvate combine to form the metabolic precursors of lysine, diaminopimelic acid, and dipicolinic acid (Figure 6-19). The latter two compounds are found only in prokaryotes. Diaminopimelic acid is a component of peptidoglycan in the cell wall, and dipicolinic acid represents a major component of endospores. Synthesis of Cell Wall Peptidoglycan The structure of peptidoglycan is shown in Figure 2-16; the pathway by which it is synthesized is shown in simplified H2C HOOC HC H C O H3C + NH2 C O Aspartate semialdehyde –2H2O COOH Pyruvate H2C HOOC HC H C N CH C –2H HOOC COOH Dihydropicolinic acid N Dipicolinic acid (spores) +2H H2 C H2C HOOC HC CoA C N H2 C Succinyl-CoA H2C CH2 COOH +H2O HOOC C O CH2 HC COOH NH Tetrahydropicolinic acid (Succ) COOH HC HC COOH HC NH2 (CH2)3 NH2 –CO2 COOH NH2 (CH2)3 H2C COOH Diaminopimelic acid (cell walls) Figure 6-19 Biosynthetic end products formed from aspartate semialdehyde and pyruvate. NH2 Lysine (proteins) 1 UDP derivatives of NAM and 3 NAM-pentapeptide is transferred NAG are synthesized (not shown). UDP to bactoprenol phosphate. They are joined by a pyrophosphate bond. NAM L-Ala L-Ala 2 Sequential addition of amino D-Glu L-Lys (DAP) 2 UDP D-Ala D-Ala NAM pentapeptide P A 3 Pi P P P P NAM 4 P P Peptidoglycan NAM NAM NAG Bactoprenol 5 Bactoprenol – 8 Peptide cross-links between peptidoglycan B Pentapeptide UDP Membrane Membrane NAG Pentapeptide chains are formed by transpeptidation (not shown). Lipid II NAG Bactoprenol Bactoprenol Periplasm UDP Pentapeptide UMP 7 – 5 The bactoprenol carrier transports the completed NAG-NAM-pentapeptide repeat unit across the membrane. Lipid I Bactoprenol Bacitracin Cycloserine D-Ala – Cytoplasm pentapeptide. If a pentaglycine interbridge is required, it is created using special glycyl-tRNA molecules, but not ribosomes. Interbridge formation occurs in the membrane. – acids to UDP-NAM to form the NAM-pentapeptide. ATP is used to fuel this, but tRNA and ribosomes are not involved in forming the peptide bonds that link the amino acids together. 4 UDP transfers NAG to the bactoprenol-NAM- 6 7 The bactoprenol carrier moves back across the membrane. As it does, it loses one phosphate, becoming bactoprenol phosphate. It is now ready to begin a new cycle. NAM P P Peptidoglycan Vancomycin NAG Pentapeptide 6 The NAG-NAM-pentapeptide is attached to the growing end of a peptidoglycan chain, increasing the chain's length by one repeat unit. Escherichia coli transpeptidation ••• NAG NAM ••• D Ala NAM ••• D Ala D Ala L Ala DAP D Glu DAP DAP D Glu DAP D Glu D Ala L Ala D Ala L Ala H 2N D Glu D Ala ••• NAM ••• NAG Staphylococcus aureus transpeptidation NAG NAG D Ala L Ala ••• ••• NAM ••• D Ala D Ala L Ala L D-GluNH2 D GluNH2 L Lys D Ala D Ala Lys H2N (Gly)5 L Ala ••• NAG NAM ••• ••• NAG NAM ••• Penicillins ••• NAG NAM ••• D Ala L Ala D Ala D GluNH2 L Lys L Lys (Gly)5 D GluNH2 L Ala D Ala ••• NAG NAM ••• Figure 6-20 A: Peptidoglycan synthesis. The pentapeptide contains l-lysine in Staphylococcus aureus peptidoglycan and diaminopimelic acid (DAP) in Escherichia coli. Inhibition by bacitracin, cycloserine, and vancomycin is also shown. The numbers correspond to six of the eight stages discussed in the text. Stage eight is depicted in B. NAM, N-acetylmuramic acid; NAG, N-acetylglucosamine; UDP, uridine diphosphate. B: Transpeptidation. The transpeptidation reactions in the formation of the peptidoglycans of Escherichia coli and Staphylococcus aureus. (Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ: Prescott, Harley, & Klein’s Microbiology, 7th ed, McGraw-Hill, 2008. © The McGraw-Hill Companies, Inc.) CHAPTER 6 Microbial Metabolism 91 have transpeptidase or carboxypeptidase activities, their relative rates perhaps controlling the degree of cross-linking in peptidoglycan (a factor important in cell septation). The biosynthetic pathway is of particular importance in medicine because it provides a basis for the selective antibacterial action of several chemotherapeutic agents. Unlike their host cells, bacteria are not isotonic with the body fluids. Their contents are under high osmotic pressure, and their viability depends on the integrity of the peptidoglycan lattice in the cell wall being maintained throughout the growth cycle. Any compound that inhibits any step in the biosynthesis of peptidoglycan causes the wall of the growing bacterial cell to be weakened and the cell to lyse. The sites of action of several antibiotics are shown in Figure 6-20A and B. Synthesis of Cell Envelope Lipopolysaccharide The general structure of the antigenic lipopolysaccharide of gram-negative cell envelopes is shown in Figure 2-20. The biosynthesis of the repeating end-group, which gives the cell envelope its antigenic specificity, is shown in Figure 6-21. Note the resemblance to peptidoglycan synthesis: In both cases, a series of subunits is assembled on a lipid carrier in the membrane and then transferred to open ends of the growing polymer. Synthesis of Extracellular Capsular Polymers The capsular polymers, a few examples of which are listed in Table 2-1, are enzymatically synthesized from activated subunits. No membrane-bound lipid carriers have been implicated in this process. The presence of a capsule is often environmentally determined: Dextrans and levans, for example, can only be synthesized using the disaccharide sucrose (fructose–glucose) as the source of the appropriate subunit, and their synthesis thus depends on the presence of sucrose in the medium. Synthesis of Reserve Food Granules When nutrients are present in excess of the requirements for growth, bacteria convert certain of them to intracellular reserve food granules. The principal ones are starch, glycogen, poly-β-hydroxybutyrate, and volutin, which consists mainly of inorganic polyphosphate (see Chapter 2). The type of granule formed is species specific. The granules are degraded when exogenous nutrients are depleted. PATTERNS OF MICROBIAL ENERGY-YIELDING METABOLISM As described in Chapter 5, there are two major metabolic mechanisms for generating the energy-rich acid pyrophosphate bonds in ATP: substrate phosphorylation (the direct transfer of a phosphate anhydride bond from an organic donor to ADP) and phosphorylation of ADP by inorganic phosphate. The latter reaction is energetically unfavorable and must be driven by a transmembrane electrochemical gradient, the proton motive force. In respiration, the electrochemical BP- P - P -(gal-rha-man)n–1 GDP BP- P - P -gal-rha-man BP- P - P GDP-man BP- P - P -gal-rha BP- P - P -(gal-rha-man)n Core polysaccharide TDP P TDP-rha BP- P - P -gal BP- P - P Core polysaccharide(gal-rha-man)n BP- P UMP UDP-gal Pi Figure 6-21 Synthesis of the repeating unit of the polysaccharide side chain of Salmonella newington and its transfer to the lipopolysaccharide core. BP, bactroprenol; GDP, guanosine diphosphate; TDP, thymidine diphosphate; UDP, uridine diphosphate; UMP, uridine monophosphate. 92 SECTION I Fundamentals of Microbiology gradient is created from externally supplied reductant and oxidant. Energy released by transfer of electrons from the reductant to the oxidant through membrane-bound carriers is coupled to the formation of the transmembrane electrochemical gradient. In photosynthesis, light energy generates membrane-associated reductants and oxidants; the proton motive force is generated as these electron carriers returns to the ground state. These processes are discussed below. Pathways of Fermentation membrane by an enzyme system in the cytoplasmic membrane that phosphorylates extracellular glucose at the expense of phosphoenolpyruvate, producing intracellular G6PD and pyruvate. The latter process is an example of vectorial metabolism, a set of biochemical reactions in which both the structure and the location of a substrate are altered. It should be noted that the choice of ATP or phosphoenolpyruvate as a phosphorylating agent does not alter the ATP yield of fermentation because phosphoenolpyruvate is used as a source of ATP in the later stages of fermentation (see Figure 6-8). A. Strategies for Substrate Phosphorylation C. The Embden-Meyerhof Pathway In the absence of respiration or photosynthesis, cells are entirely dependent on substrate phosphorylation for their energy: Generation of ATP must be coupled to chemical rearrangement of organic compounds. Many compounds can serve as fermentable growth substrates, and many pathways for their fermentation have evolved. These pathways have the following three general stages: (1) Conversion of the fermentable compound to the phosphate donor for substrate phosphorylation. This stage often contains metabolic reactions in which NAD+ is reduced to NADH. (2) Phosphorylation of ADP by the energy-rich phosphate donor. (3) Metabolic steps that bring the products of the fermentation into chemical balance with the starting materials. The most frequent requirement in the last stage is a mechanism for oxidation of NADH, generated in the first stage of fermentation, to NAD+ so that the fermentation may proceed. In the following sections, examples of each of the three stages of fermentation are considered. This pathway (Figure 6-22), a commonly encountered mechanism for the fermentation of glucose, uses a kinase and an aldolase (see Figure 6-6) to transform the hexose (C6) phosphate to two molecules of triose (C3) phosphate. Four substrate phosphorylation reactions accompany the conversion of the triose phosphate to two molecules of pyruvate. Thus, taking into account the two ATP pyrophosphate bonds required to form triose phosphate from glucose, the EmbdenMeyerhof pathway produces a net yield of two ATP pyrophosphate bonds. Formation of pyruvate from triose phosphate is an oxidative process, and the NADH formed in the first metabolic step (Figure 6-22) must be converted to NAD+ for the fermentation to proceed; two of the simpler mechanisms for achieving this goal are illustrated in Figure 6-23. Direct reduction of pyruvate by NADH produces lactate as the end product of fermentation and thus results in acidification of the medium. Alternatively, pyruvate may be decarboxylated to acetaldehyde, which is then used to oxidize NADH, resulting in production of the neutral product ethanol. The pathway taken is determined by the evolutionary history of the organism and, in some microorganisms, by the growth conditions. B. Fermentation of Glucose The diversity of fermentative pathways is illustrated by consideration of some of the mechanisms used by microorganisms to achieve substrate phosphorylation at the expense of glucose. In principle, the phosphorylation of ADP to ATP can be coupled to either of two chemically balanced transformations: Glucose −−−→ 2 Lactic acid (C6H12O6) (C3H6O3) or Glucose −−−→ 2 Ethanol + 2 Carbon dioxide (C6H12O6) (C2H6O) (CO2) The biochemical mechanisms by which these transformations are achieved vary considerably. In general, the fermentation of glucose is initiated by its phosphorylation to G6PD. There are two mechanisms by which this can be achieved: (1) Extracellular glucose may be transported across the cytoplasmic membrane into the cell and then phosphorylated by ATP to yield G6PD and ADP. (2) In many microorganisms, extracellular glucose is phosphorylated as it is being transported across the cytoplasmic D. The Entner-Doudoroff and Heterolactate Fermentations Alternative pathways for glucose fermentation include some specialized enzyme reactions, and these are shown in Figure 6-24. The Entner-Doudoroff pathway diverges from other pathways of carbohydrate metabolism by a dehydration of 6-phosphogluconate followed by an aldolase reaction that produces pyruvate and triose phosphate (Figure 6-24A). The heterolactate fermentation and some other fermentative pathways depend upon a phosphoketolase reaction (Figure 6-24B) that phosphorolytically cleaves a ketosephosphate to produce acetyl phosphate and triose phosphate. The acid anhydride acetyl phosphate may be used to synthesize ATP or may allow the oxidation of two NADH molecules to NAD+ as it is reduced to ethanol. The overall outlines of the respective Entner-Doudoroff and heterolactate pathways are shown in Figures 6-25 and 6-26. The pathways yield only a single molecule of triose phosphate from glucose, and the energy yield is correspondingly low: Unlike the Embden-Meyerhof pathway, the Entner-Doudoroff CHAPTER 6 Microbial Metabolism 93 Glucose 1C C C C C C Glucose is phosphorylated at the expense of one ATP, creating glucose 6-phosphate, a precursor metabolite and the starting molecule for the pentose phosphate pathway. ATP ADP PO4 Glucose 6-phosphate C C C C C C Isomerization of glucose 6-phosphate (an aldehyde) to fructose 6-phosphate (a ketone and a precursor PO4 metabolite). 1 Fructose 6-phosphate 1 C C C C C ATP ATP is consumed to phosphorylate C1 of fructose. The cell is spending some of its energy currency in order to earn more in the next part ADP PO4 of glycolysis. Fructose 1, 6-bisphosphate 1C PO4 C C C C 6 C phase C Dihydroxacetone PO 4 phosphate Fructose 1, 6-bisphosphate is split into two 3-carbon molecules, one of which is a precursor metabolite. H C C C C C PO4 Glyceraldehyde 3-phosphate NAD1 e2 Glyceraldehyde 3-phosphate is oxidized and simultaneously phosphorylated, creating a high-energy NADH 1 H1 molecule. The electrons released reduce NAD1 to Pi NADH. 1, 3-bisphosphoglycerate PO4 PO4 C C C PO4 C C Glyceraldehyde 3-phosphate NAD1 e2 NADH 1 H1 Pi 1, 3-bisphosphoglycerate ADP ATP is made by substrate-level phosphorylation. Another precursor metabolite is made. C C C 3 C phase ADP ATP ATP 3-phosphoglycerate 3-phosphoglycerate PO4 2-phosphoglycerate C C C Another precursor metabolite is made. H2O H2O PO4 Phosphoenolpyruvate C C C The oxidative breakdown of one glucose results in the formation of two pyruvate molecules. Pyruvate is one of the most important precursor metabolites. 2-phosphoglycerate Phosphoenolpyruvate ADP ADP ATP ATP Pyruvate C C C Pyruvate Figure 6-22 The Embden-Meyerhof pathway. This is one of three glycolytic pathways used to catabolize glucose to pyruvate and it can function during aerobic respiration, anaerobic respiration, and fermentation. When used during a respiratory process, the electrons accepted by NAD+ (nicotinamide adenine dinucleotide phosphate) are transferred to an electron transport chain and are ultimately accepted by an exogenous electron acceptor. When used during fermentation, the electrons accepted by NAD+ are donated to an endogenous electron acceptor (eg, pyruvate). The Embden-Meyerhof pathway is also an important amphibolic pathway because it generates several precursor metabolites (shown in blue). ADP, adenosine diphosphate; ATP, adenosine triphosphate. (© The McGraw-Hill Companies, Inc. Reproduced with permission from Willey JM, Sherwood LM, Woolverton CJ: Prescott, Harley, & Klein’s Microbiology, 7th ed, McGraw-Hill, 2008.) and heterolactate pathways yield only a single net substrate phosphorylation of ADP per molecule of glucose fermented. Why have the alternative pathways for glucose fermentation been selected in the natural environment? In answering this question, two facts should be kept in mind. First, in direct growth competition between two microbial species, the rate of substrate utilization can be more important than the amount of growth. Second, glucose is but one of many carbohydrates encountered by microorganisms in their natural environment. Pentoses, for example, can be fermented quite efficiently by the heterolactate pathway. E. Additional Variations in Carbohydrate Fermentations Pathways for carbohydrate fermentation can accommodate many more substrates than described here, and the end products may be far more diverse than suggested thus far. 94 SECTION I Fundamentals of Microbiology For example, there are numerous mechanisms for oxidation of NADH at the expense of pyruvate. One such pathway is the reductive formation of succinate. Many clinically significant bacteria form pyruvate from glucose via the EmbdenMeyerhof pathway, and they may be distinguished on the basis of reduction products formed from pyruvate, reflecting the enzymatic constitution of different species. The major products of fermentation, listed in Table 6-1, form the basis for many diagnostic tests. CO2– C O CH3 Pyruvate CO2 + NADH+H H + NAD O C F. Fermentation of Other Substrates CH3 Carbohydrates are by no means the only fermentable substrates. Metabolism of amino acids, purines, and pyrimidines may allow substrate phosphorylations to occur. For example, arginine may serve as an energy source by giving rise to carbamoyl phosphate, which can be used to phosphorylate ADP to ATP. Some organisms ferment pairs of amino acids, using one as an electron donor and the other as an electron acceptor. Acetaldehyde CO2– + NADH+H CHOH + NAD CH3 Lactate CH2OH CH3 Ethanol Figure 6-23 Two biochemical mechanisms by which pyruvate can oxidize NADH (nicotinamide adenine dinucleotide hybrid). Left: Direct formation of lactate, which results in net production of lactic acid from glucose. Right: Formation of the neutral products carbon dioxide and ethanol. Patterns of Respiration Respiration requires a closed membrane. In bacteria, the membrane is the cell membrane. Electrons are passed from a chemical reductant to a chemical oxidant through a specific CO2– A CO2– CO2– H2O HCOH C HOCH C O CH3 Pyruvate CH2 HCOH HCOH HCOH HCOH CHO HCOH CH2OPO32– CH2OPO32– 6-Phosphogluconate CH2OPO32– 2-Keto-3-deoxy6-phosphogluconate B O Glyceraldehyde 3-phosphate O CH2OH C Pi O CH3COPO32– Acetyl phosphate HOCH CHO HCOH 2– CH2OPO3 HCOH CH2OPO32– Xylulose 5-phosphate Glyceraldehyde 3-phosphate Figure 6-24 Reactions associated with specific pathways of carbohydrate fermentation. A: Dehydratase and aldolase reactions used in the Entner-Doudoroff pathway. B: The phosphoketolase reaction. This reaction, found in several pathways for fermentation of carbohydrates, generates the mixed acid anhydride acetyl phosphate, which can be used for substrate phosphorylation of adenosine diphosphate (ADP). CHAPTER 6 Microbial Metabolism 95 Glucose Glucose ATP ATP ADP ADP Glucose 6-phosphate Glucose 6-phosphate NAD+ + NAD + (See Figures 6–5 and 6–6) + NADH+H NADH+H 6-Phosphogluconate NAD+ NADH+H+ CO2 H2O (See Figure 6–23A) Pentose 5-phosphate (See Figure 6–23B) Triose phosphate Pyruvate + NADH+H NAD NAD+ NADH+H+ Lactate Triose phosphate O + NAD+ CH3COPO32– Acetyl phosphate NADH+H+ + NADH+H ADP ADP (See Figure 6–7) + (See Figure 6–7) NAD ATP ATP + NADH+H NAD+ CH3CH2OH ADP ADP Ethanol ATP Pyruvate ATP Pyruvate NADH+H+ NADH+H+ NAD+ NAD+ Lactate Figure 6-25 The Entner-Doudoroff pathway. ADP, adenosine diphosphate; ATP, adenosine triphosphate. set of electron carriers within the membrane, and as a result, the proton motive force is established (Figure 6-27); return of protons across the membrane is coupled to the synthesis of ATP. As suggested in Figure 6-27, the biologic reductant for respiration frequently is NADH, and the oxidant often is oxygen. Tremendous microbial diversity is exhibited in the sources of reductant used to generate NADH, and many microorganisms can use electron acceptors other than oxygen. Organic growth substrates are converted to focal metabolites that may reduce NAD+ to NADH either by the hexose monophosphate shunt (see Figure 6-7) or by the tricarboxylic acid cycle (see Figure 6-11). Additional reductant Lactate Figure 6-26 The heterolactic fermentation of glucose. ADP, adenosine diphosphate; ATP, adenosine triphosphate. may be generated during the breakdown of some growth substrates, such as fatty acids (see Figure 6-10). Some bacteria, called chemolithotrophs, are able to use inorganic reductants for respiration. These energy sources include hydrogen, ferrous iron, and several reduced forms of sulfur and nitrogen. ATP derived from respiration and NADPH generated from the reductants can be used to drive the Calvin cycle (see Figure 6-13). Compounds and ions other than O2 may be used as terminal oxidants in respiration. This ability, the capacity for anaerobic respiration, is a widespread microbial trait. Suitable electron acceptors include nitrate, sulfate, and carbon dioxide. Respiratory metabolism dependent on carbon 96 SECTION I Fundamentals of Microbiology TABLE 6-1 Microbial Fermentations Based on the Embden-Meyerhof Pathway Fermentation a Organisms Products Ethanol Some fungi (notably some yeasts) Ethanol, CO2 Lactate (homofermentation) Streptococcus Some species of Lactobacillus Lactate (accounting for at least 90% of the energy source carbon) Lactate (heterofermentation) Enterobacter, Aeromonas, Bacillus polymyxa Ethanol, acetoin, 2,3-butylene glycol, CO2, lactate, acetate, formate (total acids = 21 mola) Propionate Clostridium propionicum, Propionibacterium, Corynebacterium diphtheriae Some species of Neisseria, Veillonella, Micromonospora Propionate, acetate, succinate, CO2 Mixed acid Escherichia, Salmonella, Shigella, Proteus Lactate, acetate, formate, succinate, H2, CO2, ethanol (total acids = 159 mola) Butanol-butyrate Butyribacterium, Zymosarcina maxima Some species of Clostridium Butanol, butyrate, acetone, isopropanol, acetate, ethanol, H2, CO2 Per 100 mol of glucose fermented. dioxide as an electron acceptor is a property found among representatives of a large microbial group, the archaebacteria. Representatives of this group possess, for example, the ability to reduce carbon dioxide to acetate as a mechanism for generating metabolic energy. Membrane NADH + H+ 2H 2H+ Medium 2H+ NAD+ 2e– 2H+ Cytoplasm 2e– 2H+ 2H+ 2e– / O2 + 2H+ 1 2 H2O ADP + Pi H+ e s Pa AT ATP H+ Figure 6-27 The coupling of electron transport in respiration to the generation of adenosine triphosphate (ATP). The indicated movements of protons and electrons are mediated by carriers (flavoprotein, quinone, cytochromes) associated with the membrane. The flow of protons down their electrochemical gradient, via the membrane ATPase, furnishes the energy for the generation of ATP from adenosine diphosphate (ADP) and inorganic phosphate (Pi). See text for explanation. Bacterial Photosynthesis Photosynthetic organisms use light energy to separate electronic charge to create membrane-associated reductants and oxidants as a result of a photochemical event. Transfer of electrons from the reductant to the oxidant creates a proton motive force. Many bacteria carry out a photosynthetic metabolism that is entirely independent of oxygen. Light is used as a source of metabolic energy, and carbon for growth is derived either from organic compounds (photoheterotroph) or from a combination of an inorganic reductant (eg, thiosulfate) and carbon dioxide (photolithotroph). These bacteria possess a single photosystem that, although sufficient to provide energy for the synthesis of ATP and for the generation of essential transmembrane ionic gradients, does not allow the highly exergonic reduction of NADP+ at the expense of water. This process, essential for oxygen-evolving photosynthesis, rests upon additive energy derived from the coupling of two different photochemical events driven by two independent photochemical systems. Among prokaryotes, this trait is found solely in the Cyanobacteria (blue-green bacteria). Among eukaryotic organisms, the trait is shared by algae and plants in which the essential energy-providing organelle is the chloroplast. REGULATION OF METABOLIC PATHWAYS In their normal environment, microbial cells generally regulate their metabolic pathways so that no intermediate is made in excess. Each metabolic reaction is regulated not only with respect to all others in the cell but also with respect to the concentrations of nutrients in the environment. Thus, when a sporadically available carbon source suddenly becomes abundant, the enzymes required for its catabolism increase in both amount and activity; conversely, when a building block (eg, an amino acid) suddenly becomes abundant, the enzymes required for its biosynthesis decrease in both amount and activity. CHAPTER 6 Microbial Metabolism 97 The regulation of enzyme activity as well as enzyme synthesis provides both fine control and coarse control of metabolic pathways. For example, the inhibition of enzyme activity by the end product of a pathway constitutes a mechanism of fine control because the flow of carbon through that pathway is instantly and precisely regulated. The inhibition of enzyme synthesis by the same end product, on the other hand, constitutes a mechanism of coarse control. The preexisting enzyme molecules continue to function until they are diluted out by further cell growth, although unnecessary protein synthesis ceases immediately. The mechanisms by which the cell regulates enzyme activity are discussed in the following section. The regulation of enzyme synthesis is discussed in Chapter 7. The Regulation of Enzyme Activity A. Enzymes as Allosteric Proteins In many cases, the activity of an enzyme catalyzing an early step in a metabolic pathway is inhibited by the end product of that pathway. Such inhibition cannot depend on competition for the enzyme’s substrate binding site, however, because the structures of the end product and the early intermediate (substrate) are usually quite different. Instead, inhibition depends on the fact that regulated enzymes are allosteric: Each enzyme possesses not only a catalytic site, which binds substrate but also one or more other sites that bind small regulatory molecules, or effectors. The binding of an effector to its site causes a conformational change in the enzyme such that the affinity of the catalytic site for the substrate is reduced (allosteric inhibition) or increased (allosteric activation). Allosteric proteins are usually oligomeric. In some cases, the subunits are identical, each subunit possessing both a catalytic site and an effector site; in other cases, the subunits are different, one type possessing only a catalytic site and the other only an effector site. B. Feedback Inhibition The general mechanism that has evolved in microorganisms for regulating the flow of carbon through biosynthetic pathways is the most efficient that one can imagine. The end product in each case allosterically inhibits the activity of the first—and only the first—enzyme in the pathway. For example, the first step in the biosynthesis of isoleucine not involving any other pathway is the conversion of l-threonine to a-ketobutyric acid, catalyzed by threonine deaminase. Threonine deaminase is allosterically and specifically inhibited by l-isoleucine and by no other compound (Figure 6-28); the other four enzymes of the pathway are not affected (although their synthesis is repressed). L-Threonine L-Threonine deaminase α-Ketobutyrate α-Aceto-αhydroxybutyrate E1 Pyruvate α,β-Dihydroxyβ-methylvalerate E2 α-Acetolactate α-Keto-βmethylvalerate E3 α,β-Dihydroxyisovalerate L-Isoleucine E4 α-Ketoisovalerate L-Valine Figure 6-28 Feedback inhibition of l-threonine deaminase by l-isoleucine (dashed line). The pathways for the biosynthesis of isoleucine and valine are mediated by a common set of four enzymes, as shown. 98 SECTION I Fundamentals of Microbiology Glucose Glucose 6-phosphate Glucose 1-phosphate ADP-Glucose Glycogen Fructose 6-phosphate ADP Fructose 1,6-diphosphate 3-Phosphoglycerate Phosphoenolpyruvate AMP Pyruvate Figure 6-29 Regulation of glucose utilization by a combination of allosteric activation () and allosteric inhibition (). AMP, adenosine monophosphate; ATP, adenosine triphosphate. (Reproduced with permission from Stanier RY, Adelberg EA, Ingraham JL: The Microbial World, 4th ed. Prentice-Hall, 1976.) C. Allosteric Activation In some cases, it is advantageous to the cell for an end product or an intermediate to activate rather than inhibit a particular enzyme. In the breakdown of glucose by E coli, for example, overproduction of the intermediates G6PD and phosphoenolpyruvate signals the diversion of some glucose to the pathway of glycogen synthesis; this is accomplished by the allosteric activation of the enzyme converting glucose 1-phosphate to ADP-glucose (Figure 6-29). D. Cooperativity Many oligomeric enzymes, possessing more than one substrate binding site, show cooperative interactions of substrate molecules. The binding of substrate by one catalytic site increases the affinity of the other sites for additional substrate molecules. The net effect of this interaction is to produce an exponential increase in catalytic activity in response to an arithmetic increase in substrate concentration. E. Covalent Modification of Enzymes The regulatory properties of some enzymes are altered by covalent modification of the protein. For example, the response of glutamine synthetase to metabolic effectors is altered by adenylylation, the covalent attachment of ADP to a specific tyrosyl side chain within each enzyme subunit. The enzymes controlling adenylylation also are controlled by covalent modification. The activity of other enzymes is altered by their phosphorylation. F. Enzyme Inactivation The activity of some enzymes is removed by their hydrolysis. This process can be regulated and sometimes is signaled by covalent modification of the enzyme targeted for removal. CHAPTER SUMMARY • • • • etabolism consists of two components, catabolism and M anabolism. Catabolism consists of processes that harvest energy from the breakdown of compounds and using that energy to synthesize ATP. Anabolism (or biosynthesis) consists of processes that use the energy stored in ATP to synthesize the subunits (or building blocks) of macromolecules that make up the cell. The biosynthetic origins of the building blocks can be traced to relatively few precursors, called focal metabolites. Peptidoglycan biosynthesis is unique to bacteria. Some antibiotics kill bacteria by selectively inhibiting steps in peptidoglycan biosynthesis. The Embden-Meyerhof, Entner-Doudoroff, and heterolactate pathways are three pathways used for glucose catabolism in bacteria. The pattern of end products is a characteristic used in the identification of bacterial species. CHAPTER 6 Microbial Metabolism 99 • • • I n the absence of respiration or photosynthesis, bacteria are entirely dependent on substrate phosphorylation for their energy. Reductive assimilation of molecular nitrogen (or nitrogen fixation) is required for continuation of life on our planet. It is an energy-intensive process accomplished by a variety of bacteria and Cyanobacteria using a multicomponent nitrogenase enzyme complex. The regulation of enzyme activity provides both fine control and coarse control of metabolic pathways so that no intermediate is made in excess. REVIEW QUESTIONS 1. The synthesis of which of the following cell components is dependent on a template? (A) Lipopolysaccharide (B) Peptidoglycan (C) Capsular polysaccharide (D) Deoxyribonucleic acid (E) Phospholipids 2. The synthesis of which of the following cell components is determined entirely by enzyme specificities? (A) DNA (B) Ribosomal RNA (C) Flagella (D) Lipopolysaccharide (E) Protein 3. The steps leading to the synthesis of peptidoglycan occur in the cytoplasm, on the cytoplasmic membrane, and extracellularly. Which antibiotic inhibits an extracellular step in peptidoglycan biosynthesis? (A) Cycloserine (B) Rifampin (C) Penicillin (D) Bacitracin (E) Streptomycin 4. Amino acids are found in the protein, peptidoglycan, and capsule of bacteria. Which of the following amino acids is found only in peptidoglycan? (A) l-Lysine (B) Diaminopimelic acid (C) d-Glutamate (D) l-Alanine (E) None of the above 5. The ability to use compounds and ions other than oxygen as terminal oxidants in respiration is a widespread microbial trait. This capacity is called (A) Photosynthesis (B) Fermentation (C) Anaerobic respiration (D) Substrate phosphorylation (E) Nitrogen fixation 6. The primary route of carbon assimilation used by organisms that can use CO2 as a sole source of carbon is (A) Hexose monophosphate shunt (B) Entner-Doudoroff pathway (C) Embden-Meyerhof pathway (D) Glyoxalate cycle (E) Calvin cycle 7. The peptidoglycan biosynthetic pathway is of particular importance in medicine because it provides a basis for selective antibacterial action of several chemotherapeutic agents. All of the following antibiotics inhibit steps in peptidoglycan biosynthesis EXCEPT (A) Cycloserine (B) Vancomycin (C) Bacitracin (D) Streptomycin (E) Penicillin 8. The regulation of enzyme activity provides fine control of metabolic pathways. Which of the following regulatory mechanisms provides fine control of a biosynthetic pathway? (A) Catabolite repression (B) Induction (C) Feedback inhibition (D) Attenuation (E) None of the above 9. The biosynthetic origin of building blocks and coenzymes can be traced back to relatively few precursors called focal metabolites. Which of the following are focal metabolites? (A) a-Ketoglutarate (B) Oxaloacetate (C) Phosphoenolpyruvate (D) Glucose 6-phosphate (E) All of the above 10. Which of the following is NOT a component of peptidoglycan? (A) N-Acetyl muramic acid (B) N-Acetyl glucosamine (C) Lipid A (D) Pentaglycine (E) Diaminopimelic acid Answers 1. D B D E D C C C C E REFERENCES Atlas RM, Bartha R: Microbial Ecology: Fundamentals and Applications, 4th ed. Benjamin Cummings, 1998. Downs DM: Understanding microbial metabolism. Annu Rev Microbiol 2006;60:533. Fuchs G: Alternative pathways of carbon dioxide fixation: Insights into the early evolution of life? Annu Rev Microbiol 2011;65:631. Gibson J, Harwood CS: Metabolic diversity in aromatic compound utilization by anaerobic microbes. Annu Rev Microbiol 2002;56:345. Hillen W, Stülke: Regulation of carbon catabolism in Bacillus species. Annu Rev Microbiol 2000;54:849. Hurst CJ et al (editors): Manual of Environmental Microbiology, 2nd ed. ASM Press, 2002. 100 SECTION I Fundamentals of Microbiology Ishihama A: Functional modulation of Escherichia coli RNA polymerase. Annu Rev Microbiol 2000;54:499. Leigh JA, Dodsworth JA: Nitrogen regulation in bacteria and archaea. Annu Rev Microbiol 2007;61:349. Moat AG, Foster JW: Microbial Physiology, 4th ed. Wiley-Liss, 2002. Neidhardt FC et al (editors): Escherichia coli and Salmonella. Cellular and Molecular Biology, vols 1 and 2, 2nd ed. ASM Press, 1996. Peters JW, Fisher K, Dean DR: Nitrogenase structure and function. Annu Rev Microbiol 1995;49:335. Roberts IS: The biochemistry and genetics of capsular polysaccharide production in bacteria. Annu Rev Microbiol 1996;50:285. Russell JB, Cook GM: Energetics of bacterial growth: Balance of anabolic and catabolic reactions. Microbiol Rev 1995;59:48. Schaechter M, Ingraham JL, Neidhardt FC: Microbe. ASM Press, 2006. C Microbial Genetics The science of genetics defines and analyzes heredity in the vast array of structural and physiologic functions that form the properties of organisms. The basic unit of heredity is the gene, a segment of deoxyribonucleic acid (DNA) that encodes in its nucleotide sequence information for a specific physiologic property. The traditional approach to genetics has been to identify genes on the basis of their contribution to phenotype, or the collective structural and physiologic properties of an organism. A phenotypic property, be it eye color in humans or resistance to antibiotics in a bacterium, is generally observed at the level of the organism. The chemical basis for variation in phenotype is change in genotype, or alteration in the DNA sequence, within a gene or within the organization of genes. DNA as the fundamental element of heredity was suggested in the 1930s from a seminal experiment performed by Frederick Griffith. In this experiment (Figure 7-1), killed virulent Streptococcus pneumoniae type III-S (possessing a capsule), when injected into mice along with living but nonvirulent type II-R pneumococci (lacking a capsule), resulted in a lethal infection from which viable type III-S pneumococci were recovered. The implication was that some chemical entity transformed the live, nonvirulent strain to the virulent phenotype. A decade later, Avery, MacLeod, and McCarty discovered that DNA was the transforming agent. This formed the foundation for molecular biology as we understand it today. Subsequent investigations with bacteria revealed the presence of restriction enzymes, proteins that cleave DNA at specific sites, giving rise to DNA restriction fragments. Plasmids were identified as small genetic elements carrying genes and capable of independent replication in bacteria and yeasts. The introduction of a DNA restriction fragment into a plasmid allows the DNA fragment to be amplified many times. Amplification of specific regions of DNA also can be achieved with bacterial enzymes using polymerase chain reaction (PCR) or other enzyme-based method of nucleic acid amplification. DNA amplified by these sources and digested with appropriate restriction enzymes can be inserted into plasmids. Genes can be placed under control of high-expression bacterial promoters that allow encoded proteins to be expressed at increased levels. Bacterial genetics have fostered the development of genetic engineering not only in prokaryotes but also 7 H A P T E R in eukaryotes. This technology is responsible for the tremendous advances in the field of medicine realized today. ORGANIZATION OF GENES The Structure of DNA and rNA Genetic information in bacteria is stored as a sequence of DNA bases (Figure 7-2). In bacteriophages and viruses, genetic information can be stored as sequences of ribonucleic acid (RNA) (see Chapter 29). Most DNA molecules are double stranded, with complementary bases (A-T; G-C) paired by hydrogen bonding in the center of the molecule (Figure 7-3). The orientation of the two DNA strands is antiparallel: One strand is chemically oriented in a 5′→3′ direction, and its complementary strand runs 3′→5′. The complementarity of the bases enables one strand (template strand) to provide the information for copying or expression of information in the other strand (coding strand). The base pairs are stacked within the center of the DNA double helix (see Figure 7-2), and they determine its genetic information. Each turn of the helix has one major groove and one minor groove. Certain proteins have the capacity to bind DNA and regulate gene expression by interacting predominately with the major groove, where atoms comprising the bases are more exposed. Each of the four bases is bonded to phospho-2′-deoxyribose to form a nucleotide. The negatively charged phosphodiester backbone of DNA faces the solvent. The length of a DNA molecule is usually expressed in thousands of base pairs, or kilobase pairs (kbp). Whereas a small virus may contain a single DNA molecule of less than 0.5 kbp, the single DNA genome that encodes Escherichia coli is greater than 4000 Kbp. In either case, each base pair is separated from the next by about 0.34 nm, or 3.4 × 10−7 mm, so that the total length of the E coli chromosome is roughly 1 mm. Because the overall dimensions of the bacterial cell are roughly 1000-fold smaller than this length, it is evident that a substantial amount of folding, or supercoiling, contributes to the physical structure of the molecule in vivo. RNA most frequently occurs in single-stranded form. The base uracil (U) replaces thymine (T) in DNA, so the complementary bases that determine the structure of RNA are 101 102 SECTION I Fundamentals of Microbiology Experiment A Injection Mice Dead mice Mice Live mice Mice Live mice Live encapsulated strain A Experiment B Injection Killed encapsulated strain A Experiment C Injection Live nonencapsulated strain B Experiment D Injection + Killed encapsulated strain B Mice Live nonencapsulated strain B Dead mice Isolate live bacteria from dead mice = Encapsulated pneumococci Figure 7-1 Griffiths’ experiment demonstrating evidence for a transforming factor, later identified as DNA. In a series of experiments, mice were injected with live or killed encapsulated or nonencapsulated Streptococcus pneumoniae, as indicated in experiments A through D. The key experiment is D, showing that the killed encapsulated bacteria could supply a factor that allowed the nonencapsulated bacteria to kill mice. Besides providing key support for the importance of the capsule for pneumococcal virulence, experiment D also illustrates the principle of DNA as the fundamental basis of genetic transformation. (Reproduced by permission from McClane and Mietzner, Microbial Pathogenesis: A Principles-Oriented Approach. Fence Creek Publishing, 1999.) A-U and C-G. The overall structure of single-stranded RNA molecules is determined by pairing between bases within the strand-forming loops, with the result that single-stranded RNA molecules assume a compact structure capable of expressing genetic information contained in DNA. The most general function of RNA is communication of DNA gene sequences in the form of messenger RNA (mRNA) to ribosomes. These processes are referred to as transcription and translation. mRNA (referred to as +ssRNA) is transcribed as the RNA complement to the coding DNA strand. This mRNA is then translated by ribosomes. The ribosomes, which contain both ribosomal RNA (rRNA) and proteins, translate this message into the primary structure of proteins via aminoacyl-transfer RNAs (tRNAs). RNA molecules range in size from the small tRNAs, which contain fewer than 100 bases, to mRNAs, which may carry genetic messages extending to several thousand bases. Bacterial ribosomes contain three kinds of rRNA, with respective sizes of 120, 1540, and 2900 bases and a number of proteins (Figure 7-4). Corresponding rRNA molecules in eukaryotic ribosomes are somewhat larger. The need for expression of individual gene changes in response to physiologic demand, and requirements for flexible gene expression are reflected in the rapid metabolic turnover of most mRNAs. On the other hand, tRNAs and rRNAs—which are associated with the universally required function of protein synthesis—tend to be stable and together account for more than 95% of the total RNA in a bacterial cell. A few RNA molecules have been shown to function as enzymes (ribozymes). For example, the 23S RNA in the 50S ribosomal subunit (see Figure 7-4) catalyzes the formation of the peptide bond during protein synthesis. Recently, a new class of RNA molecules called small CHAPTER 7 Microbial Genetics 103 3’ H N 5’ N A G Major groove 7 HC 8 T C T N 1 helical turn = 3.4 nm T 7 HC 8 T A N A C H 9 (dR) C 6 3 2 C4 Sugar-phosphate backbone C C G 2 1 6 CH N (dR) H N H CH 4 5 2 1 N3 6 CH C H O N (dR) Figure 7-3 Normal base-pairing in DNA. Top: Adeninethymidine (A-T) pairing; bottom: guanine-cytosine (G-C) pair. Hydrogen bonds are indicated by dotted lines. Note that the G-C pairing shares three sets of hydrogen bonds, but the A-T pairing has only two. Consequently, a G-C interaction is stronger than an A-T interaction. dR, deoxyribose of the sugar-phosphate DNA backbone. T A G C Base rRNA T T H C N H A 5 C 1N N 4 O C 5 C N3 C O G G H 2 N T C 1N N CH3 O 6 C4 3 Hydrogen bond C H C A T Minor groove 5 9 (dR) A C A 23S (2.9 kb) A 3’ + 5S (0.12 kb) 16S (1.54 kb) 5’ Figure 7-2 A schematic drawing of the Watson-Crick structure Proteins 31 (L1– L31) 21 (S1– S21) of DNA, showing helical sugar-phosphate backbones of the two strands held together by hydrogen bonding between the bases. (Redrawn with permission from Snyder L, Champness W: Molecular Genetics of Bacteria, Washington, DC: ASM Press, 2nd ed. 2002.) interfering RNA (siRNA) was described in plants. siRNAs are double-stranded RNA molecules, 20–25 nucleotides in length, that play a variety of roles in biology. Some have been shown to function as regulators by either binding near the 5′ end of an mRNA, preventing ribosomes from translating that message, or base pairing directly with a strand of DNA near the promoter, preventing transcription. Subunits 50S 30S 70S The Eukaryotic Genome The genome is the totality of genetic information in an organism. Almost all of the eukaryotic genome is carried on two or more linear chromosomes separated from the cytoplasm within the membrane of the nucleus. Diploid eukaryotic cells contain two homologues (divergent evolutionary copies) of each chromosome. Mutations, or genetic changes, frequently Figure 7-4 The composition of a ribosome containing one copy each of the 16S, 23S, and 5S RNAs as well as many proteins. The proteins of the large 50S subunit are designated L1–L31. The proteins of the small 30S subunit are designated S1–S21. (Redrawn with permission from Snyder L, Champness W: Molecular Genetics of Bacteria, Washington, DC: ASM Press, 2nd ed. 2002.) 104 SECTION I Fundamentals of Microbiology cannot be detected in diploid cells because the contribution of one gene copy compensates for changes in the function of its homologue. Whereas a gene that does not achieve phenotypic expression in the presence of its homologue is recessive, a gene that overrides the effect of its homologue is dominant. The effects of mutations can be most readily discerned in haploid cells, which carry only a single copy of most genes. Yeast cells (which are eukaryotic) are frequently investigated because they can be maintained and analyzed in the haploid state. Eukaryotic cells contain mitochondria and, in in the case of plants, chloroplasts. Within each of these organelles is a circular molecule of DNA that contains a few genes whose function relates to that particular organelle. Most genes associated with organelle function, however, are carried on eukaryotic chromosomes. Many yeast contain an additional genetic element, an independently replicating 2-μm circle containing about 6.3 kbp of DNA. Such small circles of DNA, termed plasmids or episomes, are frequently associated with prokaryotes. The small size of plasmids renders them amenable to genetic manipulation and, after their alteration, may allow their introduction into cells. Therefore, plasmids are commonly used in genetic engineering. Repetitive DNA, which occurs in large quantities in eukaryotic cells, has been increasingly identified in prokaryotes. In eukaryotic genomes, repetitive DNA is infrequently associated with coding regions and is located primarily in extragenic regions. These short-sequence repeats (SSRs) or short tandemly repeated sequences (STRs) occur in several to thousands of copies dispersed throughout the genome. The presence of prokaryotic SSRs and STRs is well-documented, and some show extensive-length polymorphisms. This variability is thought to be caused by slipped-strand mispairing and is an important prerequisite for bacterial phase variation and adaptation. Many eukaryotic genes are interrupted by introns, intervening sequences of DNA that are missing in processed mRNA when it is translated. Introns have been observed in archaebacterial genes but with a few rare exceptions are not found in eubacteria (see Table 3-3). The Prokaryotic Genome Most prokaryotic genes are carried on the bacterial chromosome. And with few exceptions, bacterial genes are haploid. Genome sequence data from more than 340 microbial genomes demonstrate that most prokaryotic genomes (>90%) consist of a single circular DNA molecule containing from 580 kbp to more than 5220 kbp of DNA (Table 7-1). A few bacteria (eg, Brucella melitensis, Burkholderia pseudomallei, and Vibrio cholerae) have genomes consisting of two circular DNA molecules. Many bacteria contain additional genes on plasmids that range in size from several to 100 kbp. Covalently closed DNA circles (bacterial chromosomes and plasmids), which contain genetic information necessary for their own replication, are called replicons. Because TABLE 7-1 Comparison of Genome Sizes in Selected Prokaryotes, Bacteriophages, and Viruses Organism Size (kbp) Methanococcus jannaschii 1660 Archaeoglobus fulgidus 2180 Mycoplasma genitalium 580 Mycoplasma pneumoniae 820 Borrelia burgdorferi 910 Prokaryotes Archae Eubacteria Chlamydia trachomatis 1040 Rickettsia prowazekii 1112 Treponema pallidum 1140 Chlamydia pneumoniae 1230 Helicobacter pylori 1670 Haemophilus infl uenzae 1830 Francisella tularensis 1893 Coxiella burnetii 1995 Neisseria meningitides serogroup A 2180 Neisseria meningitides serogroup B 2270 Brucella melitensisa 2117 + 1178 Mycobacterium tuberculosis 4410 Escherichia coli 4640 Bacillus anthracis 5227 Burkholderia pseudomalleia 4126 + 3182 Bacteriophage Lambda 48 Viruses Ebola 19 Variola major 186 Vaccinia 192 Cytomegalovirus 229 prokaryotes do not contain a nucleus, a membrane does not separate bacterial genes from cytoplasm as in eukaryotes. Some bacterial species are efficient at causing disease in higher organisms because they possess specific genes for pathogenic determinants. These genes are often clustered together in the DNA and are referred to as pathogenicity islands. These gene segments can be quite large (up to 200 kbp) and encode a collection of virulence genes. Pathogenicity islands (1) have a different G + C content from the rest of the genome; (2) are closely linked on the chromosome to tRNA genes; (3) are flanked by direct repeats; and (4) contain diverse genes important for pathogenesis, including, antibiotic resistance, adhesins, invasins, and exotoxins. as well as genes that can be involved in genetic mobilization. Genes essential for bacterial growth (often referred to as “housekeeping genes”) are carried on the chromosome and plasmids carry genes associated with specialized functions (Table 7-2). Many plasmids also encode genetic sequences (eg, those involved with sex pili) that mediate their transfer from CHAPTER 7 Microbial Genetics 105 TABLE 7-2 Examples of Metabolic Activities Determined by Plasmids Organism Activity Pseudomonas species Degradation of camphor, toluene, octane, salicylic acid Bacillus stearothermophilus α-Amylase Alcaligenes eutrophus Utilization of H2 as oxidizable energy source Escherichia coli Sucrose uptake and metabolism, citrate uptake Klebsiella species Nitrogen fixation Streptococcus (group N) Lactose utilization, galactose phosphotransferase system, citrate metabolism Rhodospirillum rubrum Synthesis of photosynthetic pigment Flavobacterium species Nylon degradation one organism to another as well as others associated with genetic acquisition or rearrangement of DNA. Therefore, genes with independent evolutionary origins may be assimilated by plasmids that are widely disseminated among bacterial populations. A consequence of such genetic events has been observed in the swift spread among bacterial populations of plasmid-borne resistance to antibiotics after their liberal use in hospitals. Transposons are genetic elements that contain several genes, including those necessary for their migration from one genetic locus to another. In doing so, they create insertion mutations. The involvement of relatively short transposons (0.75–2.0 kbp long), known as insertion elements, produce the majority of insertion mutations. These insertion elements (also known as insertion sequence [IS] elements) carry only the genes for enzymes needed to promote their own transposition to another genetic locus but cannot replicate on their own. Almost all bacteria carry IS elements, with each species harboring its own characteristic ones. Related IS elements can sometimes be found in different bacteria, implying that at some point in evolution they have crossed species barriers. Plasmids also carry IS elements, which are important in the formation of high-frequency recombinant (Hfr) strains (see below). Complex transposons carry genes for specialized functions such as antibiotic resistance and are flanked by insertion sequences. Transposons do not carry the genetic information required to couple their own replication to cell division, and therefore their propagation depends on their physical integration with a bacterial replicon. This association is fostered by enzymes that confer the ability of transposons to form copies of themselves; these enzymes may allow the transposons to integrate within the same replicon or an independent replicon. The specificity of sequence at the insertion site is generally low, so that transposons often seem to insert in a random pattern, but they tend to favor regions encoding tRNAs. Many plasmids are transferred among bacterial cells, and insertion of a transposon into such a plasmid is a vehicle that leads to the transposon’s dissemination throughout a bacterial population. The Viral Genome Viruses are capable of survival, but not growth, in the absence of a cell host. Replication of the viral genome depends on the metabolic energy and the macromolecular synthetic machinery of the host. Frequently, this form of genetic parasitism results in debilitation or death of the host cell. Therefore, successful propagation of the virus requires (1) a stable form that allows the virus to survive in the absence of its host, (2) a mechanism for invasion of a host cell, (3) genetic information required for replication of the viral components within the cell, and (4) additional information that may be required for packaging the viral components and liberating the resulting virus from the host cell. Distinctions are frequently made between viruses associated with eukaryotes and viruses associated with prokaryotes, the latter being termed bacteriophage or phage. With more than 5000 isolates of known morphology, phages constitute the largest of all viral groups. Much of our understanding of viruses—indeed, many fundamental concepts of molecular biology—has emerged from investigation of bacteriophages. Bacteriophages occur in more than 140 bacterial genera and in many different habitats. The nucleic acid molecule of bacteriophages is surrounded by a protein coat. Considerable variability is found in the nucleic acid of phages. Many phages contain double-stranded DNA; others contain double-stranded RNA, single-stranded RNA, or single-stranded DNA. Unusual bases such as hydroxymethylcytosine are sometimes found in the phage nucleic acid. Bacteriophages exhibit a wide variety of morphologies. Many phages contain specialized syringe-like structures (tails) that bind to receptors on the cell surface and inject the phage nucleic acid into a host cell (Figure 7-5). Head (nucleic acid present) Hollow core Sheath (expanded) Empty head Sheath (contracted) Tail fiber Base plate Figure 7-5 Illustrations of phage T2 with or without nucleic acid. Note that when the phage is loaded with nucleic acid, it takes on a different form than when the nucleic acid is absent. These diagrams are redrawn from electron micrographic observations. 106 SECTION I Fundamentals of Microbiology Phages can be distinguished on the basis of their mode of propagation. Lytic phages produce many copies of themselves as they kill their host cell. The most thoroughly studied lytic phages, the T-even (eg, T2, T4) phages of E coli, demonstrate the need for precisely timed expression of viral genes to coordinate events associated with phage formation. Temperate phages are able to enter a nonlytic prophage (indicating that they have inserted into the bacterial chromosome) state in which replication of their nucleic acid is linked to replication of host cell DNA. Bacteria carrying prophages are termed lysogenic because a physiologic signal can trigger a lytic cycle resulting in death of the host cell and liberation of many copies of the phage. The best characterized temperate phage is the E coli phage λ (lambda). Genes that determine the lytic or lysogenic response to λ infection have been identified and their complex interactions explored in detail. Filamentous phages, exemplified by the well-studied E coli phage M13, are exceptional in several respects. Their filaments contain single-stranded DNA complexed with protein and are extruded from their hosts, which are debilitated but not killed by the phage infection. Engineering of DNA into phage M13 has provided single strands that are valuable sources for DNA analysis and manipulation. REPLICATION Double-stranded DNA is synthesized by semiconservative replication. As the parental duplex unwinds, each strand serves as a template (ie, the source of sequence information) for DNA replication. New strands are synthesized with their bases in an order complementary to that in the preexisting strands. When synthesis is complete, each daughter molecule contains one parental strand and one newly synthesized strand. Bacterial DNA The replication of bacterial DNA begins at one point and moves in both directions (ie, bidirectional replication). In the process, the two old strands of DNA are separated and used as templates to synthesize new strands (semiconservative replication). The structure where the two strands are separated and the new synthesis is occurring is referred to as the replication fork. Replication of the bacterial chromosome is tightly controlled, and the number of each chromosome (when more than one is present) per growing cell falls between one and four. Some bacterial plasmids may have as many as 30 copies in one bacterial cell, and mutations causing relaxed control of plasmid replication can result in 10-fold higher copy numbers. The replication of circular double-stranded bacterial DNA begins at the ori locus and involves interactions with several proteins. In E coli, chromosome replication terminates in a region called ter. The origin (ori) and termination sites (ter) for replication are located at opposite points on the circular DNA chromosome. The two daughter chromosomes are separated, or resolved, before cell division, so that each progeny cell gets one of the daughter DNAs. This is accomplished with the aid of recombination and topoisomerases, enzymes that alter the supercoiling of double-stranded DNA. (In supercoiling the DNA molecule coils up like a telephone cord, which shortens the molecule.) The topoisomerases act by transiently cutting one or both strands of the DNA to relax the coil and extend the DNA molecule. Because bacterial topoisomerases are essential and unique, they are targets of antibiotics (eg, quinolones). Similar processes lead to the replication of plasmid DNA except that in some cases, replication is unidirectional. Phage Bacteriophages exhibit considerable diversity in the nature of their nucleic acid, and this diversity is reflected in different modes of replication. Fundamentally different propagation strategies are exhibited by lytic and temperate phages. Lytic phages produce many copies of themselves in a single burst of growth. Temperate phages establish themselves as prophages either by becoming part of an established replicon (chromosome or plasmid) or by forming an independent replicon. The dsDNA of many lytic phages is linear, and the first stage in their replication is the formation of circular DNA. This process depends upon cohesive ends, complementary single-stranded tails of DNA that hybridize. Ligation, formation of a phosphodiester bond between the 5′ and 3′ DNA ends, gives rise to covalently closed circular DNA that may undergo replication in a manner similar to that used for other replicons. Cleavage of the circles produces linear DNA that is packaged inside protein coats to form daughter phages. The ssDNA of filamentous phages is converted to a circular double-stranded replicative form. One strand of the replicative form is used as a template in a continuous process that produces single-stranded DNA. The template is a rolling circle, and the ssDNA it produces is cleaved and packaged with protein for extracellular extrusion. ssRNA phages are among the smallest extracellular particles containing information that allows for their own replication. The RNA of phage MS2, for example, contains (in fewer than 4000 nucleotides) three genes that can act as mRNA following infection. One gene encodes the coat protein, and another encodes an RNA polymerase that forms a dsRNA replicative form. ssRNA produced from the replicative form is the core of new infective particles. Some temperate bacteriophages, exemplified by E coli phage P1, can be established in a prophage state as a plasmid. The dsDNA of other temperate bacteriophages is established as a prophage by its insertion into the host chromosome. The site of insertion may be quite specific, as typified by integration of E coli phage λ at a single int locus on the bacterial chromosome. The specificity of integration is determined by identity of the shared DNA sequence by the int chromosomal locus and a corresponding region of the phage genome. Other CHAPTER 7 Microbial Genetics 107 temperate phages, such as E coli phage Mu, integrate in any of a wide range of chromosomal sites and in this aspect resemble transposons. Prophages contain genes required for lytic replication (also called vegetative replication), and expression of these genes is repressed during maintenance of the prophage state. A manifestation of repression is that an established prophage frequently confers cellular immunity against lytic infection by similar phage. A cascade of molecular interactions triggers derepression (release from repression), so that a prophage undergoes vegetative replication, leading to formation of a burst of infectious particles. Stimuli such as ultraviolet (UV) light may cause derepression of the prophage. The switch between lysogeny—propagation of the phage genome with the host—and vegetative phage growth at the expense of the cell may be determined in part by the cell’s physiologic state. A nonreplicating bacterium will not support vegetative growth of phage, but a vigorously growing cell contains sufficient energy and building blocks to support rapid phage replication. TRANSFER OF DNA The haploid nature of the bacterial genome might be presumed to limit the genomic plasticity of a bacterium. However, the ubiquity of diverse bacteria in the environment provides a fertile gene pool that contributes to their remarkable genetic diversity through mechanisms of genetic exchange. Bacterial genetic exchange is typified by transfer of a relatively small fragment of a donor genome to a recipient cell followed by genetic recombination. Bacterial genetic recombination is quite unlike the fusion of gametes observed with eukaryotes; it demands that this donor DNA be replicated in the recombinant organism. Replication can be achieved either by integration of the donor DNA into the recipient’s chromosome or by establishment of donor DNA as an independent replicon. Restriction and Other Constraints on Gene Transfer Restriction enzymes (restriction endonucleases) provide bacteria with a mechanism to distinguish between their own DNA and DNA from other biologic sources. These enzymes hydrolyze (cleave) DNA at restriction sites determined by specific DNA sequences ranging from 4 to 13 bases. Each bacterial strain that possesses a restriction system is able to disguise these recognition sites in its own DNA by modifying them through methylation of adenine or cytosine residues within the site. These restriction-modification systems fall into two broad classes: type I systems, in which the restriction and modification activities are combined in a single multisubunit protein, and type II systems, which consist of separate endonucleases and methylases. A direct biologic consequence of restriction can be cleavage of donor DNA before it has an opportunity to become established as part of a recombinant replicon, rendering the bacterial “immune” to incoming DNA. Some plasmids exhibit a narrow host range and are able to replicate only in a closely related set of bacteria. Other plasmids, exemplified by some drug resistance plasmids, replicate in a broad range of bacterial genera. In some cases, two or more plasmids can stably coexist in a cell, but other pairs will interfere with the replication or partitioning. If two such plasmids are introduced into the same cell, one or the other will be lost at a higher than normal rate when the cell divides. The phenomenon is called plasmid incompatibility; two plasmids that cannot stably coexist belong to the same incompatibility (Inc) group, and two plasmids that can stably coexist belong to different Inc groups. Mechanisms of Recombination Donor DNA that does not carry information necessary for its own replication must recombine with recipient DNA to become established in a recipient strain. The recombination may be homologous, a consequence of close similarity in the sequences of donor and recipient DNA, or nonhomologous, the result of enzyme-catalyzed recombination between two dissimilar DNA sequences. Homologous recombination almost always involves exchange between genes that share common ancestry. The process requires a set of genes designated rec. Nonhomologous recombination depends on enzymes encoded by the integrated DNA and is most clearly exemplified by the insertion of DNA into a recipient to form a copy of a donor transposon. The mechanism of recombination mediated by rec gene products is reciprocal: Introduction of a donor sequence into a recipient is mirrored by transfer of the homologous recipient sequence into the donor DNA. Increasing scientific attention is being paid to the role of gene conversion—the nonreciprocal transfer of DNA sequences from donor to recipient—in the acquisition of genetic diversity. Mechanisms of Gene Transfer The DNA composition of microorganisms is remarkably fluid. DNA can be transferred from one organism to another, and that DNA can be stably incorporated in the recipient, permanently changing its genetic composition. This process is called horizontal gene transfer to differentiate it from the inheritance of parental genes, a process called vertical inheritance. Three broad mechanisms mediate efficient movement of DNA between cells—conjugation, transduction, and transformation. Conjugation requires donor cell-to-recipient cell contact to transfer only one strand of DNA (Figure 7-6). The recipient completes the structure of double-stranded DNA by synthesizing the strand that complements the strand acquired from the donor. In transduction, donor DNA is carried in a phage coat and is transferred into the recipient by the mechanism used for phage infection. Transformation, the direct uptake 108 SECTION I Fundamentals of Microbiology Donor Recipient incorporate extracellular plasmids by transformation is fundamental to genetic engineering. A. Conjugation Formation of mating pairs o r iT Plasmids are most frequently transferred by conjugation. Genetic functions required for transfer are encoded by the tra genes, which are carried by self-transmissible plasmids. Some self-transmissible plasmids can mobilize other plasmids or portions of the chromosome for transfer. In some cases, mobilization is achieved because the tra genes provide functions necessary for transfer of an otherwise nontransmissible plasmid (Figures 7-7 and 7-8). In other cases, the Single-strand nick at oriT and strand displacement tra F 5' Self-transmissible plasmid encodes tra functions that allow cell contact Strand transfer and replication Separation of mating pair Donor Transconjugant mo b F Nick made at oriT of mobilizable plasmid Mobilizable plasmid transferred Figure 7-6 Mechanism of DNA transfer during conjugation. The donor cell produces a pilus, which is encoded by the plasmid and contacts a potential recipient cell that does not contain the plasmid. Retraction of the pilus brings the cells into close contact, and a pore forms in the adjoining cell membranes. Formation of the mating pair signals the plasmid to begin transfer from a single-stranded nick at oriT. The nick is made by plasmid encoded tra functions. The 5′ end of a single strand of the plasmid is transferred to the recipient through the pore. During transfer, the plasmid in the donor is replicated, its DNA synthesis being primed by the 3′ OH of the oriT nick. Replication of the single strand in the recipient proceeds by a different mechanism with RNA primers. Both cells now contain double-stranded plasmids, and the mating pair separates. (Redrawn with permission from Snyder L, Champness W: Molecular Genetics of Bacteria, Washington, DC: ASM Press, 2nd ed. 2002.) of “naked” donor DNA by the recipient cell, may be natural or forced. Forced transformation is induced in the laboratory, where, after treatment with high salt and temperature shock, many bacteria are rendered competent for the uptake of extracellular plasmids. The capacity to force bacteria to Mobilizable plasmid is replicated in the recipient Figure 7-7 Mechanism of plasmid mobilization. The donor cell carries two plasmids, a self-transmissible plasmid, F, which encodes the tra functions that promote cell contact and plasmid transfer, and a mobilizable plasmid. The mob functions encoded by the mobilizable plasmid make a single-stranded nick at oriT in the mob region. Transfer and replication of the mobilizable plasmid then occur. The self-transmissible plasmid may also transfer. (Redrawn with permission from Snyder L, Champness W: Molecular Genetics of Bacteria, Washington, DC: ASM Press, 2nd ed. 2002.) CHAPTER 7 Microbial Genetics 109 A B C Figure 7-8 A: A male and a female cell joined by an F pilus (sex pilus). B: Mating pairs of Escherichia coli cells. Hfr cells are elongated. C: Electron micrograph of a thin section of a mating pair. The cell walls of the mating partners are in intimate contact in the “bridge” area. (Photograph [A]: Courtesy of Carnahan J and Brinton C. Photographs [B] and [C] reproduced with permission from Gross JD and Caro LG: DNA transfer in bacterial conjugation. J Mol Biol 1966;16:269.) self-transmissible plasmid integrates with the DNA of another replicon and, as an extension of itself, carries a strand of this DNA into a recipient cell. Genetic analysis of E coli was greatly advanced by elucidation of fertility factors carried on a plasmid designated F+. This plasmid confers certain donor characteristics upon cells; these characteristics include a sex pilus, an extracellular multimeric protein extrusion that attaches donor cells to recipient organisms lacking the fertility factor. A bridge between the cells allows a strand of the F+ plasmid, synthesized by the donor, to pass into the recipient, where the complementary strand of DNA is formed. The F+ fertility factor can integrate into numerous loci in the chromosome of donor cells. The integrated fertility factor creates highfrequency recombination (Hfr) donors from which chromosomal DNA is transferred (from the site of insertion) in a direction determined by the orientation of insertion (Figure 7-9). The rate of chromosomal transfer from Hfr cells is constant, and compilation of results from many conjugation experiments has allowed preparation of an E coli genetic map in which distances between loci are measured in number of minutes required for transfer in conjugation. A similar map has been constructed for the related coliform (E coli–like) bacterium Salmonella typhimurium, and comparison of the two maps shows related patterns of gene organizationcies. Analogous procedures with other plasmids have enabled researchers to map the circular chromosomes of members of distant bacterial genera; for example, drug resistance plasmids, termed R factors, can promote chromosomal transfer from diverse bacteria, including Pseudomonas spp. Comparison of chromosomal maps of Pseudomonas aeruginosa and Pseudomonas putida shows that few, albeit significant, genetic rearrangements accompanied divergence of these two closely related species. Pseudomonas maps have little in common with those of the biologically distant coliform bacteria. Integration of chromosomal DNA into a conjugal plasmid can produce a recombinant replicon—an F (fertility) prime, or R (resistance) prime, depending on the plasmid—in which the integrated chromosomal DNA can be replicated on the plasmid independently of the chromosome. This occurs when the integrated plasmid (eg, F) is bracketed by two copies of an IS element. Bacteria carrying gene copies, a full set on the chromosome and a partial set on a prime, are partial diploids, or merodiploids, and are useful for complementation studies. A wild-type gene frequently complements its mutant homologue, and selection for the wild-type phenoty pe can allow maintenance of merodiploids in the laborat ory. Such strains can allow analysis of interactions between different alleles, genetic variants of the same gene. Merodiploids frequently are genetically unstable because recombination between the plasmid and the homologous chromosome can result in loss or exchange of mutant or wildtype alleles. This problem can frequently be circumvented by maintenance of merodiploids in a genetic background in which recA, a gene required for recombination between homologous segments of DNA, has been inactivated. Homologous genes from different organisms may have diverged to an extent that prevents homologous recombination between them but does not alter the capacity of one gene to complement the missing activity of another. For example, the genetic origin of an enzyme required for amino acid biosynthesis is unlikely to influence catalytic activity in the cytoplasm of a biologically distant host. A merodiploid carrying a gene for such an enzyme would also carry flanking genes derived from the donor organism. Therefore, conventional microbial genetics, based on selection of prime plasmids, can be used to isolate genes from fastidious organisms in E coli or P aeruginosa. The significance of this technology lies in its ability to simplify or to circumvent the relatively expensive procedures demanded by genetic engineering. 110 SECTION I Fundamentals of Microbiology F– Hfr F The F plasmid encodes tra functions, including pili o riT A nick at oriT initiates transfer Replication occurs in the donor as one strand is transferred The transferred fragment undergoes recombination in the recipient F Hfr phages are preferred vehicles for gene transfer because infection of recipient bacteria under conditions that favor lysogeny minimizes cell lysis and thus favors survival of recombinant strains. Indeed, a recipient bacterium carrying an appropriate prophage may form a repressor that renders the cell immune to lytic infection; such cells may still take up bacterial DNA from transducing particles. Transducing mixtures carrying donor DNA can be prepared under conditions that favor the lytic phage cycle. The size of DNA in transducing particles is usually no more than several percent of the bacterial chromosome, and therefore co-transduction—transfer of more than one gene at a time—is limited to linked bacterial genes. This process is of particular value in mapping genes that lie too close together to be placed in map order using the gross method of conjugal transfer. Pathogenicity islands are often transported by phages. For example, two phages transport pathogenicity islands responsible for converting a benign form of Vibrio cholerae into the pathogenic form responsible for epidemic cholera (see Chapter 17). These phage encode genes for cholera toxin (responsible for symptoms) and bundle-forming pili (responsible for attachment). The speed with which phages recombine and replicate has made them central subjects for study of these processes, and many generalizations concerning the underlying mechanisms have emerged from phage genetics. The capacity of phages to make rapid replicas of their DNA makes them valuable to genetic engineering. Of particular value are recombinant phages engineered so that they contain DNA inserts from another biologic source. Inserted DNA can be replicated with the swiftness that characterizes phage DNA and regained in a form useful for manipulation. F– Figure 7-9 Transfer of chromosomal DNA by an integrated plasmid. Formation of mating pairs, nicking of the F oriT sequence, and transfer of the 5′ end of a single strand of F DNA proceed as in transfer of the F plasmid. Transfer of a covalently linked chromosomal DNA will also occur as long as the mating pair is stable. Complete chromosome transfer rarely occurs, and so the recipient cell remains F-, even after mating. Replication in the donor usually accompanies DNA transfer. Some replication of the transferred single strand may also occur. Once in the recipient cell, the transferred DNA may recombine with homologous sequences in the recipient chromosome. (Redrawn with permission from Snyder L, Champness W: Molecular Genetics of Bacteria, Washington, DC: ASM Press, 2nd ed. 2002.) B. Transduction Transduction is phage-mediated genetic recombination in bacteria. In simplest terms, a transducing particle might be regarded as bacterial nucleic acid in a phage coat. Even a lytic phage population may contain some particles in which the phage coat surrounds DNA derived from the bacterium rather than from the phage. Such populations have been used to transfer genes from one bacterium to another. Temperate C. Transformation Direct uptake of donor DNA by recipient bacteria depends on their competence for transformation. Natural competence is unusual among bacteria, and some of these strains are transformable only in the presence of competence factors, produced only at a specific point in the growth cycle. Other strains readily undergo natural transformation, and these organisms offer promise for genetic engineering because of the ease with which they incorporate modified DNA into their chromosomes. Naturally competent transformable bacteria are found in several genera and include Bacillus subtilis, Haemophilus influenzae, Neisseria gonorrhoeae, Neisseria meningitidis, and Streptococcus pneumoniae. DNA fragments containing genes from such organisms can be readily identified on the basis of their ability to transform mutant cells to the wild type. These techniques represent a substantial advance over the laborious procedures used by Avery and his colleagues to demonstrate that the pneumococcal transforming principle was DNA (see Figure 7-1). Genetic transformation is recognized as a major force in microbial evolution. Natural transformation is an active CHAPTER 7 Microbial Genetics 111 process demanding specific proteins produced by the recipient cell. For Neisseria and Haemophilus spp. specific DNA sequences (uptake sequences) are required for uptake of the DNA. These uptake sequences are species specific, thus restricting genetic exchange to a single species. The DNA that is not incorporated can be degraded and used as a source of nutrients to support microbial growth. Most bacteria are unable to undergo natural transformation. In these cases, transformation can be forced by treatment with calcium chloride and temperature shock. Transformation with engineered recombinant plasmids by this procedure is a cornerstone of modern molecular biology because it enables DNA from diverse biologic sources to be established as part of well-characterized bacterial replicons. MUTATION AND GENE REARRANGEMENT Spontaneous Mutations Spontaneous mutations for a given gene in a wild-type background generally occur with a frequency of 10−6–10−8 in a population derived from a single bacterium (depending on the bacterial species and conditions used to identify the mutation). The mutations include base substitutions, deletions, insertions, and rearrangements. Base substitutions can arise as a consequence of mispairing between complementary bases during replication. In E coli, this occurs about once every 1010 times it incorporates a nucleotide; a remarkably rare process. Occurrence of a mispaired base is minimized by enzymes associated with mismatch repair, a mechanism that essentially proofreads a newly synthesized strand to ensure that it perfectly complements its template. Mismatch repair enzymes distinguish the newly synthesized strand from the preexisting strand on the basis of methylation of adenine in GATC sequences of the preexisting strand. When DNA damage is too extensive, a special DNA repair system, the SOS response, rescues cells in which DNA has been damaged. The SOS response is a postreplication DNA repair system that allows DNA replication to bypass lesions or errors in the DNA. Many base substitutions escape detection at the phenotypic level because they do not significantly disrupt the function of the gene product. For example, missense mutations, which result in substitution of one amino acid for another, may be without discernible phenotypic effect. Nonsense mutations terminate synthesis of proteins and thus result in a protein truncated at the site of mutation. The gene products of nonsense mutations are inactive. Rearrangements are the result of deletions that remove large portions of genes or even sets of genes. These large deletions involve recombination between directly repeated sequences (eg, IS elements) and almost never revert. Other mutations cause duplication, frequently in tandem, of comparable lengths of DNA. Such mutations usually are unstable and readily revert. Still other mutations can invert lengthy DNA sequences or transpose such sequences to new loci. Comparative gene maps of related bacterial strains have shown that such rearrangements can be fixed in natural populations. These observations point to the fact that linear separation of DNA fragments does not completely disrupt possibilities for physical and chemical interaction among them. Mutagens The frequency of mutation is greatly enhanced by exposure of cells to mutagens. Ultraviolet light is a physical mutagen that damages DNA by linking neighboring thymine bases to form dimers. Sequence errors can be introduced during enzymatic repair of this genetic damage. Chemical mutagens may act by altering either the chemical or the physical structure of DNA. Reactive chemicals alter the structure of bases in DNA. For example, nitrous acid (HNO2) substitutes hydroxyl groups for amino groups. The resulting DNA has altered template activity during subsequent rounds of replication. Frameshift mutations—introduction or removal of a single base pair from DNA—are caused by slight slippage of DNA strands. This slippage is favored by exposure to acridine dyes (eg, acridine orange), which can intercalate between bases. In general, the direct effect of chemical or physical mutagens is damage to DNA. The resulting mutations are introduced by the replication process and escape the repair enzymes described above. Mutations that change the activity of replication or repair enzymes can make a bacterium more susceptible to biologic mutagens and are referred to as mutator strains. Reversion and Suppression Regaining an activity lost as a consequence of mutation, termed phenotypic reversion, may or may not result from restoration of the original DNA sequence, as would be demanded by genotypic reversion. Frequently, a mutation at a second locus, called a suppressor mutation, restores the lost activity. In intragenic suppression, after a primary mutation has changed an enzyme’s structure so that its activity has been lost, a second mutation at a different site in the enzyme’s gene restores the structure required for activity. Extragenic suppression is caused by a second mutation lying outside the originally affected gene. GENE EXPRESSION The tremendous evolutionary separation of eukaryotic and prokaryotic genomes is illustrated by comparing their mechanisms of gene expression, which share only a small subset of properties. In both groups, genetic information is encoded in DNA, transcribed into mRNA, and translated on ribosomes through tRNA into the structure of proteins. The triplet 112 SECTION I Fundamentals of Microbiology nucleotide codons used in translation are generally shared, and many enzymes associated with macromolecular synthesis in the two biologic groups have similar properties. The mechanism by which the sequence of nucleotides in a gene determines the sequence of amino acids in a protein is largely similar in prokaryotes and eukaryotes and is as follows: 1. RNA polymerase forms a single polyribonucleotide strand, called messenger RNA (mRNA), using DNA as a template; this process is called transcription. The mRNA has a nucleotide sequence complementary to a template strand in the DNA double helix if read in the 3′–5′ direction. Thus, an mRNA is oriented in a 5′–3′ direction. 2. Amino acids are enzymatically activated and transferred to specific adapter molecules of RNA, called transfer RNA (tRNA). Each adapter molecule has a triplet of bases (anticodon) complementary to a triplet of bases on mRNA, and at one end its specific amino acid. The triplet of bases on mRNA is called the codon specific for that amino acid. 3. mRNA and tRNA come together on the surface of the ribosome. As each tRNA finds its complementary nucleotide triplet on mRNA, the amino acid that it carries is put into peptide linkage with the amino acid of the preceding tRNA molecule. The enzyme peptidyltransferase (which is actually the 23S rRNA, ie, a ribozyme) catalyzes the formation of the peptide bond. The ribosome moves along the mRNA, the polypeptide growing sequentially until the entire mRNA molecule has been translated into a corresponding sequence of amino acids. This process, called translation, is diagrammed in Figure 7-10. In prokaryotes, genes associated with related functions are typically clustered in operons. Because there is no nucleus, transcription and translation is coupled, meaning that the nascent mRNA attaches to a ribosome and is translated at the same time it is transcribed. This coupled transcription and translation allows for the rapid response to changes in the environment. Likewise, the mRNA is rapidly turned over, having a half-life on the order of seconds to minutes. In eukaryotes, clustering of related genes is unusual. Enhancer sequences are regions of eukaryotic DNA that increase transcription and may lie distantly upstream from the transcribed gene. Eukaryotic genes carry introns, DNA insertions that are not found in prokaryotic genes. Introns separate exons, the coding regions of eukaryotic genes. Transcribed introns are removed from eukaryotic transcripts during RNA processing, a series of enzymatic reactions that takes place in the nucleus. The mRNA of eukaryotes is polyadenylated at the 3′ end, protecting it from exonucleases so that it can traverse the nuclear membrane into the cytosol, where the ribosomes are located; in this case, translation is uncoupled from transcription. Because of this polyadenylation, eukaryotic mRNAs have half-lives on the order of hours to days. Eukaryotic and prokaryotic ribosomes differ in many respects. Eukaryotic ribosomes are larger and have a sedimentation coefficient of 80S compared with the 70S sedimentation coefficient of prokaryotic ribosomes. The 40S and 60S eukaryotic ribosomal subunits are larger than the corresponding 30S and 50S ribosomal subunits of prokaryotes, and the eukaryotic ribosomes are relatively rich in protein. Significant differences are inherent in the sensitivity of the ribosomal activities to antibiotics (eg, tetracycline), many of which selectively inhibit protein synthesis in prokaryotic but not in eukaryotic cytoplasms (see Chapter 9). It should be remembered, however, that mitochondrial ribosomes in eukaryotes resemble those from prokaryotes. Regulation of Prokaryotic Gene Expression Specific proteins, the products of regulatory genes, govern expression of structural genes that encode enzymes. Transcription of DNA into mRNA begins at the promoter, the DNA sequence that binds RNA polymerase. The level of gene expression is determined by the ability of a promoter to bind the polymerase, and the intrinsic effectiveness of promoters differs widely. Further controls over gene expression are exerted by regulatory proteins that can bind to regions of DNA near promoters. Many prokaryotic structural genes that encode a related series of metabolic reactions are clustered on operons. This contiguous series of genes are expressed as a single mRNA transcript, and expression of the transcript may be governed by a single regulatory gene. For example, five genes associated with tryptophan biosynthesis are clustered in the trp operon of E coli. Gene expression is governed by attenuation, as described below, and is also controlled by repression: Binding of the amino acid tryptophan by a repressor protein gives it a conformation that allows it to attach to the trp operator, a short DNA sequence that helps to regulate gene expression. Binding of the repressor protein to the operator prevents transcription of the trp genes because the bacteria sense that there is sufficient tryptophan present and making more would not be in the best interests of the organism’s metabolic resources. Repression can be viewed as a course-control mechanism, an all-or-none approach to gene regulation. This form of control is independent of attenuation, a fine-tuning mechanism that also is used to govern trp gene expression. Attenuation is a regulatory mechanism of some bio synthetic pathways (eg, the tryptophan biosynthetic pathway) that controls the efficiency of transcription after transcription has been initiated but before mRNA synthesis of the operon’s genes takes place, especially when the end product of the pathway is in short supply. For example, under normal growth conditions, most trp mRNA transcripts terminate before they reach the structural genes of the trp operon. However, during conditions of severe tryptophan starvation, the premature termination of transcription is abolished, allowing expression of the operon at 10-fold higher levels than under normal conditions. The explanation for this phenomenon resides in CHAPTER 7 Microbial Genetics 113 F NH NH AA 2 O O O Site A O C C AA1 NH 2 F tRNA2 Site B AA1 NH2 AA2 C OC O O 1 2 O Anticodon 2 mRNA 1 Codon 2 3 mRNA 4 F F NH NH2 AA1 C O C NH AA 3 3 4 AA2 O O C O O C tRNA3 O NH C AA2 tRNA1 4 AA1 NH 2 OH 3 OH O 1 2 O Anticodon 3 mRNA 1 2 3 4 mRNA Figure 7-10 Four stages in the lengthening of a polypeptide chain on the surface of a 70S ribosome. Top left: A tRNA molecule bearing the anticodon complementary to codon 1 at one end and AA1 at the other binds to site A. AA1 is attached to the tRNA through its carboxyl group; its amino nitrogen bears a formyl group (F). Top right: A tRNA molecule bearing AA2 binds to site B; its anticodon is complementary to codon 2. Bottom right: An enzyme complex catalyzes the transfer of AA1 to the amino group of AA2, forming a peptide bond. (Note that transfer in the opposite direction is blocked by the prior formylation of the amino group of AA1.) Bottom left: The ribosome moves to the right, so that sites A and B are now opposite codons 2 and 3; in the process, tRNA1 is displaced and tRNA2 moves to site A. Site B is again vacant and is ready to accept tRNA3 bearing AA3. (When the polypeptide is completed and released, the formyl group is enzymatically removed.) (Redrawn and reproduced by permission of Stanier RY, Doudoroff M, Adelberg EA: The Microbial World, 3rd ed. Copyright © 1970. Prentice-Hall, Inc., Englewood Cliffs, NJ.) the 162 bp regulatory sequence in front of the trp structural genes (Figure 7-11) referred to as the leader sequence or trpL. The trp leader sequence can be transcribed into mRNA and subsequently translated into a 14 amino acid polypeptide with two adjacent tryptophan residues, a very rare occurrence. At the end of trpL and upstream of the regulatory signals that control translation of the trp structural genes is a Rho-independent terminator. The DNA sequence of this region suggests that the encoded mRNA has a high probability of forming stem loop secondary structures. These have been named the pause loop (1:2), the terminator loop (3:4), and the antiterminator loop (2:3). Attenuation of the trp operon uses the secondary structure of the mRNA to sense the amount of tryptophan in the cell (as trp-tRNA) according to the model shown in Figure 7-11. Prevention of transcription by a repressor protein is called negative control. The opposite form of transcriptional regulation—initiation of transcription in response to binding of an activator protein—is termed positive control. Both forms of control are exerted over expression of the lac operon, genes associated with fermentation of lactose in E coli. The operon contains three structural genes. Transport of lactose 114 SECTION I Fundamentals of Microbiology Figure 7-11 The predictions of the attenuation model. (1) Coupled transcription/translation takes place as for any bacterial gene. (2) RNA polymerase pauses and a 1:2 stem loop forms. (3) The ribosome disrupts the 1:2 stem loop and encounters the two trp codons. (4) If enough tryptophan is present, charged trp-tRNAs will be present and the ribosomes will translate trpL. This causes the RNA polymerase to stop at the Rho-independent terminator composed of a 3:4 stem loop. (Alternate 4) If tryptophan is limiting (no trpt-RNA), the ribosome stalls at the two trp codons, while RNA polymerase continues. The 2:3 stem loop forms. (Alternate 5) The 3:4 terminator cannot form and the RNA polymerase continues transcribing into the trp structural genes. This exposes the ribosome binding site (RBS) upstream of trpE, allowing translation. (Reproduced with permission from Trun N, Trempy J: Fundamental Bacterial Genetics. Blackwell Science Ltd, 2004.) into the cell is mediated by the product of the lacY gene. Betagalactosidase, the enzyme that hydrolyzes lactose to galactose and glucose, is encoded by the lacZ gene. The product of the third gene (lacA) is a transacetylase; the physiologic function of this enzyme has not been clearly elucidated. As a byproduct of its normal function, β-galactosidase produces allolactose, a structural isomer of lactose. Lactose itself does not influence transcriptional regulation; rather, this function is served by allolactose, which is the inducer of the lac operon because it is the metabolite that most directly CHAPTER 7 Microbial Genetics 115 elicits gene expression. In the absence of allolactose, the lac repressor, a product of the independently controlled lacI gene, exerts negative control over transcription of the lac operon by binding to the lac operator. In the presence of the inducer, the repressor is released from the operator, and transcription takes place. Expression of the lac operon and many other operons associated with energy generation is enhanced by the binding of cyclic AMP–binding protein (CAP) to a specific DNA sequence near the promoter for the regulated operon. The protein exerts positive control by enhancing RNA polymerase activity. The metabolite that triggers the positive control by binding to CAP is 3′,5′-cyclic AMP (cAMP). This compound, formed in energy-deprived cells, acts through CAP to enhance expression of catabolic enzymes that give rise to metabolic energy. Cyclic AMP is not alone in its ability to exert control over unlinked genes in E coli. A number of different genes respond to the nucleotide ppGpp (in which “p” denotes phosphodiester and “G” denotes guanine) as a signal of amino acid starvation, and unlinked genes are expressed as part of the SOS response to DNA damage. Yet another set of unlinked genes is called into play in response to heat shock. GENETIC ENGINEERING Engineering is the application of science to social needs. Over the past 4 decades, engineering based on bacterial genetics has transformed biology. Specified DNA fragments can be isolated and amplified, and their genes can be expressed at high levels. The nucleotide specificity required for cleavage by restriction enzymes allows fragments containing genes or parts of genes to be ligated (incorporated) into plasmids (“vectors”) that can in turn be used to transform bacterial cells. Bacterial colonies or clones carrying specified genes can be identified by hybridization of DNA or RNA with labeled probes (similar to that shown in Figure 3-4). Alternatively, protein products encoded by the genes can be recognized either by enzyme activity or by immunologic techniques. The latter procedures have been greatly enhanced by the remarkable selectivity with which monoclonal antibodies (see Chapter 8) bind to specific antigenic determinants in proteins. Thus, genetic engineering techniques can be used to isolate virtually any gene, and many of these genes can be expressed so that a biochemically recognizable property can be studied or exploited. Isolated genes can be used for a variety of purposes. Site-directed mutagenesis can identify and alter the DNA sequence of a gene. Nucleotide residues essential for gene function can thus be determined and, if desired, altered. With hybridization techniques, DNA can be used as a probe that recognizes nucleic acids corresponding to the complementary sequence of its own DNA. For example, a latent virus in animal tissue can be detected with a DNA probe even in the absence of overt viral infection. The protein products of isolated viral genes offer great promise as vaccines because they can be prepared without genes that encode the replication of viral nucleic acid. Moreover, proteins such as insulin that have useful functions can be prepared in large quantities from bacteria that express cloned genes. Preparation of DNA Fragments with Restriction Enzymes The genetic diversity of bacteria is reflected in their extensive range of restriction enzymes, which possess remarkable selectivity that allows them to recognize specific regions of DNA for cleavage. DNA sequences recognized by restriction enzymes are predominantly palindromes (inverted sequence repetitions). A typical sequence palindrome, recognized by the frequently used restriction enzyme EcoR1, is GAATTC; the inverted repetition, inherent in the complementarity of the G-C and A-T base pairs, results in the 5′ sequence TTC being reflected as AAG in the 3′ strand. The length of DNA fragments produced by restriction enzymes varies tremendously because of the individuality of DNA sequences. The average length of the DNA fragment is determined in large part by the number of specific bases recognized by an enzyme. Most restriction enzymes recognize four, six, or eight base sequences; however, other restriction enzymes recognize 10, 11, 12, or 15 base sequences. Recognition of four bases yields fragments with an average length of 250 base pairs and therefore is generally useful for analysis or manipulation of gene fragments. Complete genes are frequently encompassed by restriction enzymes that recognize six bases and produce fragments with an average size of about 4 kbp. Restriction enzymes that recognize eight bases produce fragments with a typical size of 64 kbp and are useful for analysis of large genetic regions. Restriction enzymes that recognize more than 10 bases are useful for construction of a physical map and for molecular typing by pulsed field gel electrophoresis. Physical Separation of Differently Sized DNA Fragments Much of the simplicity underlying genetic engineering techniques lies in the fact that gel electrophoresis permits DNA fragments to be separated on the basis of size (Figure 7-12F): The smaller the fragment, the more rapid the rate of migration. The overall rate of migration and optimal range of size for separation are determined by the chemical nature of the gel and by the degree of its crosslinking. Highly cross-linked gels optimize the separation of small DNA fragments. The dye ethidium bromide forms brightly fluorescent adducts as it binds to DNA, so that small amounts of separated DNA fragments can be visualized on gels (Figure 7-12A). Specific DNA fragments can be recognized by probes containing complementary sequences (Figure 7-12B and C). 116 SECTION I Fundamentals of Microbiology A. Restriction fragments Size of fragment (kbp) C. Hybridization restriction fragments Enzymes E E/H E/S E/H/S Size of fragment (kbp) 4 4 3 3 2 2 1 1 0.5 0.5 Enzymes E E/H E/S E/H/S B. Restriction sites 1 Length (kbp) Enzyme site E H 2 3 4 S E Probe Figure 7-12 A: Separation of DNA fragments on the basis of size by electrophoresis through a gel. Smaller fragments migrate more rapidly than large fragments, and over a range determined by the properties of the gel, the distance migrated is roughly proportionate to the logarithm of the size of the fragment. DNA fragments can be visualized on the basis of their fluorescence after staining with a dye. B: The size of restriction fragments is determined by the location of restriction sites within the DNA. In this example, a 4.0-kilobase pair (kbp) fragment formed by restriction enzyme EcoR1 (E) contains respective sites for restriction enzymes HindIII (H) and SalI (S) at positions corresponding to 1.0 and 3.5 kbp. The electrophoretic pattern in A reveals that restriction enzyme E does not cut the 4.0-kbp fragment (first lane); cleavage with restriction enzyme H produces fragments of 3.0 and 1.0 kbp (second lane); cleavage with restriction enzyme S yields fragments of 3.5 and 0.5 kbp (third lane); and cleavage with both H and S forms fragments of 2.5, 1.0, and 0.5 kbp (fourth lane). The 0.5-kbp fragment lying between the S and E sites was selected as a probe to determine DNA with hybridizing sequences as shown in C. C: Identification of hybridizing fragments. Restriction fragments were separated as in A. The hybridization procedure reveals those fragments that hybridized with the 0.5-kbp probe. These are the 4.0-kbp fragment formed by restriction enzyme E, the 3.0-kbp fragment lying between the E and H sites, and the 0.5-kbp fragment lying between the S and H sites. Pulsed field gel electrophoresis allows the separation of DNA fragments containing up to 100 kbp that are separated on high-resolution polyacrylamide gels. Characterizations of such large fragments have allowed construction of a physical map for the chromosomes from several bacterial species and have been invaluable in fingerprinting bacterial isolates associated with infectious disease outbreaks. Cloning of DNA Restriction Fragments Many restriction enzymes cleave asymmetrically and produce DNA fragments with cohesive (sticky) ends that may hybridize with one another. This DNA can be used as a donor with plasmid recipients to form genetically engineered recombinant plasmids. For example, cleavage of DNA with EcoR1 produces DNA containing the 5′ tail sequence AATT and the complementary 3′ tail sequence TTAA (Figure 7-13). Cleavage of a plasmid (a circular piece of DNA) with the same restriction enzyme produces a linear fragment with cohesive ends that are identical to one another. Enzymatic removal of the free phosphate groups from these ends ensures that they will not be ligated to form the original circular plasmid. Ligation in the presence of other DNA fragments containing free phosphate groups produces recombinant plasmids, which have DNA fragments as inserts in covalently closed circular DNA. Plasmids must be in a circular form to replicate in a bacterial host. Recombinant plasmids may be introduced into a bacterial host, frequently E coli, by transformation. Alternatively, electroporation is a recently developed procedure that introduces DNA into bacteria using an electrical gradient. Transformed cells may be selected on the basis of one or more drug resistance factors encoded by plasmid genes. The resulting bacterial population contains a library of recombinant plasmids carrying various cloned inserted restriction fragments derived from the donor DNA. Hybridization techniques may be used to identify bacterial colonies carrying specific DNA fragments, or, if the plasmid expresses the CHAPTER 7 Microbial Genetics 117 Recipient vector ampR Donor DNA EcoR1 EcoR1 restriction site restriction site EcoR1 restriction site 5 GAATTC GAATTC 3 3 CTTAAG CTTAAG 5 GAATTC CTTAAG H2O RESTRICTION H2O RESTRICTION 5 G 3 CTTAA P ampR P AATTC 3 G 5 G CTTAA P G P AATTC CTTAA P G Phosphatase P AATTC G H2O Pi ampR HYBRIDIZATION OF STICKY ENDS AATTC G CTTAA G ampR G C P AAT T C TTAA G ATP G C C G AATT P TTAA LIGATION ampR G C P AATT C TTAAG G AATT CTTAA P C G Recombinant (or chimeric) plasmid Figure 7-13 Formation of a recombinant, or chimeric, plasmid from donor DNA and a recipient vector. The vector, a plasmid that carries an EcoR1 restriction site, is cleaved by the enzyme and prepared for ligation by removal of the terminal phosphate groups. This step prevents the sticky ends of the plasmid from being ligated in the absence of an insert. The donor DNA is treated with the same restriction enzyme, and covalently bound circles are formed by ligation. A drug resistance marker, shown as ampR on the plasmid, can be used to select the recombinant plasmids after their transformation into Escherichia coli. Enzymes of the host bacterium complete covalent bonding of the circular DNA and mediate its replication. 118 SECTION I Fundamentals of Microbiology Transfer to filter Fix DNA Add labeled DNA probe Wash off unbound label Autoradiograph Figure 7-14 Use of probes to identify clones containing a specific fragment of DNA. Colonies may be transferred to a filter and baked so that the cells lyse and the DNA adheres to the filter. The filter can then be treated with a solution containing a suitably labeled DNA probe, which specifically hybridizes to the desired clones. Subsequent autoradiography of the filter identifies these clones (dark circles). Alternatively, the clones may be probed with antibodies to determine whether they have synthesized a specific protein product. inserted gene, colonies can be screened for the gene product (Figure 7-14). CHARACTERIZATION OF CLONED DNA Restriction Mapping Manipulation of cloned DNA requires an understanding of its nucleic acid sequence. Preparation of a restriction map is the first step in gaining this understanding. A restriction map is constructed similar to a jigsaw puzzle from fragment sizes produced by single digests, which are prepared with individual restriction enzymes, and by double digests, which are formed with pairs of restriction enzymes. Restriction maps are also the initial step toward DNA sequencing because they identify fragments that will provide subclones (relatively small fragments of DNA) that may be subjected to more rigorous analysis, which may involve DNA sequencing. In addition, restriction maps provide a highly specific information base that allows DNA fragments, identified on the basis of size, to be associated with specific gene function. Sequencing DNA sequencing displays gene structure and enables researchers to deduce the amino acid sequence of gene products. In turn, this information makes it possible to manipulate genes to understand or alter their function. In addition, DNA sequence analysis reveals regulatory regions that control gene expression and genetic “hot spots” particularly susceptible to mutation. Comparison of DNA sequences reveals evolutionary relationships that provide a framework for unambiguous classification of organisms and viruses. Such comparisons may facilitate identification of conserved regions that may prove particularly useful as specific hybridization probes to detect the organisms or viruses in clinical samples. The original method of DNA sequence determination used the Maxam-Gilbert technique, which relies on the relative chemical liability of different nucleotide bonds. The field now has largely moved to the Sanger (dideoxy termination) method, which interrupts elongation of DNA sequences by incorporating dideoxynucleotides into the sequences. Both techniques produce a nested set of oligonucleotides starting from a single origin and entail separation on a sequencing gel of DNA strands that differ by the increment of a single nucleotide. A polyacrylamide sequencing gel separates strands that differ in length from one to several hundred nucleotides and reveals DNA sequences of varying lengths. Four parallel lanes on the same gel reveal the relative length of strands undergoing dideoxy termination at adenine, cytidine, guanidine, and thymidine. Comparison of four lanes containing reaction mixes that differ solely in the method of chain termination makes it possible to determine DNA sequence by the Sanger method (Figure 7-15). The relative simplicity of the Sanger method has led to its more general use, but the Maxam-Gilbert technique is widely used because it can expose regions of DNA that are protected by specific binding proteins against chemical modification. DNA sequencing is greatly facilitated by genetic manipulation of E coli bacteriophage M13, which contains CHAPTER 7 Microbial Genetics 119 Termination at A C G T Sequence: CACGTG Figure 7-15 Determination of a DNA sequence by the Sanger (dideoxy termination) method. Enzymatic elongation of DNA is interrupted by inclusion of dideoxy analogs of the trinucleotides corresponding to A, C, G, and T separately in parallel reaction mixes. The resulting sets of interrupted elongated strands are separated on a sequencing gel, and the sequence can be deduced by noting the base corresponding to each increment of chain length. The sequencing gel is read from the bottom up; each band corresponds to an increase of one base. single-stranded DNA. The replicative form of the phage DNA is a covalently closed circle of double-stranded DNA that has been engineered so that it contains a multiple cloning site that permits integration of specific DNA fragments that have been previously identified by restriction mapping. Bacteria infected with the replicative form secrete modified phages containing, within their protein coat, single-stranded DNA that includes the inserted sequence. This DNA serves as the template for elongation reactions. The origin for elongation is determined by a DNA primer, which can be synthesized by highly automated machines for chemical oligonucleotide synthesis. Such machines, which can produce DNA strands containing 75 or more oligonucleotides in a predetermined sequence, are essential for sequencing and for the modification of DNA by site-directed mutagenesis. Chemically synthesized oligonucleotides can serve as primers for the PCR, a procedure that allows amplification and sequencing of DNA lying between the primers. Thus, in many instances, DNA need not be cloned in order to be sequenced or to be made available for engineering. The study of biology has been revolutionized by the development of technology that allows sequencing and analysis of entire genomes, ranging from viruses to unicellular prokaryotic and eukaryotic microorganisms to humans. This has been facilitated by use of the procedure known as shotgunning. In this procedure, the DNA is broken into random smaller fragments to create a fragment library. These unordered fragments are sequenced by automated DNA sequencers and reassembled in the correct order using powerful computer software. A sufficient number of fragments are sequenced to ensure adequate coverage of the genome so that when they are assembled, most of the genome is represented without leaving too many gaps. (To achieve this, the entire genome is usually covered five- to eightfold, leaving about 0.1% of the total DNA unsequenced.) After the random fragments have been assembled by areas of overlapping sequence, any remaining gaps can be identified and closed. Advanced data processing permits annotation of the sequence data in which putative coding regions, operons, and regulatory sequences are identified. Already, the genomes of a number of important microorganisms have been sequenced. The continued analysis of sequence data from important human pathogens combined with studies on molecular pathogenesis will facilitate our understanding of how these organisms cause disease and, ultimately, will lead to better vaccines and therapeutic strategies. SITE-DIRECTED MUTAGENESIS Chemical synthesis of oligonucleotides enables researchers to perform controlled introduction of base substitutions into a DNA sequence. The specified substitution may be used to explore the effect of a predesigned mutation on gene expression, to examine the contribution of a substituted amino acid to protein function, or—on the basis of prior information about residues essential for function—to inactivate a gene. Single-stranded oligonucleotides containing the specified mutation are synthesized chemically and hybridized to singlestranded phage DNA, which carries the wild-type sequence as an insert (Figure 7-16). The resulting partially dsDNA is enzymatically converted to the fully double-stranded replicative form. This DNA, which contains the wild-type sequence on one strand and the mutant sequence on the other, is used to infect a bacterial host by transformation. Replication results in segregation of wild-type and mutant DNA, and the double-stranded mutant gene can be isolated and subsequently cloned from the replicative form of the phage. ANALYSIS WITH CLONED DNA: HYBRIDIZATION PROBES Hybridization probes (Southern blotting; see Figure 3-4) are used routinely in the cloning of DNA. The amino acid sequence of a protein can be used to deduce the DNA sequence from which a probe may be constructed and used to detect a bacterial colony containing the cloned gene. Complementary DNA, or cDNA, encoded by mRNA, can be used to detect the gene that encoded that mRNA. Hybridization of DNA to RNA by Northern blots can provide quantitative information about RNA synthesis. Specific DNA sequences in restriction fragments separated on gels can be revealed by Southern blots, a method that uses hybridization of DNA to DNA. These blots can be used to detect overlapping restriction fragments. Cloning of these fragments makes it possible to isolate flanking regions of DNA by a technique known as chromosomal walking. With Western blots, another frequently used 120 SECTION I Fundamentals of Microbiology Mutated primer G GTGC CGTG C G T G CA C CA Wild-type sequence Template G Replica tio of temp n late GC CGTG GT GTG CA C C CA Transformation into host bacterium Replicative heteroduplex GCA CGTG GT CGTGCAC CA C G CG GC T G T CG C GC G A A C Mutant replicative form Wild-type replicative form Figure 7-16 Site-directed mutagenesis. A chemically synthesized primer containing mutation G (in box) is hybridized to a wild-type sequence inserted in DNA from a single-stranded phage. Polymerization reactions are used to form the double-stranded heteroduplex carrying the mutation on one strand. Introduction of the heteroduplex into a host bacterium followed by segregation produces derivation strains carrying replicative forms with either the wild-type insert or an insert that has acquired the chemically designed mutation. detection technique, antibodies are used to detect cloned genes by binding to their protein products. Probes can be used in a broad range of analytic procedures. Some regions of human DNA exhibit substantial variability in the distribution of restriction sites. This variability is termed restriction fragment length polymorphism (RFLP). Oligonucleotide probes that hybridize with RFLP DNA fragments can be used to trace DNA from a small sample to its human donor. Thus, the technique is valuable to forensic science. Applications of RFLP to medicine include identification of genetic regions that are closely linked to human genes with dysfunctions coupled to genetic disease. This information has been and will continue to be a valuable aid in genetic counseling. DNA probes offer the promise of techniques for rapidly identifying fastidious organisms in clinical specimens that are difficult to grow in a microbiology laboratory. Furthermore, extensions of the technique afford opportunities to identify pathogenic agents rapidly and directly in infected tissue. Kits for identification of many bacterial and viral pathogens are commercially available. Application of diagnostic DNA probes requires an appreciation of (1) the probes themselves, (2) systems used to detect the probes, (3) targets (the DNA to which the probes hybridize), and (4) the conditions of hybridization. Probes may be relatively large restriction fragments derived from cloned DNA or oligonucleotides corresponding to a specific region of DNA. Larger probes may provide greater accuracy because they are less sensitive to single base changes in target DNA. On the other hand, hybridization reactions occur more rapidly with small probes, and they can be designed against conserved regions of DNA in which base substitutions are unlikely to have occurred. Amplification of a target by PCR followed by a detection of the amplified product after hybridization to a probe has proved more sensitive than direct detection methods. Recently, significant improvements have occurred in molecular diagnostic testing methods, especially those that incorporate nucleic acid amplification technologies such as PCR. Several commercial instruments have become available that combine PCR amplification of target DNA with detection of amplicons in the same closed vessel. This technology has been referred to as real-time PCR, implying that PCR amplicons can be detected in real time. In actuality, “real time” refers to the detection of amplicons after each PCR cycle. Probe detection formats involve detecting fluorophores. Results are semi-quantitative and can be obtained in considerably less time than it takes to perform a conventional PCR assay. MANIPULATION OF CLONED DNA Genetic engineering techniques permit separation and entirely independent expression of genes associated with pathogens. Vaccines prepared with engineered genes afford previously unattainable measures of safety. For example, a vaccine might be prepared against a viral coat protein that CHAPTER 7 Microbial Genetics 121 was produced in the absence of any genes associated with replicative viral functions; inoculation with such a vaccine would therefore entail no risk of introducing functional virus. Potential difficulties in the development of such vaccines stem from the ease with which viral mutations may produce genetic variants that are not recognized by the immune defense system of a vaccinated individual. Ultimately, vaccines now (and in the future will) contain a range of proteins that anticipate the genetic response of pathogens. Recombinant Strains in the Environment Major scientific advances have sometimes elicited adverse public reactions, so it is prudent to consider the potential consequences of genetic engineering. Of most immediate concern are known pathogens that have undergone relatively slight genetic modification. These have been and should be investigated in laboratories specially designed to contain them. The need for containment diminishes after genes for specific functions, such as protein coats, are separated from genes associated with replication or toxicity of a pathogen. For the most part, standard precautions associated with microbiology laboratories should be observed if for no other reason than they foster habits that are valuable if a potential pathogen should enter the laboratory. Interesting exceptions to this general rule are engineered organisms that may provide a social benefit if introduced into the environment. Many such organisms derive from nonpathogenic bacteria that occur naturally with a frequency as high as 105/g of soil. The available evidence suggests that predation and competition rapidly eliminate engineered bacterial strains after they are introduced into the environment. The primary challenge would ideally be to maintain biologically beneficial, engineered organisms in the environment rather than to eliminate them. However, this is not without social consequence. Among the examples of engineered organisms are Pseudomonas strains that produce a protein favoring formation of ice crystals. The value of these wild-type organisms is appreciated by ski slope owners, who have deliberately introduced the bacteria into the environment without arousing any public concern. An unfortunate side effect of the introduction of these organisms is that the ice crystals they promote can injure sensitive crops such as lettuce during seasons in which light frost is likely. Mutant bacteria that do not form ice crystals were designed by microbiologists who hoped that the mutant organisms might protect lettuce crops by temporarily occupying the niche normally inhabited by the ice-forming strains; however, attempts to use the mutant organisms in field studies were met with substantial protest, and studies were conducted only after lengthy and expensive legal delays. The legal precedents that have emerged from this and more recent related applications will establish guidelines for the progressive and beneficial use of genetic engineering techniques and facilitate determination of situations in which extreme caution is justified. OBJECTIVES 1. D escribe the basic structure of a nucleotide, base pairing, the linear and three-dimensional structure of double stranded DNA. 2. Understand the differences between RNA and DNA with regard to structure, complexity, and relative sizes. 3. K now the different functions of RNA, eg, mRNA, rRNA, tRNA, and ribozymes. 4. Be able to detail the basic differences between a prokaryotic and eukaryotic chromosome. 5. Specifically explain the terms associated with bacterial recombination and genetic transfer—transposons, conjugation, transformation, and transduction. 6. Describe the mechanisms of bacterial mutation and gene rearrangement. 7. Be able to articulate the fundamental means by which bacterial genes are transcribed, including the concepts of coupled transcription and translation, activator, repressor, and attenuation. 8. Appreciate the differences between eukaryotic versus prokaryotic ribosomes and describe the steps in prokaryotic ribosomal translation. 9. Understand the concept of genetic engineering and discuss the important tools involved in this process (eg, restriction enzymes, ligation, cloning, and expression). 10. Describe the tools involved in the characterization of DNA—restriction mapping, sequencing, mutagenesis, hybridization, and other detection methods. 11. Appreciate the benefits and possible negative aspects of recombinant bacteria in the environment. REVIEW QUESTIONS 1. Mutations in bacteria can occur by which of the following mechanisms? (A) Base substitutions (B) Deletions (C) Insertions (D) Rearrangements (E) All of the above 2. The form of genetic exchange in which donor DNA is introduced to the recipient by a bacterial virus is (A) Transformation (B) Conjugation (C) Transfection (D) Transduction (E) Horizontal transfer 3. The form of genetic exchange in bacteria that is most susceptible to the activity of deoxyribonuclease during the process of DNA uptake is (A) Transformation (B) Conjugation (C) Transfection (D) Transduction (E) All of the above 122 SECTION I Fundamentals of Microbiology 4. Replication of which of the following requires physical integration with a bacterial replicon? (A) Single-stranded DNA bacteriophage (B) Double-stranded DNA bacteriophage (C) Single-stranded RNA bacteriophage (D) Plasmid (E) Transposon 5. The formation of a mating pair during the process of conjugation in Escherichia coli requires (A) Lysis of the donor (B) A sex pilus (C) Transfer of both strands of DNA (D) A restriction endonuclease (E) Integration of a transposon Answers 1. E A D E B REFERENCES Alberts B, et al: Molecular Biology of the Cell, 4th ed. Garland, 2002. Ausubel FM, et al: Current Protocols in Molecular Biology. Wiley, 1987. Avery O, Mcleod C, McCarty M. Studies on the chemical nature of the substance inducing transformation of pneumococcal types: Induction of transformation by a desoxyribonucleic acid fraction isolated from pneumococcus type III. J Exp Med 1944;79(2):137. Bushman F: Lateral DNA Transfer. Mechanisms and Consequences. Cold Spring Harbor Laboratory Press, 2002. Charlebois RL (editor): Organization of the Prokaryotic Genome. American Society for Microbiology, 1999. Condon C: RNA processing and degradation in Bacillus subtilis. Microbiol Mol Biol Rev 2003;67:157. Drlica K, Riley M (editors): The Bacterial Chromosome. American Society for Microbiology, 1990. Fraser CM, Read TD, Nelson KE (editors): Microbial Genomes. Humana Press, 2004. Grohmann E, Muuth G, Espinosa M: Conjugative plasmid transfer in gram-positive bacteria. Microbiol Mol Biol Rev 2003;67:277. Hatfull GF: Bacteriophage genomics. Curr Opin Microbiol 2008;5:447. Koonin EV, Makarova KS, Aravind L: Horizontal gene transfer in prokaryotes: Quantification and classification. Annu Rev Microbiol 2001;55:709. Kornberg A, Baker T: DNA Replication, 2nd ed. Freeman, 1992. Lengler JW, Drews G, Schlegel HG (editors): Biology of the Prokaryotes. Blackwell Science, 1999. Liebert CA, Hall RM, Summers AO: Transposon Tn21, flagship of the floating genome. Microbiol Mol Biol Rev 1999;63:507. Murray NE: Type I restriction systems: Sophisticated molecular machines (a legacy of Bertani and Weigle). Microbiol Mol Biol Rev 2000;64:412. Ptashne M: A Genetic Switch: Phage Lambda and Higher Organisms, 2nd ed. Blackwell, 1992. Rawlings DE, Tietze E: Comparative biology of IncQ and IncQ-like plasmids. Microbiol Mol Biol Rev 2001;65:481. Reischl U, Witter C, Cockerill F (editors): Rapid Cycle Real-Time PCR—Methods and Applications. Springer, 2001. Rhodius V, Van Dyk TK, Gross C, LaRossa RA: Impact of genomic technologies on studies of bacterial gene expression. Annu Rev Microbiol 2002;56:599. Riley MA, Wertz JE: Bacteriocins: Evolution, ecology, and application. Annu Rev Microbiol 2002;56:117. Sambrook J, Russell NO: Molecular Cloning: A Laboratory Manual, 3rd ed. Cold Spring Harbor Laboratory, 2001. Singleton P, Sainsbury D: A Dictionary of Microbiology and Molecular Biology, 3rd ed. Wiley, 2002. Snyder L, Champness W: Molecular Genetics of Bacteria. ASM Press, 1997. Trun N, Trempy J: Fundamental Bacterial Genetics. Blackwell Science Ltd, 2004. van Belkum A, Scherer S, van Alphen L, Verbrugh H: Shortsequence DNA repeats in prokaryotic genomes. Microbiol Mol Biol Rev 1998;62:275. Zimmer C, Störl K, Störl J: Microbial DNA topoisomerases and their inhibition by antibiotics. J Basic Microbiol 1990;30:209–224. SECTION II IMMUNOLOGY C Immunology OVERVIEW The daunting role of the immune system is to afford protection. It serves as a host defense system against infectious diseases and foreign (nonself) antigens. To accomplish this goal, the immune system is equipped with a rapid response mechanism, exquisite specificity, adaptability, an intricate regulatory network, and memory. Over the past several decades, dramatic progress has taken place in the field of immunology. As a consequence, significant advances have been realized not only in the research realm but also in the diagnostic and clinical arena. These advances have allowed us to better understand how the immune system works and have provided insight into a variety of immune disorders, such as infectious diseases, allergy, autoimmunity, immunodeficiency, cancer, and transplantation. This information has led to better diagnosis, new treatment strategies, and improved management for patients with these disorders. This chapter presents the basic principles of immunology, particularly as they relate to response to infection. More detailed discussions on the various aspects of the immune system are available in the reference section. The Immune Response As the immune system defends the host against pathogens, it uses different recognition systems to effectively eliminate the invading pathogen or its products. A response generated 8 H A P T E R against a potential pathogen is called an immune response. The first line of defense, which is nonspecific to the invading pathogen, is rapidly mobilized at the initial site of infection but lacks immunologic memory and is called innate immunity. The second defense system is called adaptive immunity. It is specific for the pathogen and can confer protective immunity to reinfection with that pathogen. Adaptive immunity can specifically recognize and destroy the pathogen because of lymphocytes carrying specialized cellular receptors and specific antibodies. A protein that is produced in response to a particular pathogen is called antibody, and the substance that induces the production of antibodies is called the antigen. In summary, the innate immune response is effective and critical in eliminating most pathogens. However, if this initial mechanism fails, the adaptive immune response is induced that specifically confronts the pathogen and establishes immunity to that invading pathogen. Hence, both systems interact and collaborate to achieve the final goal of destroying the pathogen. INNATE IMMUNITY Innate immunity is an immediate response to the pathogen that does not confer long-lasting protective immunity. It is a nonspecific defense system and includes barriers to infectious agents, such as the skin (epithelium) and mucous membranes. It also includes many immune components important in the 123 124 SECTION II Immunology adaptive immune response, including phagocytic cells, natural killer (NK) cells, Toll-like receptors (TLRs), cytokines, and complement. For example, in the adult vagina, an acid pH is maintained by normal lactobacilli, inhibiting establishment of yeasts, anaerobes, and gram-negative bacteria. Physiologic Barriers Mechanisms of Innate Immunity A. Skin Although innate immunity does not generate antigen specific protective immunity and does not rely on specific recognition of the pathogen; nevertheless, it provides a powerful line of defense. This innate system has both cells and cytokines at its disposal. Phagocytic leukocytes, such as polymorphonuclear neutrophilic leukocytes (PMNs) and macrophages along with natural killer (NK) cells are the primary cellular components to combat microbes. The interaction of the invading microbe with these cells and other cells throughout the body triggers the release of complement and numerous cytokines and chemokines. Many of these are the proinflammatory cytokines, such as interleukin-1 (IL-1), tumor necrosis factoralpha (TNF-α), IL-6, and interferon-gamma (IFN-γ), that are induced through TLR interactions. Armed with these tools, the host initiates its defense against the invading pathogen. Few microorganisms are capable of penetrating intact skin, but many can enter sweat or sebaceous glands and hair follicles and establish themselves there. Sweat and sebaceous secretions—by virtue of their acid pH and certain chemical substances (especially fatty acids)—have antimicrobial properties that tend to eliminate pathogenic organisms. Lysozyme, an enzyme that dissolves some bacterial cell walls, is present on the skin and can help provide protection against some microorganisms. Lysozyme is also present in tears and in respiratory and cervical secretions. In addition, the skin produces a variety of antimicrobial agents, including a protein with antibacterial properties known as psoriasin. Therefore, the skin provides a physiologic barrier to the entry of the pathogen as well as anti-microbial agents to stop the pathogen at its first attempt to invade. B. Mucous Membranes In the respiratory tract, a film of mucus covers the surface and is constantly being driven upward by ciliated cells toward the natural orifices. Bacteria tend to stick to this film. In addition, mucus and tears contain lysozyme and other substances with antimicrobial properties. For some microorganisms, the first step in infection is their attachment to surface epithelial cells by means of adhesive bacterial surface proteins (eg, the pili of gonococci and Escherichia coli). If such cells have IgA antibody on their surfaces—a host resistance mechanism— attachment may be prevented. (The organism can overcome this resistance mechanism by breaking down the antibody with a protease.) When organisms enter the body via mucous membranes, they tend to be taken up by phagocytes and are transported into regional lymphatic vessels that carry them to lymph nodes. The phagocytes act as barriers to the further spread of large numbers of bacteria. The mucociliary apparatus for removal of bacteria in the respiratory tract is aided by the pulmonary macrophages. Special protective mechanisms in the respiratory tract include the hairs at the nares and the cough reflex, which prevents aspiration. In the gastrointestinal tract, several systems function to inactivate bacteria: Saliva contains numerous hydrolytic enzymes; the acidity of the stomach kills many ingested bacteria (eg, Vibrio cholerae), and the small intestine contains many proteolytic enzymes and active macrophages. Both factors can destroy microorganisms in the small intestine. It must be remembered that most mucous membranes of the body carry a constant normal microbiota that itself opposes establishment of pathogenic microorganisms (“bacterial interference”) and has important physiologic functions. A. Microbial Sensors When a pathogen enters the skin, it is confronted with macrophages and other phagocytic cells possessing “microbial sensors.” There are three major groups of microbial sensors: (1) TLRs, (2) NOD-like receptors (NLRs), and (3) RIG-1 like helicases and MDA-5. The best studied of the microbial sensors are the TLR. The TLRs are a family of evolutionary conserved pattern recognition receptors (PRRs) that recognize pathogen-associated molecular patterns (PAMPs). They constitute a first line of defense against a variety of pathogens and play a critical role in initiating the innate immune response. TLRs are type 1 transmembrane proteins with an extracellular domain, a single transmembrane α-helix, and a cytoplasmic domain. TLR recognition of these specific microbial patterns leads to a signal transduction cascade that generates a rapid and robust inflammatory response marked by cellular activation and cytokine release. To date, 10 human TLRs have been identified, and each receptor appears to be involved in the recognition of a unique set of microbial patterns. For example, TLR2 recognizes various ligands (eg, lipoteichoic acid) expressed by gram-positive bacteria, whereas TLR3 engages dsRNA in viral replication. TLR1 and TLR6 recognize multiple diacyl peptides (eg, mycoplasma), while TLR4 is specific for gram-negative lipopolysaccharides (LPS). TLR5, on the other hand, recognizes bacterial flagellin, and TLR7 and TLR8 interact with ssRNA in viral replication and TLR-9 binds bacterial DNA. At present, TLR-10 remains an orphan receptor. Another large family of innate receptors, NOD-like receptors, are located in the cytoplasm and serve as intracellular sensors for microbial products. They activate the nuclear factor kappa-light chain–enhancer of activated B cells (NF-κB) pathway and drive inflammatory responses similar to the TLRs. CHAPTER 8 Immunology 125 The third group of microbial sensors is the RIG-1–like helicases and MDA-5. These are cytoplasmic sensors of viral ssRNA. The engagement of ssRNA with these sensors triggers type I IFN production. These IFNs are highly effective inhibitors of viral replication. B. Phagocytosis During infections, the number of circulating phagocytic cells often increases. The main functions of phagocytic cells include chemotaxis, migration, ingestion, and microbial killing. Microorganisms and other exogenous antigens that enter the lymphatics, lung, or bloodstream are engulfed by a variety of phagocytic cells. When a pathogen makes its way through the epithelial barrier and replicates within tissues, it will encounter a tissue phagocytic cell. Phagocytes in the immune system consist of (1) monocytes and macrophages; (2) granulocytes, including PMNs, eosinophils, and basophils; and (3) dendritic cells. Monocytes are small leukocytes that circulate in the blood and mature into macrophages that can be found in almost all tissues. For example, they are known as Kupffer cells in the liver and microglial cells in the nervous tissue. Macrophages are critical cells that engulf and kill pathogens, process and present antigen, and regulate immune reactivity by producing cytokines and chemokines. Granulocytes are leukocytes that contain densely staining granules. PMNs have a short half-life and are important phagocytic cells that destroy pathogens within intracellular vesicles. Eosinophils and basophils are less abundant and contain granules containing enzymes and toxic proteins that can be released upon activation of the cells. They are important in defense against parasites. Dendritic cells are phagocytic and can degrade pathogens; however, their main role is to activate T cells in the adaptive immune response by acting as antigen-presenting cells (APCs) and by producing regulatory cytokines. The key elements of effective innate immunity are responses that are rapid, nonspecific, and of short duration. These features are the hallmark of the phagocytic process. Phagocytosis is the process whereby a phagocytic cell, especially the PMN, recognizes the pathogen, ingests it, and then destroys the engulfed organism. This is a multistep process that begins with the rolling of a PMN along the wall of postcapillary venules. If the pathogen enters the blood, the PMN will encounter it there. If the pathogen now invades the tissue, PMNs will migrate to the site of infection. This migration is dependent on the release of chemoattractant signals produced by either the cells of the host or the pathogen itself. One such chemoattractant is IL-8, a potent chemokine that attracts PMNs. In the initial stages of the migration process, PMNs attach to the endothelial cell surface by means of adhesion molecules, such as P-selectin. PMNs follow the chemokine attraction and migrate from the circulation through the endothelium into the tissues to the site of infection. Here PMNs recognize the pathogen and engulf it and internalize the pathogen into an endocytic vesicle called a phagosome. Once inside the PMN, the pathogen is killed. There are several antimicrobial mechanisms used by phagocytes. For example, (1) acidification occurs within the phagosome. The phagosome pH is 3.5–4.0, and this level of acidity is bacteriostatic or bactericidal. (2) Toxic oxygenderived products are generated and include superoxide O2-, hydrogen peroxide H2O2, and singlet oxygen O2. (3) Toxic nitrogen oxides are also produced, and nitric oxide NO is formed. (4) Antimicrobial peptides participate in killing. In the macrophage, cathelicidin and macrophage elastasederived peptides are found. The PMN, on the other hand, is rich in α-defensins, β-defensin, cathelicidin, and lactoferricin. All of these mechanisms are used by the phagocytes to destroy the pathogen. When the PMN completes its mission, it undergoes apoptosis and dies. As already mentioned, phagocytosis can occur without antibody. However, phagocytosis is made more efficient by the presence of antibodies that coat the surface of bacteria and facilitate their ingestion by phagocytes. The process is called opsonization, and it can occur by three mechanisms: (1) antibody alone can act as opsonin; (2) antibody plus antigen can activate complement via the classic pathway to yield opsonin; and (3) opsonin may be produced when the alternative pathway is activated and C3 is generated (see Figure 8-8). Macrophages have receptors on their membranes for the Fc portion of an antibody and for the C3 component of complement. These receptors aid in the phagocytosis of antibodycoated particles. C. Natural Killer Cells Natural killer cells are large, granular lymphocytes morphologically related to T cells, which make up 10–15% of leukocytes in the blood. NK cells contribute to innate immunity by providing protection against viruses and other intracellular pathogens. NK cells have the ability to recognize virus-infected cells and tumor cells and to respond by killing these cells. NK cells have two types of surface receptors: (1) lectin-like NK-cell receptors that bind proteins not carbohydrates and (2) killer immunoglobulin-like receptors (KIRs) that recognize the major histocompatibility complex (MHC) class I molecules, human leukocyte antigen B (HLA-B) or HLA-C. These NK-cell receptors have both activation and inhibition properties. The NK cell can lyse target cells that have undergone malignant transformation and may play a role in immune surveillance against tumor establishment. Furthermore, they can kill certain virus-infected cells with altered levels of MHC class I molecules. NK cells contain large amounts of granzyme and perforin, substances that mediate the cytotoxic actions of NK cells. In addition, when antibody production is initiated in the adaptive immune response, NK cells play a critical role in antibody-dependent cellular cytotoxicity (ADCC). In this process, specific antibody binds to the target cell surface. The NK cell has Fc receptors that bind to the attached antibody 126 SECTION II Immunology and kill the cell. This property allows the NK cell another opportunity to inhibit the replication of viruses and intracellular bacteria. NK cells and the IFN system are both integral parts of innate immunity that communicate with each other. NK cells are primary sources of IFN-γ, a potent antiviral and immunoregulating cytokine. Moreover, the lytic activity of NK cells is enhanced by the type 1 IFNs, IFN-α, and IFN-β. These two cytokines are actually induced by the invading virus. Finally, NK cell killing acts through MHC class I molecules, which are upregulated by the IFNs. D. Complement The complement system is another key component of innate immunity. The complement system consists of 30 proteins found in the serum or on the membrane of selected cells. The complement proteins target pathogens for destruction by lysis or for engulfment by phagocytes. As described later in this chapter, there are three complement pathways, classical, alternative, and lectin. Even though each has a different initiating mechanism, they all result in the lysis of the offending invader. When complement is activated, it initiates a cascade of biochemical reactions that ultimately culminate in cellular lysis or destruction of the pathogen. Another consequence of complement activation is the release of complement fragments that can interact directly with T and B lymphocytes. During this event, lymphocytes can produce cytokines, particularly inflammatory cytokines. For example, the alternative pathway is important as a first line of defense against infection by microorganisms. As shown in Figure 8-8, the alternative complement pathway can be activated by microbial surfaces and proceeds in the absence of antibody. There are several antimicrobial properties of complement proteins that contribute to host defense, including opsonization, lysis of bacteria, and amplification of inflammatory responses through the anaphylatoxins C5a and C3a. Some microorganisms have developed mechanisms to interfere with the complement system and in this way evade the immune response. For example, vaccinia virus encodes a soluble protein that functions as a complement control protein by blocking both major pathways of complement activation through binding to C3b and C4b. E. Mediators of Inflammation: Cytokines Any injury to tissue, such as that following the establishment and multiplication of microorganisms, elicits an inflammatory response. The innate immune response of macrophages includes the release of cytokines, including IL-1 and TNF-α. The other mediators released from activated macrophages include prostaglandins and leukotrienes. These inflammatory mediators begin to elicit changes in local blood vessels. This begins with dilation of local arterioles and capillaries, from which plasma escapes. Edema fluid accumulates in the area of injury, and fibrin forms a network and occludes the lymphatic channels, limiting the spread of organisms. A second effect of the mediators is to induce changes in expression of various adhesion molecules on endothelial cells and on leukocytes. Adhesion molecules such as the selectins and integrins cause leukocytes to attach to the endothelial cells of the blood vessels and thereby promote their movement across the vessel wall. Thus, PMNs in the capillaries stick to the walls and then migrate out (extravasation) of the capillaries toward the irritant. This migration (chemotaxis) is stimulated by substances in the inflammatory exudate, including some small polypeptides called chemokines. Chemokines, which belong to the cytokine family, stimulate leukocyte movement and are synthesized by a variety of cell types including macrophages and endothelial cells. One such chemokine is IL-8 (Table 8-1). These chemokines function mainly to recruit monocytes and neutrophils from the blood into sites of infection. Phagocytes engulf the microorganisms, and intracellular digestion begins. Soon the pH of TABLE 8-1 Properties of Human Immunoglobulins IgG IgA IgM IgD IgE Heavy chain symbol γ α μ δ ε Valence 2 4 5 2 2 143,000-160,000 159,000-447,000 900,000 177,000-185,000 188,000-200,000 8–16 1.4–4.0 0.4–2.0 0.03 Trace amounts Serum half-life (days) 21 7 7 2 2 Percentage of total immunoglobulins in serum 80 15 5 0.2 0.002 Yes (+) No Yes (++) No No + - - - - Molecular weight (daltons) Serum concentration (mg/ml)(adult) Complement fixing capacity Placental transfer to fetus c b a a In secretions, eg, saliva, milk, and tears and in respiratory, intestinal, and genital tract secretions, IgA is generally found as a dimer or a tetramer but in serum IgA exists primarily as a monomer. a b Subclasses 1,2,4. Subclass 3 has half-life of 7 days. c Primarily subclasses IgG1 and IgG3, but all the subclasses have been detected. CHAPTER 8 Immunology 127 the inflamed area becomes more acid, and cellular proteases induce lysis of the leukocytes. Large mononuclear macrophages arrive on the site and in turn engulf leukocytic debris as well as microorganisms and pave the way for resolution of the local inflammatory process. Fever is the most common systemic manifestation of the inflammatory response and is a cardinal symptom of infectious disease. The ultimate regulator of body temperature is the thermoregulatory center in the hypothalamus. Among the substances capable of inducing fever (pyrogens) are endotoxins of gram-negative bacteria and cytokines released from lymphoid cells, such as IL-1. Various activators can act upon mononuclear phagocytes and other cells and induce them to release IL-1. Among these activators are microbes and their products; toxins, including endotoxins; antigen–antibody complexes; inflammatory processes; and many others. IL-1 is carried by the blood to the thermoregulatory center in the hypothalamus, where physiologic responses are initiated that result in fever (eg, increased heat production, reduced heat loss). Other effects of IL-1 are mentioned throughout this chapter. The interferons (IFNs) are critical cytokines that play a key role in defense against virus infections and other intracellular organisms, such as Toxoplasma gondii. Although the IFNs were first identified in 1957 as antiviral proteins, they are now recognized as critical immunoregulating proteins capable of altering various cellular processes, such as cell growth, differentiation, gene transcription, and translation. The IFN family consists of three groups. Type I IFNs comprise numerous genes and primarily include IFN-α and IFN-β. Type II IFN consists of a single gene that produces IFN-γ. IFN-λ is a third group of IFN-like cytokines that have more recently been described. Virus infection itself triggers the production of type I IFNs, usually through TLR-3, -7, or -9. IFN-γ is produced by activated NK cells in innate immune responses and by specifically sensitized T cells in adaptive immune responses. Moreover, the cytokines IL-2 and IL-12 can trigger T cells to produce IFN-γ. The IFN system consists of a series of events leading to protection of a cell from virus replication. The IFNs bind to their cellular receptor and activate the signal transducers, JAK and activators of transcription, STAT, signaling pathways. This process triggers activation of genes containing IFN-stimulated response elements or an IFN-γ–activated sequence. Some of these activated genes initiate production of selected proteins that inhibit virus replication. The different IFNs have overlapping biological activities such as antiviral actions, antiproliferative actions, and immunoregulatory actions (see Table 8-1). However, non-overlapping functions also exist. For example, IFN-β is used successfully to treat patients with multiple sclerosis, whereas IFN-γ has been shown to exacerbate this disease. These potent actions of the IFNs and the advances in biotechnology are the underlying factors that have identified the clinical relevance of the IFNs. In fact, many of the IFNs have been approved by the U.S. Food and Drug Administration (FDA) for the treatment of infections, malignancies, autoimmunity, and immunodeficiency. In addition to cytokines, derivatives of arachidonic acid, including prostaglandins and leukotrienes, are mediators of the inflammatory response. Drugs that inhibit synthesis of prostaglandins (by blocking the enzyme cyclooxygenase) act as anti-inflammatory agents. ADAPTIVE IMMUNITY The adaptive immune response can be antibody mediated (humoral), cell mediated (cellular), or both. Unlike innate immunity, adaptive immunity is highly specific, has immunologic memory, and can respond rapidly and vigorously to a second antigen exposure. An overview of the components of the adaptive immune response is outlined below, and details are presented throughout this chapter. Cellular Basis of the Adaptive Immune Response During embryonic development, blood cell precursors (hematopoietic stem cells) are found in fetal liver and other tissues; in postnatal life, the stem cells reside in bone marrow. They can differentiate in several ways. Stem cells may differentiate into cells of the myeloid series or into cells of the lymphoid series. Lymphoid progenitor cells evolve into two main lymphocyte populations, B cells and T cells. B cells are lymphocytes that develop in the bone marrow in mammals. In birds, they develop in the bursa of Fabricius, a gut appendage. They rearrange their immunoglobulin genes and express a unique receptor for antigen on their cell surface. At this point, they migrate to a secondary lymphoid organ (eg, the spleen) and may be activated by an encounter with antigen to become antibody-secreting plasma cells. T cells are lymphocytes that are produced in the bone marrow but travel to the thymus to mature. Here they undergo variable diverse joining (VDJ) recombination of their beta chain TCR DNA and then their alpha chain TCR DNA. Once TCR rearrangement has occurred and positive and negative selection has terminated, these cells form T-cell subclasses with specific functions (eg, CD4 T cell, CD8 T cells). They are the source of cell-mediated immunity, discussed later. Figure 8-1 presents an overview of immunologically active lymphocytes and their interactions. The two arms of the immune response, cell mediated and antibody mediated, develop concurrently. In the antibody-mediated arm, helper (CD4) T lymphocytes recognize the pathogen’s antigens complexed with class II MHC molecules on the surface of an APC (eg, macrophage, B cell) and produce cytokines that activate B cells expressing antibodies that specifically match the antigen. The B cells undergo clonal proliferation and differentiate into plasma cells, which then produce specific immunoglobulins (antibodies). Major host defense functions of antibodies include neutralization of toxins and viruses, ADCC, and opsonization of the pathogen, all of which facilitate elimination of the 128 SECTION II Immunology T cell Thymus T cell T cell Cytokines Macrophage activation Cytokines Inflammation via PMN, etc Cytokines Differentiation of cytotoxic T cells T cell Bone marrow stem cell Antigenspecific interaction Cytokines T cell Antigenpresenting cell (APC, eg, a macrophage) T cell B cell Antigenspecific interaction Cytokines Antibodies Plasma cell FIGURE 8-1 Schematic diagram of the cellular interactions in the immune response. pathogen. Antibody-mediated defense is important against pathogens that produce toxins (eg, Clostridium tetani) or have polysaccharide capsules that interfere with phagocytosis (eg, the pneumococci). Thus, this arm of the immune response is excellent for combating extracellular pathogens and their toxins. In the cell-mediated arm, the antigen–MHC class II complex is recognized by helper (CD4) T lymphocytes, while the antigen–MHC class I complex is recognized by cytotoxic (CD8) T lymphocytes. Each class of T cells produces cytokines, becomes activated, and expands by clonal proliferation. T cells can differentiate into effector cells. Helper (CD4) T-cell activity, in addition to stimulating B cells to produce antibodies, promotes the development of delayed hypersensitivity and thereby also serves in the defense against intracellular agents, including intracellular bacteria (eg, mycobacteria), fungi, protozoa, and viruses. Cytotoxic (CD8) T-cell activity is aimed mainly at the destruction of cells in tissue grafts, tumor cells, or cells infected by viruses. The net result of effective immunity (humoral and cell mediated) is the host’s resistance to microbial and other pathogens and foreign cells. Antigens An antigen is substance that can provoke the production of an antibody. There are a wide variety of features that largely determine immunogenicity. They include the following: 1. Foreignness: Generally, molecules recognized as “self” are not immunogenic; for immunogenicity to occur, molecules must be recognized as “nonself.” 2. Size: The most potent immunogens are usually large proteins. In most cases, molecules with a molecular weight less than 10,000 are weakly immunogenic, and very small ones (eg, amino acids) are nonimmunogenic. Certain small molecules (eg, haptens) become immunogenic only when linked to a carrier protein. 3. Chemical and structural complexity: A certain amount of chemical complexity is required. For example, amino CHAPTER 8 Immunology 129 acid homopolymers are less immunogenic than heteropolymers containing two or three different amino acids. 4. Genetic constitution of the host: Two strains of the same species of animal may respond differently to the same antigen because of a different composition of genes involved in the immune response (eg, different MHC alleles). 5. Dosage, route, and timing of antigen administration: Since the degree of the immune response depends on the amount of antigen given, the immune response can be optimized by carefully defining the dosage (including number of doses), route of administration, and timing of administration (including intervals between doses). Finally, it should be noted that it is possible to enhance the immunogenicity of a substance by combining it with an adjuvant. Adjuvants are substances that stimulate the immune response by facilitating uptake into APCs. Antigen Recognition Molecules For the immune system to respond to nonself (ie, foreign antigen), a recognition system capable of precisely distinguishing self from nonself had to evolve. This section of the chapter outlines the molecules that are used to recognize foreign antigens. First, molecules of the MHC and antigen presentation are reviewed followed by an overview of the structure and function of antibodies. Then the chapter focuses on the membrane-bound receptors for antigen (ie, the B-cell receptor and the T-cell receptor for antigen). The Major Histocompatibility Complex The major histocompatibility complex (MHC) was first detected as the genetic locus encoding the glycoprotein molecules (transplantation antigens) responsible for the rapid rejection of tissue grafts transplanted between genetically nonidentical individuals. It is now known that MHC molecules bind peptide antigens and present them to T cells. Thus, these transplantation antigens are responsible for antigen recognition by the T-cell receptor.In this respect, the T-cell receptor is different from antibody. Antibody molecules interact with antigen directly, whereas the T-cell receptor only recognizes peptide antigens presented by MHC molecules on the APC. The T-cell receptor is specific for antigen, but the antigen must be presented on a self-MHC molecule. The T-cell receptor is also specific for the MHC molecule. If the antigen is presented by another allelic form of the MHC molecule in vitro (normally in an experimental situation), there is no recognition by the T-cell receptor. This phenomenon is known as MHC restriction. In humans, the MHC is a cluster of extensively studied genes located on chromosome 6. Among the many important genes in the human MHC, also known as HLA, are those that encode the class I, class II, and class III MHC proteins. As outlined in Table 8-2, MHC class I proteins are encoded by the HLA-A, -B, and -C genes. These proteins are made up of two chains: (1) a transmembrane glycoprotein of MW 45,000, noncovalently associated with (2) a non–MHC-encoded polypeptide of MW 12,000 that is known as β2-microglobulin. Class I molecules are to be found on virtually all nucleated cells in the body. Key exceptions are observed on cells in the retina and brain. Class II proteins are encoded by the HLA-D region. As shown in Table 8-2, there are three main families: the DP-, DQ-, and DR-encoded molecules. This locus retains control of immune responsiveness and different allelic forms of these genes confer striking differences in the ability to mount an immune response against a given antigen. The HLA-D locus-encoded proteins are made up of two noncovalently associated transmembrane glycoproteins of approximately MW 33,000 and MW 29,000. Unlike class I proteins, they have a restricted tissue distribution and are chiefly found on macrophages, dendritic cells, B cells, and other APCs. However, their expression on other cells (eg, endothelial cells or epithelial cells) is induced by IFN-γ. The class I MHC locus also includes genes encoding proteins involved in antigen processing (eg, transporters associated with antigen processing [TAPs]) (Figure 8-2). The class III MHC locus encodes complement proteins and several cytokines. The genes of the MHC exhibit a remarkable genetic variability. The MHC is polygenic in that there are several genes for each class of molecule. The MHC is also polymorphic. Thus, a large number of alleles exist in the population for each of the genes. Each individual inherits a restricted set of alleles from his or her parent. Sets of MHC genes tend to be inherited as a block or haplotype. There are relatively infrequent cross-over events at this locus. Much is known about the structural organization and sequence of MHC genes and proteins. Perhaps the most important information, however, has come from the x-ray analysis of crystals of MHC proteins. These elegant studies helped to clearly explain the function of the MHC proteins. The x-ray analysis (Figure 8-3) shows that the domains of the class I MHC molecule farthest away from the membrane are composed of two parallel α helices above a platform created by a β-pleated sheet. The whole structure undoubtedly looks like a cleft whose sides are formed by the α helices and whose floor is shaped by the β sheets. The x-ray analysis also showed that the cleft was occupied by a peptide. In essence, the T-cell receptor sees the peptide antigen bound in a cleft provided by the MHC protein. A simplified diagram of this interaction is provided in Figure 8-4A. Major histocompatibility class proteins show a broad specificity for peptide antigens, and many different peptides can be presented by any given MHC allele (one peptide is bound at a time). The α helices that form the binding cleft are the site of the amino acid residues that are polymorphic in MHC proteins (ie, those that vary among alleles). This means that different alleles can bind and present different peptide antigens. For all these reasons, MHC polymorphism has a major effect on antigen recognition. 130 SECTION II Immunology TABLE 8-2 SELECTED CYTOKINES: PRODUCTION AND ACTIVITIES Cytokine Family Primary Cell Type Activity Interferons Alpha Leukocytes Antiviral, immunoregulatory, (enhance MHC class I, NK cell activity), anti-proliferative Beta Fibroblasts, epithelial cells Antiviral, immunoregulatory, (enhance MHC class I, NK cell activity), anti-proliferative Gamma T cell, NK cells Antiviral, immunoregulatory, (enhance MHC class I and II and macrophage activation) anti-proliferative Alpha Macrophage, lymphocytes Activate macrophages and cytotoxic cells, induce cachexia, acute phase proteins, induces cytokines such as IL-1 and IL-6. Beta T cells Activate macrophages, induces cytokines (IL-1, IL-6) IL-1 Most cells, macrophages, dendritic cells Induces inflammation, fever and sepsis, activate TNF-α IL-2 T cells Induces proliferation and maturation of T cells IL-6 Most cells B cell stimulation, mediator of acute phase reactions IL-10 T cells, monocytes/macrophages Inhibits IFN-γ and IL-12 production IL-11 Bone marrow stromal cells, mesenchymal cells Synergistic effects on hematopoiesis and thrombopoiesis, cytoprotective effects on epithelial cells, induces immunosuppression IL-12 Dendritic cells, macrophages, B cells Induces production of IFN-γ , TNF-α, and IL-2 by resting and activated T and NK cells IL-15 T cells, atrocytes, microglia, fibroblasts, epithelial cells Biological activities similar to IL-2, induces proliferation of peripheral blood, mononuclear cells, maturation of NK cells (IL-1, IFN-γ, TNF-α) IL-17 (6 members) (IL-17 A-F) Th17 cells Stimulates epithelial, endothelial, and fibroblastic cells to produce IL-6, IL-8, G-CSF, and ICAM-1 IL-23 Macrophages, dendritic cells Similar to IL-12 (induces IFN-γ) helps to differentiate CD4 T cells to TH17 M-CSF Monocytes Proliferation of macrophage precursors G-CSF Macrophages Proliferation, differentiation, and activation of neutrophils GM-CSF T cells, macrophages Proliferation of granulocytes and macrophages precursors Stem cell factor Bm stromal cells, fibroblasts, fetal liver cells Proliferation and differentiation of early myeloid and lymphoid cells (synergizes with other cytokines) TGF-β Most cells Anti-inflammatory, drives differentiation of CD4 T cells to T regs; in presence of IL-6 drives CD4 T cells to Th17 VEGF-A Most cells Stimulates vasculogenesis and angiogenesis IL-8 (CXCL8) Most cells Neutrophil activation and chemotoxis Rantes (CCL5) Most cells Chemotactic for T cells monocytes, eosinophils and basophils CXCL9, CXCL10, CXCL11 Most cells Chemotactic for Th1 cells (CXCR3 positive T cells) and induced by the IFNs ICAM-1 Endothelial cells Adhesion and migration VCAM-1 Leukocytes Adhesion and migration E-selectin Endothelial cells Adhesion and migration TNF Interleukins Growth Factors Chemokines Adhesion Molecules This list is not inclusive; primary cells have been identified. CHAPTER 8 Immunology 131 Intact viral protein Viral protein synthesis Proteasome Class II MHC pathway Nucleus Class II MHC Endogenous peptides Ii αβ Peptide transporter (TAP) α Class I MHC pathway β2m RER α Exocytic vesicle β2m Golgi Vesicle fusion Class I MHC Vesicle Peptides To cell surface To cell surface Endosome Processing Endocytosis Surface Ag–class II MHC complex Exogenous antigen Cell surface Surface Ag–class I MHC complex FIGURE 8-2 Antigen-processing pathways. (MHC Class I and Class II). (Modified and reproduced with permission from Parslow TG, et al [editors]: Medical Immunology, 10th ed. McGraw-Hill, 2001. © The McGraw-Hill Companies, Inc.) Analysis of the function of T cells with respect to interaction with MHC molecules reveals that peptide antigens associated with class I MHC molecules are recognized by CD8-positive cytotoxic T lymphocytes, whereas MHC class II–associated peptide antigens are recognized by CD4positive helper T cells. For more details on this topic, see Murphy et al (2011). Antigen Processing and Presentation Antigen processing and presentation are the means by which antigens become associated with self-MHC molecules for presentation to T cells with appropriate receptors. Proteins from exogenous antigens, such as bacteria, are internalized via endocytic vesicles into APCs such as the various types of dendritic cells and macrophages. Then, as illustrated in Figure 8-2, they are exposed to cellular proteases in intracellular vesicles. Peptides, approximately 10–30 amino acid residues in length, are generated in endosomal vesicles. The endosomal vesicles can then fuse with exocytic vesicles containing class II MHC molecules. The class II MHC molecules are synthesized, as are other membrane glycoproteins, in the rough endoplasmic reticulum and then they proceed out through the Golgi apparatus. A third polypeptide, the invariant chain (Ii), protects the binding site of the class II αβ dimer until the lowered pH of the compartment created after fusion with an endosomal vesicle causes a dissociation of the Ii chain. The MHC class 132 SECTION II Immunology α2 α-Helix Eight-strand β-pleated sheet Peptidebinding groove α1 N N C C β2m α3 FIGURE 8-3 Diagrammatic structure of a class I HLA molecule. (Reproduced with permission from Bjorkman PJ et al: Structure of the human class I histocompatibility antigen, HLA-A2. Nature 1987;329:506.) II–peptide antigen complex is then transported to the cell surface for display and recognition by a T-cell receptor of a CD4 T cell. The CD4 T cells can now provide help to other T cells and B cells. The interaction of endogenous antigens within the viralinfected cell and the class I MHC molecules is outlined in Figure 8-2. In brief, cytosolic proteins are broken down by a proteolytic complex known as the proteasome. The cytosolic peptides gain access to nascent MHC class I molecules in the rough endoplasmic reticulum via peptide transporter systems (TAPs). The TAP genes are also encoded in the MHC. Within the lumen of the endoplasmic reticulum, peptide antigens approximately 8–10 residues in length complex with nascent MHC class I proteins and cooperate with β2-microglobulin to create a stable, fully folded MHC class I–peptide antigen complex that is then transported to the cell surface for display and recognition by CD8 cytotoxic T cells. The binding groove of the class I molecule is more constrained than that of the class II molecule, and for that reason, shorter peptides are found in class I than in class II MHC molecules. Once the cytotoxic T cell recognizes the MHC class I peptide antigen, it can now kill the virus-infected cell. Several viruses attempt to defeat the immune response by interfering with the antigen-processing pathways. For example, an HIV Tat protein is able to inhibit expression of class I MHC molecules. A herpesvirus protein binds to the TAPs, preventing transport of viral peptides into the endoplasmic reticulum, where class I molecules are being synthesized. A consequence of these inhibitory mechanisms is that the infected cells are not recognized by cytotoxic lymphocytes. Some superantigens are able to bind to MHC molecules outside the peptide-binding cleft. One consequence is that whereas an individual peptide complexed to an MHC molecule will normally stimulate only a small percentage of the T cells in an individual, superantigens cause up to 10% of T cells to be nonspecifically activated. Examples of superantigens include certain bacterial toxins, including the staphylococcal enterotoxins, toxic shock syndrome toxin, and group A streptococcal pyrogenic exotoxin A. These antigens bind to the “outside” of the MHC protein and to the T-cell receptor (Figure 8-4B). They are active at very low concentrations (10 –9 mol/L) and cause T cells expressing particular Vβ APC or target cell APC MHC Class II MHC Vα Vβ Cα Peptide antigen Vα Cβ Cα T cell A Vβ Superantigen Cβ T cell B FIGURE 8-4 Binding of antigen by MHC and T-cell receptor. In panel A, a model of the interaction between peptide antigen, MHC, and the T-cell receptor is shown. The Vα and Vβ regions of the TCR are shown interacting with the α helices that form the peptide-binding groove of MHC. In panel B, a model of the interaction between a superantigen, MHC, and the T-cell receptor is shown. The superantigen interacts with the Vβ region of the TCR and with class II MHC outside the peptide-binding groove. (Adapted with permission from Stites DG, et al [editors]: Medical Immunology, 9th ed. McGraw-Hill, 1997. © The McGraw-Hill Companies, Inc.) CHAPTER 8 Immunology 133 sequences to be stimulated and to release large amounts of cytokines, including IL-1 and TNF. It is the release of large amounts of cytokines (cytokine storm) from activated T lymphocytes that explain in part the pathogenesis of diseases caused by organisms expressing superantigens. Understanding the details of antigen processing has helped to clarify our thinking about T-cell function. An explanation for why T cells do not respond to carbohydrate antigens may be due to their inability to fit well in the groove. Moreover, why T cells recognize only linear antigenic determinants may be explained by the fact that T cells respond only to proteolytically processed antigen. Therefore, whether an antigen is destined for class I or class II presentation depends on the intracellular compartments it traverses. B Cells and Antibodies Humoral immunity is mediated by antibodies, which are produced by the B-cell component of the immune response. Each individual has a large pool of different B lymphocytes (~1011) that have a life span of days or weeks and are found in the bone marrow, lymph nodes, and gut-associated lymphoid tissues (eg, tonsils, Peyer patches, appendix). A. B-Cell Receptor for Antigen B cells display a single homogenous clonal immunoglobulin molecule (~105 copies/cell) on their surface. These immunoglobulins serve as receptors (B-cell receptors [BCRs]) for a specific antigen, so that each B cell can respond to only one antigen or a closely related group of antigens. All immature B cells carry IgM immunoglobulins on their surface, and most also carry IgD. B cells also have surface receptors for the Fc portion of immunoglobulins and for several complement components. An antigen interacts with the B lymphocyte that shows the best “fit” by virtue of its immunoglobulin surface receptor. The antigen binds to this BCR, and the B cell is stimulated to divide and form a clone (clonal selection). Such selected B cells differentiate to become plasma cells and secrete antibody. Since each person can make approximately 1011 different antibody molecules, there is an antigen-binding site on a B cell to fit almost any antigenic determinant. The initial step in antibody formation begins with the binding of antigen to the surface immunoglobulin via the BCR. Then the following steps ensue: (1) The BCR with its bound antigen is internalized by the B cell and the antigen is degraded to yield peptides that are then returned to the cell surface bound to MHC class II molecules. (2) MHC class II– peptide complex on B cells is recognized by antigen-specific helper (CD4) T cells. These T cells have already interacted with antigen-presenting dendritic cells and have differentiated in response to the same pathogen. This interaction can occur because the B cell and the T cell that have encountered antigen migrate toward the boundaries between B- and T-cell areas in the secondary lymphoid tissue. (3) Chemokines, such as CXCL13 and its receptor, CXCR5, play an important role in this migration process. (4) CD40 ligand on T cells binds to CD40 on B cells, and the T cell produces cytokines, such as, IL-4, IL-5, and IL-6, which induce B-cell proliferation. (5) Finally, the activated B cells migrate into follicles and proliferate to form germinal centers, here somatic hypermutation and class switching occur. Germinal center B cells that survive this process now differentiate into either antibody-producing plasma cells or memory B cells. Additional details on this topic can be found in the chapter reference, see Murphy et al (2011). It should be noted that some bacterial antigens can directly stimulate this antibody production and do not require T cell help to activate B cells. These antigens are usually bacterial polysaccharides and lipopolysaccharides. These thymus independent antigens induce B cell responses with limited class switching and do not induce memory B cells. B. Antibody Structure and Function Antibodies are immunoglobulins, which react specifically with the antigen that stimulated their production. They make up about 20% of plasma proteins. Antibodies that arise in an animal in response to a single complex antigen are heterogeneous because they are formed by many different clones of cells, each expressing an antibody capable of reacting with a different antigenic determinant on the complex antigen. These antibodies are said to be polyclonal. Immunoglobulins that arise from a single clone of cells, eg, in a plasma cell tumor (myeloma), are homogeneous and are referred to as monoclonal. Monoclonal antibodies can be produced by fusing a myeloma cell with an antibody-producing lymphocyte. Such hybridomas can produce unlimited quantities of monoclonal antibodies in vitro. Important information about the structure and function of antibodies has been derived from the study of monoclonal immunoglobulins. Immunoglobulin (Ig) molecules share common structural features, for example, all Ig are composed of light and heavy polypeptide chains. The terms light and heavy refer to their molecular weight. The light chains have a molecular weight of approximately 25,000, whereas heavy chains have a molecular weight of approximately 50,000. Each Ig molecule consists of two identical light (L) chains and two identical heavy (H) chains linked by disulfide bridges. The L chains can be either κ (kappa) or λ (lambda) and their classification is made based on amino acid differences in their constant regions (Figure 8-5). Both light chain types occur in all classes of immunoglobulins (IgG, IgM, IgA, IgD, and IgE), but any one immunoglobulin molecule contains only one type of L chain. The amino terminal portion of each L chain contains part of the antigen-binding site. Heavy (H) chains are distinct for each of the five immunoglobulin classes and are designated γ (gamma), μ (mu), α (alpha), δ (delta), and d (epsilon) (Table 8-3). The amino terminal portion of each H chain participates in the antigen-binding site; the other (carboxyl) terminal forms the Fc fragment (see Figure 8-5), which has 134 SECTION II Immunology bl e r Va ia Heavy chain t Va r ia st an an st e bl t Co n ia n Co bl e e bl Va r ia r Va t Co n an st st an t n Co Fab fragments Binds to antigen Light chain Amino terminal Constant Constant Fc fragment Constant Constant Hinge region Carbohydrate Activates complement and phagocytes Carboxyl terminal S–S bonds FIGURE 8-5 Schematic representation of an IgG molecule, indicating the location of the constant and the variable regions on the light and heavy chains. Fab fragment is fragment antigen binding, Fc fragment is fragment crystallizable. various biologic activities (eg, complement activation and binding to cell surface receptors). Therefore, an individual antibody molecule always consists of identical H chains and identical L chains. The simplest antibody molecule has a Y shape (see Figure 8-5) and consists of four polypeptide chains: two H chains and two L chains. The four chains are covalently linked by disulfide bonds. If such an antibody molecule is treated with a proteolytic enzyme (eg, papain), peptide bonds in the hinge region are broken. This breakage produces two identical Fab fragments, which carry the antigen-binding sites, and one Fc fragment, which is involved in placental transfer, complement fixation, attachment to various cells, and other biologic activities. L and H chains are subdivided into variable regions and constant regions. The regions are composed of threedimensionally folded, repeating segments called domains. By using high resolution x-ray crystallography the structure of these domains has been determined. An L chain consists of one variable domain (VL) and one constant domain (CL). Most H chains consist of one variable domain (VH) and three or more constant domains (CH). Each domain is approximately 110 amino acids long. Variable regions are responsible for antigen binding; constant regions are responsible for the biologic functions described below. Within the variable regions of both L and H chains are subregions consisting of extremely variable (hypervariable) amino acid sequences that cooperate in space to form the antigen-binding site. The hypervariable regions form the area of the antibody molecule complementary in structure to the antigenic determinant or epitope and are therefore also known as complementarity-determining regions (CDRs). Only 5–10 amino acids in each hypervariable region TABLE 8-3 Important Features of Human MHC Class I and Class II Gene Products Class I Class II Genetic loci (partial list) HLA-A, -B, and -C HLA-DP, -DQ, and -DR Polypeptide composition MW 45,000 + β2M (MW 12,000) α chain (MW 33,000), β chain (MW 29,000), Ii chain (MW 30,000) Cell distribution Most nucleated somatic cells, except cells of the brain and retina Antigen-presenting cells (macrophages, dendritic cells, B cells, etc), and IFN-γ–activated cells Present peptide antigens to CD8 T cells CD4 T cells Size of peptide bound 8–10 residues 10–30 or more residues CHAPTER 8 Immunology 135 constitute the antigen-binding site. Antigen binding is noncovalent, involving van der Waals and electrostatic as well as other weak forces. Immunoglobulin Classes A. IgG Each IgG molecule consists of two L chains and two H chains (molecular formula H2L2). Because it has two identical antigen-binding sites, it is divalent. The IgG antibodies are made up of four subclasses (IgG1, IgG2, IgG3, IgG4). This subclass distinction is based on amino acid sequence differences in the H chain constant region and on the number and location of disulfide bonds. IgG1 is 65% of the total IgG. IgG2 is directed against polysaccharide antigens and may be an important host defense against encapsulated bacteria. IgG3 is an effective activator of complement due to its rigid hinge region, while IgG4 does not activate complement due to its compact structure. IgG is the predominant antibody in secondary immune responses and constitutes an important defense against bacteria and viruses. It readily crosses the placenta and therefore is the most abundant immunoglobulin in newborns. It also mediates opsonization of antigen through binding of antigen-antibody complexes to Fc receptors on macrophages and other cells. B. IgM IgM is the main immunoglobulin produced early in the primary immune response. IgM is present on the surface of virtually all uncommitted B cells. It is composed of five H2L2 units (each similar to one IgG unit) and one molecule of J (joining) chain (Figure 8-6). The pentamer (MW 900,000) has S S- S-S S- S S–S S-S S-S S-S S–S - - S S S S S-S S-S S-S SS S-S S– S S S– S S- S-S S-S S S- C. IgA IgA is the main immunoglobulin responsible for mucosal immunity and it is found in secretions such as milk, saliva, and tears and in other secretions of the respiratory, intestinal, and genital tracts. It protects mucous membranes from attack by bacteria and viruses. Each secretory IgA molecule consists of two H2L2 units and one molecule each of J chain and secretory component. The latter is a protein derived from cleavage of the poly-Ig receptor. This receptor binds IgA dimers and facilitates their transport across mucosal epithelial cells. IgA exists in serum as a monomer H2L2 as well as larger polymers in low concentration. There are at least two subclasses, IgA1 and IgA2. Some bacteria (eg, Neisseria spp.) can destroy IgA1 by producing a protease and can thus overcome antibody-mediated resistance on mucosal surfaces. D. IgE The Fc region of IgE binds to its high-affinity receptor on the surface of mast cells, basophils, and eosinophils. This bound IgE acts as a receptor for the antigen that stimulates its production, and the resulting antigen–antibody complex triggers allergic responses of the immediate (anaphylactic) type through the release of mediators. E. IgD IgD acts as an antigen receptor when present on the surface of certain B lymphocytes. In serum it is present only in trace amounts. At the present time, the function of IgD is unclear. S S- S-S S-S S–S J chain a total of 10 identical antigen-binding sites and thus a valence of 10. It is the most efficient immunoglobulin in agglutination, complement fixation, and other antigen–antibody reactions and is important also in defense against bacteria and viruses. Since its interaction with antigen can involve all 10 binding sites, it has the highest binding capacity and cross linking of all immunoglobulins. Evaluating the presence of serum IgM may be useful in the diagnosis of certain infectious diseases. For example, IgM does not cross the placenta and the presence of the IgM antibody in the fetus or newborn provides evidence of intrauterine infection. FIGURE 8-6 Schematic diagram of the pentameric structure of human IgM. The IgM monomers are connected to each other and the J chain by disulphide bonds. Immunoglobulin Genes and Generation of Diversity Special genetic mechanisms have evolved to produce the very large number of immunoglobulin molecules (approximately 1011) that develop in the host in response to antigenic stimulation without requiring excessive numbers of genes. Thus, immunoglobulin genes (as reviewed below, T-cell receptor genes) undergo somatic recombination to produce the enormous diversity of antibody specificities. Each immunoglobulin chain consists of a variable (V) and a constant (C) region. For each type of immunoglobulin 136 SECTION II Immunology Immunoglobulin Class Switching Initially, all B cells matched to an antigen carry IgM specific for that antigen and produce IgM in response to this exposure to the antigen. Later, gene rearrangement permits elaboration of antibodies of the same antigenic specificity but of different immunoglobulin classes. In class switching, the same assembled VH gene can sequentially associate with different CH genes, so that the immunoglobulin produced later (IgG, IgA, or IgE) has the same specificity as the original IgM but different biologic characteristics. Class switching is dependent on cytokines released from T cells and also happens after antigenic stimulation. The Primary Response When an individual encounters an antigen for the first time, antibody to that antigen is detectable in the serum within days or weeks depending on the nature and dose of the antigen and the route of administration (eg, oral, parenteral). Secondary response to A Amount of antibody in serum chain—ie, kappa light chain (κ), lambda light chain (λ), and the five heavy chains (γH, μH, αH, δH, and εH)—there is a separate pool of gene segments located on different chromosomes. In humans the multigene families are found on the following chromosomes: λ, chromosome 22; κ, chromosome 2; and the heavy chain family, chromosome 14. Each of the three gene loci contains a set of different V gene segments separated from C gene segments. During B cell differentiation, the DNA is rearranged to bring the selected gene segments adjacent to each other in the genome. A family of enzymes known as the V(D)J recombinases is responsible for this gene rearrangement process. The variable region of each L chain is encoded by two gene segments: V and J. The variable region of each H chain is encoded by three gene segments: V, D, and J. The segments are united into one functional V-variable gene by DNA rearrangement. Each assembled V-variable gene is then transcribed with the appropriate C-constant gene to produce a messenger RNA (mRNA) that encodes for the complete peptide chain. L and H chains are synthesized separately on polysomes and finally assembled in the cytoplasm to form H2L2 units by means of disulfide bonds. The carbohydrate moiety is then added during passage through the membrane components of the cell (eg, Golgi apparatus), and the immunoglobulin molecule is released from the cell. This gene rearrangement mechanism permits the assembly of an enormous variety of immunoglobulin molecules. Antibody diversity depends on (1) multiple V, D, and J gene segments; (2) combinatorial association, that is, the association of any V gene segment with any D or J segment; (3) the random combining of different L and H chains; (4) somatic hypermutation; and (5) junctional diversity, created by imprecise joining during rearrangement with the addition of nucleotides by the enzyme terminal deoxynucleotidyl transferase. Time (months) Primary response to A 1 Injection of antigen A 2 Primary response to B 3 4 5 6 Second injection of antigen A, injection of antigen B FIGURE 8-7 Rate of antibody production following initial antigen administration and a second “booster” injection. The serum antibody concentration continues to rise for several weeks and then declines; it may drop to very low levels (Figure 8-7). The first antibodies formed are IgM, followed by IgG, IgA, or both. IgM levels tend to decline sooner than IgG levels. The Secondary Response In the event of a second encounter with the same antigen (or a closely related “cross-reacting” antigen) months or years after the primary response, the antibody response is more rapid and rises to higher levels than during the primary response. This change in response is attributed to the persistence of antigen-sensitive “memory cells” following the first immune response. In the secondary response, the amount of IgM produced is qualitatively similar to that produced after the first contact with the antigen; however, much more IgG is produced, and the level of IgG tends to persist much longer than in the primary response. Furthermore, such antibody tends to bind antigen more firmly (ie, to have higher affinity) and thus to dissociate less easily. Protective Functions of Antibodies Because of the close structural complementarity between antibodies and the antigen that elicited them, the two tend to bind to each other whenever they meet, in vitro or in vivo. This binding is non-covalent and involves electrostatic, van der Waals, and other weak forces. Antibodies can produce resistance to infection by five major mechanisms. Enhanced Phagocytosis—Antibodies produce resis- tance by opsonizing (coating) organisms, which make them more readily ingested by phagocytes. In addition, antibodymediated immunity against bacteria is most effective when directed against microbial infections in which virulence is related to polysaccharide capsules (eg, pneumococcus, CHAPTER 8 Immunology 137 Haemophilus spp., Neisseria spp.). In such infections, antibodies complex with the capsular antigens and make the organisms susceptible to ingestion by phagocytic cells and destruction within the cells. Virus Neutralization—Antibodies directed against specific viral proteins can bind to the virus and block the ability of the virus particle to attach to its cellular receptor. Since the virus cannot invade the cell, it cannot replicate. 3. Neutralization of Toxins—Antibodies can neutralize toxins of microorganisms (eg, diphtheria, tetanus, and botulism) and inactivate their harmful effects. Complement-Mediated Lysis—The attachment of antibodies to viral proteins on virus infected cells or to a microbial cell can activate the complement system leading to cell lysis. 5. Antibody-Dependent Cell Cytotoxicity (ADCC)— The attachment of antibodies to viral proteins on the virusinfected cell can lead to the interaction of the antibody-coated cells with a killer cell, leading to lysis. Since antibodies are protective, strategies have been developed to induce their production (Active Immunity) or to administer preformed antibodies to the host (Passive Immunity). A. Active Immunity Active immunity is induced after contact with foreign antigens (eg, microorganism or their products). This contact may consist of clinical or subclinical infection, immunization with live or killed infectious agents or their antigens, exposure to microbial products (eg, toxins, toxoids), or transplantation of foreign cells. In all these instances the host actively produces antibodies. However, protection is delayed until antibody production reaches an effective level. B. Passive Immunity Passive immunity is generated by administration of preformed antibodies. The primary advantage of passive immunization with preformed antibodies is the prompt availability of large amounts of antibody. This approach can be useful against certain viruses (eg, hepatitis B virus) after a needlestick injury to someone who has not been vaccinated. In addition to antibody-mediated protective effects, harmful effects can be seen. In passive immunity it is possible to initiate hypersensitivity reactions if the antibody is from another species. In active immunity, the binding of antibodies to antigen leads to the formation of circulating immune complexes. The deposition of these complexes may be an important feature in the development of organ dysfunction. For example, immune complexes may deposit in the kidney and induce glomerulonephritis which can result following streptococcal infections. T Cells A. Cell-Mediated Immunity Antibody-mediated immunity is important in toxin-induced disorders, in microbial infections in which polysaccharide capsules determine virulence, and as a part of the host defense response to some viral infections. However, cellmediated immunity is central in host defense against a variety of pathogens. In cell-mediated immunity, T cell subsets and APCs recognize and respond to the pathogen. In this section we review T-cell development, T-cell proliferation and differentiation, and finally effector T cell functions. Development of T Cells—Within the thymus, T cell progenitor cells undergo differentiation. Here, under the influence of thymic hormones Tcells differentiate into committed cells expressing a specific T-cell receptor (TCR). These TCR have undergone VDJ recombination of their beta-chain TCR DNA followed by rearrangement of their alpha-chain TCR DNA creating a functional alpha-beta TCR complex. Most cells contain α, β TCR, however, some T cells express gamma-delta TCR (γ, δ T cells). These TCR cells now become positive for the expression of both the CD4 and CD8, co-receptor molecules. At this point, T cells now possess a functional TCR. Next, the T cells undergo positive and negative selection processes that result in the retention of only those cells with the most useful antigen receptors, that is those that are nonself (foreign) antigen-specific. These antigens can be recognized only in the context of self MHC (self MHC-restricted ). Those clones that are potentially anti-self antigen are either deleted or functionally inactivated (made anergic). A consequence of the selection process is that about 95% of T cells die in the thymus. As the cells undergo a final maturation, they lose their expression of either CD4 or CD8 and become single positive T cells which now express TCR/CD3 complex and either CD4 or CD8. Only a minority of developing T cells express the appropriate receptors to be retained and to exit into the periphery where they may mature into effector T cells. 2. T-Cell Receptor for Antigen—The T-cell receptor is a transmembrane heterodimeric protein composed of two disulfide-linked chains. As mentioned above, there are two different classes of T cell receptors, α and β in one class and γ and δ in the other. The αβ T cells make up the predominant T cell phenotype and are subdivided by their expression of other cell surface markers, CD4 and CD8. T cells that express γδ are relatively infrequent in humans and seem to be predisposed toward recognition of frequently encountered bacterial antigens-for example, certain glycolipid and phosphorylated lipid moieties. The γδ T cells are primarily located in the epithelial linings of the reproductive and gastrointestinal tracts. The T-cell receptor proteins have variable and constant regions similar to antibodies. The variable regions are located at the amino terminals of the polypeptide chain farthest away from the cell membrane. Both chains contribute to the 138 SECTION II Immunology variable domain that has been shown to interact with antigen presented by self-MHC complex. The T-cell receptor genes closely resemble immunoglobulin genes, and the generation of diversity in the T-cell receptor is accomplished in a fashion largely analogous to that described earlier for immunoglobulins. Thus, there are multiple variable region segments, contributing a repertoire of different antigen specificities; multiple V, D, and J segments that can combine in different ways. There are more J and D segments for T-cell receptor genes than for immunoglobulin genes. In all functional antigen-specific T cells, the two T-cell receptor chains are noncovalently associated with six other polypeptide chains composed of four different proteins that make up the CD3 complex. The invariant proteins of the CD3 complex are responsible for transducing the signal received by the T-cell receptor on recognition of antigen to the inside of the cell. All four different proteins of the CD3 complex are transmembrane proteins that can interact with cytosolic tyrosine kinases on the inside of the membrane. It is this interaction that begins the biochemical events of signal transduction leading to gene transcription, cell activation, and initiation of the functional activities of T cells. The CD4 and CD8 molecules that differentiate the two major functional classes of T cells function as co-receptor molecules on the T cell surface. During recognition of antigen, the CD4 and CD8 molecules interact with the T-cell receptor complex and with MHC molecules on the APC. CD4 binds to MHC class II molecules, and CD8 binds to MHC class I molecules. T-Cell Proliferation and Differentiation—T-cell proliferation depends on a series of events. In MHC class II presentation two signals are required for the naïve CD4 T cell activation to occur. The first signal comes from the T-cell receptor on the T cell interacting with an MHC-peptide complex presented on the APC. The CD4 glycoprotein on the naïve T cell acts as a co-receptor, binding to MHC class II molecules. This binding event ensures stability between the T cell and the APC. The second signal (costimulation) that is required for T cell activation is derived from the interaction of the B7 family costimulatory molecules (B7-1/B7-2 also identified as CD80 and CD86) on the APC with CD28 on the T cell. These are the major costimulatory molecules. Upon completion of these two stimulation steps (eg, T-cell receptor binding to MHC class II–peptide complex and CD28 binding to B7-1/B7-2), a set of biochemical pathways are triggered in the cell that results in DNA synthesis, mitosis, and proliferation. During these events, the T cell secretes cytokines, mainly IL-2 and IFN-γ, and increases the expression of IL-2 receptors. These T cells are able to proliferate and differentiate into effector cells. CD 8 T-cell activation occurs when the T-cell receptor interacts with the MHC class I–peptide complex on the infected cell. The CD8 glycoprotein on the T cell acts as a co-receptor, binding to MHC class I molecule on the APC. Again, this interaction keeps the two cells bound together during antigen-specific activation. Once activated, the cytotoxic T cell produces IL-2 and IFN-γ, growth and differentiation factors for T cells. Unlike CD4 cell activation, CD8 T-cell activation in most cases is independent of co-stimulation, and the virus-infected cell is destroyed through cytotoxic granules released from the CD 8 T cell. B. T-Cell Effector Functions 1. CD4 Effector Cells—Proliferating CD4 T cells can become one of four main categories of effector T cells: Th1 cells, Th2 cells, Th17 cells, or regulatory T (Treg) cells. In an environment of IFN-γ, Th1 cells dominate and either activate macrophages or cause B cells to switch to produce different subclasses of IgG. In either case, this can promote bacterial clearance either by direct destruction in the IFN-γ activated macrophage or by destruction after phagocytosis of opsonized particles. These Th1 cells also produce IL-2 and IFN-γ. In an environment where IL-4 is being produced, Th2 cells predominate and activate mast cells and eosinophils and cause B cells to synthesize IgE. This aids in the response to helminths. The Th2 cells secrete IL-4, IL-5, IL-9, and IL-13. When TGF-β and IL-6 are present together, CD4 T cells can become Th17 cells. These cells produce IL-17 and IL-22. IL-17 is a cytokine that induces stromal and epithelial cells to produce IL-8. IL-8 is a potent chemokine that is responsible for the recruitment of neutrophils and macrophages to infected tissues. CD4 T cells can become Tregs when they are exposed to TGF-β alone. Treg cells function by suppressing T-cell responses. They are identified by expression of CD4 and CD25 molecules on their surface and the transcription factor, Foxp3. T reg cells produce TGF-β and IL-10, which can suppress immune responses. CD8 Effector Cells—CD8 cells differentiate into effector cytotoxic cells by engagement of their TCR and recognition of class I MHC–peptide complex on the surface on an infected cell. Following recognition, the CD8 T cell proceeds to kill the infected cell. The primary method of killing is through cytotoxic granules containing perforin, the family of granzymes, and a third protein recently identified, granulysin. The CD8 T cell releases perforin that helps granzyme and granulysin enter the infected cell. Granzyme initiates apoptosis (programmed cell death) by activating cellular caspases. It should be noted that a similar process occurs with CD8 T-cell recognition of tumor cells. For additional information on this topic, see Murphy et al (2011). COMPLEMENT The complement system includes serum and membranebound proteins that function in both innate and adaptive host defense systems. These proteins are highly regulated and interact via a series of proteolytic cascades. Several complement components are proenzymes, which must be cleaved to form active enzymes. The components of the classical pathway are numbered from C1 to C9, and the reaction sequence is C1-C4-C2-C3-C5-C6-C7-C8-C9. CHAPTER 8 Immunology 139 Biologic Effects of Complement Activation of complement results in four major outcomes: (1) cytolysis, (2) chemotaxis, (3) opsonization, and (4) anaphylatoxins. 1. Cytolysis is the lysis of cells, such as bacteria, virusinfected cells, and tumor cells. This process occurs through the development of the membrane attack complex (MAC) (C5b, 6, 7, 8, 9), which inserts into the membrane of an organism or cell. The MAC leads to loss of osmotic integrity and cell lysis. 2. Chemotaxis is a process in which an immune cell, usually a phagocyte, is attracted to and moves toward a soluble factor. For example, C5a is a potent chemotactic agent that stimulates movement of neutrophils and monocytes toward sites of antigen deposition. 3. Opsonization is a process in which microorganisms and antigen–antibody complexes become coated with molecules that bind to receptors on phagocytes. Phagocytosis is more efficient in the presence of C3b because of the presence of C3b receptors on phagocytes. 4. Anaphylatoxins promote vasodilation and increased vascular permeability. Both C3a and C5a are potent promoters of vasodilation and vascular permeability. These two complement components also stimulate mast cells and basophils to release histamine. This function of complement results in an increased blood flow to the site of infection, allowing more complement, antibodies, and immune cells to enter the site of infection. Complement Pathways There are three major pathways that activate complement, the classical pathway, alternative pathway, and mannan-binding lectin (MBL) pathway (Figure 8-8). Each of these pathways results in the formation of the MAC. All three pathways lead to the release of C5 convertase, which breaks down C5 into C5a and C5b. As mentioned previously, C5a is an anaphylatoxin as well as a chemotactic factor. C5b binds to C6 and C7 to form a complex that inserts into the membrane bilayer. C8 then binds to the C5b–C6–C7 complex followed by polymerization of up to 16 C9 molecules to produce the MAC. The MAC now generates a channel or pore in the membrane and causes cytolysis by allowing free passage of water across the cell membrane. The Classical Pathway C1, which binds to the Fc region of an immunoglobulin, is composed of three proteins, C1q, C1r, and C1s. C1q is an aggregate of polypeptides that bind to the Fc portion of IgG and IgM. The antibody–antigen immune complex bound to C1 activates C1s, which cleaves C4 and C2 to form C4b2b. The latter is an active C3 convertase, which cleaves C3 molecules into two fragments, C3a and C3b. C3a is an anaphylatoxin. C3b forms a complex with C4b2b, producing a new MB Lectin pathway Classic pathway Microbial surfaces Immune complex Microbial surfaces MBL Activated C1 C4 C2 Alternative pathway [C4b2b] C3 [C3bBb] Factor B Factor D Properdin C3 convertases C3 C3a C3b [C4b2bC3b] [C3bBbC3b] Anaphylatoxins C5 convertases C5 C5a C5b C6,C7,C8,C9 C5b–C9 Membrane attack complex Cell lysis FIGURE 8-8 Complement reaction sequence. enzyme, C5 convertase, which cleaves C5 to form C5a and C5b. This leads to MAC and then to cell lysis. Only IgM and IgG activate or fix complement via the classical pathway. Furthermore, only IgG subclass 1, 2, and 3 fix complement; IgG4 does not. An example of the classical complement pathway in action can be observed in Herpes simplex virus (HSV) infections. HSV replication within cells is accompanied by the insertion of virus proteins on the cell surface. A specific antiHSV antibody can bind to the surface of the infected cell by its Fab site. The Fc portion of the antigen–antibody complex is now exposed and ready for C1 to attach. The classical pathway is now activated and the infected cell is destroyed by MAC. The Alternative Pathway Infectious agents can activate the complement system by triggering the cellular production of factors B, D, and properdin. These factors cleave C3 and generate C3 convertase. C3 140 SECTION II Immunology convertase (C3bBb) that was generated during the alternative pathway produces more C3b. The additional C3b binds to the C3 convertase to form C3bBbC3b. This enzyme is the alternative pathway C5 convertase that generates C5b, leading to production of the MAC described earlier. Mannan-Binding Lectin Pathway In recent years, the concept of an additional pathway of complement activation has emerged, and a third pathway has been identified as the MBL pathway. Its main constituent is a plasma protein MBL. MBL binds to sugar residues such as mannose found in microbial surface polysaccharides such as LPS. The MBL complex, when bound to microbial surfaces, can activate C4 and C2. The remaining steps in the pathway are the same as the classic pathway of complement activation. A. Regulation of the Complement System Several serum proteins regulate the complement system at different stages: (1) C1 inhibitor binds to and inactivates the serine protease activity of C1r and C1s; (2) factor I cleaves C3b and C4b, thereby reducing the amount of C5 convertase available; (3) factor H enhances the effect of factor I on C3b; and (4) factor P (properdin) protects C3b and stabilizes the C3 convertase of the alternative pathway. Regulation is also provided by proteins that have the ability to accelerate the decay of the complement proteins, such as decay-accelerating factor, a membrane-bound protein found on most blood cell surfaces that can act to accelerate dissociation of the C3 convertases of all three pathways. Complement Deficiencies and Pathogen Evasion Many genetic deficiencies of complement proteins have been described, and these generally lead to enhanced susceptibility to infectious disease (eg, C2 deficiency frequently leads to serious pyogenic bacterial infections). Deficiency in components of the MAC greatly enhances susceptibility to neisserial infections. Deficiencies in components of the alternative pathway are also known (eg, properdin deficiency is associated with greater susceptibility to meningococcal disease). There are also deficiencies in complement regulating proteins. For example, lack of the C1 inhibitor protein leads to hereditary angioedema. The complement system is an important host protective system. Some pathogens have evolved a mechanism to evade it. For example, some microbes have developed surfaces that interfere with opsonization by C3b or interfere with insertion of the MAC. Complement activation can also be inhibited by the presence of microbial generated proteins, such as, protein A, and protein C, that bind IgG Fc. Finally, microbes can generate enzymes that degrade complement components. Organisms that possess these inhibitory properties are usually more pathogenic. CYTOKINES Over the last two decades, we have witnessed an explosion in cytokine biology. Cytokines are potent, low-molecularweight protein cell regulators produced transiently and locally by numerous cell types. Today we recognize that cytokines are multifunctional proteins whose biological properties suggest a key role in hematopoiesis, immunity, infectious disease, tumorigenesis, homeostasis, tissue repair, and cellular development and growth. Cytokines usually act as signaling molecules by binding to their own glycoprotein receptors on cell membranes. This initial interaction is followed by a relay of the signal to the cell nucleus. Signal transduction is mediated as in many hormone-receptor systems via kinase-mediated phosphorylation of cytoplasmic proteins. In fact, tyrosine kinase activity is intrinsic to many cytokine receptors. Classification and Functions Cytokines can be categorized into groups based on their common functions. Examples of functional categories are immunoregulatory, proinflammatory, anti-inflammatory, and growth and differentiation. Because of its major role in antigen presentation, an important immunoregulatory cytokine is IFN-γ. Proinflammatory cytokines are commonly seen in infectious diseases, and they include IL-1, IL-6, TNFα, and the IFNs. The anti-inflammatory cytokines include TGF-β, IL-10, IL-11, and IFN-β. These may be required to dampen or downregulate an overactive inflammatory response. Cytokines that have a key role in growth and differentiation include the colony stimulation factors (CSFs) and stem cell factor. Selected cytokines, their sources, and their main activities are identified in Table 8-1. We have also seen that T cells use cytokines for differentiation into T-cell subsets. Whereas Th1 cells are generated in the presence of IFN-γ, Th2 cells are differentiated in the presence of IL-4. Th17 cells are produced in the presence of TGF-β and IL-6, but Treg cells are formed in the presence of TGF-β alone. Each of these T-cell subsets now secretes its own set of cytokines that have distinct regulatory properties. Thus, cytokines orchestrate the type of immune response that is generated. Clinical Applications Today there are at least four major clinical applications of cytokines. First, cytokines can serve as biomarkers of disease and provide clues for mechanisms of disease. For example, the proinflammatory cytokines TNF-α, IL-1, and IL-6 can be detected in the sera of patients with septic shock. These cytokines appear to play a critical role in the development of septic shock, and tracking their presence may be of prognostic value in severe sepsis. Second, the measurement of cytokine production in vitro is a useful monitor of immune status. T-cell function can be monitored by the ability of the T cells to produce IFN-γ. This is currently being used to CHAPTER 8 Immunology 141 identify tuberculosis (TB) reactivity and is discussed later. Third, recombinant cytokines are key therapeutic agents. An example of this is seen with the IFN molecules. The FDA has approved the use of IFN-α for hepatitis C infections, IFN-β for multiple sclerosis, and IFN-γ for CGD. Fourth, cytokines can be targets of therapeutics. Recently, cytokine receptor antagonists and anti-cytokine monoclonal antibodies both which downregulate pathogenic responses to exaggerated cytokine production have been used as effective treatments. Examples of this approach are the inhibitors of TNF-α used to manage rheumatoid arthritis (RA) and inhibitors of IL-2 and IL-15 used in transplantation and cancer. HYPERSENSITIVITY The term hypersensitivity denotes a condition in which an immune response results in exaggerated or inappropriate reactions that are harmful to the host. In a given individual, such reactions typically occur after the second contact with a specific antigen (allergen). The first contact is a necessary preliminary event that induces sensitization to that allergen. There are four main types of hypersensitivity reactions. Types I, II, and III are antibody mediated, and type IV is T-cell mediated. Type I: Immediate Hypersensitivity (Allergy) Type I hypersensitivity manifests itself in tissue reactions occurring within seconds after the antigen combines with the specific IgE antibody. Its symptoms may manifest as a systemic anaphylaxis (eg, after intravenous administration of heterologous proteins) or as a local reaction (eg, an atopic allergy involving rhinitis such as occurs with hay fever). The general mechanism of immediate hypersensitivity involves the following steps. An antigen induces the formation of IgE antibody, which binds firmly by its Fc portion to high-affinity IgE receptors on mast cells, basophils, and possibly eosinophils. Some time later, a second exposure of the individual with the same antigen results in the antigen’s binding to cell-bound IgE, cross-linking of IgE molecules, and inducing the release of pharmacologically active mediators from cells within seconds to minutes. Cyclic nucleotides and calcium are essential in the release of mediators. There may also be a secondary “late phase” response that lasts for several days and involves infiltration of tissues with leukocytes, particularly eosinophils. Some important mediators of type I hypersensitivity and their main effects are listed below. Histamine—Histamine exists in a preformed state in platelets and in granules of mast cells, basophils, and eosinophils. Its release causes vasodilation, increased capillary permeability, and smooth muscle contraction (eg, bronchospasm). Antihistamine drugs can block histamine receptor sites and are relatively effective in allergic rhinitis. Histamine is one of the primary mediators of a type I reaction. Prostaglandins and leukotrienes—Prostaglandins and leukotrienes are derived from arachidonic acid via the cyclooxygenase pathway. Prostaglandins chiefly produce bronchoconstriction. Leukotrienes chiefly cause increased permeability of capillaries. These mediators, along with cytokines such as TNF-α and IL-4, are referred to as secondary mediators of a type I reaction. A. Treatment and Prevention of Anaphylactic Reactions Treatment aims to reverse the action of mediators by maintaining the airway, providing artificial ventilation if necessary, and supporting cardiac function. One or more of the following may be given: epinephrine, antihistamines, and corticosteroids. Prevention relies on identification of the allergen (often by skin test or IgE antibody serology) and subsequent avoidance. B. Atopy Atopic hypersensitivity disorders exhibit a strong familial predisposition and are associated with elevated IgE levels. Predisposition to atopy is clearly genetic, but symptoms are induced by exposure to specific allergens. These antigens are typically environmental (eg, respiratory allergy to pollens, ragweed, or house dust) or foods (eg, intestinal allergy to shellfish). Common clinical manifestations include hay fever, asthma, eczema, and urticaria. Many patients experience immediate-type reactions to skin tests (injection, patch, scratch) involving the offending antigen. Type II: Hypersensitivity Type II hypersensitivity involves the binding of IgG antibodies to cell surface antigens or extracellular matrix molecules. Antibody directed at cell surface antigens can activate complement (or other effectors) to damage the cells. The result may be complement-mediated lysis, which occurs in hemolytic anemias, ABO transfusion reactions, and Rh hemolytic disease. Drugs such as penicillin can attach to surface proteins on red blood cells and initiate antibody formation. Such autoimmune antibodies may then combine with the cell surface, with resulting hemolysis. In Goodpasture syndrome, antibody forms against basement membranes of kidney and lung, resulting in severe damage to the membranes through activity of complement-attracted leukocytes. In some cases, antibodies to cell surface receptors alter function without cell injury (eg, in Graves disease, an autoantibody binds to the thyroid-stimulating hormone receptor, stimulates the thyroid, and causes hyperthyroidism). 142 SECTION II Immunology Type III: Immune Complex Hypersensitivity When antibody combines with its specific antigen, immune complexes are formed. Normally, they are promptly removed, but occasionally, they persist and are deposited in tissues, resulting in several disorders. In persistent microbial or viral infections, immune complexes may be deposited in organs (eg, the kidneys), resulting in dysfunction. In autoimmune disorders, “self” antigens may elicit antibodies that bind to organ antigens or are deposited in organs and tissues as complexes, especially in joints (arthritis), kidneys (nephritis), and blood vessels (vasculitis). Finally, environmental antigens such as fungal spores and certain drugs can cause immune complex formation with disease. Wherever immune complexes are deposited, they activate the complement system, and macrophages and neutrophils are attracted to the site, where they cause inflammation and tissue injury. There are two major forms of immune complex-mediated hypersensitivity. One is local (Arthus reaction) and typically elicited in the skin when a low dose of antigen is injected and immune complexes form locally. IgG antibodies are involved, and the resulting activation of complement leads to activation of mast cells and neutrophils, mediator release, and enhanced vascular permeability. This typically occurs in about 12 hours. A second form of type III hypersensitivity involves systemic immune complex disease. There are several examples, including diseases such as acute poststreptococcal glomerulonephritis. Acute poststreptococcal glomerulonephritis is a wellknown immune complex disease. Its onset occurs several weeks after a group A β-hemolytic streptococcal infection, particularly of the skin, and often occurs with infection due to nephritogenic types of streptococci. The complement level is typically low, suggesting an antigen–antibody reaction with consumption of complement. Lumpy deposits of immunoglobulin and complement component C3 are seen along glomerular basement membranes stained by immunofluorescence, suggesting antigen–antibody complexes. It is likely that streptococcal antigen–antibody complexes are filtered out by glomeruli, fix complement, and attract neutrophils. This series of events results in an inflammatory process that damages the kidney. Type IV: Cell-Mediated (Delayed) Hypersensitivity Cell-mediated hypersensitivity is a function not of antibody but of specifically sensitized T lymphocytes that activate macrophages to cause an inflammatory response. The response is delayed— it usually starts 2 or 3 days after contact with the antigen and often lasts for days. A. Contact Hypersensitivity Contact hypersensitivity occurs after sensitization with simple chemicals (eg, nickel, formaldehyde), plant materials (poison ivy, poison oak), topically applied drugs (eg, sulfonamides, neomycin), some cosmetics, soaps, and other substances. In all cases, small molecules enter the skin and then, acting as haptens, attach to body proteins to serve as complete antigens. Cell-mediated hypersensitivity is induced, particularly in the skin. When the skin again comes in contact with the offending agent, the sensitized person develops erythema, itching, vesication, eczema, or necrosis of skin within 12-48 hours. Patch testing on a small area of skin can sometimes identify the offending antigen. Subsequent avoidance of the material will prevent recurrences. The APC in contact sensitivity is probably the Langerhans cell in the epidermis, which interacts with CD4 Th1 cells that drive the response. B. Tuberculin-Type Hypersensitivity Delayed hypersensitivity to antigens of microorganisms occurs in many infectious diseases and it has been used as an aid in diagnosis. It is typified by the tuberculin reaction. When a small amount of tuberculin is injected into the epidermis of a patient previously exposed to Mycobacterium tuberculosis, there is little immediate reaction. Gradually, however, induration and redness develop and reach a peak in 24–72 hours. Mononuclear cells accumulate in the subcutaneous tissue, and there are CD4 Th1 cells in abundance. A positive skin test indicates that the person has been infected with the agent but does not imply the presence of current disease. However, a recent change of skin test response from negative to positive suggests recent infection and possible current activity. A positive skin test response assists in diagnosis. For example, in leprosy, a positive skin test indicates tuberculoid disease with active cell-mediated immunity, whereas a negative test suggests lepromatous leprosy with weak cell-mediated immunity. DEFICIENCIES OF THE IMMUNE RESPONSE Immunodeficiency Diseases Immunodeficiency can be divided into two categories, primary immunodeficiency diseases and secondary immunodeficiency diseases. Primary immunodeficiency diseases consist of disorders of the immune system in which the defect is intrinsic to the cells of the immune system. Secondary immunodeficiency diseases consist of disorders of the immune system in which the defect is induced by external factors, such as viruses, malignancy, and drugs. This section is clearly relevant to medical microbiology because the primary immunodeficiency diseases are usually identified first by the type, duration, and frequency of infectious diseases. In contrast, secondary immunodeficiency diseases are frequently induced by microorganisms. A. Primary Immunodeficiencies Primary immunodeficiencies usually have a genetic basis, and more than 150 genetically based diseases have been CHAPTER 8 Immunology 143 identified. The genetic defect results in the loss of number or function of B cells, T cells, or phagocytic cells, complement components, cytokines, or TLRs. Clearly, the loss of these functional elements leads to increased susceptibility to infections. One example, is chronic granulomatous disease (CGD), which is an impairment of phagocytic cell function. Patients have normal levels of immunoglobulins, T and B cells, and phagocytic cells. However, the phagocytic cells do not kill microbes a due to genetic defect in cytochrome b-558. This leads to a metabolic defect in the ability of phagocytic cells to produce peroxide and superoxide. The phagocytic defect can be identified by using the nitroblue tetrazolium (NBT) test. These cells are unable to efficiently kill some bacteria or fungi, such as, staphylococcus, Escherichia coli, and Aspergillus spp. Unless treated, this disease is usually fatal within the first decade of life. IFN-γ has been shown to restore phagocytic function in these cells. Therefore, in most cases, administration of IFN-γ or bone marrow transplantation are effective treatments. A second example is severe combined immunodeficiency (SCID). This syndrome is now known to be the final expression of several different genetic defects leading to defects in both B- and T-cell function. These individuals are susceptible to infection by virtually any microbe, and if untreated, they will die within the first year of life. B. Secondary Immunodeficiencies Secondary immunodeficiencies are a major predisposing cause of infection. Secondary immunodeficiency states are associated with infections, malignancies, and drugs. C. Infections Infections can also induce immunosuppression in the host. It is well known that measles virus infection and Epstein-Barr virus (EBV) infection in mononucleosis result in a depression of the delayed-type hypersensitivity (DTH) skin test. Analysis of EBV replication has revealed a possible mechanism for this immunosuppression. EBV infects B cells, resulting in a transformed cell that proliferates indefinitely. Interestingly, the EBV genome codes for a human IL-10 analog. IL-10 is an immunosuppressive cytokine that inhibits Th1 cells from proliferating and producing cytokines, such as IFN-γ. This may account for the negative DTH skin test. The most obvious example of a virus-induced immunodeficiency is HIV infection and the resulting disease, AIDS. HIV primarily infects CD4 T cells. This is possible because the virus uses the CD4 molecule itself as the virus receptor and the chemokine receptor, CCR5, as a co-receptor to enter the cell. HIV replication in CD4 T cells leads to a progressive loss of CD4 T cells and the development of AIDS. As a consequence of this infection, HIV patients develop multiple opportunistic infections. As noted in this chapter, CD4 T cells are critically important in generating Th1, Th2, Th17, and Treg. They also provide help to B cells during antibody production and serve as a source of IL-2 and IFN-γ. Replication of a cytotoxic virus in this cell is devastating to the immune response. D. Malignancy Selected leukemias, lymphomas, multiple myeloma, and other cancers can lead to immunodeficiency and increased infections. For example, patients with leukemia can have a deficiency in neutrophils, which results in loss of phagocytosis and increased infections with bacteria and fungi. Some tumors secrete high levels of TGF-β that can suppress a variety of responses, including Th1 responses. E. Drugs Cytotoxic drugs used to treat cancer (eg, cisplatin), immunosuppressive drugs (eg, cyclosporine) that are used to manage transplant patients, and newer anti-cytokine (anti–TNF-α) drugs used to treat autoimmune diseases (eg, RA) can lead to increased risk of infection. CLINICAL IMMUNOLOGY LABORATORY (DIAGNOSTIC TESTING) Exciting discoveries in molecular biology, recombinant DNA and proteins, cytokine biology, and human genetics have enhanced our understanding of immune-mediated diseases. With these advancements, clinical laboratory immunology has matured, and its applications have increased extensively. Hence, the scope of the clinical immunology laboratory now extends to a wide variety of disorders, such as transplantation, rheumatology, oncology, dermatology, infectious disease, allergy, and immunodeficiencies. The goal of the clinical immunology laboratory is to provide laboratory testing to support the diagnosis and monitoring of patients with immune disorders. A variety of technologies are used to evaluate both the antibody and cellular components of the immune response. For a comprehensive review of current test systems used the clinical immunology hospital setting, see Detrick et al (2006). Selected assays are highlighted below: Antibody Evaluation Assays A. Enzyme-Linked Immunosorbent Assay Enzyme immunoassay, which has many variations, depends on the conjugation of an enzyme to an antibody. The enzyme is detected by assaying for enzyme activity with its substrate. To measure antibody, known antigens are fixed on a solid phase (eg, plastic microtiter plate), incubated with test antibody dilutions, washed, and reincubated with an antiimmunoglobulin labeled with an enzyme (eg, horseradish peroxidase). Enzyme activity, measured by adding the specific substrate and estimating the color reaction, is a direct function of the concentration of antibody bound. This serologic test is used to detect antibodies to a number of infectious diseases, such as antibodies to HIV proteins in blood samples or antibodies to the syphilis organism, Treponema pallidum. This assay system is also widely used to detect autoantibodies present in the circulation of patients with systemic and organ-specific 144 SECTION II Immunology autoimmune diseases (eg, antibodies in systemic lupus erythematosus, scleroderma, or Sjogren’s syndrome). Variations of the enzyme-linked immunosorbent assay (ELISA) include some of the newer technologies, such as chemiluminescence assay (CIA) and multiplex particle-based assays. B. Immunofluorescence Fluorescent dyes (eg, fluorescein, rhodamine) can be covalently attached to antibody molecules and made visible by ultraviolet light in the fluorescence microscope. Such labeled antibody can be used to identify antigens (eg, on the surfaces of bacteria such as streptococci or treponema) or in cells in histologic section or other specimens. A direct immunofluorescence reaction occurs when known labeled antibody interacts directly with unknown antigen. An indirect immunofluorescence reaction occurs when a two-stage process is used (eg, a known antigen is attached to a slide, unknown serum is added, and the preparation is washed). If the unknown serum antibody binds to the antigen, it will remain fixed to it on the slide and can be detected by adding a fluorescent-labeled anti-immunoglobulin or other antibody-specific reagent such as staphylococcal protein A or G and examining the slide by ultraviolet microscopy. This assay historically has been used to detect antibodies to certain microorganisms (eg, T pallidum) and is the standard procedure for the detection of autoantibodies in autoimmune diseases (eg, antinuclear antibodies). C. Immunoblotting Immunoblotting (sometimes called Western blotting) is a method for identifying a particular antigen in a complex mixture of proteins. The complex mixture of proteins is subjected to sodium dodecyl sulfate (SDS)–polyacrylamide gel electrophoresis (PAGE). This separates the proteins according to charge and molecular size. The gel is then covered with a membrane (often a sheet of nitrocellulose), and the proteins are “transferred” by electrophoresis to the membrane. The nitrocellulose membrane (blot) acquires a replica of the proteins separated by SDS-PAGE. During the transfer, the SDS is largely removed from the proteins and, at least for some proteins, there is refolding, and enough conformation is restored so that antibodies can react with the proteins on the membrane. The nitrocellulose membrane is then reacted with an enzyme-labeled antibody in a direct test or in an indirect test, with antibody followed by an enzyme-labeled antiimmunoglobulin. The protein antigen then becomes visible as a band on the membrane. None of the other proteins in the mixture are detected. This technique is used, such as to confirm an HIV-positive ELISA test, by demonstrating the presence of antibodies to specific HIV proteins in a patient’s serum. It is also widely used as a secondary test for HCV and Lyme disease. More recently, this technology is being applied to identification of autoantibodies in selected autoimmune diseases (eg, polymyositis). Variations of the immunoblotting techniques include dot or slot blot assays, both of which use purified antigens. D. Other Laboratory Assays Other technologies often available in the clinical immunology laboratory include protein electrophoresis and immunofixation electrophoresis, which are essential tests used to identify abnormal immunoglobulin production in the serum or urine of patients with myeloma. Nephelometry is another laboratory test that quantifies a wide variety of analytes in serum or plasma. This is the method of choice for quantitating complement components, immunoglobulins, and other serum analytes. These assays can also be used to evaluate other abnormalities associated with selected infectious diseases (eg, HCV can be associated with a monoclonal protein and the presence of cryoglobulins). Evaluation of Cellular Responses A. Flow Cytometry Another use of fluorescent-tagged antibody molecules is to count and classify cells by flow cytometry using a fluorescence-activated cell sorter (FACS). Flow cytometry analyzes a single-cell suspension flowing through a set of laser beams to measure the relative amount of light scattered by microscopic particles (providing information on relative size and granularity) and the relative fluorescence of those particles. For a mixture of white blood cells, it is relatively easy to separate the cells in this mixture into major classes, such as small lymphocytes separated from granulocytes that are larger and contain more granules (scatter more light). With the availability of panels of monoclonal antibodies (that can be detected by fluorescent anti-immunoglobulin) to cell surface proteins, it is also possible to count subpopulations of cells, such as CD4 expressing helper T cells from CD8 expressing cytotoxic T cells or antibody expressing B cells from T cells. This technology is widely used both in clinical laboratory and in biomedical research (eg, to enumerate CD4 T cells in HIV-positive patients or to distinguish tumor cells from normal white blood cells). B. Functional Cellular Assays In order to measure T-cell function in vitro, the cells ability to proliferate or produce specific cytokines, such as IFN-γ, are analyzed. This assay is the in vitro counterpart of type IV hypersensitivity reactions, with TB skin test as a model. In the skin, the administered TB antigen interacts with specific T cells to proliferate, produce IFN-γ, and give a positive skin reaction. In this in vitro assay, peripheral blood leukocytes (PBLs) are incubated with a specific antigen for 24–72 hours. When specifically sensitized T cells in the PBLs interact with their specific antigen (eg, TB antigen), the cells will proliferate and produce IFN-γ. Proliferation can be measured by H3 thymidine incorporation, or IFN-γ production can be CHAPTER 8 Immunology 145 monitored by ELISA or flow cytometry. This assay can be used to assess the immune status of an individual, particularly patients who are immunocompromised because of an infectious disease, malignancy, or drug therapy. • CHAPTER SUMMARY • • • • • • Innate immunity is an immediate, nonspecific response to the pathogen. The components of this response include phagocytic cells (macrophages and neutrophils), NK cells, TLRs, cytokines, and complement. Protective functions of phagocytic cells: Phagocytosis, primarily by macrophages and PMNs, is a major mechanism of detecting and destroying pathogens. The process includes the following steps: chemotaxis, migration, ingestion, and microbial killing. Adaptive immunity can be antibody mediated, cell mediated, or both. It is specific for the pathogen and can confer protective immunity to re-infection with that pathogen. Antigen presentation is a critical component of adaptive immunity. Proteins from exogenous antigens are processed by APCs and then returned to the cell surface as an MHC class II–peptide complex. This complex is recognized by a T-cell receptor on a CD4 T cell. CD4 molecule acts as a co-receptor. A second signal required for Tcell activation is derived from the interaction of CD80 on the APC with CD28 on the T cell. T cells now proliferate and differentiate into effector T cells. Endogenous antigens are processed by APCs via an MHC class I peptide complex. This complex is recognized by a TCR on CD8 T cells. Antibody production: B cells rearrange immunoglobulin genes and express a receptor (BCR) for antigen. When antigen interacts with BCR, the B cell is stimulated to divide and form a clone. The B cell differentiates to become plasma cells and secrete antibody or memory B cells. Functions of antibody: Antibody can enhance phagocytosis, induce neutralization of viruses and bacterial toxins, and participate in complement-mediated lysis and ADCC. • • • Functions of T cells. (1) CD4 T cells can become Th1, Th2, Th17, or Treg cells. Th1 cells can produce cytokines (IL-2, IFN-γ), activate macrophages, or trigger B cell switching to IgG synthesis. Th2 cells activate mast cells and eosinophils and trigger B-cell switching to IgE synthesis. Th17 cells can produce IL-17 triggering production of IL-8 and recruitment of neutrophils and macrophages. Treg cells produce TGF-β and IL-10, which can suppress immune responses. (2) CD8 T cells function as cytotoxic T cells. There are three major ways to activate the complement cascade. Each pathway results in the formation of MAC, leading to cell lysis. Complement provides protection from pathogens by four mechanisms: (1) cytolysis, (2) chemotaxis (3) opsonization, and (4) vasodilation and vascular permeability. Cytokines are critical molecules in immune reactivity, driving cellular responses for macrophages, PMNs, NK cells, T cells, and B cells. IFNs are potent antiviral and immunoregulatory molecules. Hypersensitivity Reactions: ç Type I, Immediate: IgE antibody is induced by the allergen and binds via its Fc receptor to mast cells and eosinophils. After encountering the antigen again, the fixed IgE becomes cross-linked, which induces degranulation and release of mediators, especially histamine. ç Type II: Antigens on a cell surface combine with antibody, which leads to complement-mediated lysis (eg, transfusion or Rh reactions) or other cytotoxic membrane damage (eg, autoimmune hemolytic anemia). Type III, Immune Complex: Antigen-antibody ç immune complexes are deposited in tissues, complement is activated, and PMNs are attracted to the site, causing tissue damage. ç Type IV, Delayed: T lymphocytes, sensitized by an antigen, release cytokines upon second contact with the same antigen. The cytokines induce inflammation and activate macrophages. 146 SECTION II Immunology G LOSSARY 1 Allele: A variant form of a single genetic locus. Anaphylatoxins: Inflammatory fragments of complement proteins released during activation (ie, C5a, C3a). They are recognized by specific receptors and result in increased vascular permeability and attract leukocytes. Antibody (Ab): A protein produced as a result of interaction with an antigen. The protein has the ability to combine with the antigen that stimulated its production. Antibodies are produced by plasma cells. Histocompatible: Sharing major histocompatibility complex (MHC) antigens. Humoral Immunity: Pertaining to immunity in a body fluid and used to denote immunity mediated by antibody. Hypersensitivity Reactions: • Type I, Immediate: IgE antibody is induced by an allergen and binds via its Fc receptor to mast cells and eosinophils. After encountering the antigen again, the fixed IgE becomes cross-linked, inducing degranulation and release of mediators, especially histamine. • Type II: Antigens on a cell surface combine with antibody, which leads to complement-mediated lysis (eg, transfusion or Rh reactions) or other cytotoxic membrane damage (eg, autoimmune hemolytic anemia). • Type III, Immune Complex: Antigen–antibody immune complexes are deposited in tissues, complement is activated, and PMNs are attracted to the site, causing tissue damage. • Type IV, Delayed: T lymphocytes, sensitized by an antigen, release cytokines upon second contact with the same antigen. The cytokines induce inflammation and activate macrophages. Antigen (Ag): A molecule that reacts with an antibody. B cell (also B lymphocyte): B cells are lymphocytes that mature in the bone marrow. They rearrange their immunoglobulin genes and express a unique receptor for antigen on their surface. They mature into antibodyproducing plasma cells. Cell adhesion molecules (CAMs): Cell surface proteins that mediate the binding of cells to other cells to extracellular matrix molecules (eg, the integrins and selectins). Cell-mediated (cellular) immunity: An adaptive immune response in which T cells and macrophages play a predominant role. Chemokines: Low-molecular-weight proteins that stimulate leukocyte movement. Chemotaxis: A process whereby phagocytic cells are attracted to the vicinity of invading pathogens in response to a chemokine. Complement: A set of plasma proteins that is the primary mediator of antigen–antibody reactions. Activation of complement can involve the classical, alternative, and lectin pathways. Cytokine: Potent low-molecular, cell-signaling molecules produced transiently and locally by numerous cell types and involved in a variety of immune responses. Immunity: • Innate immunity: Nonspecific host defense not acquired through contact with an antigen. It includes skin and mucous membrane barriers to infectious agents and a variety of nonspecific immunologic factors (eg, phagocytic cells, NK cells, complement, TLRs, and cytokines). • Adaptive immunity: Protection acquired by deliberate introduction of an antigen into a responsive host. Active immunity is specific and is mediated by either antibody or lymphoid cells or both. Cytolysis: The lysis of bacteria or of cells such as tumor or red blood cells. This can occur as complement-mediated lysis or as CD8 cytotoxicity. Immunoglobulin: A glycoprotein, composed of H and L chains, that functions as antibody. All antibodies are immunglobulins but not all immunoglobulins have antibody function. Cytotoxic T cell: T cells that can kill other cells (eg, cells infected with intracellular pathogens). Most cytotoxic T cells are MHC class I restricted CD8 T cells that play a main role in the defense against intracellular pathogens. Immunoglobulin class: A subdivision of immunoglobulin molecule based on structural (amino acid sequence) differences. In humans, there are five immunoglobulin classes: IgG, IgM, IgA, IgD, and IgE. Endotoxins: Bacterial toxins released from damaged cells (eg, LPS). Immunoglobulin subclass: A subdivision of immuno globulin molecule based on structural differences in the H chains. For human IgG, there are four subclasses, IgG1, IgG2, IgG3, and IgG4. Epitope: Site on an antigen recognized by an antibody. Also known as an antigenic determinant. Hapten: A molecule that is not immunogenic by itself but can react with specific antibody after being joined to a suitable carrier molecule. Inflammation: Local accumulation of fluid and cells after injury, infection, or local immune response. 1 Modified and reproduced, with permission, from Stites DP, Stobo JD, Wells JV (editors): Basic & Clinical Immunology, 6th ed. Originally published by Appleton & Lange. Copyright © 1987 by the McGraw-Hill Companies, Inc. CHAPTER 8 Immunology 147 G LOSSAry (continued) Interferons: A heterogeneous group of low-molecularweight proteins that belong to the cytokine family. There are two major types of IFNs. The type I IFNs (α and β) are made by virus-infected cells. The type II IFN is IFN-γ. It is made by activated T cells and NK cells. The IFNs have antiviral, immunoregulatory, and antiproliferative activities. Monocyte: A circulating phagocytic blood cell that develops into a tissue macrophage. Leukocyte: General term for a white blood cell. Opsonin: A substance capable of enhancing phagocytosis. Antibodies and complement are the two main opsonins. Lymphocyte: A mononuclear cell, 7–12 μm in diameter, containing a nucleus with densely packed chromatin and small rim of cytoplasm. Lymphocytes include the T cells and B cells, which have primary roles in immunity. Macrophage: A phagocytic mononuclear cell derived from bone marrow monocytes and found in many tissues of the body and at the site of inflammation. Macrophages serve accessory roles in immunity, particularly as antigenpresenting cells (APCs). Major histocompatibility complex (MHC): A cluster of genes located in close proximity, such as on human chromosome 6 that encode the histocompatibility antigens (MHC molecules). Membrane attack complex (MAC): The end product of activation of the complement cascade, which contains C5, C6, C7, C8, and C9. The MAC produces holes in the membranes of gram-negative bacteria, killing them and, in red blood or other cells, resulting in lysis. Monoclonal antibodies: Each B lymphocyte produces antibody of a single specificity. However, normal B cells do not grow indefinitely. If B cells are fused to a myeloma cell by somatic cell hybridization and fused cells that secrete the desired antibody specificity are selected, an immortalized antibody-producing cell line, known as a hybridoma, is obtained. These hybrid cells produce monoclonal antibodies. REVIEW QUESTIONS 1. What is a characteristic of the adaptive immune response and not of the innate response? (A) Physical barriers (B) Chemical barriers (C) Clonal expansion of effector cells (D) Inflammatory mediators (E) Phagocytosis 2. Which genetic mechanism increases the number of different antibody molecules during an immune response without increasing the diversity of the pool of antigen receptor specificities? (A) V gene segment recombination (B) Class switching Natural killer (NK) cells: Large granular lymphoid cells with no known antigen-specific receptors. They are able to recognize and kill certain virally infected cells and activate the innate response. They produce IFN-γ. Opsonization: The coating of an antigen or particle (eg, infectious agent) by substances, such as antibodies, complement components, fibronectin, and so forth, that facilitate uptake of the foreign particle into phagocytic cell. Plasma cell: A terminally differentiated B cell that secretes antibody. Polymorphonuclear cell (PMN): Also known as neutrophil. A PMN is characterized by a multilobed nucleus. PMNs migrate from the circulation to a site of inflammation by chemotaxis and are phagocytic for bacteria and other particles. T cell (also T lymphocyte): A thymus-derived cell that participates in a variety of cell-mediated immune reactions. Toll-like receptors (TLRs): A family of evolutionary conserved pattern recognition receptors that recognize pathogen-associated molecular patterns on microbes and serve as the first line of defense in the innate immune response. Thymocytes: Developing T cells found in the thymus. Vaccination: Induction of immunity by injecting a dead or attenuated form of a pathogen. (C) Somatic hypermutation (D) Junctional variability due to imprecise V, D, and J joining (E) Gene duplication ie, multiple V, D, and J gene segments 3. What is the principal function of the class I and class II MHC molecules? (A) They are mediators of T-independent B-cell responses. (B) They bind peptide antigens for presentation to antigenspecific receptors on B cells. (C) They help in endocytosis of antigens by phagocytic cells. (D) They bind carbohydrate antigens directly for presentation on T cells. (E) They display peptide antigens for review by antigenspecific receptors on T cells. 148 SECTION II Immunology 4. MHC class I molecules need to bind peptide antigens to fold properly and to be expressed at the cell surface. What would you expect to be the most common health problem in a child with a defect in the function of the peptide transporter (TAP) found in the endoplasmic reticulum? (A) Chronic upper respiratory viral infections (B) Parasitic infections (C) Infections with encapsulated bacteria (D) Pronounced allergies to household pets (E) Autoimmune disease 5. Which major antibody molecule has the ability to cross the placenta? (A) IgG (B) IgA (C) IgM (D) IgE (E) IgD 6. A man in his twenties presents in the emergency room with shortness of breath and fatigue. He is also very pale. Two days earlier he was given penicillin for an infection. He had penicillin previously without problems and stated that he had “no allergy” to penicillin. Laboratory testing shows that antibodies to penicillin are present in the patient’s serum and that he is breaking down his own red blood cells. He is diagnosed with immune hemolytic anemia. The patient has which type of hypersensitivity reaction? (A) Type I (B) Type II (C) Type III (D) Type IV (DTH) 7. Which one of the following cell types expresses receptors for IgE on its cell surface that stimulate the cell to mount a response to parasites such as worms? (A) T cells (B) B cells (C) Promonocytes (D) NK cells (E) Mast cells 8. Which immunologic test is widely used to precisely enumerate and collect cells expressing an antigen bound by a fluorescencetagged monoclonal antibody? (A) ELISA (B) Direct immunofluorescence (C) Western blotting (D) Fluorescence-activated cell sorting (E) Indirect immunofluorescence 9. In any given immunoglobulin molecule, the light chains are: (A) Identical to each other in their antigenic determinants (B) Identical to each other (C) Identical to each other except in their hypervariable regions (D) Of related but different amino acid sequences (E) Identical to each other except in their overall domain structure 10. Antigen–antibody complexes are phagocytosed more effectively in the presence of which complement component? (A) C3a and C5a (B) C3b (C) C56789 complex (D) MBL (E) Properdin 11. NK cells express a killer immunoglobulin-like receptor, which recognizes: (A) MHC class I molecules (B) MHC class II molecules (C) Cell adhesion molecules (D) Glycophospholipid molecules (E) CD40 molecules 12. A cell that plays a critical role in the innate immune response and kills virus infected cells is: (A) T cell (B) Neutrophil (C) NK cell (D) Macrophage (E) B cell 13. A cytokine that activates cells to express MHC class II antigens and protects cells from virus replication is: (A) Interferon-α (B) IL-6 (C) Interferon-γ (D) TNF-α (E) IL-10 14. IgE-mediated histamine release is classified as what type of hypersensitivity reaction? (A) Type 1 (B) Type 2 (C) Type 3 (D) Type 4 15. The interaction of a pathogen molecule with its specific TLR directly results in which of the following? (A) Presentation of pathogen molecule to helper T cells (B) Cell activation and production of cytokines and chemokines (C) IgG production (D) Immunoglobulin class switching (E) Phagocytosis Answers 1. C A B C B B B A E E A B A D C REFERENCES Abbas AK, Lichtman AH, Pillai S: Cellular and Molecular Immunology, 7th ed. Saunders Elsevier, 2012. Detrick B, Hamilton RG, Folds JD: Manual of Molecular and Clinical Laboratory Immunology, 7th ed. ASM Press, 2006. Murphy K, Travers P, Wolport M: Janeway’s Immunobiology, 8th ed. Garland Science, 2011. Nairn R, Helbert M: Immunology for Medical Students, 2nd ed. Mosby/Elsevier, 2007. O’Gorman MRG, Donnenberg AD: Handbook of Human Immunology, 2nd ed. CRC Press, 2008. Paul WE (editor): Fundamental Immunology, 6th ed. Wolters Kluwer/Lippincott Williams & Wilkins, 2008. SECTION III BACTERIOLOGY C Pathogenesis of Bacterial Infection The pathogenesis of bacterial infection includes initiation of the infectious process and the mechanisms that lead to the development of signs and symptoms of disease. The biochemical, structural, and genetic factors that play important roles in bacterial pathogenesis are introduced in this chapter and may be revisited in the organism-specific sections. Characteristics of bacteria that are pathogens include transmissibility, adherence to host cells, persistence, invasion of host cells and tissues, toxigenicity, and the ability to evade or survive the host’s immune system. Resistance to antimicrobials and disinfectants can also 9 H A P T E R contribute to virulence, or an organism’s capacity to cause disease. Many infections caused by bacteria that are commonly considered to be pathogens are inapparent or asymptomatic. Disease occurs if the bacteria or immunologic reactions to their presence cause sufficient harm to the person. Terms frequently used in describing aspects of pathogenesis are defined in the Glossary (see below). Refer to the Glossary in Chapter 8 for definitions of terms used in immunology and in describing aspects of the host’s response to infection. G LOSSARY Adherence (adhesion, attachment): The process by which bacteria stick to the surfaces of host cells. After bacteria have entered the body, adherence is a major initial step in the infection process. The terms adherence, adhesion, and attachment are often used interchangeably. Invasion: The process whereby bacteria, animal parasites, fungi, and viruses enter host cells or tissues and spread in the body. Carrier: A person or animal with asymptomatic infection that can be transmitted to another susceptible person or animal. Nonpathogen: A microorganism that does not cause disease; may be part of the normal microbiota. Infection: Multiplication of an infectious agent within the body. Multiplication of the bacteria that are part of the normal flora of the gastrointestinal tract, skin, and so on is generally not considered an infection; on the other hand, multiplication of pathogenic bacteria (eg, Salmonella species)—even if the person is asymptomatic—is deemed an infection. Opportunistic pathogen: An agent capable of causing disease only when the host’s resistance is impaired (ie, when the patient is “immunocompromised”). Microbiota: Microbial flora harbored by normal, healthy individuals. Pathogen: A microorganism capable of causing disease. Pathogenicity: The ability of an infectious agent to cause disease. (See also virulence.) 149 150 SECTION III Bacteriology Superantigens: Protein toxins that activate the immune system by binding to major histocompatibility complex (MHC) molecules and T-cell receptors (TCR) and stimulate large numbers of T cells to produce massive quantities of cytokines. Virulence: The quantitative ability of an agent to cause disease. Virulent agents cause disease when introduced into the host in small numbers. Virulence involves adherence, persistence, invasion, and toxigenicity (see above). Toxigenicity: The ability of a microorganism to produce a toxin that contributes to the development of disease. IDENTIFYING BACTERIA THAT CAUSE DISEASE Humans and animals have abundant normal microbiota that usually do not produce disease (see Chapter 10) but achieve a balance that ensures the survival, growth, and propagation of both the bacteria and the host. Some bacteria that are important causes of disease are cultured commonly with the normal flora (eg, Streptococcus pneumoniae, Staphylococcus aureus). Sometimes bacteria that are clearly pathogens (eg, Salmonella serotype Typhi) are present, but infection remains latent or subclinical, and the host is a “carrier” of the bacteria. It can be difficult to show that a specific bacterial species is the cause of a particular disease. In 1884, Robert Koch proposed a series of postulates that have been applied broadly to link many specific bacterial species with particular diseases. Koch’s postulates are summarized in Table 9-1. Koch’s postulates have remained a mainstay of micro biology; however, since the late 19th century, many microorganisms that do not meet the criteria of the postulates have been shown to cause disease. For example, Treponema pallidum (syphilis) and Mycobacterium leprae (leprosy) cannot be grown in vitro; however, there are animal models of infection with these agents. In another example, Neisseria gonorrhoeae (gonorrhea), there is no animal model of infection even though the bacteria can readily be cultured in vitro; experimental infection in humans has been produced that substitutes for an animal model. In other instances, Koch’s postulates have been at least partially satisfied by showing bacterial pathogenicity in an in vitro model of infection rather than in an animal model. For example, some forms of Escherichia coli–induced diarrhea (see Chapter 15) have been defined by the interaction of the E coli with host cells in tissue culture. TABLE 9-1 Guidelines for Establishing the Causes of Infectious Diseases Koch’s Postulates Molecular Koch’s Postulates The microorganism should be found in all cases of the disease in question, and its distribution in the body should be in accordance with the lesions observed. The phenotype or property under investigation should be significantly associated with pathogenic strains of a species and not with nonpathogenic strains. The microorganism should be grown in pure culture in vitro (or outside the body of the host) for several generations. Specific inactivation of the gene or genes associated with the suspected virulence trait should lead to a measurable decrease in pathogenicity or virulence. When such a pure culture is inoculated into susceptible animal species, the typical disease must result. 4. The microorganism must again be isolated from the lesions of such experimentally produced disease. Reversion or replacement of the mutated gene with the wild-type gene should lead to restoration of pathogenicity or virulence. Molecular Guidelines for Establishing Microbial Disease Causation 1. The nucleic acid sequence of a putative pathogen should be present in most cases of an infectious disease and preferentially in anatomic sites where pathology is evident. 2. The nucleic acid sequence of a putative pathogen should be absent from most healthy control participants. If the sequence is detected in healthy control participants, it should be present with a lower prevalence as compared with patients with disease and in lower copy numbers. 3. The copy number of a pathogen-associated nucleic acid sequence should decrease or become undetectable with resolution of the disease (eg, with effective treatment) and should increase with relapse or recurrence of disease. 4. The presence of a pathogen-associated nucleic acid sequence in healthy subjects should help predict the subsequent development of disease. 5. The nature of the pathogen inferred from analysis of its nucleic acid sequence should be consistent with the known biologic characteristics of closely related organisms and the nature of the disease. The significance of a detected microbial sequence is increased when microbial genotype predicts microbial morphology, pathology, clinical features of disease, and host response CHAPTER 9 Pathogenesis of Bacterial Infection 151 The host’s immune responses also should be considered when an organism is being investigated as the possible cause of a disease. Thus, development of a rise in specific antibody during recovery from disease is an important adjunct to Koch’s postulates. Modern-day microbial genetics has opened new frontiers to study pathogenic bacteria and differentiate them from nonpathogens. Molecular cloning has allowed investigators to isolate and modify specific virulence genes and study them with models of infection. The ability to study genes associated with virulence has led to a proposed form of molecular Koch’s postulates. These postulates are summarized in Table 9-1. Some pathogens are difficult or impossible to grow in culture, and for that reason, it is not possible with Koch’s postulates or the molecular Koch’s postulates to establish the cause of their associated diseases. The polymerase chain reaction is used to amplify microorganism-specific nucleic acid sequences from host tissues or fluids. The sequences are used to identify the infecting organisms. The molecular guidelines for establishing microbial disease causation are listed in Table 9-1. This approach has been used to establish the causes of several diseases, including Whipple disease (Tropheryma whipplei), bacillary angiomatosis (Bartonella henselae), human monocytic ehrlichiosis (Ehrlichia chaffeensis), hantavirus pulmonary syndrome (Sin Nombre virus), and Kaposi sarcoma (human herpesvirus 8). Analysis of infection and disease through the application of principles such as Koch’s postulates leads to classification of bacteria as pathogens, opportunistic pathogens, or nonpathogens. Some bacterial species are always considered to be pathogens, and their presence is abnormal; examples include Mycobacterium tuberculosis (tuberculosis) and Yersinia pestis (plague). Such bacteria readily meet the criteria of Koch’s postulates. Other species are commonly part of the normal microbiota of humans (and animals) but also can frequently cause disease. For example, E coli is part of the gastrointestinal microbiota of normal humans but is also a common cause of urinary tract infections, traveler’s diarrhea, and other diseases. Strains of E coli that cause disease are differentiated from those that do not by determining (1) whether they are virulent in animals or in vitro models of infection and (2) whether they have a genetic makeup that is significantly associated with production of disease. Other bacteria (eg, Pseudomonas species, Stenotrophomonas maltophilia, and many yeasts and molds) only cause disease in immunosuppressed and debilitated persons and are opportunistic pathogens. TRANSMISSION OF INFECTION Bacteria (and other microorganisms) can adapt to a variety of environments that include external sources such as soil, water and organic matter or internal milieu as found within insect vectors, animals and humans, where they normally reside and subsist. In doing so, the bacteria ensure their survival and enhance the possibility of transmission. By producing asymptomatic infection or mild disease rather than death of the host, microorganisms that normally live in people enhance the possibility of transmission from one person to another. Some bacteria that commonly cause disease in humans exist primarily in animals and incidentally infect humans. For example, Salmonella and Campylobacter species typically infect animals and are transmitted in food products to humans. Other bacteria produce infection of humans that is inadvertent, a mistake in the normal life cycle of the organism; the organisms have not adapted to humans, and the disease they produce may be severe. For example, Y pestis (plague) has a well-established life cycle in rodents and rodent fleas, and transmission by the fleas to humans is inadvertent; Bacillus anthracis (anthrax) lives in the environment, occasionally infects animals, and is transmitted to humans by products such as raw hair from infected animals. The Clostridium species are ubiquitous in the environment and are transmitted to humans by ingestion (eg, C perfringens gastroenteritis and C botulinum [botulism]) or when wounds are contaminated by soil (eg, C perfringens [gas gangrene] and C tetani [tetanus]). Both Bacillus anthracis and the Clostridium species elaborate spores to protect the organisms’ nucleic acid from harsh environmental factors such as ultraviolet light, desiccation, chemical detergents, and pH extremes. These spores ensure survival in external environments including foods ingested by humans. After being ingested or inoculated, the spores germinate into the vegetative, metabolically active form of the pathogen. The clinical manifestations of diseases (eg, diarrhea, cough, genital discharge) produced by microorganisms often promote transmission of the agents. Examples of clinical syndromes and how they enhance transmission of the causative bacteria are as follows: Vibrio cholerae can cause voluminous diarrhea, which may contaminate salt and fresh water; drinking water or seafood such as oysters and crabs may be contaminated; ingestion of contaminated water or seafood can produce infection and disease. Similarly, contamination of food products with sewage containing E coli that cause diarrhea results in transmission of the bacteria. M tuberculosis (tuberculosis) naturally infects only humans; it produces respiratory disease with cough and production of aerosols, resulting in transmission of the bacteria from one person to another. Many bacteria are transmitted from one person to another on hands. A person with S aureus carriage in the anterior nares may rub his nose, pick up the staphylococci on the hands, and spread the bacteria to other parts of the body or to another person, where infection results. Many opportunistic pathogens that cause nosocomial infections are transmitted from one patient to another on the hands of hospital personnel. Handwashing is thus an important component of infection control. The most frequent portals of entry of pathogenic bacteria into the body are the sites where mucous membranes 152 SECTION III Bacteriology meet with the skin, which are the respiratory (upper and lower airways), gastrointestinal (primarily mouth), genital, and urinary tracts. Abnormal areas of mucous membranes and skin (eg, cuts, burns, and other injuries) are also frequent sites of entry. Normal skin and mucous membranes provide the primary defense against infection. To cause disease, pathogens must overcome these barriers. THE INFECTIOUS PROCESS In the body, most bacteria that cause disease do so first by attaching or adhering to host cells, usually epithelial cells. After the bacteria have established a primary site of infection, they multiply and spread directly through tissues or via the lymphatic system to the bloodstream. This infection (bacteremia) can be transient or persistent. Bacteremia allows bacteria to spread widely in the body and permits them to reach tissues particularly suitable for their multiplication. Pneumococcal pneumonia is an example of the infectious process. S pneumoniae can be cultured from the nasopharynx of 5–40% of healthy people. Occasionally, pneumococci from the nasopharynx are aspirated into the lungs; aspiration occurs most commonly in debilitated people and in settings such as coma when normal gag and cough reflexes are diminished. Infection develops in the terminal air spaces of the lungs in persons who do not have protective antibodies against that particular pneumococcal capsular polysaccharide type. Multiplication of the pneumococci and resultant inflammation lead to pneumonia. The pneumococci enter the lymphatics of the lung and move to the bloodstream. Between 10% and 20% of persons with pneumococcal pneumonia have bacteremia at the time the diagnosis of pneumonia is made. When bacteremia occurs, the pneumococci can spread to secondary sites of infection (eg, cerebrospinal fluid, heart valves, and joint spaces). The major complications of pneumococcal pneumonia are meningitis, septic arthritis, and rarely endocarditis. The infectious process in cholera involves ingestion of V cholerae, chemotactic attraction of the bacteria to the gut epithelium, motility of the bacteria by a single polar flagellum, and penetration of the mucous layer on the intestinal surface. The V cholerae adherence to the epithelial cell surface is mediated by pili and possibly other adhesins. Production of cholera toxin results in flow of chloride and water into the lumen of the gut, causing diarrhea and electrolyte imbalance. GENOMICS AND BACTERIAL PATHOGENICITY Bacteria are haploid (see Chapter 7) and limit genetic interactions that might change their chromosomes and potentially disrupt their adaptation and survival in specific environmental niches. The Clonal Nature of Bacterial Pathogens One important result of the conservation of chromosomal genes in bacteria is that the organisms are clonal. For most pathogens, there are only one or a few clonal types that are spread in the world during a period of time. For example, epidemic serogroup A meningococcal meningitis occurs in Asia, the Middle East, and Africa and occasionally spreads into Northern Europe and the Americas. On several occasions, over a period of decades, single clonal types of serogroup A Neisseria meningitidis have been observed to appear in one geographic area and subsequently spread to others with resultant epidemic disease. There are many types of Haemophilus influenzae, but only clonal H influenzae type B is commonly associated with disease. There are two clonal types of Bordetella pertussis, both associated with disease. Similarly, Salmonella serotype Typhi (typhoid fever) from patients is of two clonal types. There are, however, mechanisms that bacteria use, or have used a long time in the past, to transmit virulence genes from one to another. Mobile Genetic Elements Primary mechanisms for exchange of genetic information between bacteria include natural transformation and transmissible mobile genetic elements such as plasmids, transposons, and bacteriophages (often referred to as “phages”). Transformation occurs when DNA from one organism is released into the environment and is taken up by a different organism that is capable of recognizing and binding DNA. In other cases, the genes that encode many bacterial virulence factors are carried on plasmids, transposons, or phages. Plasmids are extrachromosomal pieces of DNA and are capable of replicating. Transposons are highly mobile segments of DNA that can move from one part of the DNA to another. This can result in recombination between extrachromosomal DNA and the chromosome (illegitimate or nonhomologous recombination; Chapter 7). If this recombination occurs, the genes coding for virulence factors may become chromosomal. Finally, bacterial viruses or phages are another mechanism by which DNA can be moved from one organism to another. Transfer of these mobile genetic elements between members of one species or, less commonly, between species can result in transfer of virulence factors, including antimicrobial resistance genes. A few examples of plasmid- and phage-encoded virulence factors are given in Table 9-2. Pathogenicity Islands Large groups of genes that are associated with pathogenicity and are located on the bacterial chromosome are termed pathogenicity islands (PAIs). They are large organized groups of genes, usually 10–200 kb in size. The major properties of PAIs are as follows: they have one or more virulence genes; they are present in the genome of pathogenic members of CHAPTER 9 Pathogenesis of Bacterial Infection 153 TABLE 9-2 Examples of Virulence Factors Encoded by Genes on Mobile Genetic Elements Genus and Species Plasmid encoded Escherichia coli Escherichia coli Escherichia coli and Shigella species Bacillus anthracis Phage encoded Clostridium botulinum Corynebacterium diphtheriae Vibrio cholerae Virulence Factor and Disease Heat-labile and heat-stable enterotoxins that cause diarrhea Hemolysin (cytotoxin) of invasive disease and urinary tract infections Adherence factors and gene products involved in mucosal invasion Capsule essential for virulence (on one plasmid) Edema factor, lethal factor, and protective antigen are all essential for virulence (on other plasmids) Botulinum toxin that causes paralysis Diphtheria toxin that inhibits human protein synthesis Cholera toxin that can cause a severe watery diarrhea a species but absent in the nonpathogenic members; they are large; they typically have a different guanine plus cytosine (G + C) content than the rest of the bacterial genome; they are commonly associated with tRNA genes; they are often found with parts of the genome associated with mobile genetic elements; they often have genetic instability; and they often represent mosaic structures with components acquired at different times. Collectively, the properties of PAIs suggest that they originate from gene transfer from foreign species. A few examples of PAI virulence factors are provided in Table 9-3. REGULATION OF BACTERIAL VIRULENCE FACTORS Pathogenic bacteria (and other pathogens) have adapted both to saprophytic or free-living states, possibly environments outside of the body, and to the human host. In the adaptive process, pathogens husband their metabolic needs and products. They have evolved complex signal transduction systems to regulate the genes important for virulence. Environmental signals often control the expression of the virulence genes. Common signals include temperature, iron availability, osmolality, growth phase, pH, and specific ions (eg, Ca2+) or nutrient factors. A few examples are presented in the following paragraphs. The gene for diphtheria toxin from Corynebacterium diphtheriae is carried on temperate bacteriophages. Toxin is produced only by strains lysogenized by the phages. Toxin production is greatly enhanced when C diphtheriae is grown in a medium with low iron. Expression of virulence genes of B pertussis is enhanced when the bacteria are grown at 37°C and suppressed when they are grown at lower temperatures or in the presence of high concentrations of magnesium sulfate or nicotinic acid. The virulence factors of V cholerae are regulated on multiple levels and by many environmental factors. Expression of the cholera toxin is higher at a pH of 6.0 than at a pH of 8.5 and higher also at 30°C than at 37°C. TABLE 9-3 A Few Examples of the Very Large Number of Pathogenicity Islands of Human Pathogens Genus and Species PAI Name Escherichia coli PAI I536, II536 Alpha hemolysin, fimbriae, adhesions, in urinary tract infections Escherichia coli PAI IJ96 Alpha hemolysin, P-pilus in urinary tract infections Escherichia coli (EHEC) O157 Macrophage toxin of enterohemorrhagic Escherichia coli Salmonella serotype Typhimurium SPI-1 Invasion and damage of host cells; diarrhea Yersinia pestis HPI/pgm Genes that enhance iron uptake Vibrio cholerae El Tor O1 VPI-1 Neuraminidase, utilization of amino sugars Staphylococcus aureus SCC mec Methicillin and other antibiotic resistance Staphylococcus aureus SaPI1 Toxic shock syndrome toxin-1, enterotoxin Enterococcus faecalis NPm Cytolysin, biofilm formation PAI, pathogenicity island SPI, Salmonella pathogenicity island HPI, high pathogenicity island VPI, Vibrio pathogenicity island SCC, staphylococcal cassette chromosome mec SaPI, Staphylococcus aureus pathogenicity island NP, non-protease Virulence Characteristics 154 SECTION III Bacteriology Osmolality and amino acid composition also are important. As many as 20 other genes of V cholerae are similarly regulated. Y pestis produces a series of virulence plasmid-encoded proteins. One of these is an antiphagocytic fraction 1 capsular protein that results in antiphagocytic function. This protein is expressed maximally at 35–37°C, the host temperature, and minimally at 20–28°C, the flea temperature at which antiphagocytic activity is not needed. The regulation of other virulence factors in Yersinia species also is influenced by environmental factors. Motility of bacteria enables them to spread and multiply in their environmental niches or in patients. Yersinia enterocolitica and Listeria monocytogenes are common in the environment where motility is important to them. Presumably, motility is not important in the pathogenesis of the diseases caused by these bacteria. Y enterocolitica is motile when grown at 25°C but not when grown at 37°C. Similarly, Listeria is motile when grown at 25°C and not motile or minimally motile when grown at 37°C. BACTERIAL VIRULENCE FACTORS Many factors determine bacterial virulence or the ability to cause infection and disease. Adherence Factors When bacteria enter the body of the host, they must adhere to cells of a tissue surface. If they did not adhere, they would be swept away by mucus and other fluids that bathe the tissue surface. Adherence, which is only one step in the infectious process, is followed by development of microcolonies and subsequent steps in the pathogenesis of infection. The interactions between bacteria and tissue cell surfaces in the adhesion process are complex. Several factors play important roles, including surface hydrophobicity and net surface charge, binding molecules on bacteria (ligands), and host cell receptor interactions. Bacteria and host cells commonly have net negative surface charges and therefore repulsive electrostatic forces. These forces are overcome by hydrophobic and other more specific interactions between bacteria and host cells. In general, the more hydrophobic the bacterial cell surface, the greater the adherence to the host cell. Different strains of bacteria within a species may vary widely in their hydrophobic surface properties and ability to adhere to host cells. Bacteria also have specific surface molecules that interact with host cells. Many bacteria have pili, thick rodlike appendages or fimbriae, shorter “hairlike” structures that extend from the bacterial cell surface and help mediate adherence of the bacteria to host cell surfaces. For example, some E coli strains have type 1 pili, which adhere to epithelial cell receptors; adherence can be blocked in vitro by addition of d-mannose to the medium. E coli organisms that cause urinary tract infections commonly do not have d-mannose– mediated adherence but have P-pili, which attach to a portion of the P blood group antigen; the minimal recognition structure is the disaccharide α-d-galactopyranosyl-(1–4)β-d-galactopyranoside (GAL–GAL binding adhesion). The E coli that cause diarrheal diseases (see Chapter 15) have pilus (fimbriae)-mediated adherence to intestinal epithelial cells. The type of pili and specific molecular mechanisms of adherence appear to be different depending on the form of the E coli that induce the diarrhea. Other specific ligand-receptor mechanisms have evolved to promote bacterial adherence to host cells, illustrating the diverse mechanisms used by bacteria. Group A streptococci (Streptococcus pyogenes) (see Chapter 14) also have hairlike appendages, termed fimbriae, that extend from the cell surface. Lipoteichoic acid, protein F, and M protein are found on the fimbriae. The lipoteichoic acid and protein F cause adherence of the streptococci to buccal epithelial cells; this adherence is mediated by fibronectin, which acts as the host cell receptor molecule. M protein acts as an antiphagocytic molecule and is a major virulence factor. Antibodies that act against the specific bacterial ligands that promote adherence (eg, pili and lipoteichoic acid) can block adherence to host cells and protect the host from infection. After adherence occurs, conformational changes in the host cell ensue that can lead to cytoskeletal changes allowing organism uptake by the cell. Sometimes changes in the adhesin molecule after attachment may trigger activation of virulence genes that promote invasion or that result in other pathogenic changes as described below. Invasion of Host Cells and Tissues For many disease-causing bacteria, invasion of the host’s epithelium is central to the infectious process. Some bacteria (eg, Salmonella species) invade tissues through the junctions between epithelial cells. Other bacteria (eg, Yersinia species, N gonorrhoeae, Chlamydia trachomatis) invade specific types of the host’s epithelial cells and may subsequently enter the tissue. When inside the host cell, bacteria may remain enclosed in a vacuole composed of the host cell membrane, or the vacuole membrane may be dissolved and bacteria may be dispersed in the cytoplasm. Some bacteria (eg, Shigella species) multiply within host cells, but other bacteria do not. Invasion is the term commonly used to describe the entry of bacteria into host cells, implying an active role for the organisms and a passive role for the host cells. In many infections, the bacteria produce virulence factors that influence the host cells, causing them to engulf (ingest) the bacteria. The host cells play a very active role in the process. Toxin production and other virulence properties are generally independent of the ability of bacteria to invade cells and tissues. For example, C diphtheriae is able to invade the epithelium of the nasopharynx and cause symptomatic sore throat even when the C diphtheriae strains are nontoxigenic. CHAPTER 9 Pathogenesis of Bacterial Infection 155 In vitro studies with cells in tissue culture have helped characterize the mechanisms of invasion for some pathogens; however, the in vitro models have not necessarily provided a complete picture of the invasion process. Full understanding of the invasion process, as it occurs in naturally acquired infection, has required study of genetically engineered mutants and their ability to infect susceptible animals and humans. Thus, understanding of eukaryotic cell invasion by bacteria requires satisfying much of Koch’s postulates and the molecular Koch’s postulates. The following paragraphs contain examples of bacterial invasion of host cells as part of the infectious process. Shigella species adhere to host cells in vitro. Commonly, HeLa cells are used; these undifferentiated unpolarized cells were derived from a cervical carcinoma. The adherence causes actin polymerization in the nearby portion of the HeLa cell, which induces the formation of pseudopods by the HeLa cells and engulfment of the bacteria. Adherence and invasion are mediated at least in part by products of genes located on a large plasmid common to many shigellae. There are multiple proteins, including the invasion plasmid antigens (IpA-D), that contribute to the process. Inside the HeLa cells, the shigellae either are released or escape from the phagocytic vesicle, where they multiply in the cytoplasm. Actin polymerization propels the shigellae within a HeLa cell and from one cell into another. In vivo the shigellae adhere to integrins on the surface of M cells in Peyer’s patches and not to the polarized absorptive cells of the mucosa. M cells normally sample antigens and present them to macrophages in the submucosa. The shigellae are phagocytosed by the M cells and pass through the M cells into the underlying collection of macrophages. Shigellae inside the M cells and macrophages can cause these cells to die by activating the normal cell death process (apoptosis). The shigellae spread to adjacent mucosal cells in a manner similar to the in vitro model by actin polymerization that propels the bacteria. From studies using cells in vitro, it appears that the adherence-invasion process with Y enterocolitica is similar to that of Shigella. Yersiniae adhere to the host cell membrane and cause it to extrude protoplasmic projections. The bacteria are then engulfed by the host cell with vacuole formation. Invasion is enhanced when the bacteria are grown at 22°C rather than at 37°C. When yersiniae have entered the cell, the vacuolar membrane dissolves and the bacteria are released into the cytoplasm. In vivo, the yersiniae are thought to adhere to and invade the M cells of Peyer’s patches rather than the polarized absorptive mucosal cells, much like shigellae. L monocytogenes from the environment is ingested in food. Presumably, the bacteria adhere to and invade the intestinal mucosa, reach the bloodstream, and disseminate. The pathogenesis of this process has been studied in vitro. L monocytogenes adheres to and readily invades macro phages and cultured undifferentiated intestinal cells. The listeriae induce engulfment by the host cells. Proteins, called internalins, have a primary role in this process. The engulfment process, movement within a cell and movement between cells, requires actin polymerization to propel the bacteria, as with shigellae. Legionella pneumophila infects pulmonary macrophages and causes pneumonia. Adherence of the legionellae to the macrophage induces formation of a long, thin pseudopod that then coils around the bacteria, forming a vesicle (coiling phagocytosis). The vesicle remains intact, phagolysosome fusion is inhibited, and the bacteria multiply within the vesicle. N gonorrhoeae uses pili as primary adhesins and opacity associated proteins (Opa) as secondary adhesins to host cells. Certain Opa proteins mediate adherence to polymorphonuclear cells. Some gonococci survive after phagocytosis by these cells. Pili and Opa together enhance the invasion of cells cultured in vitro. In uterine (fallopian) tube organ cultures, the gonococci adhere to the microvilli of nonciliated cells and appear to induce engulfment by these cells. The gonococci multiply intracellularly and migrate to the subepithelial space by an unknown mechanism. Toxins Toxins produced by bacteria are generally classified into two groups: exotoxins and endotoxins. Exotoxins are proteins that are most often excreted from the cell. However some exotoxins accumulate inside the cell and are either injected directly into the host or are released by cell lysis. Endotoxins are lipid molecules that are components of the bacterial cell membrane. The primary features of the two groups are listed in Table 9-4. A. Exotoxins Many gram-positive and gram-negative bacteria produce exotoxins of considerable medical importance. Some of these toxins have had major roles in world history. For example, tetanus caused by the toxin of C tetani killed as many as 50,000 soldiers of the Axis powers in World War II; the Allied forces, however, immunized military personnel against tetanus, and very few died of that disease. Vaccines have been developed for some of the exotoxin-mediated diseases and continue to be important in the prevention of disease. These vaccines—called toxoids—are made from exotoxins, which are modified so that they are no longer toxic. Many exotoxins consist of A and B subunits. The B subunit generally mediates adherence of the toxin complex to a host cell and aids entrance of the exotoxin into the host cell. The A subunit provides the toxic activity. Examples of some pathogenetic mechanisms associated with exotoxins are given below. Other toxins of specific bacteria are discussed in the chapters covering those bacteria. C diphtheriae is a gram-positive rod that can grow on the mucous membranes of the upper respiratory tract or in minor skin wounds (see Chapter 12). Strains of C diphtheriae that carry a lysogenic, temperate corynebacteriophage (β-phage or ω-phage) with the structural gene for the 156 SECTION III Bacteriology TABLE 9-4 Characteristics of Exotoxins and Endotoxins (Lipopolysaccharides) Exotoxins Endotoxins Excreted by living cell; high concentrations in liquid medium Integral part of the cell wall of gram-negative bacteria; released on bacterial death and in part during growth; may not need to be released to have biologic activity Produced by both gram-positive and gram-negative bacteria Found only in gram-negative bacteria Polypeptides with a molecular weight of 10,000–900,000 Lipopolysaccharide complexes; lipid A portion probably responsible for toxicity Relatively unstable; toxicity often destroyed rapidly by heating at temperatures above 60°C Relatively stable; withstand heating at temperatures above 60°C for hours without loss of toxicity Highly antigenic; stimulate formation of high-titer antitoxin; antitoxin neutralizes toxin Weakly immunogenic; antibodies are antitoxic and protective; relationship between antibody titers and protection from disease is less clear than with exotoxins Converted to antigenic, nontoxic toxoids by formalin, acid, heat, and so on; toxoids are used to immunize (eg, tetanus toxoid) Not converted to toxoids Highly toxic; fatal to animals in microgram quantities or less Moderately toxic; fatal for animals in tens to hundreds of micrograms Usually bind to specific receptors on cells Specific receptors not found on cells Usually do not produce fever in the host Usually produce fever in the host by release of interleukin-1 and other mediators Frequently controlled by extrachromosomal genes (eg, plasmids) Synthesis directed by chromosomal genes toxin are toxigenic and produce diphtheria toxin and cause diphtheria. Many factors regulate toxin production; when the availability of inorganic iron is the factor limiting the growth rate, then maximal toxin production occurs. The toxin molecule is secreted as a single polypeptide molecule (molecular weight [MW], 62,000). This native toxin is enzymatically degraded into two fragments, A and B, linked together by a disulfide bond. Fragment B (MW, 40,700) binds to specific host cell receptors and facilitates the entry of fragment A (MW, 21,150) into the cytoplasm. Fragment A inhibits peptide chain elongation factor EF-2 by catalyzing a reaction that attaches an adenosine diphosphate–ribosyl group to EF-2, yielding an inactive adenosine diphosphate–ribose–EF-2 complex. Arrest of protein synthesis disrupts normal cellular physiologic functions. Diphtheria toxin is very potent. C tetani is an anaerobic gram-positive rod that causes tetanus (see Chapter 11). C tetani from the environment contaminates wounds, and the spores germinate in the anaerobic environment of the devitalized tissue. Infection often is minor and not clinically apparent. The vegetative forms of C tetani produce the toxin tetanospasmin (MW, 150,000) that is cleaved by a bacterial protease into two peptides (MW, 50,000 and 100,000) linked by a disulfide bond. The toxin initially binds to receptors on the presynaptic membranes of motor neurons. It then migrates by the retrograde axonal transport system to the cell bodies of these neurons to the spinal cord and brainstem. The toxin diffuses to terminals of inhibitory cells, including both glycinergic interneurons and γ-aminobutyric acid (GABA)–secreting neurons from the brainstem. The toxin degrades synaptobrevin, a protein required for docking of neurotransmitter vesicles on the presynaptic membrane. Release of the inhibitory glycine and GABA is blocked, and the motor neurons are not inhibited. Spastic paralysis results. Extremely small amounts of toxin can be lethal for humans. Tetanus is totally preventable in immunologically normal people by immunization with tetanus toxoid. C botulinum causes botulism. This anaerobic, grampositive spore-forming organism is found in soil or water and may grow in foods (eg, canned, vacuum packed) if the environment is appropriately anaerobic. An exceedingly potent toxin (the most potent toxin known) is produced. It is heat labile and is destroyed by sufficient heating. There are seven distinct serologic types of toxin. Types A, B, E, and F are most commonly associated with human disease. The toxin is very similar to tetanus toxin, with a 150,000 MW protein that is cleaved into 100,000-MW and 50,000-MW proteins linked by a disulfide bond. Botulinum toxin is absorbed from the gut and binds to receptors of presynaptic membranes of motor neurons of the peripheral nervous system and cranial nerves. Proteolysis, by the light chain of botulinum toxin, of target proteins in the neurons inhibits the release of acetylcholine at the synapse, resulting in lack of muscle contraction and flaccid paralysis. Spores of C perfringens are introduced into wounds by contamination with soil or feces. In the presence of necrotic tissue (an anaerobic environment), spores germinate, and vegetative cells can produce several different toxins. Many of these are necrotizing and hemolytic and—together with distention of tissue by gas formed from carbohydrates and CHAPTER 9 Pathogenesis of Bacterial Infection 157 interference with blood supply—favor the spread of gas gangrene. The alpha toxin of C perfringens is a lecithinase that damages cell membranes by splitting lecithin to phosphorylcholine and diglyceride. Theta toxin also has a necrotizing effect. Collagenases and DNAses are produced by clostridiae as well. Some S aureus strains growing on mucous membranes (eg, the vagina in association with menstruation) or in wounds, elaborate toxic shock syndrome toxin-1 (TSST-1), which causes toxic shock syndrome (Chapter 13). The illness is characterized by shock, high fever, and a diffuse red rash that later desquamates; multiple other organ systems are involved as well. TSST-1 is a super antigen and stimulates T-cells to produce large amounts of interleukin-2 (IL-2) and tumor necrosis factor (TNF) (see Chapter 8). The major clinical manifestations of the disease appear to be secondary to the effects of the cytokines. Many of the systemic effects of TSST-1 are similar to those of toxicity caused by lipopolysaccharide (LPS; see discussion below). Some strains of group A β-hemolytic streptococci produce pyrogenic exotoxin A that is similar to or the same as streptococcal erythrogenic toxin, which results in scarlet fever. Rapidly progressive soft tissue infection by streptococci that produce the pyrogenic exotoxin A has many clinical manifestations similar to those of staphylococcal toxic shock syndrome. The pyrogenic exotoxin A also is a super antigen that acts in a manner similar to TSST-1. B. Exotoxins Associated with Diarrheal Diseases and Food Poisoning Exotoxins associated with diarrheal diseases are frequently called enterotoxins. (See also Table 48-3.) Characteristics of some important enterotoxins are discussed below. V cholerae has produced epidemic diarrheal disease (cholera) in many parts of the world (see Chapter 17) and is another toxin-produced disease of historical and current importance. After entering the host via contaminated food or drink, V cholerae penetrates the intestinal mucosa and attaches to microvilli of the brush border of gut epithelial cells. V cholerae, usually of the serotype O1 (and O139), can produce an enterotoxin with a MW of 84,000. The toxin consists of two subunits—A, which is split into two peptides, A1 and A2, linked by a disulfide bond, and B. Subunit B has five identical peptides and rapidly binds the toxin to cell membrane ganglioside molecules. Subunit A enters the cell membrane and causes a large increase in adenylate cyclase activity and in the concentration of cAMP. The net effect is rapid secretion of electrolytes into the small bowel lumen, with impairment of sodium and chloride absorption and loss of bicarbonate. Life-threatening massive diarrhea (eg, 20–30 L/day) can occur, and acidosis develops. The deleterious effects of cholera are due to fluid loss and acid–base imbalance; treatment, therefore, is by electrolyte and fluid replacement. Some strains of S aureus produce enterotoxins while growing in meat, dairy products, or other foods. In typical cases, the food has been recently prepared but not properly refrigerated. There are at least seven distinct types of the staphylococcal enterotoxin. After the preformed toxin is ingested, it is absorbed in the gut, where it stimulates vagus nerve receptors. The stimulus is transmitted to the vomiting center in the central nervous system. Vomiting, often projectile, results within hours. Diarrhea is less frequent. Staphylococcal food poisoning is the most common form of food poisoning. S aureus enterotoxins are super antigens. Enterotoxins are also produced by some strains of Y enterocolitica (see Chapter 19), Vibrio parahaemolyticus (see Chapter 17), Aeromonas species (see Chapter 17), and other bacteria, but the role of these toxins in pathogenesis is not as well defined. The enterotoxin produced by C perfringens is discussed in Chapter 11. C. Lipopolysaccharides of Gram-Negative Bacteria The LPS (endotoxin) of gram-negative bacteria are bacterial cell wall components that are often liberated when the bacteria lyse. The substances are heat-stable, have MWs between 3000 and 5000 (lipooligosaccharides, LOS) and several million (lipopolysaccharides) and can be extracted (eg, with phenol-water). They have three main regions (see Figure 2-19). The pathophysiologic effects of LPS are similar regardless of their bacterial origin except for those of Bacteroides species, which have a different structure and are less toxic (see Chapter 21). LPS in the bloodstream is initially bound to circulating proteins, which then interact with receptors on macrophages neutrophils and other cells of the reticuloendothelial system. Proinflammatory cytokines such as IL-1, IL-6, IL-8, TNF-α, and other cytokines are released, and the complement and coagulation cascades are activated. The following can be observed clinically or experimentally: fever, leukopenia, and hypoglycemia; hypotension and shock resulting in impaired perfusion of essential organs (eg, brain, heart, kidney); intravascular coagulation; and death from massive organ dysfunction. Injection of LPS produces fever after 60–90 minutes, the time needed for the body to release IL-1. Injection of IL-1 produces fever within 30 minutes. Repeated injection of IL-1 produces the same fever response each time, but repeated injection of LPS causes a steadily diminishing fever response because of tolerance partly caused by reticuloendothelial blockade and partly caused by IgM antibodies to LPS. Injection of LPS produces early leukopenia, as does bacteremia with gram-negative organisms. Secondary leukocytosis occurs later. The early leukopenia coincides with the onset of fever caused by liberation of IL-1. LPS enhances glycolysis in many cell types and can lead to hypoglycemia. Hypotension occurs early in gram-negative bacteremia or after injection of LPS. There may be widespread arteriolar and venular constriction followed by peripheral vascular dilatation, increased vascular permeability, decrease in venous return, lowered cardiac output, stagnation in the 158 SECTION III Bacteriology microcirculation, peripheral vasoconstriction, shock, and impaired organ perfusion and its consequences. Disseminated intravascular coagulation (DIC) also contributes to these vascular changes. LPS is among the many different agents that can activate the alternative pathway of the complement cascade, precipitating a variety of complement-mediated reactions (eg, anaphylatoxins, chemotactic responses, membrane damage) and a drop in serum levels of complement components (C3, C5–C9). Disseminated intravascular coagulation is a frequent complication of gram-negative bacteremia and can also occur in other infections. LPS activates factor XII (Hageman factor)—the first step of the intrinsic clotting system—and sets into motion the coagulation cascade, which culminates in the conversion of fibrinogen to fibrin. At the same time, plasminogen can be activated by LPS to plasmin (a proteolytic enzyme), which can attack fibrin with the formation of fibrin split products. Reduction in platelet and fibrinogen levels and detection of fibrin split products are evidence of DIC. Heparin can sometimes prevent the lesions associated with DIC. LPS causes platelets to adhere to vascular endothelium and occlusion of small blood vessels, causing ischemic or hemorrhagic necrosis in various organs. Endotoxin levels can be assayed by the limulus test: A lysate of amebocytes from the horseshoe crab (limulus) gels or coagulates in the presence of 0.0001 μg/mL of endotoxin. This test is rarely used in clinical laboratories because it is difficult to perform accurately. D. Peptidoglycan of Gram-Positive Bacteria The peptidoglycan of gram-positive bacteria is made up of cross-linked macromolecules that surround the bacterial cells (see Chapter 2 and Figure 2-15). Vascular changes leading to shock may also occur in infections caused by grampositive bacteria that contain no LPS. Gram-positive bacteria have considerably more cell wall–associated peptidoglycan than do gram-negative bacteria. Peptidoglycan released during infection may yield many of the same biologic activities as LPS, although peptidoglycan is invariably much less potent than LPS. but have been difficult to prove, especially those of individual enzymes. For example, antibodies against the tissue-degrading enzymes of streptococci do not modify the features of streptococcal disease. In addition to lecithinase, C perfringens produces the proteolytic enzyme collagenase, which degrades collagen, the major protein of fibrous connective tissue, and promotes spread of infection in tissue. S aureus produces coagulase, which works in conjunction with blood factors to coagulate plasma. Coagulase contributes to the formation of fibrin walls around staphylococcal lesions, which helps them persist in tissues. Coagulase also causes deposition of fibrin on the surfaces of individual staphylococci, which may help protect them from phagocytosis or from destruction within phagocytic cells. Hyaluronidases are enzymes that hydrolyze hyaluronic acid, a constituent of the ground substance of connective tissue. They are produced by many bacteria (eg, staphylococci, streptococci, and anaerobes) and aid in their spread through tissues. Many hemolytic streptococci produce streptokinase (fibrinolysin), a substance that activates a proteolytic enzyme of plasma. This enzyme is then able to dissolve coagulated plasma and probably aids in the rapid spread of streptococci through tissues. Streptokinase has been used in treatment of acute myocardial infarction to dissolve fibrin clots. Many bacteria produce substances that are cytolysins— that is, they dissolve red blood cells (hemolysins) or kill tissue cells or leukocytes (leukocidins). Streptolysin O, for example, is produced by group A streptococci and is lethal for mice and hemolytic for red blood cells from many animals. Streptolysin O is oxygen labile and can therefore be oxidized and inactivated, but it is reactivated by reducing agents. It is antigenic. The same streptococci also produce oxygen-stable, serum-inducible streptolysin S, which is not antigenic. Clostridia produce various hemolysins, including the lecithinase described earlier. Hemolysins are produced by most strains of S aureus; staphylococci also produce leukocidins. Most gram-negative rods isolated from sites of disease produce hemolysins. For example, whereas E coli strains that cause urinary tract infections typically produce hemolysins, strains that are part of the normal gastrointestinal flora may or may not produce hemolysins. Enzymes Many species of bacteria produce enzymes that are not intrinsically toxic but do play important roles in the infectious process. Some of these enzymes are discussed below. A. Tissue-Degrading Enzymes Many bacteria produce tissue-degrading enzymes. The best-characterized are enzymes from C perfringens (see Chapter 11), and, to a lesser extent, anaerobic bacteria (see Chapter 21), S aureus (see Chapter 13), and group A streptococci (see Chapter 14). The roles of tissue-degrading enzymes in the pathogenesis of infections appear obvious B. IgA1 Proteases Immunoglobulin A is the secretory antibody on mucosal surfaces. It has two primary forms, IgA1 and IgA2, that differ near the center, or hinge region of the heavy chains of the molecules (see Chapter 8). IgA1 has a series of amino acids in the hinge region that are not present in IgA2. Some bacteria that cause disease produce enzymes, IgA1 proteases, that split IgA1 at specific proline–threonine or proline–serine bonds in the hinge region and inactivate its antibody activity. IgA1 protease is an important virulence factor of the pathogens N gonorrhoeae, N meningitidis, H influenzae, and S pneumoniae. CHAPTER 9 Pathogenesis of Bacterial Infection 159 The enzymes are also produced by some strains of Prevotella melaninogenica, some streptococci associated with dental disease, and a few strains of other species that occasionally cause disease. Nonpathogenic species of the same genera do not have genes coding for the enzyme and do not produce it. Production of IgA1 protease allows pathogens to inactivate the primary antibody found on mucosal surfaces and thereby eliminate protection of the host by the antibody. Antiphagocytic Factors Many bacterial pathogens are rapidly killed after they are ingested by polymorphonuclear cells or macrophages. Some pathogens evade phagocytosis or leukocyte microbicidal mechanisms by adsorbing normal host components to their surfaces. For example, S aureus has surface protein A, which binds to the Fc portion of IgG. Other pathogens have surface factors that impede phagocytosis (eg, S pneumoniae, N meningitidis) many other bacteria have polysaccharide capsules. S pyogenes (group A streptococci) has M protein. N gonorrhoeae (gonococci) has pili. Most of these antiphagocytic surface structures show much antigenic heterogeneity. For example, there are more than 90 pneumococcal capsular polysaccharide types and more than 150 M protein types of group A streptococci. Antibodies against one type of the antiphagocytic factor (eg, capsular polysaccharide, M protein) protect the host from disease caused by bacteria of that type but not from those with other antigenic types of the same factor. A few bacteria (eg, Capnocytophaga and Bordetella species) produce soluble factors or toxins that inhibit chemotaxis by leukocytes and thus evade phagocytosis by a different mechanism. Intracellular Pathogenicity Some bacteria (eg, M tuberculosis, Listeria monocytogenes, Brucella species, and Legionella species) live and grow in the hostile environment within polymorphonuclear cells, macrophages, or monocytes. The bacteria accomplish this feat by several mechanisms: they may avoid entry into phagolysosomes and live within the cytosol of the phagocyte; they may prevent phagosome–lysosome fusion and live within the phagosome; or they may be resistant to lysosomal enzymes and survive within the phagolysosome. Many bacteria can live within nonphagocytic cells (see previous section, Invasion of Host Cells and Tissues). Antigenic Heterogeneity The surface structures of bacteria (and of many other microorganisms) have considerable antigenic heterogeneity. Often these antigens are used as part of a serologic classification system for the bacteria. The classification of the 2000 or so different salmonellae is based principally on the types of the O (LPS side chain) and H (flagellar) antigens. Similarly, there are more than 150 E coli O types and more than 100 E coli K (capsule) types. The antigenic type of the bacteria may be a marker for virulence, related to the clonal nature of pathogens, although it may not actually be the virulence factor (or factors). V cholerae O antigen type 1 and O antigen type 139 typically produce cholera toxin, but very few of the many other O types produce the toxin. Only some of the group A streptococcal M protein types are associated with a high incidence of poststreptococcal glomerulonephritis. N meningitidis capsular polysaccharide types A and C are associated with epidemic meningitis. In the examples cited earlier and in other typing systems that use surface antigens in serologic classification, antigenic types for a given isolate of the species remain constant during infection and on subculture of the bacteria. Some bacteria and other microorganisms have the ability to make frequent shifts in the antigenic form of their surface structures in vitro and presumably in vivo. One wellknown example is Borrelia recurrentis, which causes relapsing fever. A second widely studied example is N gonorrhoeae (see Chapter 20). The gonococcus has three surface-exposed antigens that switch forms at very high rates of about one in every 1000: lipooligosaccharide, 6–8 types; pili, innumerable types; and Opa, 10–12 types for each strain. The number of antigenic forms is so large that each strain of N gonorrhoeae appears to be antigenically distinct from every other strain. Switching of forms for each of the three antigens appears to be under the control of different genetic mechanisms. It is presumed that frequent switching of antigenic forms allows gonococci to evade the host’s immune system; gonococci that are not attacked by the immune system survive and cause disease. Bacterial Secretion Systems Bacterial secretion systems are important in the pathogenesis of infection and are essential for the interaction of bacteria with the eukaryotic cells of the host. The gram-negative bacteria have cell walls with cytoplasmic membranes and outer membranes; a thin layer of peptidoglycan is present. Grampositive bacteria have a cytoplasmic membrane and a very thick layer of peptidoglycan (see Chapter 2). Some gramnegative bacteria and some gram-positive bacteria have capsules as well. The complexity and rigidity of the cell wall structures necessitate mechanisms for the translocation of proteins across the membranes. These secretion systems are involved in cellular functions such as the transport of proteins that make pili or flagella and in the secretion of enzymes or toxins into the extracellular environment. The differences in cell wall structure between gram-negative and gram-positive bacteria result in some differences in the secretion systems. The basic mechanisms of the different bacterial secretion systems are discussed in Chapter 2. (Note: The specific bacterial secretion systems were named in the order of their discovery and not by their mechanisms of action.) Both gram-negative and gram-positive bacteria have a general secretion pathway (Sec) as the major mechanism for 160 SECTION III Bacteriology protein secretion. This pathway is involved in the insertion of most of the bacterial membrane proteins and provides the major pathway for proteins crossing the bacterial cytoplasmic membrane. Gram-negative organisms have an additional six mechanisms, secretion systems (SS) 1–6 (sometimes denoted I–VI), for protein secretion. These can be further characterized as Sec dependent (types 2 and 5) and Sec independent (types 1, 3, 4, 6). Type 2 SS use the general Sec to transport the proteins to the periplasm and then create an outer membrane channel made by a special pore-forming protein complex. This type 2 SS is used to secrete portions of bacterial A B type toxins, such as cholera toxin. Similarly, the type 5 SS, uses the general Sec to export an autotransporter to the periplasm; from there it transports itself across the outer membrane. An example of this type of SS includes the IgA proteases secreted by Haemophilus influenzae. The sec-independent pathways include the type 1 secretion system or ABC secretion system (ATP binding cassette) and the type 3 secretion system. The type 1 and type 3 pathways do not interact with proteins that have been transported across the cytoplasmic membrane by the Sec system. Instead, these systems translocate proteins across both the cytoplasmic and outer membranes. The type 3, which is activated upon contact with a eukaryotic host cell, promotes transport of proteins directly from inside the bacterium to the inside of the host cell using a needlelike structure called an injectosome; when in the host cell cytoplasm, the transported proteins can manipulate host cell function. The type 4 secretion pathway consists of a protein complex that forms a “tunnel” that is able to directly transport proteins or DNA. The most recent SS to be discovered is the type 6 SS. This SS plays a role in the secretion of virulence proteins in V cholerae and Pseudomonas aeruginosa among other gramnegative pathogens. A seventh SS has been described in M tuberculosis and is not well understood. Its function appears to be transport of membrane proteins required for virulence. Some other examples of the secretion systems and their roles in pathogenesis are shown in Table 9-5. These examples are but a small sample designed to illustrate the roles of the large number of molecular secretion activities used by bacteria to provide nutrients and facilitate their pathogenesis. The Requirement for Iron Iron is an essential nutrient for the growth and metabolism of nearly all microorganisms and is an essential cofactor of numerous metabolic and enzymatic processes. The availability of iron in humans for microbial assimilation is limited because the iron is sequestered by the high-affinity iron-binding proteins transferrin in serum and lactoferrin on mucosal surfaces. The ability of a microbial pathogen to efficiently obtain iron from the host environment is critical to its ability to cause disease. The requirement for iron, how bacteria acquire iron, and bacterial iron metabolism are discussed in Chapter 5. Iron availability affects the virulence of many pathogens. For example, iron is an essential virulence factor in P aeruginosa. The use of animal models in Listeria monocytogenes infection has demonstrated that increased iron results in enhanced susceptibility to infection, but iron depletion results in prolonged survival; iron supplementation therapy yields an increase in lethal infections. Decreased iron availability can also be important in pathogenesis. For example, the gene for diphtheria toxin resides on a lysogenic bacteriophage, and only strains of C diphtheriae that carry the lysogenic bacteriophage are toxigenic. In the presence of low iron availability, there is increased production of diphtheria toxin and potentially more severe disease. The virulence of N meningitidis for mice is increased 1000-fold or more when the bacteria are grown under iron-limited conditions. Human iron deficiency also plays a role in the infectious process. Iron deficiency affects hundreds of millions of people worldwide. Iron deficiency can affect multiple organ systems, including the immune system, and can result in impaired cell-mediated immunity and decreased polymorphonuclear cell function. Providing iron therapy during an active infection probably should be delayed because many pathogenic microorganisms can use the small amounts of supplemental iron, resulting in an increase in virulence. The Role of Bacterial Biofilms A biofilm is an aggregate of interactive bacteria attached to a solid surface or to each other and encased in an exopolysaccharide matrix. This is distinct from planktonic or free-living bacteria, in which interactions of the microorganisms do not occur in the same way. Biofilms form a slimy coat on solid surfaces and occur throughout nature. A single species of bacteria may be involved or more than one species may coaggregate to form a biofilm. Fungi, including yeasts, are occasionally involved. After a biofilm is formed, quorum-sensing molecules produced by the bacteria in the biofilm accumulate, resulting in a modification of the metabolic activity of the bacteria. The basic biology of biofilm exopolysaccharide (glycocalyx) is discussed in Chapter 2; the quorum-sensing molecules are discussed in Chapter 1. The bacteria in the exopolysaccharide matrix may be protected from the host’s immune mechanisms. This matrix also functions as a diffusion barrier for some antimicrobials, but other antimicrobials may bind to it. Some of the bacteria within the biofilm show marked resistance to antimicrobials compared with the same strain of bacteria grown free living in broth, which helps to explain why it is so difficult to treat infections associated with biofilms. Biofilms are important in human infections that are persistent and difficult to treat. A few examples include Staphylococcus epidermidis and S aureus infections of central venous catheters, eye infections such as that occur with contact lenses and intraocular lenses, in dental plaque, and in prosthetic joint infections. Perhaps the most profound example of a biofilm in human infection is in P aeruginosa airway infections in cystic fibrosis patients. CHAPTER 9 Pathogenesis of Bacterial Infection 161 TABLE 9-5 Examples of Molecules Translocated by Bacterial Secretion Systems and Their Relevance to Pathogenesis Secretion System Genus Species Substrate and Role in Pathogenesis Type 1 (Sec-independent) Escherichia coli Proteus vulgaris Morganella morganii Bordetella pertussis Pseudomonas aeruginosa Serratia marcescens α Hemolysin makes holes in cell membranes Hemolysin Hemolysin Adenylate cyclase which catalyzes synthesis of cAMP Alkaline protease Zn protease yields host cell damage Type 2 (Sec dependent) Pseudomonas aeruginosa Legionella pneumophila Vibrio cholera Serratia marcescens Elastase, exotoxin A, phospholipase C, others Acid phosphatase, lipase, phospholipase, protease, RNAse Cholera toxin Hemolysin Type 3 (Sec-independent; contact-dependent) Yersinia species Vibrio parahaemolyticus Ysc-Yop system; toxins that block phagocytosis and induce apoptosis Cytotoxin Controls host cell signaling, invasion, and death Effectors from Salmonella pathogenicity islands I and II (SPI1 and SPI2), which promote attachment to and invasion of host cells Enterohemorrhagic (EHEC) and enteropathogenic (EPEC); disruption of epithelial barriers and tight junctions Direct cytotoxicity Bordetella pertussis Helicobacter pylori Neisseria gonorrhoeae Helicobacter pylori Pertussis toxin Cytotoxin DNA export system DNA uptake and release system Neisseria gonorrhoeae Haemophilus influenzae Escherichia coli Shigella flexneri Serratia marcescens Bordetella species Bordetella pertussis Yersinia pestis IgA1 protease splits IgA1 in hinge region and destroys antibody activity (sec-dependent) IgA1 protease, adhesins Serine protease, adhesins, type 1 pili, P-pili Serine protease Proteases Adhesins Filamentous hemagglutinin Capsular antigen Type 6 (Sec Independent) Pseudomonas aeruginosa Vibrio cholerae Pore-forming toxin Hcp1 Virulence proteins Type 7 (Sec dependent) Mycobacterium tuberculosis CFP-10, ESAT-6 T-cell antigen target Pseudomonas aeruginosa Shigella species Salmonella enterica subspecies enterica serotypes Choleraesuis, Dublin, Paratyphi, Typhi, Typhimurium, and so on Escherichia coli Type 4 (Sec-dependent and Sec-independent) Protein substrates DNA substrates Type 5 (Sec dependent) CFP, culture filtrate protein 10 kDa ESAT-6, early secretory antigenic target-6 kDa CHAPTER SUMMARY • • • • • • nimals and human are colonized with abundant A microbiota, normal commensals that do not cause disease and are protective to the host. Virulent bacteria cause disease through the elaboration of factors that facilitate adherence, persistence, invasion, and toxigenicity. Genes that encode virulence factors may be carried on mobile genetic elements such as plasmids or bacteriophages or are found on large pathogenicity islands on bacterial chromosomes. Pili and fimbriae are rodlike or hairlike structures, respectively, that facilitate attachment to host cells. • • Invasion of host cells is a complex mechanism that involves elaboration of proteins that facilitate entry. Bacterial toxins may be extracellular (exotoxins) or are a component of the bacterial cell membrane (endotoxin, LPS) and are among the most powerful toxins in nature (eg, botulinum toxin). Other mechanisms important to bacterial survival and virulence include tissue-degrading enzymes, antiphagocytic factors, IgA proteases, antigenic heterogeneity, and the ability to chelate iron. There are at least seven known bacterial secretion systems, protein complexes, or channels that ensure transport of structural and toxigenic proteins through the bacterial cell after translation. 162 SECTION III Bacteriology REVIEW QUESTIONS 1. A 22-year-old woman who works in a plant nursery presents with a history of fever and cough for 2 months. Over this period of time she has lost 5 kg. Chest radiography shows bilateral upper lobe infiltrates with cavities. A stain of her sputum shows acid-fast bacilli. The likely means by which the patient acquired her infection is (A) Sexual activity (B) Ingesting the microorganisms in her food (C) Holding onto contaminated hand rails when she takes public transportation (D) Handling potting soil (E) Breathing aerosolized droplets containing the micro organism 2. During a pandemic of a well-characterized disease, a group of 175 airline passengers flew from Lima, Peru, to Los A ngeles. Lunch on the plane included crab salad, which was eaten by about two-thirds of the passengers. After landing in Los Angeles, many of the passengers transferred to other flights with destinations in other parts of California and other Western states. Two of the passengers who stayed in Los Angeles developed severe watery diarrhea. The status of the other passengers was unknown. The likely cause of the diarrhea in the two passengers is (A) Escherichia coli O157:H7 (lipopolysaccharide O antigen 157; flagellar antigen 7) (B) Vibrio cholerae type O139 (lipopolysaccharide O antigen 139) (C) Shigella dysenteriae type 1 (D) Campylobacter jejuni (E) Entamoeba histolytica 3. A 65-year-old woman has a long-term central venous catheter for intravenous therapy. She develops fever and subsequently has multiple blood cultures positive for Staphylococcus epidermidis. All of the S epidermidis isolates have the same colony morphology and antimicrobial susceptibility pattern, suggesting that they are the same strain. A S epidermidis biofilm is thought to be present on the catheter. Which one of the following statements about such an infection is correct? (A) The biofilm containing the S epidermidis is likely to wash off the catheter. (B) Production of an extracellular polysaccharide inhibits growth of the S epidermidis, limiting the infection. (C) The S epidermidis in the biofilm are likely to be more susceptible to antimicrobial therapy because the bacteria have decreased rates of metabolism. (D) The quorum-sensing ability of S epidermidis results in increased susceptibility to antimicrobial therapy. (E) The complex molecular interactions within the biofilm make it difficult to provide effective antimicrobial therapy, and it is likely the catheter will have to be removed to cure the infection. 4. The first microorganism to satisfy Koch’s postulates (in the late 19th century) was (A) Treponema pallidum (B) Stenotrophomonas maltophilia (C) Mycobacterium leprae (D) Bacillus anthracis (E) Neisseria gonorrhoeae Which of the following statements about lipopolysaccharide is correct? (A) It interacts with macrophages and monocytes yielding release of cytokines. (B) The toxic component is the O side chain. (C) It forms holes in red blood cell membranes yielding hemolysis. (D) It causes hypothermia. (E) It causes paralysis. 6. A 27-year-old man had a rhinoplasty. A nasal tampon was placed to control the bleeding. Approximately 8 hours later, he developed headache, muscle aches, and abdominal cramps with diarrhea. He then developed an erythematous rash (resembling sunburn) over much of his body, including the palms and soles. His blood pressure is 80/50 mm Hg. The nasal tampon remained in place. His liver enzyme tests were elevated, and there was evidence of moderate renal failure. This patient’s illness was likely to be caused by which of the following? (A) Lipopolysaccharide (B) Peptidoglycan (C) A toxin that is a superantigen (D) A toxin that has A and B subunits (E) Lecithinase (alpha toxin) 7. The organism most likely to be responsible for the patient’s disease (Question 6) is (A) Escherichia coli (B) Corynebacterium diphtheriae (C) Clostridium perfringens (D) Neisseria meningitidis (E) Staphylococcus aureus 8. Which of the following is most likely to be associated with the formation of a bacterial biofilm? (A) Airway colonization in a cystic fibrosis patient with a mucoid (alginate-producing) strain of Pseudomonas aeruginosa (B) Urinary tract infection with Escherichia coli (C) Meningitis with Neisseria meningitidis (D) Tetanus (E) Impetigo caused by Staphylococcus aureus 9. Regarding bacterial type III secretions systems, which of the following statements is correct? (A) They are commonly found in gram-positive commensal bacteria. (B) They play an important role in the pathogenesis of toxininduced diseases of Clostridium species, tetanus, botulism, gas gangrene, and pseudomembranous colitis. (C) They cause release of effectors of pathogenesis into the extracellular environment, promoting bacterial colonization and multiplication. (D) They directly inject bacterial proteins into host cells across bacterial and host cell membranes, promoting pathogenesis of infections. (E) Mutations that prevent the bacterial type III secretion from functioning enhance pathogenesis 10. Which of the following statements is correct? (A) Lipopolysaccharide is part of the cell wall of Escherichia coli. (B) Cholera toxin is attached to the flagella of Vibrio cholerae. (C) The lecithinase of Clostridium perfringens causes diarrhea. (D) Toxic shock syndrome toxin-1 is produced by hemolytic stains of Staphylococcus epidermidis. CHAPTER 9 Pathogenesis of Bacterial Infection 163 11. A 15-year-old Bangladeshi girl develops severe watery diarrhea. The stool looks like “rice water.” It is voluminous—more than 1 L in the last 90 minutes. She has no fever and seems otherwise normal except for the effects of loss of fluid and electrolytes. The most likely cause of her illness is (A) Clostridium difficile enterotoxin (B) A toxin with A and B subunits (C) Shigella dysenteriae type 1 that produces Shiga toxin (D) Enterotoxigenic Escherichia coli that produces heat-labile and heat-stable toxins (E) Staphylococcal enterotoxin F 12. The most important thing that can be done to treat the patient (Question 11) is (A) To give her ciprofloxacin (B) To give her a toxoid vaccine (C) To give her the appropriate antitoxin (D) To treat her with fluid and electrolyte replacement (E) To culture her stool to make the correct diagnosis and then treat specifically 13. A 23-year-old woman has a history of recurrent urinary tract infections, including at least one episode of pyelonephritis. Blood typing shows the P blood group antigen. Which of the following is likely to be the primary cause of her infections? (A) Escherichia coli that produce heat-stable toxin (B) Escherichia coli with K1 (capsular type 1) antigen (C) Escherichia coli O139 (lipopolysaccharide O antigen 139) (D) Escherichia coli with P-pili (fimbriae) (E) Escherichia coli O157:H7 (lipopolysaccharide O antigen 157; flagellar antigen 7) 14. A 55-year-old man presents with gradually increasing weight loss, abdominal pain, diarrhea, and arthropathy. During the evaluation process, a small bowel biopsy is done. After processing, examination of the specimen by light microscopy reveals periodic acid-Schiff–positive inclusions in the bowel wall. Which of the following tests could be done to confirm the diagnosis of Whipple disease, caused by Tropheryma whipplei? (A) Culture on agar media (B) Polymerase chain reaction amplification and sequencing of an appropriate segment of DNA (C) Cocultivation with Escherichia coli (D) In situ hybridization (E) Direct fluorescent antibody test Which of the following best describes the mechanism of action of diphtheria toxin? (A) Forms pores in red blood cells causing hemolysis (B) Degrades lecithin in eukaryotic cell membranes (C) Causes release of tumor necrosis factor (D) Inhibits elongation factor 2 (E) Causes increased adenylate cyclase activity Answers 1. E 5. A 9. D D B 6. C A B E 7. E B D D 8. A D REFERENCES Barton LL: Structural and Functional Relationships in Prokaryotes. Springer, 2005. Coburn B, Sekirov, Finlay BB: Type III secretion systems and disease. Clin Microbiol Rev 2007;20:535. Costerton JW, Stewart PS, Greenberg EP: Bacterial biofilms: A common cause of persistent infections. Science 1999;284:1318. Fredricks DN, Relman DA: Sequence-based identification of microbial pathogens: A reconsideration of Koch’s postulates. Clin Microbiol Rev 1996;9:18. Götz F: MicroReview: Staphylococcus and biofilms. Mol Microbiol 2002;43:1367. Nickerson CA, Schurr MJ (editors): Molecular Paradigms of Infectious Disease: A Bacterial Perspective. Springer, 2006. Relman DA, Falkow S: A molecular perspective of microbial pathogenicity. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. Schmidt H, Hensel M: Pathogenicity islands in bacterial pathogenesis. Clin Microbiol Rev 2004;17:14. Schroeder GN, Hilbi H: Molecular pathogenesis of Shigella spp.: Controlling host cell signaling, invasion, and death by type III secretion. Clin Microbiol Rev 2008;21:134. Wilson BA, Salyers AA, Whitt DD, Winkler ME: Bacterial Pathogenesis, 3rd ed. American Society for Microbiology, 2011. This page intentionally left blank 10 C Normal Human Microbiota The term “normal microbial flora” denotes the population of microorganisms that inhabit the skin and mucous membranes of healthy normal persons. The microorganisms that live inside and on humans (now referred to as the normal microbiota) are estimated to outnumber human somatic and germ cells by a factor of 10. The genomes of these microbial symbionts are collectively defined as the microbiome. Research has shown that the “normal microbiota” provides a first line of defense against microbial pathogens, assist in digestion, play a role in toxin degradation, and contribute to maturation of the immune system. Shifts in the normal microbiota or stimulation of inflammation by these commensals may cause diseases such as inflammatory bowel disease. HUMAN MICROBIOME PROJECT In a broad attempt to understand the role played by resident microbial ecosystems in human health and disease, in 2007, the National Institutes of Health launched the Human Microbiome Project. One of the main goals of this project is to understand the range of human genetic and physiologic diversity, the microbiome, and the factors that influence the distribution and evolution of the constituent microorganisms. One aspect of this project involves having several research groups simultaneously embark upon surveying the microbial communities on human skin and in mucosal areas such as the mouth, esophagus, stomach, colon, and vagina using small-subunit (16S) ribosomal RNA gene sequencing. Among the questions that will be addressed by this project are How stable and resilient is an individual’s microbiota throughout one day and during his or her lifespan? How similar are the microbiomes between members of a family or members of a community or across communities in different environments? Do all humans have an identifiable “core” microbiome, and if so, how is it acquired and transmitted? What affects the genetic diversity of the microbiome, and how does this diversity affect adaptation by the microorganisms and the host to markedly different lifestyles and to various physiological or pathophysiological states? Numerous observations have already been made. For example, it has been determined H A P T E R that there are large differences among individuals in terms of the numbers and types of species of microorganisms inhabiting the colon and that obesity may be correlated with the types of microbes involved in specific metabolic pathways in the gastrointestinal tract. Readers should be aware that this field is rapidly evolving, and our understanding of the human microbiota will necessarily change as more information about resident microbial communities becomes available through the Human Microbiome Project. ROLE OF THE RESIDENT MICROBIOTA The skin and mucous membranes always harbor a variety of microorganisms that can be arranged into two groups: (1) the resident microbiota consists of relatively fi xed types of microorganisms regularly found in a given area at a given age; if disturbed, it promptly reestablishes itself; and (2) the transient microbiota consists of nonpathogenic or potentially pathogenic microorganisms that inhabit the skin or mucous membranes for hours, days, or weeks. The transient microbiota is derived from the environment, does not produce disease, and does not establish itself permanently on the surface. Members of the transient microbiota are generally of little significance so long as the normal resident flora remains intact. However, if the resident microbiota is disturbed, transient microorganisms may colonize, proliferate, and produce disease. Organisms frequently encountered in specimens obtained from various areas of the human body—and considered normal microbiota—are listed in Table 10-1. The classification of anaerobic normal bacterial flora is discussed in Chapter 21. It is likely that microorganisms that can be cultured in the laboratory represent only a fraction of those that are part of the normal resident or transient microbiota. When the broad range polymerase chain reaction (PCR) is used to amplify bacterial 16S rDNA, many previously unidentified bacteria can be detected, as in secretions from patients with bacterial vaginosis. The number of species that make up the normal microbiota has been shown to be much greater than previously recognized. Thus, the understanding of normal 165 166 SECTION III Bacteriology TABLE 10–1 Normal Bacterial Microbiota Skin Staphylococcus epidermidis Staphylococcus aureus (in small numbers) Micrococcus species α-Hemolytic and nonhemolytic streptococci (eg, Streptococcus mitis) Corynebacterium species Propionibacterium species Peptostreptococcus species Acinetobacter species Small numbers of other organisms (Candida species, Pseudomonas aeruginosa, etc) Nasopharynx Any amount of the following: diphtheroids, nonpathogenic Neisseria species, α-hemolytic streptococci; S epidermidis, nonhemolytic streptococci, anaerobes (too many species to list; varying amounts of Prevotella species, anaerobic cocci, Fusobacterium species, etc) Lesser amounts of the following when accompanied by organisms listed above: yeasts, Haemophilus species, pneumococci, S aureus, gramnegative rods, Neisseria meningitidis Gastrointestinal tract and rectum Various Enterobacteriaceae except Salmonella, Shigella, Yersinia, Vibrio, and Campylobacter species Glucose non-fermenting gram-negative rods Enterococci α-Hemolytic and nonhemolytic streptococci Diphtheroids Staphylococcus aureus in small numbers Yeasts in small numbers Anaerobes in large numbers (too many species to list) Genitalia Any amount of the following: Corynebacterium species, Lactobacillus species, α-hemolytic and nonhemolytic streptococci, nonpathogenic Neisseria species The following when mixed and not predominant: enterococci, Enterobacteriaceae and other gram-negative rods, Staphylococcus epidermidis, Candida albicans, and other yeasts Anaerobes (too many to list); the following may be important when in pure growth or clearly predominant: Prevotella, Clostridium, and Peptostreptococcus species microbiota is in transition. As already mentioned, the relationship of previously unidentified microorganisms, which are potentially part of the normal microbiota, to disease is likely to change. The microorganisms that are constantly present on body surfaces are commensals. Their flourishing in a given area depends on physiologic factors of temperature, moisture, and the presence of certain nutrients and inhibitory substances. Their presence is not essential to life because “germ-free” animals can be reared in the complete absence of a normal microbiota. Yet the resident flora of certain areas plays a definite role in maintaining health and normal function. Members of the resident microbiota in the intestinal tract synthesize vitamin K and aid in the absorption of nutrients. On mucous membranes and skin, the resident microbiota may prevent colonization by pathogens and possible disease through “bacterial interference.” The mechanism of bacterial interference may involve competition for receptors or binding sites on host cells, competition for nutrients, mutual inhibition by metabolic or toxic products, mutual inhibition by antibiotic materials or bacteriocins, or other mechanisms. Suppression of the normal microbiota clearly creates a partial local void that tends to be filled by organisms from the environment or from other parts of the body. Such organisms behave as opportunists and may become pathogens. On the other hand, members of the normal microbiota may themselves produce disease under certain circumstances. These organisms are adapted to a noninvasive mode of life defined by the limitations of the environment. If forcefully removed from the restrictions of that environment and introduced into the bloodstream or tissues, these organisms may become pathogenic. For example, streptococci of the viridans group are the most common resident organisms of the upper respiratory tract. If large numbers of them are introduced into the bloodstream (eg, after tooth extraction or oral surgery), they may settle on deformed or prosthetic heart valves and produce infective endocarditis. Small numbers occur transiently in the bloodstream with minor trauma (eg, dental scaling or vigorous brushing). Bacteroides species are the most common resident bacteria of the large intestine CHAPTER 10 Normal Human Microbiota 167 and are quite harmless in that location. However, if introduced into the peritoneal cavity or into pelvic tissues along with other bacteria as a result of trauma, they cause suppuration and bacteremia. There are many other examples, but the important point is that the normal resident microbiota is harmless and may be beneficial in their normal location in the host and in the absence of coincident abnormalities. They may produce disease if introduced into foreign locations in large numbers and if predisposing factors are present. NORMAL MICROBIOTA OF THE SKIN The skin is the human body’s largest organ, colonized by a diverse array of microorganisms, most of which are harmless or even beneficial to the host. Because of its constant exposure to and contact with the environment, the skin is particularly apt to contain transient microorganisms. Nevertheless, there is a constant and well-defined resident flora, modified in different anatomic areas by secretions, habitual wearing of clothing, or proximity to mucous membranes (mouth, nose, and perineal areas) ( Figure 10-1). The predominant resident microorganisms of the skin are aerobic and anaerobic diphtheroid bacilli (eg, Corynebacterium, Propionibacterium); nonhemolytic aerobic and anaerobic staphylococci (Staphylococcus epidermidis and other coagulase-negative staphylococci, occasionally Staphylococcus aureus, and Peptostreptococcus species); grampositive, aerobic, spore-forming bacilli that are ubiquitous in air, water, and soil; α-hemolytic streptococci (viridans streptococci) and enterococci (Enterococcus species); and gram-negative coliform bacilli and Acinetobacter. Fungi and yeasts are often present in skin folds; acid-fast, nonpathogenic mycobacteria occur in areas rich in sebaceous secretions (genitalia, external ear). Among the factors that may be important in eliminating nonresident microorganisms from the skin are the low pH, the fatty acids in sebaceous secretions, and the presence of lysozyme. Neither profuse sweating nor washing and bathing can eliminate or significantly modify the normal resident flora. The number of superficial microorganisms may be diminished by vigorous daily scrubbing with soap containing hexachlorophene or other disinfectants, but the flora is rapidly replenished from sebaceous and sweat glands even when contact with other skin areas or with the environment is completely excluded. Placement of an occlusive dressing on the skin tends to result in a large increase in the total microbial population and may also produce qualitative alterations in the flora. Anaerobes and aerobic bacteria often join to form synergistic infections (gangrene, necrotizing fasciitis, and cellulitis) of skin and soft tissues. The bacteria are frequently part of the normal microbial flora. It is usually difficult to pinpoint one specific organism as being responsible for the progressive lesion because mixtures of organisms are usually involved. In addition to being a physical barrier, the skin is an immunologic barrier. Keratinocytes continuously sample the microbiota colonizing the skin surface through pattern recog nition receptors (eg, Toll-like receptors, mannose receptors, NOD-like receptors). The activation of keratinocyte pattern recognition receptors by pathogen-associated molecular patterns initiates the innate immune response, resulting in the secretion of antimicrobial peptides, cytokines, and chemokines. Despite being constantly exposed to large numbers of microorganisms, the skin can distinguish between harmless commensals and harmful pathogenic microorganisms. The mechanism for this selectivity is unclear. NORMAL MICROBIOTA OF THE MOUTH AND UPPER RESPIRATORY TRACT The flora of the nose consists of prominent corynebacteria, staphylococci (S epidermidis, S aureus), and streptococci. In direct contrast to the highly differentiated communities of their mothers, neonates harbored bacterial communities that were undifferentiated across multiple body habitats, regardless of delivery mode. Thus, at its earliest stage of community development (0.5 mm) colonies Pharyngitis, pyogenic infections similar to group A streptococci Enterococcus faecalis (and other enterococci) D None, α Colon Growth in presence of bile, hydrolyze esculin, growth in 6.5% NaCl, PYR positive Abdominal abscess, urinary tract infection, endocarditis Streptococcus bovis group D None Colon, biliary tree Growth in presence of bile, hydrolyze esculin, no growth in 6.5% NaCl, degrades starch Endocarditis, common blood isolate in colon cancer, biliary disease Streptococcus anginosus group (S anginosus, S intermedius, S constellatus, S milleri group) F (A, C, G) and untypeable α, β, none Throat, colon, female genital tract Small (150,000) of one of six types (a–f). The capsular antigen of type b is a polyribitol ribose phosphate (PRP). Encapsulated H influenzae can be typed by slide agglutination, coagglutination with staphylococci, or agglutination of latex particles coated with type-specific antibodies. 265 266 SECTION III Bacteriology TABLE 18–1 Characteristics and Growth Requirements of the Haemophilus and Aggregatibacter Species Important to Humans Requires Species X V Hemolysis Haemophilus influenzae (H aegyptius) + + - Haemophilus parainfluenzae - + - Haemophilus ducreyi + - - + + + Aggregatibacter aphrophilus - +/– Haemophilus paraphrophaemolyticus - + + Aggregatibacter segnisb - + - Haemophilus haemolyticus a a Formerly Haemophilus aphrophilus and Haemophilus paraphrophilus. b Formerly Haemophilus segnis. X, heme; V, nicotinamide-adenine dinucleotide. A capsule swelling test with specific antiserum is analogous to the quellung test for pneumococci. Typing can also be done by immunofluorescence. Most H influenzae organisms in the normal microbiota of the upper respiratory tract are not encapsulated. The somatic antigens of H influenzae consist of outer membrane proteins. Lipooligosaccharides (endotoxins) share many structures with those of neisseriae. Pathogenesis H influenzae produces no exotoxin. The nonencapsulated organism is a regular member of the normal respiratory microbiota of humans. The capsule is antiphagocytic in the absence of specific anticapsular antibodies. The polyribose phosphate capsule of type b H influenzae is the major virulence factor. The carrier rate in the upper respiratory tract for H influenzae type b was 2–4% in the prevaccine era and is now less than 1%. The carrier rate for nontypeable H influenzae is 50–80% or higher. Type b H influenzae causes meningitis, pneumonia and empyema, epiglottitis, cellulitis, septic arthritis, and occasionally other forms of invasive infection. Nontypeable H influenzae tends to cause chronic bronchitis, otitis media, sinusitis, and conjunctivitis after breakdown of normal host defense mechanisms. The carrier rate for the encapsulated types a and c to f is low (1–2%), and these capsular types rarely cause disease. Although type b can cause chronic bronchitis, otitis media, sinusitis, and conjunctivitis, it does so much less commonly than nontypeable H influenzae. Similarly, nontypeable H influenzae only occasionally causes invasive disease (~5% of cases). The blood of many persons older than age 3–5 years is bactericidal for H influenzae, and clinical infections are less frequent in such individuals. However, bactericidal antibodies have been absent from 25% of adults in the United States, and clinical infections have occurred in adults. Clinical Findings H influenzae type b enters by way of the respiratory tract. There may be local extension with involvement of the sinuses or the middle ear. H influenzae, mostly nontypeable, and pneumococci are two of the most common etiologic agents of bacterial otitis media and acute sinusitis. Encapsulated organisms may reach the bloodstream and be carried to the meninges or, less frequently, may establish themselves in the joints to produce septic arthritis. Before the use of the conjugate vaccine, H influenzae type b was the most common cause of bacterial meningitis in children age 5 months to 5 years in the United States. Clinically, it resembles other forms of childhood meningitis, and diagnosis rests on bacteriologic demonstration of the organism. Occasionally, a fulminating obstructive laryngotracheitis with swollen, cherry-red epiglottis develops in infants and requires prompt tracheostomy or intubation as a lifesaving procedure. Pneumonitis and epiglottitis caused by H influenzae may follow upper respiratory tract infections in small children and old or debilitated people. Adults may have bronchitis or pneumonia caused by H influenzae. Diagnostic Laboratory Tests A. Specimens Specimens consist of expectorated sputum and other types of respiratory specimens, pus, blood, and spinal fluid for smears and cultures depending on the source of the infection. B. Direct Identification Commercial kits are available for immunologic detection of H influenzae antigens in spinal fluid. A positive test result indicates that the fluid contains high concentrations of specific polysaccharide from H influenzae type b. These antigen detection tests generally are not more sensitive than a Gram stain and therefore are not widely used, especially because the incidence of H influenzae meningitis is so low. A Gram stain of H influenzae in sputum is depicted in Figure 18-1. C. Culture Specimens are grown on IsoVitaleX-enriched chocolate agar until typical colonies appear. H influenzae is differentiated from related gram-negative bacilli by its requirements for X and V factors and by its lack of hemolysis on blood agar (see Table 18-1). Tests for X (heme) and V (nicotinamide-adenine dinucleotide) factor requirements can be done in several ways. The Haemophilus species that require V factor grow around paper strips or disks containing V factor placed on the surface of agar that has been autoclaved before the blood was added CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 267 There is a correlation between the presence of bactericidal antibodies and resistance to major H influenzae type b infections. However, it is not known whether these antibodies alone account for immunity. Pneumonia or arthritis caused by infection with H influenzae can develop in adults with such antibodies. Treatment Figure 18-1 Gram stain of Haemophilus influenzae in sputum. The organisms are very small (0.3 × 1 μm) gram-negative coccobacilli (small arrows). The large, irregularly shaped objects (large arrow) are the nuclei of polymorphonuclear cells. Mucus is faintly stained pink in the background. (V factor is heat labile). Alternatively, a strip containing X factor can be placed in parallel with one containing V factor on agar deficient in these nutrients. Growth of Haemophilus in the area between the strips indicates requirement for both factors. A better test for X factor requirement is based on the inability of H influenzae (and a few other Haemophilus species) to synthesize heme from δ-aminolevulinic acid. The inoculum is incubated with the δ-aminolevulinic acid. Haemophilus organisms that do not require X factor synthesize porphobilinogen, porphyrins, protoporphyrin IX, and heme. The presence of red fluorescence under ultraviolet light (~360 nm) indicates the presence of porphyrins and a positive test result. Haemophilus species that synthesize porphyrins (and thus heme) are not H influenzae (see Table 18-1.) Immunity Infants younger than age 3 months may have serum antibodies transmitted from their mothers. During this time, H influenzae infection is rare, but subsequently, the antibodies are lost. Children often acquire H influenzae infections, which are usually asymptomatic but may be in the form of respiratory disease or meningitis. H influenzae was the most common cause of bacterial meningitis in children from 5 months to 5 years of age until the early 1990s when the conjugate vaccines became available (see later discussion). By age 3–5 years, many unimmunized children have naturally acquired anti-PRP antibodies that promote complementdependent bactericidal killing and phagocytosis. Immuni zation of children with H influenzae type b conjugate vaccine induces the same antibodies. The mortality rate for individuals with untreated H influenzae meningitis may be up to 90%. Many strains of H influenzae type b are susceptible to ampicillin, but up to 25% produce β-lactamase under control of a transmissible plasmid and are resistant. Essentially all strains are susceptible to the thirdgeneration cephalosporins. Cefotaxime given intravenously gives excellent results. Prompt diagnosis and antimicrobial therapy are essential to minimize late neurologic and intellectual impairment. Prominent among late complications of H influenzae type b meningitis is the development of a localized subdural accumulation of fluid that requires surgical drainage. Epidemiology, Prevention, and Control Encapsulated H influenzae type b is transmitted from person to person by the respiratory route. H influenzae type b disease can be prevented by administration of Haemophilus b conjugate vaccine to children. Currently, two conjugate vaccines are available for use: PRP-OMPC (polysaccharide linked to outer membrane protein complex), the outer membrane protein complex of Neisseria meningitidis serogroup B, and PRP-T, which uses tetanus toxoid. Beginning at age 2 months, all children should be immunized with one of the conjugate vaccines. Depending on which vaccine product is chosen, the series consists of three doses at 2, 4, and 6 months of age or two doses given at 2 and 4 months of age. An additional booster dose is given sometime between 12 and 15 months of age. Both conjugate vaccines can be given at the time of other vaccine administration such as DTaP (diphtheria, tetanus, and acellular pertussis). Widespread use of H influenzae type b vaccine has reduced the incidence of H influenzae type b meningitis in children by more than 95%. The vaccine reduces the carrier rates for H influenzae type b. Contact with patients with H influenzae type b clinical infection poses little risk for adults but presents a definite risk for nonimmune siblings and other nonimmune children younger than age 4 years who are close contacts. Prophylaxis with rifampin is recommended for such children. HAEMOPHILUS AEGYPTIUS This organism was formerly called the Koch-Weeks bacillus and it is associated with highly communicable form of conjunctivitis (pinkeye) in children. H aegypticus is closely related to H influenzae biotype III, the causative agent of Brazilian purpuric fever. The latter is a disease of children 268 SECTION III Bacteriology characterized by fever, purpura, shock, and death. In the past, these infections were mistakenly attributed to H aegyptius. • AGGREGATIBACTER APHROPHILUS Organisms belonging to the species H aphrophilus and H paraphrophilus have been combined into the same species, and the name was changed to Aggregatibacter aphrophilus. H segnis and Actinobacillus actinomycetemcomitans have also been added to the genus Aggregatibacter. A aphrophilus isolates are often encountered as causes of infective endocarditis and pneumonia. These organisms are present in the oral cavity as part of the normal respiratory microbiota along with other members of the HACEK (Haemophilus species, Actinobacillus/Aggregatibacter species, Cardiobacterium hominis, Eikenella corrodens, and Kingella kingae) group (see Chapter 16) HAEMOPHILUS DUCREYI H ducreyi causes chancroid (soft chancre), a sexually transmitted disease. Chancroid consists of a ragged ulcer on the genitalia, with marked swelling and tenderness. The regional lymph nodes are enlarged and painful. The disease must be differentiated from syphilis, herpes simplex infection, and lymphogranuloma venereum. The small gram-negative rods occur in strands in the lesions, usually in association with other pyogenic microorganisms. H ducreyi requires X factor but not V factor. It is grown best from scrapings of the ulcer base that are inoculated onto chocolate agar containing 1% IsoVitaleX and vancomycin, 3 μg/mL; the agar is incubated in 10% CO2 at 33°C. There is no permanent immunity after chancroid infection. The recommended treatment by the Centers for Disease Control and Prevention is 1 g of azithromycin taken orally. Other treatment regimens include intramuscular ceftriaxone, oral ciprofloxacin, or oral erythromycin; healing results in 2 weeks. OTHER HAEMOPHILUS SPECIES Haemophilus haemoglobinophilus requires X factor but not V factor and has been found in dogs but not in human disease. H haemolyticus is the most markedly hemolytic organism of the group in vitro; it occurs both in the normal nasopharynx and in association with rare upper respiratory tract infections of moderate severity in childhood. H parainfluenzae resembles H influenzae and is a normal inhabitant of the human respiratory tract; it has been encountered occasionally in infective endocarditis and in urethritis. Concept Checks • Haemophilus species are pleomorphic gram-negative rods that require either X (hemin) or V (NAD) factors • • • or both for growth. Most of the species in this genus are colonizers of the upper respiratory tract of humans. H influenzae is the major pathogen in the group, and strains that are encapsulated, especially serotype b, are more virulent, causing invasive disease, including bacteremia and meningitis in unprotected individuals. H influenzae type b, once a significant cause of childhood morbidity and mortality, is now rare in industrialized countries that routinely vaccinate children with one of two available conjugate vaccines. H aphrophilus and H paraphrophilus have been combined into a single new genus and species, A aphrophilus. Other members of the Aggregatibacter genus include A actinomycetemcomitans and A segnis. These organisms are associated with a variety of different infections including endocarditis. H ducreyi is associated with the sexually transmitted disease chancroid. THE BORDETELLAE There are several species of Bordetella. Bordetella pertussis, a highly communicable and important pathogen of humans, causes whooping cough (pertussis). Bordetella parapertussis can cause a similar disease. Bordetella bronchiseptica (Bordetella bronchicanis) causes diseases in animals, such as kennel cough in dogs and snuffles in rabbits, and only occasionally causes respiratory disease and bacteremia in humans, primarily in immunocompromised hosts. Bordetella avium causes turkey coryza and is a rare cause of respiratory illness in humans. Newer species and their disease associations include Bordetella hinzii (bacteremia, respiratory illness, arthritis), Bordetella holmesii (bacteremia among immunosuppressed patients), and Bordetella trematum (wound infections and otitis media). B pertussis, B parapertussis, and B bronchiseptica are closely related, with 72–94% DNA homology and very limited differences in multilocus enzyme analysis; the three species might be considered three subspecies within a species. B avium is a distinct species. BORDETELLA PERTUSSIS Morphology and Identification A. Typical Organisms The organisms are minute gram-negative coccobacilli resem bling H influenzae. With toluidine blue stain, bipolar metachromatic granules can be demonstrated. A capsule is present. B. Culture Primary isolation of B pertussis requires enriched media. Bordet-Gengou medium (potato-blood-glycerol agar) that CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 269 contains penicillin G, 0.5 μg/mL, can be used; however, a charcoal-containing medium supplemented with horse blood (Regan Lowe) is preferable because of the longer shelf life. The plates are incubated at 35–37°C for 3–7 days aerobically in a moist environment (eg, a sealed plastic bag). The small, faintly staining gram-negative rods are identified by immunofluorescence staining. B pertussis is nonmotile. C. Growth Characteristics The organism is a strict aerobe and it is oxidase and catalase positive but nitrate, citrate, and urea negative, the results of which are useful for differentiating among the other species of bordetellae. It does not require X and V factors on subculture. Hemolysis of blood-containing medium is associated with virulent B pertussis. D. Variation When isolated from patients and cultured on enriched media, B pertussis is in the hemolytic and pertussis toxin-producing virulent phase. There are two mechanisms for B pertussis to shift to nonhemolytic, nontoxin-producing avirulent forms. Reversible phenotypic modulation occurs when B pertussis is grown under certain environmental conditions (eg, 28°C vs. 37°C, the presence of MgSO4). Reversible phase variation follows a low-frequency mutation in the genetic locus that controls the expression of the virulence factors (see later discussion). It is possible that these mechanisms play a role in the infectious process, but such a role has not been demonstrated clinically. Antigenic Structure, Pathogenesis, and Pathology B pertussis produces a number of factors that are involved in the pathogenesis of disease. One locus on the B pertussis chromosome acts as a central regulator of virulence genes. This locus has two Bordetella operons, bvgA and bvgS. The products of the A and S loci are similar to those of known two-component regulatory systems. bvgS responds to environmental signals, and bvgA is a transcriptional activator of the virulence genes. Filamentous hemagglutinin and fimbriae mediate adhesion to ciliated epithelial cells and are essential for tracheal colonization. Pertussis toxin promotes lymphocytosis, sensitization to histamine, and enhanced insulin secretion and has adenosine diphosphate– ribosylating activity, with an A/B structure and mechanism of action similar to that of cholera toxin. The filamentous hemagglutinin and pertussis toxin are secreted proteins and are found outside of the B pertussis cells. Adenylate cyclase toxin, dermonecrotic toxin, and hemolysin also are regulated by the bvg system. The tracheal cytotoxin inhibits DNA synthesis in ciliated cells and is not regulated by bvg. The lipopolysaccharide in the cell wall may also be important in causing damage to the epithelial cells of the upper respiratory tract. B pertussis survives for only brief periods outside the human host. There are no vectors. Transmission is largely by the respiratory route from early cases and possibly via carriers. The organism adheres to and multiplies rapidly on the epithelial surface of the trachea and bronchi and interferes with ciliary action. The blood is not invaded. The bacteria liberate the toxins and substances that irritate surface cells, causing coughing and marked lymphocytosis. Later, there may be necrosis of parts of the epithelium and polymorphonuclear infiltration, with peribronchial inflammation and interstitial pneumonia. Secondary invaders such as staphylococci or H influenzae may give rise to bacterial pneumonia. Obstruction of the smaller bronchioles by mucous plugs results in atelectasis and diminished oxygenation of the blood. This probably contributes to the frequency of convulsions in infants with whooping cough. Clinical Findings After an incubation period of about 2 weeks, the “catarrhal stage” develops, with mild coughing and sneezing. During this stage, large numbers of organisms are sprayed in droplets, and the patient is highly infectious but not very ill. During the “paroxysmal” stage, the cough develops its explosive character and the characteristic “whoop” upon inhalation. This leads to rapid exhaustion and may be associated with vomiting, cyanosis, and convulsions. The “whoop” and major complications occur predominantly in infants; paroxysmal coughing predominates in older children and adults. The white blood count is high (16,000–30,000/μL), with an absolute lymphocytosis. Convalescence is slow. B pertussis is a common cause of prolonged (4–6 weeks) cough in adults. Rarely, whooping cough is followed by the serious and potentially fatal complication of encephalitis. Several types of adenovirus and Chlamydia pneumoniae can produce a clinical picture resembling that caused by B pertussis. Diagnostic Laboratory Tests A. Specimens A saline nasal wash is the preferred specimen. Nasopharyngeal swabs or cough droplets expelled onto a “cough plate” held in front of the patient’s mouth during a paroxysm are sometimes used but are not as good as the saline nasal wash. B. Direct Fluorescent Antibody Test The fluorescent antibody (FA) reagent can be used to examine nasopharyngeal swab specimens. However, false-positive and false-negative results may occur; the sensitivity is about 50%. The FA test is most useful in identifying B pertussis after culture on solid media. C. Culture The saline nasal wash fluid is cultured on solid medium agar (see earlier discussion). The antibiotics in the media 270 SECTION III Bacteriology tend to inhibit other respiratory flora but permit growth of B pertussis. Organisms are identified by immunofluorescence staining or by slide agglutination with specific antiserum. D. Polymerase Chain Reaction Polymerase chain reaction (PCR) is the most sensitive method to diagnosis pertussis. Primers for both B pertussis and B parapertussis should be included. When available, the PCR test should replace the direct FA tests. Existing primer targets may cross-react with other Bordetella species. E. Serology Serologic tests on patients are of little diagnostic help because a rise in agglutinating or precipitating antibodies does not occur until the third week of illness. A single serum with high-titer antibodies may be helpful in diagnosing the cause of a long-term cough, one of several weeks’ duration. Immunity Recovery from whooping cough or immunization is followed by immunity. Second infections may occur but are mild; reinfections occurring years later in adults may be severe. It is probable that the first defense against B pertussis infection is the antibody that prevents attachment of the bacteria to the cilia of the respiratory epithelium. Treatment B pertussis is susceptible to several antimicrobial drugs in vitro. Administration of erythromycin during the catarrhal stage of disease promotes elimination of the organisms and may have prophylactic value. Treatment after onset of the paroxysmal phase rarely alters the clinical course. Oxygen inhalation and sedation may prevent anoxic damage to the brain. Prevention Every infant should receive three injections of pertussis vaccine during the first year of life followed by a booster series for a total of five doses. Multiple acellular pertussis vaccines are licensed in the United States and elsewhere. Use of these vaccines is recommended. The acellular vaccines have at least two of the following antigens: inactivated pertussis toxin, filamentous hemagglutinin, fimbrial proteins, and pertactin. Because different vaccines have different antigens, the same product should be used throughout an immunization series. Pertussis vaccine is usually administered in combination with toxoids of diphtheria and tetanus (DTaP). Five doses of pertussis vaccine are recommended before school entry. The usual schedule is administration of doses at 2, 4, 6, and 15–18 months of age and a booster dose at 4–6 years of age. In 2005, it was recommended by the Advisory Committee on Immunization Practices that all adolescents and adults receive a single booster dose of tetanus, diphtheria, and acellular pertussis (Tdap) to replace the booster dose of tetanus and diphtheria toxoids alone (Td). Two acellular pertussis vaccines are available in the United States for use in adolescents and adults. Prophylactic administration of erythromycin for 5 days may also benefit unimmunized infants or heavily exposed adults. Epidemiology and Control Whooping cough is endemic in most densely populated areas worldwide and occurs intermittently in epidemics. The source of infection is usually a patient in the early catarrhal stage of the disease. Communicability is high, ranging from 30–90%. Most cases occur in children younger than age 5 years; most deaths occur in the first year of life. Control of whooping cough rests mainly on adequate active immunization of all infants. BORDETELLA PARAPERTUSSIS This organism may produce a disease similar to whooping cough, but it is generally less severe. The infection is often subclinical. B parapertussis grows more rapidly than typical B pertussis and produces larger colonies. It also grows on blood agar. B parapertussis has a silent copy of the pertussis toxin gene. BORDETELLA BRONCHISEPTICA B bronchiseptica is a small gram-negative bacillus that inhabits the respiratory tracts of canines, in which it may cause “kennel cough” and pneumonitis. It causes snuffles in rabbits and atrophic rhinitis in swine. It is infrequently responsible for chronic respiratory tract infections in humans, primarily in individuals with underlying diseases. It grows on blood agar medium. B bronchiseptica has a silent copy of the pertussis toxin gene. This organism possesses a β-lactamase that renders it resistant to penicillins and cephalosporins. Concept Checks • • Bordetella species are gram-negative coccobacilli. The genus includes diverse species that range from the fastidious and virulent pathogen B pertussis, the cause of “whooping” cough, to species found primarily in animals. B pertussis elaborates numerous virulence factors that are responsible for pathogenesis—fimbriae and filamentous hemagglutinin promote adherence; a variety of toxins such as pertussis toxin, tracheal cytotoxin, hemolysin, dermonecrotic toxin mediate the severe respiratory symptoms, the hallmark lymphocytosis, and a prolonged course. CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 271 • • • • • B pertussis is fastidious and slow growing; specialized media such as Regan-Lowe agar and incubation in ambient conditions at 35–37° C for up to 7 days are required for optimum results. Nucleic acid amplification tests combined with culture are the diagnostic methods of choice. Pertussis, also known as “whooping cough,” begins with the catarrhal stage followed by the characteristic paroxysmal coughing stage that lasts weeks and ends with the convalescence stage. Treatment requires supportive care; erythromycin is given to reduce infectivity, but it does not alter the course of the disease. Disease is preventable by vaccination with an acellular vaccine. The other species of Bordetella may cause respiratory infections but are not capable of causing classic pertussis. THE BRUCELLAE The brucellae are obligate parasites of animals and humans and are characteristically located intracellularly. They are relatively inactive metabolically. Brucella melitensis typically infects goats; Brucella suis, swine; Brucella abortus, cattle; and Brucella canis, dogs. Other species are found only in animals. Although named as species, DNA relatedness studies have shown there is only one species in the genus, B melitensis, with multiple biovars. The disease in humans, brucellosis (undulant fever, Malta fever), is characterized by an acute bacteremic phase followed by a chronic stage that may extend over many years and may involve many tissues. Morphology and Identification A. Typical Organisms The appearance in young cultures varies from cocci to rods 1.2 μm in length, with short coccobacillary forms predominating. They are gram negative but often stain irregularly, and they are aerobic, nonmotile, and nonspore forming. B. Culture Small, convex, smooth colonies appear on enriched media in 2–5 days. C. Growth Characteristics Brucellae are adapted to an intracellular habitat, and their nutritional requirements are complex. Some strains have been cultivated on defined media containing amino acids, vitamins, salts, and glucose. Fresh specimens from animal or human sources are usually inoculated on trypticase-soy agar or blood culture media. Whereas B abortus requires 5–10% CO2 for growth, the other three species grow in air. Brucellae use carbohydrates but produce neither acid nor gas in amounts sufficient for classification. Catalase and oxidase are produced by the four species that infect humans. Hydrogen sulfide is produced by many strains, and nitrates are reduced to nitrites. Brucellae are moderately sensitive to heat and acidity. They are killed in milk by pasteurization. D. Variation The typical virulent organism forms a smooth, transparent colony; upon culture, it tends to change to the rough form, which is avirulent. The serum of susceptible animals contains a globulin and a lipoprotein that suppresses growth of nonsmooth, avirulent types and favors the growth of virulent types. Resistant animal species lack these factors, so that rapid mutation to avirulence can occur. D-alanine has a similar effect in vitro. Antigenic Structure Differentiation among Brucella species or biovars is made possible by their characteristic sensitivity to dyes and their production of H2S. Few laboratories have maintained the procedures for these tests, and the brucellae are seldom placed into the traditional species. Because brucellae are hazardous in the laboratory, tests to classify them should be performed only in reference public health laboratories using appropriate biosafety precautions. Pathogenesis and Pathology Although each species of Brucella has a preferred host, all can infect a wide range of animals, including humans. The common routes of infection in humans are the intestinal tract (ingestion of infected milk), mucous membranes (droplets), and skin (contact with infected tissues of animals). Cheese made from unpasteurized goats’ milk is a particularly common vehicle. The organisms progress from the portal of entry via lymphatic channels and regional lymph nodes to the thoracic duct and the bloodstream, which distributes them to the parenchymatous organs. Granulomatous nodules that may develop into abscesses form in lymphatic tissue, liver, spleen, bone marrow, and other parts of the reticuloendothelial system. In such lesions, the brucellae are principally intracellular. Osteomyelitis, meningitis, or cholecystitis also occasionally occurs. The main histologic reaction in brucellosis consists of proliferation of mononuclear cells, exudation of fibrin, coagulation necrosis, and fibrosis. The granulomas consist of epithelioid and giant cells, with central necrosis and peripheral fibrosis. The brucellae that infect humans have apparent differences in pathogenicity. B abortus usually causes mild disease without suppurative complications; noncaseating granulomas of the reticuloendothelial system are found. B canis also causes mild disease. B suis infection tends to be chronic with suppurative lesions; caseating granulomas may be present. B melitensis infection is more acute and severe. 272 SECTION III Bacteriology Persons with active brucellosis react more markedly (fever, myalgia) than normal persons to injected Brucella endotoxin. Sensitivity to endotoxin thus may play a role in pathogenesis. Placentas and fetal membranes of cattle, swine, sheep, and goats contain erythritol, a growth factor for brucellae. The proliferation of organisms in pregnant animals leads to placentitis and abortion in these species. There is no erythritol in human placentas, and abortion is not part of Brucella infection of humans. Clinical Findings The incubation period ranges from 1–4 weeks. The onset is insidious, with malaise, fever, weakness, aches, and sweats. The fever usually rises in the afternoon; its fall during the night is accompanied by drenching sweat. There may be gastrointestinal and nervous symptoms. Lymph nodes enlarge, and the spleen becomes palpable. Hepatitis may be accompanied by jaundice. Deep pain and disturbances of motion, particularly in vertebral bodies, suggest osteomyelitis. These symptoms of generalized Brucella infection generally subside in weeks or months, although localized lesions and symptoms may continue. After the initial infection, a chronic stage may develop, characterized by weakness, aches and pains, low-grade fever, nervousness, and other nonspecific manifestations compatible with psychoneurotic symptoms. Brucellae cannot be isolated from the patient at this stage, but the agglutinin titer may be high. The diagnosis of “chronic brucellosis” is difficult to establish with certainty unless local lesions are present. Diagnostic Laboratory Tests A. Specimens Blood should be taken for culture, biopsy material for culture (lymph nodes, bone, and so on), and serum for serologic tests. B. Culture Brucella agar was specifically designed to culture Brucella species bacteria. The medium is highly enriched and—in reduced form—is used primarily in cultures for anaerobic bacteria. In oxygenated form, the medium grows Brucella species bacteria very well. However, infection with Brucella species is often not suspected when cultures of a patient’s specimens are set up, and Brucella agar incubated aerobically is seldom used. Brucella species bacteria grow on commonly used media, including trypticase-soy medium with or without 5% sheep blood, brain–heart infusion medium, and chocolate agar. Blood culture media (see below) readily grow Brucella species bacteria. Liquid medium used to culture Mycobacterium tuberculosis also supports the growth of at least some strains. All cultures should be incubated in 8–10% CO2 at 35–37°C and should be observed for 3 weeks before being discarded as being negative results; liquid media cultures should be blindly subcultured during this time. Bone marrow and blood are the specimens from which brucellae are most often isolated. The method of choice for bone marrow is to use pediatric Isolator tubes, which do not require centrifugation, with inoculation of the entire contents of the tube onto solid media. Media used in semiautomated and automated blood culture systems readily grow brucellae, usually within 1 week; however, holding the cultures for 3 weeks is recommended. Negative culture results for Brucella do not exclude the disease because brucellae can be cultivated from patients only during the acute phase of the illness or during recurrence of activity. After a few days of incubation on agar media, the brucellae form colonies in the primary streak that are smaller than 1 mm in diameter. They are nonhemolytic. The observation of tiny gram-negative coccobacilli that are catalase positive and oxidase positive suggests Brucella species. All further work on such a culture should be done in a biologic safety cabinet. A Christensen’s urea slant should be inoculated and observed frequently. A positive urease test result is characteristic of Brucella species. B suis and some strains of B melitensis can yield a positive test result less than 5 minutes after inoculating the slant; other strains take a few hours to 24 hours. Bacteria that meet these criteria should be quickly submitted to a reference public health laboratory for presumptive identification. Brucella species are category B select agents. Molecular methods have been developed to rapidly differentiate among the various biovars. C. Serology Immunoglobulin M (IgM) antibody levels rise during the first week of acute illness, peak at 3 months, and may persist during chronic disease. Even with appropriate antibiotic therapy, high IgM levels may persist for up to 2 years in a small percentage of patients. IgG antibody levels rise about 3 weeks after onset of acute disease, peak at 6–8 weeks, and remain high during chronic disease. IgA levels parallel the IgG levels. The usual serologic tests may fail to detect infection with B canis. Agglutination test—To be reliable, serum agglutination tests must be performed with standardized heat-killed, phenolized, smooth Brucella antigens. IgG agglutinin titers above 1:80 indicate active infection. Individuals injected with cholera vaccine may develop agglutination titers to brucellae. If the serum agglutination test result is negative in patients with strong clinical evidence of Brucella infection, tests must be made for the presence of “blocking” antibodies. These can be detected by adding antihuman globulin to the antigen– serum mixture. Brucellosis agglutinins are cross-reactive with tularemia agglutinins, and tests for both diseases should CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 273 be done on positive sera; usually, the titer for one disease will be much higher than that for the other. Blocking antibodies—These are IgA antibodies that interfere with agglutination by IgG and IgM and cause a serologic test result to be negative in low serum dilutions (prozone), although positive in higher dilutions. These antibodies appear during the subacute stage of infection, tend to persist for many years independently of activity of infection, and are detected by the Coombs antiglobulin method. 3. ELISA assays—IgG, IgA, and IgM antibodies may be detected using enzyme-linked immunosorbent assay (ELISA), which use cytoplasmic proteins as antigens. These assays tend to be more sensitive and specific than the agglutination test especially in the setting of chronic disease Immunity An antibody response occurs with infection, and it is probable that some resistance to subsequent attacks is produced. Immunogenic fractions from Brucella cell walls have a high phospholipid content; lysine predominates among eight amino acids; and there is no heptose (thus distinguishing the fractions from endotoxin). Treatment Brucellae may be susceptible to tetracyclines, rifampin, trimethoprim–sulfamethoxazole, aminoglycosides, and some quinolones. Symptomatic relief may occur within a few days after treatment with these drugs. However, because of their intracellular location, the organisms are not readily eradicated completely from the host. For best results, treatment must be prolonged. Combined treatment with a tetracycline (eg, doxycycline) and either streptomycin for 2–3 weeks or rifampin for 6 weeks is recommended. Epidemiology, Prevention, and Control Brucellae are animal pathogens transmitted to humans by accidental contact with infected animal feces, urine, milk, or tissues. The common sources of infection for humans are unpasteurized milk, milk products, and cheese as well as occupational contact (eg, farmers, veterinarians, and slaughterhouse workers) with infected animals. Cheese made from unpasteurized goat’s milk is a particularly common vehicle for transmission of brucellosis. Occasionally, the airborne route may be important. Because of occupational contact, Brucella infection is much more frequent in men. The majority of infections remain asymptomatic (latent). Infection rates vary greatly with different animals and in different countries. Outside the United States, infection is more prevalent. Eradication of brucellosis in cattle can be attempted by test and slaughter, active immunization of heifers with avirulent live strain 19, or combined testing, segregation, and immunization. Cattle are examined by means of agglutination tests. Active immunization of humans against Brucella infection is experimental. Control rests on limitation of spread and possible eradication of animal infection, pasteurization of milk and milk products, and reduction of occupational hazards wherever possible. Concept Checks • • • • Brucella species are obligate intracellular pathogens found in animals; the disease in humans, brucellosis, known by a variety of synonyms, such as Malta fever, undulant fever, and so on, is caused primarily by contact with animals or animal products, especially unpasteurized milk or cheese. The incubation period ranges from 1–4 weeks; infection may begin abruptly with fever, chills, sweats, and malaise and progress to involve a multisystem illness with splenomegaly, lymphadenopathy, and osteomyelitis; chronic infection may persist for years. Diagnosis may be difficult and in many cases relies on serology because this fastidious organism can be difficult to cultivate even on selective media using prolonged incubation. Treatment consists of prolonged antimicrobial agents that are effective against intracellular pathogens such as rifampin, trimethoprim–sulfamethoxazole, fluoroquinolones, aminoglycosides and tetracyclines. FRANCISELLA TULARENSIS and TULAREMIA Francisella species are widely found in animal reservoirs and aquatic environments. The taxonomy of this genus has undergone numerous changes over the years. There are three recognized subspecies of Francisella tularensis: tularensis (type A), holarctica (type B), and mediasiatica. Subspecies tularensis (type A) is the most virulent among this group and the most pathogenic for humans. It is associated with wild rabbits, ticks, and tabanid flies. Subspecies holarctica strains cause milder infection and are associated with hares, ticks, mosquitoes, and tabanid flies. F tularensis is transmitted to humans by biting arthropods and flies, direct contact with infected animal tissue, inhalation of aerosols, or ingestion of contaminated food or water. The clinical presentation depends on the route of infection; six major syndromes are described (see Pathogenesis and Clinical Findings). Other species of Francisella exist and include Francisella philomiragia, Francisella novicida, and Francisella noatunensis. Infections caused by F philomiragia and F novicida are rare causes of human infections, and the former has usually been found in situations of near-drowning. F noatunensis has not been associated with human disease. These organisms will not be discussed. 274 SECTION III Bacteriology Morphology and Identification A. Typical Organisms F tularensis is a small, gram-negative coccobacillus. It is rarely seen in smears of tissue (Figure 18-2). B. Specimens Blood is taken for serologic tests. The organism may be recovered in culture from lymph node aspirates, bone marrow, peripheral blood, deep tissue, and ulcer biopsies. C. Culture Growth requires enriched media containing cysteine. In the past, glucose-cysteine blood agar was preferred, but F tularensis grows on commercially available hemin containing media such as chocolate agar, modified Thayer-Martin agar, and buffered charcoal yeast extract (BCYE) agar used to grow Legionella species. Media should be incubated in CO2 at 35–37°C for 2–5 days. Caution: To avoid laboratoryacquired infections, biosafety level three (BSL III) practices are required when working with live cultures suspected of containing F tularensis. Clinical specimens require BSL II facilities and practice. D. Serology All isolates are serologically identical, possessing a polysaccharide antigen and one or more protein antigens that crossreact with brucellae. However, there are two major biogroups of strains, called Jellison type A and type B. Type A occurs only in North America, is lethal for rabbits, produces severe illness in humans, ferments glycerol, and contains citrulline ureidase. Type B lacks these biochemical features; is not lethal for rabbits; produces milder disease in humans; and is isolated often from rodents or from water in Europe, Asia, and North America. Other biogroups are of low pathogenicity. The usual antibody response consists of agglutinins developing 7–10 days after the onset of illness. Pathogenesis and Clinical Findings F tularensis is highly infectious: Penetration of the skin or mucous membranes or inhalation of 50 organisms can result in infection. Most commonly, organisms enter through skin abrasions. In 2–6 days, an inflammatory, ulcerating papule develops. Regional lymph nodes enlarge and may become necrotic, sometimes draining for weeks (ulceroglandular tularemia). Inhalation of an infective aerosol results in peribronchial inflammation and localized pneumonitis (pneumonic tularemia). Oculoglandular tularemia can develop when an infected finger or droplet touches the conjunctiva. Yellowish granulomatous lesions on the eyelids may be accompanied by preauricular adenopathy. The other forms of the disease are glandular tularemia (lymphadenopathy but no ulcers), oropharyngeal tularemia, and typhoidal tularemia (septicemia). All affected individuals have fever, malaise, headache, and pain in the involved region and regional lymph nodes. Because of the highly infectious nature of F tularensis, this organism is a potential agent of bioterrorism and is currently classified on the select agent list as a category A agent. Laboratories that recover a suspected F tularensis should notify public health officials and should send the isolate to a reference laboratory capable of performing definitive identification. Diagnostic Laboratory Tests Although F tularensis may be recovered from the clinical specimens listed earlier, the diagnosis rests on serologic studies. Paired serum samples collected 2 weeks apart can show a rise in agglutination titer. A single serum titer of 1:160 is highly suggestive if the history and physical findings are compatible with the diagnosis. Because antibodies reactive in the agglutination test for tularemia also react in the test for brucellosis, both tests should be done for positive sera; the titer for the disease affecting the patient is usually fourfold greater than that for the other disease. Treatment Figure 18-2 Gram stain of Francisella tularensis. These bacteria are tiny gram-negative coccobacilli approximately 0.2 × 0.7 μm. Original magnification ×1000. (Courtesy of CDC Public Health Image Library.) Streptomycin or gentamicin therapy for 10 days almost always produces rapid improvement. Tetracycline may be equally effective, but relapses occur more frequently. Chloramphenicol and ciprofloxacin are other potential agents. F tularensis is resistant to all β-lactam antibiotics as a result of β-lactamase production. CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 275 Prevention and Control Humans acquire tularemia from handling infected rabbits or muskrats or from bites by an infected tick or deer-fly. Less often, the source is contaminated water or food or contact with a dog or cat that has caught an infected wild animal. Avoidance is the key to prevention. The infection in wild animals cannot be controlled. The live attenuated F tularensis vaccine (LVS) is no longer available to persons at high risk. Newer vaccines are under development. Concept Checks • • • • • Francisella tularensis is a faintly staining gram-negative coccobacillus that causes the zoonotic infection tularemia that can be mediated by vectors, such as ticks, through direct contact with animals or rarely through ingestion. There are three subspecies of F tularensis; subspecies tularensis (type A) is the most virulent and pathogenic for humans. There are several clinical manifestations of tularemia depending on the type of exposure; the glandular forms are well-localized and associated with less mortality than the septicemic or inhalational forms of the disease. Diagnosis of tularemia can be made by recovery of the organism from appropriate clinical material and by serology. Agents that have been useful in treatment include streptomycin, gentamicin, tetracyclines, chloramphenicol, and fluoroquinolones. Because of its virulence, F tularensis is considered a potential agent of bioterrorism. REVIEW QUESTIONS 1. A 68-year-old woman was seen in the clinic because she had felt feverish and had been experiencing increasing pain and swelling in her left knee during the past 3 weeks. Four years earlier, a prosthetic joint had been placed in her left knee. On examination, the knee was swollen, and fluid could be detected. An aspirate of the fluid was obtained. There were 15,000 polymorphonuclear cells/mL in the fluid. No organisms were seen on Gram stain. A routine culture was done. On the fourth day of incubation, colorless colonies smaller than 1 mm in diameter were seen on the blood and chocolate agar plates. The organism was a tiny gram-negative coccobacillus that was catalase positive and oxidase positive. A urea slant was inoculated and was positive for urease activity after overnight incubation. The patient was probably infected with which of the following microorganisms? (A) Haemophilus influenzae (B) Haemophilus ducreyi (C) Francisella tularensis (D) Brucella species (E) Staphylococcus aureus After the culture (question 1) turned positive, additional history was obtained. Approximately 4 weeks before the onset of her knee pain, the patient had visited relatives in Israel and traveled to other countries in the Mediterranean area. She had a particular fondness for one food product that was the probable vehicle for her infection. The product most likely was (A) Bananas (B) Unpasteurized goat’s cheese (C) Rare hamburger (D) Fresh orange juice (E) Green tea 3. A 55-year-old game warden in Vermont found a dead muskrat on the bank of a stream. He picked up the animal, thinking it might have been illegally trapped or shot; it was not, and the game warden buried it. Four days later, he developed a 1.5-cm painful ulcer on the index finger of his right hand, a 1-cm ulcer on his right forehead, and pain in his right axilla. Physical examination also revealed right axillary lymphadenopathy. This patient is most likely infected with (A) Brucella species (B) Rickettsia rickettsii (C) Salmonella Typhi (D) Haemophilus ducreyi (E) Francisella tularensis 4. An 18-month-old boy has been playing with a child who develops Haemophilus influenzae meningitis. The boy’s parents consult his pediatrician, who says she is comfortable that the child will be fine because he has been fully immunized with the polyribitol ribose phosphate (PRP)–protein conjugate vaccine. For what reason is it necessary to immunize infants of 2 months to 2 years of age with polysaccharide–protein conjugate vaccines? (A) The conjugate protein is diphtheria toxoid, and the goal is for the infant to develop simultaneous immunity to diphtheria. (B) Infants 2 months to 2 years of age do not immunologically respond to polysaccharide vaccines that are not conjugated to a protein. (C) The conjugate vaccine is designed for older children and adults as well as infants. (D) Maternal (transplacental) antibodies against Haemophilus influenzae are gone from the infant’s circulation by 2 months of age. (E) None of the above 5. An 11-year-old boy from Peru was referred to the Brain Tumor Institute. Three months earlier he had developed headaches and then slowly progressive right-sided weakness. A CT scan showed a mass lesion in the left hemisphere. He was thought to have a brain tumor. A lumbar puncture was not done because of concern about increased intracranial pressure and brain herniation through the tentorium cerebelli. During surgery, a mass lesion in the left hemisphere was found. Frozen sections of the tissue were done while the patient was in the operating room. Microscopy of the sections showed a granulomatous inflammatory reaction. No tumor was seen. Tissue was submitted for culture for Mycobacterium tuberculosis. Middlebrook 7H9 broth medium was used. Six days after the culture was set up, the automated machine detected that the culture result was positive. Results of an acid-fast stain and a Gram stain were both negative. Subcultures were done. Two days later, very small colonies were seen on the sheep blood agar plate. 276 SECTION III Bacteriology The organism was a tiny gram-negative coccobacillus that was catalase positive and oxidase positive. It showed urease activity after 2 hours of incubation on urea-containing medium. This child had infection with (A) Brucella species (B) Mycobacterium tuberculosis (C) Francisella tularensis (D) Haemophilus influenzae (E) Moraxella catarrhalis 6. A 3-year-old child develops Haemophilus influenzae meningitis. Therapy is begun with cefotaxime. Why is this thirdgeneration cephalosporin used rather than ampicillin? (A) About 80% of H influenzae organisms have modified penicillin-binding proteins that confer resistance to ampicillin. (B) The drug of choice, trimethoprim–sulfamethoxazole, cannot be used because the child is allergic to sulfonamides. (C) It is easier to administer intravenous cefotaxime than intravenous ampicillin. (D) There is concern that the child will rapidly develop a penicillin (ampicillin) allergy. (E) About 20% of H influenzae organisms have a plasmid that encodes for beta-lactamase. 7. A 55-year-old man with severe dental caries presented with 1 month of fever, malaise, and back pain and now presents with moderately severe shortness of breath. The examination reveals a febrile man who appears pale and dyspneic. Other physical findings include conjunctival petechiae, a grade III/VI systolic murmur, and an enlarged spleen. Blood cultures grow a pleomorphic gram-negative rod that is not hemolytic and that when tested is X and V factor negative. The most likely causative pathogen is (A) Haemophilus influenzae (B) Haemophilus ducreyi (C) Aggregatibacter aphrophilus (D) Actinobacillus hominis (E) Haemophilus parainfluenzae 8. All of the following statements regarding acellular pertussis vaccines are correct except (A) All formulations of the vaccine contain at least two antigens. (B) The acellular vaccine has replaced the whole cell vaccine in the childhood vaccine series. (C) All children should receive five doses of the vaccine before school entry. (D) The vaccine is approved only for young children and adolescents. (E) The vaccine is safer than and as immunogenic as wholecell vaccines. 9. Which of the following subspecies of Francisella tularensis is the most virulent for humans? (A) tularensis (B) holarctica (C) mediasiatica (D) novicida 10. All of the following statements regarding the etiologic agent of chancroid are correct except (A) The organism is a small gram-negative rod. (B) The organism requires X factor but not V factor. (C) The organism grows well on standard chocolate agar. (D) On Gram stain of lesions, the organism occurs in strands. (E) The organism is susceptible to erythromycin. 11. A 3-month-old infant is brought to the pediatric emergency department in severe respiratory distress. The child appears dehydrated, and there is a prominent peripheral lymphocytosis. The chest radiograph reveals perihilar infiltrates. The child’s grandmother, who watches the infant now that the mother has returned to work, has had a dry hacking cough for about 2 weeks. The most likely causative agent is (A) Haemophilus influenzae type b (B) Bordetella pertussis (C) Streptococcus agalactiae (D) Chlamydia pneumoniae (E) Bordetella bronchiseptica 12. In question 11, the factor responsible for the profound lymphocytosis is (A) A hemagglutinin (B) A polysaccharide capsule (C) An A/B structured toxin (D) A heat-labile toxin (E) A neuraminidase 13. All of the following cause zoonotic infections except (A) Francisella tularensis (B) Brucella melitensis (C) Bordetella pertussis (D) Bacillus anthracis (E) Leptospira interrogans 14. Which of the following is not a recognized virulence factor of Bordetella pertussis? (A) Heat-labile toxin (B) Filamentous hemagglutinin (C) Tracheal cytotoxin (D) Pertussis toxin (E) Dermonecrotic toxin 15. Which of the following pathogens discussed in this chapter is on the select agent list? (A) Haemophilus influenzae (B) Aggregatibacter aphrophilus (C) Bordetella pertussis (D) Francisella tularensis (E) All of the above Answers D 5. A 9. A 13. C 2. B 6. E C A E 7. C B D B 8. D C REFERENCES Broder KR, Cortese MM, Iskander JK, et al; Advisory Committee on Immunization Practices: Preventing tetanus, diphtheria, and pertussis among adolescents: Use of tetanus toxoid, reduced diphtheria toxoid and acellular pertussis vaccines Recommendations of the Advisory Committee on Immunization Practices (ACIP). MMWR Recomm Rep 2006;55(RR-3), 1–34. CHAPTER 18 Haemophilus, Bordetella, Brucella, and Francisella 277 Ledeboer NA, Doern GV: Haemophilus. In Versalovic J, Carroll KC, Funke G, et al (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Miscellaneous fastidious gram-negative bacilli. In Winn W Jr, Allen S, Janda W, et al (editors). Koneman’s Color Atlas and Textbook of Diagnostic Microbiology, 6th ed. Lippincott Williams & Wilkins, 2006. Murphy TF: Haemophilus species (including H. influenzae and chancroid). In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. Nigrovic LE, Wingerter SL. Tularemia. Infect Dis Clin North Am 2008;22:489. Pappas G, Akritidis N, Bosilkovski M, Tsianos E: Brucellosis. N Engl J Med 2005;352:2325–2336. Penn RL: Francisella tularensis (tularemia). In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. Petersen JM, Schriefer ME, Araj GF: Francisella and Brucella. In Versalovic J, Carroll KC, Funke G, et al (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Von Konig CHW, Riffelmann M, Coenye T: Bordetella and related genera. In Versalovic J, Carroll KC, Funke G,et al (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Waters V, Halperin S: Bordetella pertussis. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. Young EJ: Brucella species. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. This page intentionally left blank 19 C Yersinia and Pasteurella The organisms discussed in this chapter are short, pleomorphic gram-negative rods that can exhibit bipolar staining. They are catalase positive, oxidase negative, and microaerophilic or facultatively anaerobic. Most have animals as their natural hosts, but they can produce serious disease in humans. The genus Yersinia includes Yersinia pestis, the cause of plague; Yersinia pseudotuberculosis and Yersinia enterocolitica, important causes of human diarrheal diseases; and several others considered nonpathogenic for humans. Several species of Pasteurella are primarily animal pathogens but can also produce human disease. YERSINIA PESTIS and PLaGUE Plague is an infection of wild rodents transmitted from one rodent to another and occasionally from rodents to humans by the bites of fleas. Serious infection often results, which in previous centuries produced pandemics of “black death” with millions of fatalities. The ability of this organism to be transmitted by aerosol and the severity and high mortality associated with pneumonic plague make Y pestis a potential biological weapon. Morphology and identification Y pestis is a gram-negative rod that exhibits striking bipolar staining with special stains such as Wright, Giemsa, Wayson, or methylene blue (Figure 19-1). It is nonmotile. It grows as a facultative anaerobe on many bacteriologic media. Growth is more rapid in media containing blood or tissue fluids and fastest at 30°C. In cultures on blood agar at 37°C, colonies may be very small at 24 hours. A virulent inoculum, derived from infected tissue, produces gray and viscous colonies, but after passage in the laboratory, the colonies become irregular and rough. The organism has little biochemical activity, and this is somewhat variable. Antigenic Structure All yersiniae possess lipopolysaccharides that have endotoxic activity when released. The three pathogenic species H A P T E R produce antigens and toxins that act as virulence factors. They have type III secretion systems that consist of a membrane-spanning complex that allows the bacteria to inject proteins directly into cytoplasm of the host cells. The virulent yersiniae produce V and W antigens, which are encoded by genes on a plasmid of approximately 70 kb. This is essential for virulence; the V and W antigens yield the requirement for calcium for growth at 37°C. Compared with the other pathogenic yersiniae, Y pestis has gained additional plasmids. pPCP1 is a 9.5 kb plasmid that contains genes that yield plasminogen-activating protease that has temperaturedependent coagulase activity (20°–28°C, the temperature of the flea) and fibrinolytic activity (35°–37°C, the temperature of the host). This factor is involved in dissemination of the organism from the flea bite injection site. The pFra/pMT plasmid (80–101 kb) encodes the capsular protein (fraction F1) that is produced mainly at 37°C and confers antiphagocytic properties. In addition, this plasmid contains genes that encode phospholipase D, which is required for organism survival in the flea midgut. The three pathogenic yersiniae have a pathogenicity island (PAI) that encodes for an iron-scavenging siderophore (see Chapter 9), yersiniabactin. Among several exotoxins produced, one is lethal for mice in amounts of 1 µg. This homogeneous protein (molecular weight, 74,000) produces β-adrenergic blockade and is cardiotoxic in animals. Its role in human infection is unknown. Pathogenesis and Pathology When a flea feeds on a rodent infected with Y pestis, the ingested organisms multiply in the gut of the flea and, helped by the coagulase, block its proventriculus so that no food can pass through. Subsequently, the “blocked” and hungry flea bites ferociously, and the aspirated blood, contaminated with Y pestis from the flea, is regurgitated into the bite wound. The inoculated organisms may be phagocytosed by polymorphonuclear cells and macrophages. The Y pestis organisms are killed by the polymorphonuclear cells but multiply in the macrophages; because the bacteria are multiplying at 37°C, they produce the antiphagocytic protein and subsequently are able to resist phagocytosis. The pathogens rapidly reach 279 280 SECTION III Bacteriology Diagnostic Laboratory Tests Plague should be suspected in febrile patients who have been exposed to rodents in known endemic areas. Rapid recognition and laboratory confirmation of the disease are essential to institute lifesaving therapy. A. Specimens Blood is taken for culture and aspirates of enlarged lymph nodes for smear and culture. Acute and convalescent sera may be examined for antibody levels. In pneumonia, sputum is cultured; in possible meningitis, cerebrospinal fluid is taken for smear and culture. B. Smears Figure 19-1 Yersinia pestis (arrows) in blood, Wright-Giemsa stain. Some of the Yersinia pestis have bipolar staining, which gives them a hairpin-like appearance. Original magnification ×1000. (Courtesy of K Gage, Plague Section, Centers for Disease Control and Prevention, Ft. Collins, CO.) the lymphatics, and an intense hemorrhagic inflammation develops in the enlarged lymph nodes, which may undergo necrosis and become fluctuant. Although the invasion may stop there, Y pestis organisms often reach the bloodstream and become widely disseminated. Hemorrhagic and necrotic lesions may develop in all organs; meningitis, pneumonia, and serosanguineous pleuropericarditis are prominent features. Primary pneumonic plague results from inhalation of infective droplets (usually from a coughing patient), and it is characterized by hemorrhagic consolidation, sepsis, and death. Clinical Findings The clinical manifestations of plague depend on the route of exposure. After an incubation period of 2–7 days, there is high fever and painful lymphadenopathy, commonly with greatly enlarged, tender nodes (buboes) in the neck, groin, or axillae. This is the bubonic form of the disease. Vomiting and diarrhea may develop with the early septicemic form of disease. Later, disseminated intravascular coagulation leads to hypotension, altered mental status, and renal and cardiac failure. Terminally, signs of pneumonia and meningitis can appear, and Y pestis multiplies intravascularly and can be seen in blood smears. Primary pneumonic plague results from direct inhalation of organism into the lung. Patients often have a fulminant course with chest pain, cough, hemoptysis, and severe respiratory distress. Y pestis are small gram-negative bacilli that appear as single cells or as pairs or short chains in clinical material. Wright, Giemsa, or Wayson stains may be more useful when staining material from a suspected buboe or a positive blood culture result because of the striking bipolar appearance (safety pin shape) of the organism using these stains that is not evident on a direct Gram stain. More specific direct staining methods (possibly available through reference laboratories) include the use of fluorescent antibody stains targeting the capsular F1 antigen. C. Culture All materials are cultured on blood agar, chocolate, and MacConkey agar plates and in brain–heart infusion broth. Growth on solid media may be slow, requiring more than 48 hours, but blood culture results are often positive in 24 hours. Cultures can be tentatively identified by biochemical reactions. Y pestis produces nonlactose-fermenting colonies on MacConkey agar, and it grows better at 25°C than at 37°C. The organism is catalase positive; indole, oxidase, urease negative; and nonmotile. The last two reactions are useful in differentiating Y pestis from other pathogenic yersiniae. An organism with these characteristics should be referred to a public health laboratory for more confirmatory testing. Definite identification of cultures is best done by immunofluorescence or by lysis by a specific Y pestis bacteriophage (confirmation available through state health department laboratories and by consultation with the Centers for Disease Control and Prevention [CDC], Plague Branch, Fort Collins, CO). All cultures are highly infectious and must be handled with extreme caution inside a biological safety cabinet. D. Serology In patients who have not been previously vaccinated, a convalescent serum antibody titer of 1:16 or greater is presumptive evidence of Y pestis infection. A titer rise in two sequential specimens confirms the serologic diagnosis. CHAPTER 19 Yersinia and Pasteurella 281 Treatment Unless promptly treated, plague may have a mortality rate of nearly 50%; pneumonic plague, nearly 100%. The drug of choice is streptomycin, but the more readily available aminoglycoside gentamicin has been shown to be as effective. Doxycycline is an alternative drug as are fluoroquinolone antibiotics. These agents are sometimes given in combination with streptomycin or gentamicin. Drug resistance has rarely been noted in Y pestis. Epidemiology and Control Plague is an infection of wild rodents (field mice, gerbils, moles, skunks, and other animals) that occurs in many parts of the world. The chief enzootic areas are India, Southeast Asia (especially Vietnam), Africa, and North and South America. The western states of the United States and Mexico also contain reservoirs of infection. Epizootics with high mortality rates occur intermittently; at such times, the infection can spread to domestic rodents (eg, rats) and other animals (eg, cats), and humans can be infected by flea bites or by contact. The commonest vector of plague is the rat flea (Xenopsylla cheopis), but other fleas may also transmit the infection. The control of plague requires surveys of infected animals, vectors, and human contacts; in the United States, this is done by county and state agencies with support from the Plague Branch of the CDC and by destruction of plagueinfected animals. If a human case is diagnosed, health authorities must be notified promptly. All patients with suspected plague should be isolated, particularly if pulmonary involvement has not been ruled out. All specimens must be treated with extreme caution. Contacts of patients with suspected plague pneumonia should receive doxycycline as chemoprophylaxis. Killed whole-cell vaccines are no longer available. Because of concern for bioterrorism, numerous vaccines are currently under development. enterotoxin, but the role of this toxin in diarrhea associated with infection is not well defined. Y enterocolitica has been isolated from rodents and domestic animals (eg, sheep, cattle, swine, dogs, and cats) and waters contaminated by them. Transmission to humans probably occurs by contamination of food, drink, or fomites. Y pseudotuberculosis occurs in domestic and farm animals and birds, which excrete the organisms in feces. Human infection probably results from ingestion of materials contaminated with animal feces. Person-to-person transmission with either of these organisms is probably rare. Pathogenesis and Clinical Findings An inoculum of 108–109 yersiniae must enter the alimentary tract to produce infection. During the incubation period of 4–7 days, yersiniae multiply in the gut mucosa, particularly the ileum. This leads to inflammation and ulceration, and leukocytes appear in feces. The process may extend to mesenteric lymph nodes and, rarely, to bacteremia. Early symptoms include fever, abdominal pain, and diarrhea. Diarrhea ranges from watery to bloody and may be caused by an enterotoxin or to invasion of the mucosa. At times, the abdominal pain is severe and located in the right lower quadrant, suggesting appendicitis (more common with Y pseudotuberculosis). One to 2 weeks after onset, some patients with histocompatibility antigen HLA-B 27 develop arthralgia, arthritis, and erythema nodosum, suggesting an immunologic reaction to the infection. Very rarely, Yersinia infection produces pneumonia, meningitis, or sepsis; in most cases, it is self-limited. Y enterocolitica has also been associated with transfusion-related infections caused by contaminated red blood cells. This is a consequence of the ability of the organism, transmitted by an asymptomatic donor, to multiply at refrigeration temperatures. Diagnostic Laboratory Tests A. Specimens YERSINIA ENTEROCOLITICA and YERSINIA PSEUDOTUBERCULOSIS Specimens may be stool, blood, or material obtained at surgical exploration. Stained smears are not contributory. These are nonlactose-fermenting gram-negative rods that are urease positive and oxidase negative. They grow best at 25°C and are motile at 25°C but nonmotile at 37°C. They are found in the intestinal tracts of a variety of animals, in which they may cause disease, and are transmissible to humans, in whom they can produce a variety of clinical syndromes. Y enterocolitica exists in more than 70 serotypes; most isolates from human disease belong to serotypes O:3, O:8, and O:9. There are striking geographic differences in the distribution of Y enterocolitica serotypes. Y pseudotuberculosis exists in at least six serotypes, but serotype O:1 accounts for most human infections. Y enterocolitica can produce a heat-stable B. Culture The number of yersiniae in stool may be small and can be increased by “cold enrichment”: a small amount of feces or a rectal swab is placed in buffered saline with a pH of 7.6 and kept at 4°C for 2–4 weeks; many fecal organisms do not survive, but Y enterocolitica multiplies. Subcultures made at intervals on MacConkey agar may yield yersiniae. Alternatively, most clinical laboratories use a Yersiniaselective agar such as cefsulodin-Irgasan-novobiocin (CIN) agar incubated at room temperature for several days. Y enterocolitica colonies have a bull’s eye appearance with a red center on CIN agar. 282 SECTION III Bacteriology C. Serology • In paired serum specimens taken 2 or more weeks apart, a rise in agglutinating antibodies can be shown; however, cross-reactions between yersiniae and other organisms (vibrios, salmonellae, and brucellae) may confuse the results. • Treatment Most Yersinia infections with diarrhea are self-limited, and the possible benefits of antimicrobial therapy are unknown. Y enterocolitica is generally susceptible to aminoglycosides, chloramphenicol, tetracycline, trimethoprim–sulfamethoxazole, piperacillin, third-generation cephalosporins, and fluoroquinolones; it is typically resistant to ampicillin and to first-generation cephalosporins. Proved Yersinia sepsis or meningitis has a high mortality rate, but deaths occur mainly in immunocompromised patients. Yersinia sepsis can be successfully treated with third-generation cephalosporins (possibly in combination with an aminoglycoside) or a fluoroquinolone (possibly in combination with another antimicrobial). In cases in which clinical manifestations strongly point to either appendicitis or mesenteric adenitis, surgical exploration has been the rule unless several simultaneous cases indicate that Yersinia infection is likely. • Y enterocolitica and Y pseudotuberculosis cause gastroenteritis or mesenteric lymphadenitis after ingestion of contaminated food or water. Yersinia species can be recovered from the stool of infected patients using selective media called CIN agar incubated at room temperature. Treatment for gastroenteritis caused by Yersinia consists of trimethoprim–sulfamethoxazole, doxycycline, or a fluoroquinolone antibiotic. PASTEURELLA Contact with farm and domestic animals, their feces, or materials contaminated by them probably accounts for most human infections. Meat and dairy products have occasionally been indicated as sources of infection, and group outbreaks have been traced to contaminated food or drink. Conventional sanitary precautions are probably helpful. There are no specific preventive measures. Pasteurella species are primarily animal pathogens, but they can produce a range of human diseases. Pasteurellae are nonmotile gram-negative coccobacilli with a bipolar appearance on stained smears. They are aerobes or facultative anaerobes that grow readily on ordinary bacteriologic media at 37°C. They are all oxidase positive and catalase positive but diverge in other biochemical reactions. Pasteurella multocida occurs worldwide in the respiratory and gastrointestinal tracts of many domestic and wild animals. It is perhaps the most common organism in human wounds inflicted by bites from cats and dogs. It is one of the common causes of hemorrhagic septicemia in a variety of animals, including rabbits, rats, horses, sheep, fowl, cats, and swine. It can also produce human infections in many systems and may at times be part of normal human microbiota. Pasteurella bettyae has been recovered from infections of the human genital tract and of newborns. Its habitat is uncertain. Pasteurella pneumotropica is a normal inhabitant of the respiratory tracts and guts of mice and rats and can cause pneumonia or sepsis when the host–parasite balance is disturbed. A few human infections have followed animal bites. Concept Checks Clinical Findings Prevention and Control • • • • Yersinia species are zoonotic pathogens that cause disease in humans ranging from mild gastrointestinal infections to serious diseases with high mortality such as plague. Y pestis is transmitted to humans usually through the bite of an infected flea, although inhalation is another potential route. Y pestis possesses virulence factors transmitted by plasmids that allow it to survive in the gut of the flea and that contribute to severe clinical manifestations in humans. A bubo (an enlarged suppurative lymph node) forms close to the bite wound accompanied by fever and is the most common form of plague. From the localized lesion, the infection may disseminate causing the septicemic form of the disease. Treatment consists of supportive care and antibiotic treatment with streptomycin, gentamicin, doxycycline, or a fluoroquinolone antibiotic. The most common presentation is a history of animal bite followed within hours by an acute onset of redness, swelling, and pain. Regional lymphadenopathy is variable, and fever is often low grade. Pasteurella infections sometimes present as bacteremia or chronic respiratory infection without an evident connection with animals. P multocida is susceptible to most antibiotics. Penicillin G is considered the drug of choice for P multocida infections resulting from animal bites. Tetracyclines and fluoroquinolones are alternative drugs. REVIEW QUESTIONS 1. An 18-year-old male resident of Arizona came to the emergency department (ED) complaining of fever, pain in his left groin, and diarrhea for the past 2 days. On examination, he was afebrile, had a pulse rate of 126 beats/min, a respiratory rate of 20 breaths/min, and a blood pressure of 130/80 mm Hg. Left groin CHAPTER 19 Yersinia and Pasteurella 283 swelling and tenderness were noted. A groin muscle strain was diagnosed, attributed to a fall 2 days earlier. He was treated with nonsteroidal anti-inflammatory drugs and released. The next day, the patient reported feeling weak, had difficulty breathing, and collapsed while taking a shower. He was transported to a hospital ED and pronounced dead shortly after arrival. Cultures of blood samples obtained in the ED were positive for Yersinia pestis. An epidemiologic investigation indicated that the patient most likely became infected as a result of bites by Yersinia pestis–infected fleas while walking through a prairie dog colony (see Chapter 48.) Which of the following statements about the pathogenesis of plague is correct? (A) Y pestis produces a coagulase when incubated at 28°C. (B) There is no risk for pneumonia caused by person-to-person transmission of Y pestis. (C) Y pestis organisms multiply in polymorphonuclear cells. (D) After the bite of an infected flea, Y pestis infection seldom, if ever, disseminates beyond the site of the flea bite and the regional lymph nodes. (E) Y pestis is transmitted to animals (and humans) in flea feces excreted when the flea is feeding. 2. The drug of choice to treat the patient in question 1 would have been (A) Ampicillin (B) Cefotaxime (C) Levofloxacin (D) Erythromycin (E) Streptomycin 3. Yersinia pestis entered North America through San Francisco in the 1890s, carried by rats on ships that had sailed from Hong Kong, where a plague epidemic occurred. The current reservoir for Y pestis in the United States is (A) Urban feral cats (B) Urban rats (C) Domestic cows (D) Coyotes (E) Rural wild rodents 4. Which of the following is generally not considered a potential agent of bioterrorism and biologic warfare? (A) Yersinia pestis (B) Botulinum toxin (C) Streptococcus pyogenes (D) Brucella species (E) Bacillus anthracis 5. An 8-year-old boy was bitten by a stray cat. Two days later, the wound was red and swollen and drained purulent fluid. Pasteurella multocida was cultured from the wound. The drug of choice to treat this infection is (A) Amikacin (B) Erythromycin (C) Gentamicin (D) Penicillin G (E) Clindamycin 6. Intimate contacts of patients with suspected plague pneumonia should receive which of the following agents as chemoprophylaxis? (A) Gentamicin (B) Cefazolin (C) Rifampin (D) Penicillin (E) Doxycycline 7. In a patient who has the bubonic form of plague, all of the following specimens are acceptable for diagnosis except (A) Stool culture on hektoen enteric agar (B) Blood culture using routine laboratory media (C) Culture of a lymph node aspirate on blood and MacConkey agars (D) Acute and convalescent serology (E) Immunohistochemical staining of lymph node tissue 8. All of the following statements regarding the pFra/pMT plasmid of Yersinia pestis are true except (A) It encodes the capsular protein (fraction FI) that confers antiphagocytic properties. (B) It contains genes that yield plasminogen-activating protease that has temperature-dependent coagulase activity. (C) It contains genes that encode phospholipase D, which is required for organism survival in the flea midgut. (D) It is unique to Y pestis. (E) It encodes factors that are important for survival in both the flea and the human. 9. All of the following statements regarding the epidemiology of infections caused by Yersinia enterocolitica are correct except (A) Most human infections are caused by serotype O:1. (B) Humans acquire the infection from ingestion of food or drinks contaminated by animals or animal products. (C) Person-to-person spread is quite common. (D) A large inoculum is required to cause infection. (E) Infection is more prevalent in persons with histocompatibility antigen HLA-B27. 10. Which of the following Pasteurella species has been associated with infections of the female genital tract and of newborns? (A) Pasteurella multocida (B) Pasteurella pneumotropica (C) Pasteurella ureae (D) Pasteurella bettyae 11. Optimum recovery of Yersinia enterocolitica from the stools of patients with gastroenteritis requires which of the following specialized media? (A) Cefsulodin-Irgasan-novobiocin agar (B) Xylose-lysine decarboxylase agar (C) Hektoen-enteric agar (D) Regan-Lowe medium (E) MacConkey agar 12. Which of the following organisms is likely to cause a transfusion reaction even if the donor is asymptomatic? (A) Pasteurella multocida (B) Escherichia coli (C) Pasteurella bettyae (D) Yersinia enterocolitica (E) None of the above 13. A 25-year-old graduate student is rushed to the operating room for fever, acute abdominal pain, and leukocytosis suggestive of acute appendicitis. During surgery, the appendix appears normal, but numerous, enlarged mesenteric lymph nodes are present. A likely diagnosis is (A) Epstein-Barr virus infection causing atypical infectious mononucleosis 284 SECTION III Bacteriology (B) Mesenteric lymphadenitis caused by Y pseudotuberculosis (C) Gastrointestinal plague (D) An unusual presentation of shigellosis (E) Lymphadenitis caused by Pasteurella pneumotropica 14. A typical source of the infection in the case in question 13 is (A) Contact with the patient’s pet cat saliva (B) Accidental ingestion of prairie dog feces (C) Ingestion of contaminated water or food (D) Direct contact with another infected individual (E) Bite of an infected arthropod 15. An organism suspected of being Yersinia pestis is recovered from a patient with sepsis. The isolate has bipolar staining is catalase positive but is oxidase and urease negative and is nonmotile. At this point, what should be done? (A) Nothing; the laboratory has confirmed the diagnosis. (B) Inoculate the isolate to an identification kit or automated system for confirmation. (C) Call the police because there is a possible bioterrorism event. (D) Send the isolate to the nearest public health laboratory for confirmation. (E) Send the isolate to the hospital across town for sequencing. Answers C 1. A 5. D 13. B 2. E 6. E D C E 7. A A D C 8. B D REFERENCES Dennis DT, Mead PS: Yersinia species, including plague. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. Prentice MB, Rahalson L: Plague. Lancet 2007;369:1196–1207. Schriefer ME, Petersen JM: Yersinia. In Versalovic J, Carroll KC, Funke G, et al (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Zbinden R, von Graevenitz A: Actinobacillus, Capnocytophaga, Eikenella, Kingella, Pasteurella, and other fastidious or rarely encountered gram-negative rods. In Versalovic J, Carroll KC, Funke G, et al (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Zurlo JJ: Pasteurella species. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Elsevier, 2010. 20 C The Neisseriae The family Neisseriaceae includes the genera Neisseria, Kingella, Eikenella, Simonsiella, and Alysiella (see Chapter 16). The neisseriae are gram-negative cocci that usually occur in pairs (diplococci). Neisseria gonorrhoeae (gonococci) and Neisseria meningitidis (meningococci) are pathogenic for humans and typically are found associated with or inside polymorphonuclear cells. Some neisseriae are normal inhabitants of the human respiratory tract, rarely if ever cause disease, and occur extracellularly. Members of the group are listed in Table 20-1. Gonococci and meningococci are closely related, with 70% DNA homology, and are differentiated by a few laboratory tests and specific characteristics: Meningococci have polysaccharide capsules but gonococci do not, and meningococci rarely have plasmids but most gonococci do. Most importantly, the two species are differentiated by the usual clinical presentations of the diseases they cause: Meningococci typically are found in the upper respiratory tract and cause meningitis, but gonococci cause genital infections. The clinical spectra of the diseases caused by gonococci and meningococci overlap, however. H A P T E R C. Growth Characteristics The neisseriae grow best under aerobic conditions, but some grow in an anaerobic environment. They have complex growth requirements. Most neisseriae oxidize carbohydrates, producing acid but not gas, and their carbohydrate patterns are a means of distinguishing them (see Table 20-1). The neisseriae produce oxidase and give positive oxidase reactions; the oxidase test is a key test for identifying them. When bacteria are spotted on a fi lter paper soaked with tetramethylparaphenylenediamine hydrochloride (oxidase), the neisseriae rapidly turn dark purple. Meningococci and gonococci grow best on media containing complex organic substances such as heated blood, hemin, and animal proteins and in an atmosphere containing 5% CO2 (eg, candle jar). Growth is inhibited by some toxic constituents of the medium (eg, fatty acids or salts). The organisms are rapidly killed by drying, sunlight, moist heat, and many disinfectants. They produce autolytic enzymes that result in rapid swelling and lysis in vitro at 25°C and at an alkaline pH. NEISSERIA GONORRHOEAE Morphology and identification A. Typical Organisms The typical Neisseria is a gram-negative, nonmotile diplococcus, approximately 0.8 μm in diameter (Figures 20-1 and 20-2). Individual cocci are kidney shaped; when the organisms occur in pairs, the flat or concave sides are adjacent. B. Culture In 48 hours on enriched media (eg, modified Thayer-Martin, Martin-Lewis, GC-Lect, and New York City), gonococci and meningococci form convex, glistening, elevated, mucoid colonies 1–5 mm in diameter. Colonies are transparent or opaque, nonpigmented, and nonhemolytic. Neisseria flavescens, Neisseria cinerea, Neisseria subflava, and Neisseria lactamica may have yellow pigmentation. Neisseria sicca produces opaque, brittle, wrinkled colonies. Moraxella catarrhalis produces nonpigmented or pinkish gray opaque colonies. Gonococci oxidize only glucose and differ antigenically from the other neisseriae. Gonococci usually produce smaller colonies than those of the other neisseriae. Gonococci that require arginine, hypoxanthine, and uracil (Arg−, Hyx− , and Ura − auxotype) tend to grow most slowly on primary culture. Gonococci isolated from clinical specimens or maintained by selective subculture have typical small colonies containing piliated bacteria. On nonselective subculture, larger colonies containing nonpiliated gonococci are also formed. Opaque and transparent variants of both the small and large colony types also occur; the opaque colonies are associated with the presence of a surface-exposed protein, Opa. Antigenic Structure N gonorrhoeae is antigenically heterogeneous and capable of changing its surface structures in vitro—and presumably in vivo—to avoid host defenses. Surface structures include the following. 285 286 SECTION III Bacteriology TABLE 20-1 Biochemical Reactions of the Neisseriae and Moraxella catarrhalis Acid Formed from Growth on MTM, ML, or NYC Mediuma a Glucose Maltose Lactose Sucrose or Fructose DNAse Neisseria gonorrhoeae + + − − − − Neisseria meningitidis + + + − − − Neisseria lactamica + + + + − − Neisseria sicca − + + − + − Neisseria subflava − + + − ± − Neisseria mucosa − + + − + − Neisseria flavescens − − − − − − Neisseria cinerea ± − − − − − Neisseria polysaccharea ± + + − − − Neisseria elongata − −/w − − − − Moraxella catarrhalis − − − − − + ML, Martin-Lewis medium; MTM, modified Thayer-Martin medium; NYC, New York City medium. A. Pili (Fimbriae) Pili are the hairlike appendages that extend up to several micrometers from the gonococcal surface. They enhance attachment to host cells and resistance to phagocytosis. They are made up of stacked pilin proteins (molecular weight [MW], 17–21kDa). The amino terminal of the pilin molecule, which contains a high percentage of hydrophobic amino acids, is conserved. The amino acid sequence near the midportion of the molecule also is conserved; this portion of the molecule serves in attachment to host cells and is less prominent in the immune response. The amino acid sequence near the carboxyl terminal is highly variable; this portion of the molecule is most prominent in the immune response. The pilins of almost all strains of N gonorrhoeae are antigenically different, and a single strain can make many antigenically distinct forms of pilin. B. Por Por protein extends through the gonococcal cell membrane. It occurs in trimers to form pores in the surface through Outer membrane Peptidoglycan Cell envelope Cytoplasmic membrane Peptidoglycan Outer membrane Cytoplasmic membrane Figure 20-1 Gram stain of a urethral exudate of a patient with gonorrhea. Nuclei of many polymorphonuclear cells are seen (large arrows). Intracellular gram-negative diplococci (Neisseria gonorrhoeae) in one polymorphonuclear cell are marked by the small arrow. Pilus Pilus Figure 20-2 Collage and drawing of Neisseria gonorrhoeae showing pili and the three layers of the cell envelop. CHAPTER 20 The Neisseriae 287 which some nutrients enter the cell. Por proteins may impact intracellular killing of gonococci within neutrophils by preventing phagosome–lysosome fusion. In addition, variable resistance of gonococci to killing by normal human serum depends on whether Por protein selectively binds to complement components C3b and C4b. The MW of Por varies from 32–36 kDa. Each strain of gonococcus expresses only one of two types of Por, but the Por of different strains is antigenically different. Serologic typing of Por by agglutination reactions with monoclonal antibodies has distinguished 18 serovars of PorA and 28 serovars of PorB. (Serotyping is done only in reference laboratories.) C. Opa Proteins These proteins function in adhesion of gonococci within colonies and in attachment of gonococci to host cell receptors such as heparin-related compounds and CD66 or carcinoembryonic antigen–related cell adhesion molecules. One portion of the Opa molecule is in the gonococcal outer membrane, and the rest is exposed on the surface. The MW of Opa ranges from 20–28 kDa. A strain of gonococcus can express no, one, two, or occasionally three types of Opa, but each strain has 11 or 12 genes for different Opas. D. Rmp (Protein III) This protein (MW, 30–31 kDa) is antigenically conserved in all gonococci. It is a reduction-modifiable protein (Rmp) and changes its apparent MW when in a reduced state. It associates with Por in the formation of pores in the cell surface. E. Lipooligosaccharide In contrast to the enteric gram-negative rods (see Chapters 2 and 15), gonococcal lipopolysaccharide (LPS) does not have long O-antigen side chains and is called a lipooligosaccharide (LOS). Its MW is 3–7 kDa. Gonococci can express more than one antigenically different LOS chain simultaneously. Toxicity in gonococcal infections is largely attributable to the endotoxic effects of LOS. Specifically, in the fallopian tube explant model, LOS causes ciliary loss and mucosal cell death. In a form of molecular mimicry, gonococci make LOS molecules that structurally resemble human cell membrane glycosphingolipids. A structure is depicted in Figure 20-3. The gonococcal LOS and the human glycosphingolipid of the same structural class react with the same monoclonal antibody, indicating the molecular mimicry. The presence on the gonococcal surface of the same surface structures as human cells helps gonococci evade immune recognition. The terminal galactose of human glycosphingolipids is often conjugated with sialic acid. Sialic acid is a nine-carbon, 5-N-acetylated ketulosonic acid also called N-acetylneuraminic acid (NANA). Gonococci do not make sialic acid but do make a sialyltransferase that functions to take NANA from the human nucleotide sugar cytidine 5′-monophospho-Nacetylneuraminic acid (CMPNANA) and place the NANA on the terminal galactose of a gonococcal acceptor LOS. This sialylation affects the pathogenesis of gonococcal infection. It makes the gonococci resistant to killing by the human antibody–complement system and interferes with gonococcal binding to receptors on phagocytic cells. N meningitidis and Haemophilus influenzae make many but not all of the same LOS structures as N gonorrhoeae. The biology of the LOS for the three species and for some of the nonpathogenic Neisseria species is similar. Four of the various serogroups of N meningitidis make different sialic acid capsules (see later discussion), indicating that they also have biosynthetic pathways different from those of gonococci. These four serogroups sialylate their LOS using sialic acid from their endogenous pools. F. Other Proteins Several antigenically constant proteins of gonococci have poorly defined roles in pathogenesis. Lip (H8) is a surfaceexposed protein that is heat modifiable like Opa. The Fbp (ferric-binding protein), similar in MW to Por, is expressed when the available iron supply is limited, such as in human infection. Gonococci elaborate an IgA1 protease that splits and inactivates IgA1, a major mucosal immunoglobulin of humans. Meningococci, H influenzae, and Streptococcus pneumoniae elaborate similar IgA1 proteases. Genetics and Antigenic Heterogeneity Gonococci have evolved mechanisms for frequently switching from one antigenic form (pilin, Opa, or LPS) to another antigenic form of the same molecule. This switching takes place in one in every 102.5–103 gonococci, an extremely rapid rate of change for bacteria. Because pilin, Opa, and LPS are surface-exposed antigens on gonococci, they are important in the immune response to infection. The molecules’ rapid switching from one antigenic form to another helps the gonococci elude the host immune system. The switching mechanism for pilin, which has been the most thoroughly studied, is different from the mechanism for Opa. Gonococci have multiple genes that code for pilin, but only one gene is inserted into the expression site. Gonococci can remove all or part of this pilin gene and replace it with all or part of another pilin gene. This mechanism allows gonococci to express many antigenically different pilin molecules over time. The switching mechanism of Opa involves, at least in part, the addition or removal from the DNA of one or more of the pentameric coding repeats preceding the sequence that codes for the structural Opa gene. The switching mechanism of LPS is unknown. The antigens and heterogeneity of types are shown in Table 20-2. Gonococci contain several plasmids; 95% of strains have a small, “cryptic” plasmid (MW, 2.6mDa) of unknown 288 1–1–M 1B2 Basal oligosaccharide 2–1–L8 Lipoidal moiety Lacto-N-neotetraose Gal O O GlcNAc Glc O 2 p He CH N O H H O C H H O 1 Hep1 O O O H H H P C C N O H H H H O C H3C O O GlcNH O 2 O O C O C CH 3 O P P O O O KD lcN G NH O C O O O O O O O O Ac CH3 O O O C C O NH C C O O O NH P O GalNAc Gal O H O O O O H KD O H O O O O CH2 CHOH (CH2)8 CH3 C O CH2 CH (CH2)10 CH3 C C O (CH2)10 CH3 O C CH2 CH2 CHOH CH (CH2)8 CH3 O O O O NH O O P P GlcNH O O O O NH O O (CH2)10 CH3 C (CH2)10 CH3 Figure 20-3 Structure of gonococcal lipooligosaccharide, which has lacto-N-neotetraose and a terminal galactosamine in a structure similar to the human ganglioside glycosphingolipid series. The basal oligosaccharide is in light red, and the lacto-N-neotetraose is in dark red. (Courtesy of JM Griffiss.) O CHAPTER 20 The Neisseriae 289 TABLE 20-2 Antigenic Heterogeneity of Neisseria gonorrhoeae Antigen Number of Types Pilin Hundreds Por (protein) (U.S. system) PorA with 18 subtypes PorB with 28 subtypes Opa (protein II) Many (perhaps hundreds) Rmp (protein III) One Lipooligosaccharide Eight or more Fbp (iron-binding protein) One Lip (H8) One IgA1 protease Two function. Two other plasmids (MW, 3.4 mDa and 4.7 mDa) contain genes that code for TEM-1 type (penicillinases) β-lactamases, which causes resistance to penicillin. These plasmids are transmissible by conjugation among gonococci; they are similar to a plasmid found in penicillinaseproducing Haemophilus species and may have been acquired from Haemophilus or other gram-negative organisms. About 5–20% of gonococci contain a plasmid (MW, 24.5 × 106 kDa) with the genes that code for conjugation; the incidence is highest in geographic areas where penicillinaseproducing gonococci are most common. High-level tetracycline resistance (minimal inhibitory concentrations [MICs] of ≥16 mg/L) has developed in gonococci by the insertion of a streptococcal gene tetM coding for tetracycline resistance into the conjugative plasmid. Pathogenesis, Pathology, and Clinical Findings Gonococci exhibit several morphologic types of colonies (see earlier discussion), but only piliated bacteria appear to be virulent. Opa protein expression varies depending on the type of infection. Gonococci that form opaque colonies are isolated from men with symptomatic urethritis and from uterine cervical cultures at midcycle. Gonococci that form transparent colonies are frequently isolated from men with asymptomatic urethral infection; from menstruating women; and from patients with invasive forms of gonorrhea, including salpingitis and disseminated infection. Antigenic variation of surface proteins during infection allows the organism to circumvent host immune response. Gonococci attack mucous membranes of the genitourinary tract, eye, rectum, and throat, producing acute suppuration that may lead to tissue invasion; this is followed by chronic inflammation and fibrosis. Men usually have urethritis, with yellow, creamy pus and painful urination. The process may extend to the epididymis. As suppuration subsides in untreated infection, fibrosis occurs, sometimes leading to urethral strictures. Urethral infection in men can be asymptomatic. In women, the primary infection is in the endocervix and extends to the urethra and vagina, giving rise to mucopurulent discharge. It may then progress to the uterine tubes, causing salpingitis, fibrosis, and obliteration of the tubes. Infertility occurs in 20% of women with gonococcal salpingitis. Chronic gonococcal cervicitis and proctitis are often asymptomatic. Gonococcal bacteremia leads to skin lesions (especially hemorrhagic papules and pustules) on the hands, forearms, feet, and legs and to tenosynovitis and suppurative arthritis, usually of the knees, ankles, and wrists. Gonococci can be cultured from blood or joint fluid of only 30% of patients with gonococcal arthritis. Gonococcal endocarditis is an uncommon but severe infection. Gonococci sometimes cause meningitis and eye infections in adults; these have manifestations similar to those caused by meningococci. Complement deficiency is frequently found in patients with gonococcal bacteremia. Patients with bacteremia, especially if recurrent, should be tested for total hemolytic complement activity. Gonococcal ophthalmia neonatorum, an infection of the eye in newborns, is acquired during passage through an infected birth canal. The initial conjunctivitis rapidly progresses and, if untreated, results in blindness. To prevent gonococcal ophthalmia neonatorum, instillation of tetracycline, erythromycin, or silver nitrate into the conjunctival sac of newborns is compulsory in the United States. Gonococci that produce localized infection are often serum sensitive (ie, killed by antibody and complement). Diagnostic Laboratory Tests A. Specimens Pus and secretions are taken from the urethra, cervix, rectum, conjunctiva, throat, or synovial fluid for culture and smear. Blood culture is necessary in systemic illness, but a special culture system is helpful because gonococci (and meningococci) may be susceptible to the polyanethol sulfonate present in standard blood culture media. B. Smears Gram-stained smears of urethral or endocervical exudates reveal many diplococci within pus cells. These give a presumptive diagnosis. Stained smears of the urethral exudate from men have a sensitivity of about 90% and a specificity of 99%. Stained smears of endocervical exudates have a sensitivity of about 50% and a specificity of about 95% when examined by an experienced microscopist. Additional diagnostic testing of urethral exudates from men is not necessary when the stain result is positive, but nucleic acid amplification tests (NAATs) or cultures should be done for women. Stained smears of conjunctival exudates can also be diagnostic, but those of specimens from the throat or rectum are generally not helpful. 290 SECTION III Bacteriology C. Culture Immediately after collection, pus or mucus is streaked on enriched selective medium (eg, modified Thayer-Martin medium [MTM]) and incubated in an atmosphere containing 5% CO2 (candle extinction jar) at 37°C. To avoid overgrowth by contaminants, the selective medium contains antimicrobial drugs (eg, vancomycin, 3 μg/mL; colistin, 7.5 μg/mL; amphotericin B, 1 μg/mL; and trimethoprim, 3 μg/mL). If immediate incubation is not possible, the specimen should be placed in a CO2-containing transport-culture system. Fortyeight hours after culture, the organisms can be quickly identified by their appearance on a Gram-stained smear; by oxidase positivity; and by coagglutination, immunofluorescence staining, or other laboratory tests. The species of subcultured bacteria may be determined by oxidation of specific carbohydrates (see Table 20-1). Matrix-assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF MS) has potential to provide rapid (same-day) identification of cultured isolates. The gonococcal isolates from anatomic sites other than the genital tract or from children should be identified as to species using two different confirmatory tests because of the legal and social implications of a positive culture result. D. Nucleic Acid Amplification Tests Several Food and Drug Administration–cleared nucleic acid amplification assays are available for direct detection of N gonorrhoeae in genitourinary specimens, and these are the preferred tests from these sources. In general, these assays have excellent sensitivity and specificity in symptomatic, high-prevalence populations. Advantages include better detection, more rapid results, and the ability to use urine as a specimen source. Disadvantages include poor specificity of some assays because of cross-reactivity with nongonococcal Neisseria species. These assays are not recommended for use for the diagnosis of extragenital gonococcal infections or for infection in children. NAATs are not recommended as tests of cure because nucleic acid may persist in patient specimens for up to 3 weeks after successful treatment. E. Serology Serum and genital fluid contain immunoglobulin G (IgG) and IgA antibodies against gonococcal pili, outer membrane proteins, and LPS. Some IgM of human sera is bactericidal for gonococci in vitro. In infected individuals, antibodies to gonococcal pili and outer membrane proteins can be detected by immunoblotting, radioimmunoassay, and ELISA (enzyme-linked immunosorbent assay) tests. However, these tests are not useful as diagnostic aids for several reasons, including gonococcal antigenic heterogeneity, the delay in development of antibodies in acute infection, and a high background level of antibodies in the sexually active population. Immunity Repeated gonococcal infections are common. Protective immunity to reinfection does not appear to develop as part of the disease process, because of the antigenic variety of gonococci. Although antibodies can be demonstrated, including the IgA and IgG on mucosal surfaces, they are either highly strain specific or have little protective ability. Treatment Since the development and widespread use of penicillin, gonococcal resistance to penicillin has gradually risen, owing to the selection of chromosomal mutants, so that many strains now require high concentrations of penicillin G for inhibition (MIC ≥2 μg/mL). Penicillinase-producing N gonorrhoeae (PPNG) also have increased in prevalence (see earlier discussion). Chromosomally mediated resistance to tetracycline (MIC ≥2 μg/mL) is common. High-level resistance to tetracycline (MIC ≥32 μg/mL) also occurs. Spectinomycin resistance as well as resistance to fluoroquinolones has been noted. Single-dose fluoroquinolone treatment was recommended for treatment of gonococcal infections from 1993 until 2006. Since 2006, rates of quinolone resistance among gonococcal isolates have exceeded 5% in men who have sex with men and in heterosexual men. Because of the problems with antimicrobial resistance in N gonorrhoeae, the Centers for Disease Control and Prevention (CDC) recommends that patients with uncomplicated genital or rectal infections be treated with ceftriaxone (250 mg) given intramuscularly as a single dose or 400 mg of oral cefixime as a single dose. Additional therapy with 1 g of azithromycin orally in a single dose or with 100 mg of doxycycline orally twice a day for 7 days is recommended for possible concomitant chlamydial infections. Azithromycin has been found to be safe and effective in pregnant women, but doxycycline is contraindicated. Modifications of these therapies are recommended for other types of N gonorrhoeae infection. See the CDC’s website for the 2010 updated treatment guidelines ( mmwr/pdf/rr/rr5912.pdf). Because other sexually transmitted diseases may have been acquired at the same time as gonorrhea, steps must also be taken to diagnose and treat these diseases (see discussions of chlamydiae, syphilis, and so on). Epidemiology, Prevention, and Control Gonorrhea is worldwide in distribution. In the United States, its incidence rose steadily from 1955 until the late 1970s, when the incidence was between 400 and 500 cases per 100,000 population. Between 1975 and 1997, there was a 74% decline in the rate of reported gonococcal infections. Thereafter, the rates plateaued for 10 years, decreased from 2006–2009, but increased by 2.8% between 2009 and 2010. Gonorrhea is exclusively transmitted by sexual contact, often by women and men with asymptomatic infections. CHAPTER 20 The Neisseriae 291 The infectivity of the organism is such that the chance of acquiring infection from a single exposure to an infected sexual partner is 20–30% for men and even greater for women.The infection rate can be reduced by avoiding multiple sexual partners, rapidly eradicating gonococci from infected individuals by means of early diagnosis and treatment, and finding cases and contacts through education and screening of populations at high risk. Mechanical prophylaxis (condoms) provides partial protection. Chemoprophylaxis is of limited value because of the rise in antibiotic resistance of the gonococcus. Gonococcal ophthalmia neonatorum is prevented by local application of 0.5% erythromycin ophthalmic ointment or 1% tetracycline ointment to the conjunctiva of newborns. Although instillation of silver nitrate solution is also effective and is the classic method for preventing ophthalmia neonatorum, silver nitrate is difficult to store and causes conjunctival irritation; its use has largely been replaced by use of erythromycin or tetracycline ointment. NEISSERIA MENINGITIDIS Antigenic Structure At least 13 serogroups of meningococci have been identified by immunologic specificity of capsular polysaccharides. The most important serogroups associated with disease in humans are A, B, C, X, Y, and W-135. The group A polysaccharide is a polymer of N-acetylmannosamine phosphate, and that of group C is a polymer of N-acetylO-acetylneuraminic acid. Meningococcal antigens are found in blood and cerebrospinal fluid of patients with active disease. Outbreaks and sporadic cases in the Western hemisphere in the past decade have been caused mainly by groups B, C, W-135, and Y; outbreaks in southern Finland and S¯ao Paulo, Brazil, were caused by groups A and C; outbreaks in New Zealand have been caused by a particular B strain; and those in Africa were mainly caused by group A. Group C and, especially, group A are associated with epidemic disease. The outer membrane proteins of meningococci have been divided into classes on the basis of MW. All strains have class 1, class 2, or class 3 proteins; these are analogous to the Por proteins of gonococci and are responsible for the serotype specificity of meningococci. These proteins play a role in attachment. As many as 20 serotypes have been defined; serotypes 2 and 15 have been associated with epidemic disease. The Opa (class 5) protein is comparable to Opa of the gonococci. Meningococci are piliated, but unlike gonococci, they do not form distinctive colony types indicating piliated bacteria. Meningococcal LPS is responsible for many of the toxic effects found in meningococcal disease. The highest levels of endotoxin measured in sepsis have been found in patients with meningococcemia (50- to 100fold greater than with other gram-negative infections). Pathogenesis, Pathology, and Clinical Findings Humans are the only natural hosts for whom meningococci are pathogenic. The nasopharynx is the portal of entry. There, the organisms attach to epithelial cells with the aid of pili; they may form part of the transient flora without producing symptoms. From the nasopharynx, organisms may reach the bloodstream, producing bacteremia; the symptoms may be similar to those of an upper respiratory tract infection. Fulminant meningococcemia is more severe, with a high fever and a hemorrhagic rash; the patient may have disseminated intravascular coagulation and circulatory collapse (Waterhouse-Friderichsen syndrome). Meningitis is the most common complication of meningococcemia. It usually begins suddenly with an intense headache, vomiting, and stiff neck and progresses to coma within a few hours. During meningococcemia, there is thrombosis of many small blood vessels in many organs, with perivascular infiltration and petechial hemorrhages. There may be interstitial myocarditis, arthritis, and skin lesions. In meningitis, the meninges are acutely inflamed, with thrombosis of blood vessels and exudation of polymorphonuclear leukocytes, so that the surface of the brain is covered with a thick purulent exudate. It is not known what transforms an asymptomatic infection of the nasopharynx into meningococcemia and meningitis, but this can be prevented by specific bactericidal serum antibodies against the infecting serotype. Neisseria bacteremia is favored by the absence of bactericidal antibody (IgM and IgG), inhibition of serum bactericidal action by a blocking IgA antibody, or a complement component deficiency (C5, C6, C7, or C8). Meningococci are readily phagocytosed in the presence of a specific opsonin. Diagnostic Laboratory Tests A. Specimens Specimens of blood are taken for culture, and specimens of spinal fluid are taken for smear, culture, and chemical determinations. Nasopharyngeal swab cultures are suitable for carrier surveys. Puncture material from petechiae may be taken for smear and culture. B. Smears Gram-stained smears of the sediment of centrifuged spinal fluid or of petechial aspirate often show typical neisseriae within polymorphonuclear leukocytes or extracellularly. C. Culture Culture media without sodium polyanethol sulfonate are helpful in culturing blood specimens. Cerebrospinal fluid specimens are plated on “chocolate” agar and incubated at 37°C in an atmosphere of 5% CO2. Freshly drawn spinal fluid 292 SECTION III Bacteriology can be directly incubated at 37°C if agar culture media are not immediately available. A modified MTM with antibiotics (vancomycin, colistin, amphotericin) favors the growth of neisseriae, inhibits many other bacteria, and is used for nasopharyngeal cultures. Presumptive colonies of neisseriae on solid media, particularly in mixed culture, can be identified by Gram stain and the oxidase test. Spinal fluid and blood generally yield pure cultures that can be further identified by carbohydrate oxidative reactions (see Table 20-1) and agglutination with type-specific or polyvalent serum. D. Serology Antibodies to meningococcal polysaccharides can be measured by latex agglutination or hemagglutination tests or by their bactericidal activity. These tests are done only in reference laboratories. Immunity Immunity to meningococcal infection is associated with the presence of specific, complement-dependent, bactericidal antibodies in the serum. These antibodies develop after subclinical infections with different strains or injection of antigens and are group specific, type specific, or both. The immunizing antigens for groups A, C, Y, and W-135 are the capsular polysaccharides. For group B, a specific antigen suitable for use as a vaccine has not been defined; however, group B vaccines with mixtures of antigens have been used in many parts of the world. Currently, there are three vaccines against serogroups A, C, Y, and W-135 available in the United States. A polysaccharide tetravalent vaccine (Menomune, Sanofi Pasteur) in which each dose consists of four purified bacterial capsular polysaccharides is poorly immunogenic in children younger than age 18 months, does not confer longlasting immunity and does not cause a sustainable reduction in nasopharyngeal carriage. A tetravalent conjugate vaccine approved in 2005 (Menactra, Sanofi Pasteur) is licensed for use in persons 9 months to 55 years of age. It contains capsular polysaccharide conjugated to diphtheria toxoid. Menveo is another tetravalent conjugate vaccine in which A, C, Y, W135 oligosaccharide is conjugated to diphtheria CRM197 (Novartis, Inc.). This vaccine is approved for use in individuals 2–55 years of age. The advantage of these vaccines is that a T cell–dependent response to vaccine is induced. This enhances primary response among infants and substantially reduces asymptomatic carriage. Routine vaccination of young adolescents (ages 11–12 years) before high school using the conjugated vaccine is now recommended. Vaccination is also recommended for persons 11–55 years of age who are among the following at-risk groups: persons with functional or surgical asplenia, persons with complement deficiencies, travelers to highly endemic areas (eg, sub-Saharan Africa), “closed populations” such as college freshman living in dorms and the military, populations experiencing a community outbreak, and clinical laboratory workers (microbiologists). Treatment Penicillin G is the drug of choice for treating patients with meningococcal disease. Either chloramphenicol or a thirdgeneration cephalosporin such as cefotaxime or ceftriaxone is used in persons who are allergic to penicillins. Epidemiology, Prevention, and Control Meningococcal meningitis occurs in epidemic waves (eg, in military encampments, in religious pilgrims, and in subSaharan Africa); and a smaller number of sporadic interepidemic cases. About 5–30% of the normal population may harbor meningococci (often nontypeable isolates) in the nasopharynx during interepidemic periods. During epidemics, the carrier rate goes up to 70–80%. A rise in the number of cases is preceded by an increased number of respiratory carriers. Treatment with oral penicillin does not eradicate the carrier state. Rifampin, 600 mg orally twice daily for 2 days (or ciprofloxacin in adults, 500 mg as a single dose), can often eradicate the carrier state and serve as chemoprophylaxis for household and other close contacts. Since the appearance of many sulfonamide-resistant meningococci, chemoprophylaxis with sulfonamides is no longer reliable. Clinical cases of meningitis present only a negligible source of infection, and isolation therefore has only limited usefulness. More important is the reduction of personal contacts in a population with a high carrier rate. This is accomplished by avoidance of crowding or administration of vaccines as discussed. As mentioned, such vaccines are currently used in selected populations (eg, the military and in civilian epidemics). OTHER NEISSERIAE Neisseria lactamica very rarely causes disease but is important because it grows in the selective media (eg, modified MTM) used for cultures of gonococci and meningococci from clinical specimens. N lactamica can be cultured from the nasopharynx of 3–40% of persons and most often is found in children. Unlike the other neisseriae, it ferments lactose. N sicca, N subflava, N cinerea, N mucosa, and N flavescens are also members of the normal microbiota of the respiratory tract, particularly the nasopharynx, and very rarely produce disease. N cinerea sometimes resembles N gonorrhoeae because of its morphology and positive hydroxyprolyl aminopeptidase reaction. M catarrhalis was previously named Branhamella catarrhalis and before that Neisseria catarrhalis. It is a member of the normal microbiota in 40% to 50% of healthy school children. M catarrhalis causes bronchitis, pneumonia, sinusitis, otitis media, and conjunctivitis. It is also of concern as a cause of infection in immunocompromised patients. Most strains of M catarrhalis from clinically significant infections produce β-lactamase. M catarrhalis can be differentiated from the neisseriae by its lack of carbohydrate fermentation CHAPTER 20 The Neisseriae 293 and by its production of DNase. It produces butyrate esterase, which forms the basis for rapid fluorometric tests for identification. CHAPTER SUMMARY • • • • • • • • The genus Neisseria consists of two major pathogens, N gonorrhoeae and N meningitidis; both of them have elaborated factors that facilitate disease in otherwise healthy people. The remaining species constitute part of the normal human microbiota of the respiratory tract and may play a role in localized infections. Members of this genus are gram-negative diplococci that vary in their growth requirements. N gonorrhoeae is very fastidious, and selective enriched media containing antibiotics, amino acids, and so on are used to recover the organism in clinical cultures. The other species are less fastidious and grow on routine laboratory media. N gonorrhoeae causes the sexually transmitted disease gonorrhea and is characterized by purulent cervicitis in women and purulent urethral discharge in men. Infants born to women infected at the time of delivery may develop purulent conjunctivitis. Diagnosis is made primarily by NAATs; treatment consists of intramuscular ceftriaxone or oral cefixime plus an agent such as azithromycin or doxycycline to treat concomitant Chlamydia infections. N meningitidis is the cause of endemic and epidemic meningitis. Its major virulence factor is the thick polysaccharide capsule. There are approximately 13 capsular types, the most common of which are A, B, C, X, Y, and W-135. Meningococcal meningitis is a serious infection that carries a high morbidity and is often associated with sepsis because of its potent LOS. Penicillin is the drug of choice. Diagnosis is made by culturing the cerebrospinal fluid on chocolate agar incubated at 37°C in CO2. Prevention consists of immunization with one of two conjugate vaccines (routinely recommended for children 11–12 years of age) or the polysaccharide vaccine. REVIEW QUESTIONS 1. The inhabitants of a group of small villages in rural subSaharan Africa experienced an epidemic of meningitis. Ten percent of the people died, most of them younger than the age of 15 years. The microorganism that most likely caused this epidemic was (A) Streptococcus agalactiae (group B) (B) Escherichia coli K1 (capsular type 1) (C) Haemophilus influenzae serotype b (D) Neisseria meningitidis serogroup A (E) West Nile virus 2. A 19-year-old man presented to the clinic with a urethral discharge for the past 24 hours. Neisseria gonorrhoeae was cultured from the specimen and found to be β-lactamase positive and resistant to high levels (≥32 μg/mL) of tetracycline. Which of the following statements about these antimicrobial resistance factors is correct? (A) β-lactamase production and high-level resistance to tetracycline are both mediated by genes on plasmids. (B) Whereas β-lactamase production is mediated by a gene on the bacterial chromosome, high-level tetracycline resistance is mediated by a gene on a plasmid. (C) Whereas β-lactamase production is mediated by a gene on a plasmid, high-level tetracycline resistance is mediated by a gene on the bacterial chromosome. (D) β-lactamase production and high-level resistance to tetracycline are both mediated by genes on the bacterial chromosome. 3. A 6-year-old boy develops a fever and headache. He is taken to the emergency department, where he is noted to have a stiff neck, suggesting meningeal irritation. A lumbar puncture is done, and culture of the cerebrospinal fluid grows Neisseria meningitidis serogroup B. Which of the following should be considered for his family (household) members? (A) No prophylaxis or other steps are necessary. (B) They should be given N meningitidis pilin vaccine. (C) They should be given N meningitidis serogroup B polysaccharide capsule vaccine. (D) They should be given rifampin prophylaxis. (E) They should be given sulfonamide prophylaxis. 4. An 18-year-old woman who reports unprotected sex with a new partner 2 weeks previously develops fever and left lower quadrant abdominal pain with onset in association with her menstrual period. On pelvic examination in the emergency department, she has bilateral tenderness when the uterus is palpated. A mass 2–3 cm in diameter is felt on the left, suggestive of tubo-ovarian abscess. Subsequently, Neisseria gonorrhoeae is cultured from her endocervix. The diagnosis is gonococcal pelvic inflammatory disease. A common sequela of this infection is (A) Cancer of the cervix (B) Urethral stricture (C) Uterine fibroid tumors (D) Infertility (E) Vaginal-rectal fistula 5. A 38-year-old vice squad police officer comes to the emergency department with a chief complaint expressed as follows: “I have disseminated gonococcal infection again.” He is correct. Cultures of his urethra and knee fluid yield Neisseria gonorrhoeae. He has previously had five episodes of disseminated gonococcal infection. The patient should be evaluated for (A) Selective IgA deficiency (B) A polymorphonuclear cell chemotactic defect (C) Deficiency of a late-acting complement component C5, C6, C7, or C8 (D) Absent lymphocyte adenosine deaminase activity (E) Myeloperoxidase deficiency 6. Which of the following individuals should routinely receive vaccination with the conjugate meningococcal vaccine? (A) A healthy young adolescent entering high school (B) A healthy child entering kindergarten (C) A 60-year-old man with insulin-dependent diabetes 294 SECTION III Bacteriology (D) A healthy 40-year-old technician who works in a cancer research laboratory (E) A 65-year-old woman with coronary artery disease 7. A 25-year-old sexually active woman presents with purulent vaginal discharge and dysuria 7 days after having unprotected sexual intercourse with a new partner. Of the choices below, what is the most sensitive diagnostic method for determining the likely etiologic agent? (A) Gram stain (B) An enzyme immunoassay (C) Bacterial culture on selective media (D) A nucleic acid amplification test (E) Serology 8. What is the currently recommended treatment for gonococcal urethritis in men who have sex with men in the United States? (A) Single dose of an oral fluoroquinolone (B) Seven days of oral doxycycline (C) Ceftriaxone given intramuscularly as a single dose (D) Spectinomycin given intramuscularly as a single dose (E) Seven days of oral amoxicillin 9. Which of the following cell components produced by Neisseria gonorrhoeae is responsible for attachment to host cells? (A) Lipooligosaccharide (B) Pili (fimbriae) (C) IgA1 protease (D) Outer membrane porin protein (E) Iron-binding protein 10. A 60-year-old man with severe chronic lung disease presents with fever, cough productive of purulent sputum, and worsening hypoxemia. A sputum sample is collected, and the specimen is sent promptly to the laboratory. Microscopic examination of a Gram stain reveals numerous polymorphonuclear leukocytes and predominately gram-negative diplococci that are both intracellular and extracellular. The organism grows well on 5% SBA and chocolate agar and is positive for butyrate esterase. What is the most likely organism causing this man’s illness? (A) Neisseria gonorrhoeae (B) Neisseria lactamica (C) Moraxella catarrhalis (D) Haemophilus influenzae (E) Neisseria meningitidis 11. One major advantage of the conjugate meningococcal vaccines compared with the polysaccharide vaccine is (A) Stimulation of mucosal secretory IgA (B) Fewer side effects (C) A T cell–dependent response to vaccine is induced (D) Inclusion of serogroup B 12. A 25-year-old woman presents with septic arthritis of the knee. The fluid aspirate grows a gram-negative diplococcus on chocolate agar after 48 hours of incubation. The isolate is oxidase positive and oxidizes glucose but not maltose, lactose, or sucrose. You suspect infection with (A) Neisseria meningitidis (B) Neisseria lactamica (C) Moraxella catarrhalis (D) Neisseria gonorrhoeae (E) None of the above 13. All of the following are virulence factors associated with N gonorrhoeae except (A) Pili (B) Por (C) Lipooligosaccharide (D) Opa proteins (E) A thick polysaccharide capsule 14. The prevalence of gonococcal infections increased between 2009 and 2010. (A) True (B) False 15. A useful test to differentiate Moraxella catarrhalis from saprophytic neisseriae in respiratory samples is (A) Butyrate esterase (B) Gram stain (C) Growth on 5% sheep blood agar (D) PYR (E) Oxidase Answers 9. B 1. D 5. C E A 6. A C A D 7. D C A D 8. C D REFERENCES Apicella MA: Neisseria meningitidis. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Churchill Livingstone Elsevier, 2010. Centers for Disease Control and Prevention: Sexually Transmitted Disease Surveillance 2010. Retrieved from std/stats10/surv2010.pdf. Elias J, Frosch M, Vogel U: Neisseria. In Versalovic J, Carroll KC, Funke G, et al. (editors). Manual of Clinical Microbiology, 10th ed. ASM Press, 2011. Marrazzo JM: Neisseria gonorrhoeae. In Mandell GL, Bennett JE, Dolin R (editors). Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th ed. Churchill Livingstone Elsevier, 2010. Workowski KA, Berman S. Sexually transmitted treatment guidelines. MMWR Recommend Rep 2010;59(RR12):1–110. 21 C Infections Caused by Anaerobic Bacteria H A P T E R Medically important infections caused by anaerobic bacteria are common. The infections are often polymicrobial— that is, the anaerobic bacteria are found in mixed infections with other anaerobes, facultative anaerobes, and aerobes (see the glossary of defi nitions). Anaerobic bacteria are found throughout the human body—on the skin, on mucosal surfaces, and in high concentrations in the mouth and gastrointestinal tract—as part of the normal microbiota (see Chapter 10). Infection results when anaerobes and other bacteria of the normal microbiota contaminate normally sterile body sites. Several important diseases are caused by anaerobic Clostridium species from the environment or from normal flora: botulism, tetanus, gas gangrene, food poisoning, and pseudomembranous colitis. These diseases are discussed in Chapters 9 and 11 and briefly later in this chapter. G LOSSARY Cytochrome systems for the metabolism of O2 2. Superoxide dismutase (SOD), which catalyzes the following reaction: Aerobic bacteria: Bacteria that require oxygen as a terminal electron acceptor and will not grow under anaerobic conditions (ie, in the absence of O2). Some Micrococcus species and Nocardia asteroides are obligate aerobes (ie, they must have oxygen to survive). Anaerobic bacteria: Bacteria that do not use oxygen for growth and metabolism but obtain their energy from fermentation reactions. A functional definition of anaerobes is that they require reduced oxygen tension for growth and fail to grow on the surface of solid medium in 10% CO2 in ambient air. Bacteroides and Clostridium species are examples of anaerobes. Capnophilic bacteria: Bacteria that require carbon dioxide for growth. Facultative anaerobes: Bacteria that can grow either oxidatively, using oxygen as a terminal electron acceptor, or anaerobically, using fermentation reactions to obtain energy. Such bacteria are common pathogens. Streptococcus species and the Enterobacteriaceae (eg, Escherichia coli ) are among the many facultative anaerobes that cause disease. Often, bacteria that are facultative anaerobes are called “aerobes.” PHYSIOLOGY AND GROWTH CONDITIONS FOR ANAEROBES Anaerobic bacteria do not grow in the presence of oxygen and are killed by oxygen or toxic oxygen radicals (see later discussion). pH and oxidation-reduction potential (Eh) are also important in establishing conditions that favor growth of anaerobes. Anaerobes grow at a low or negative Eh. Aerobes and facultative anaerobes often have the metabolic systems listed below, but anaerobic bacteria frequently do not. − − O2 + O2 + 2H+ → H2O2 + O2 3. Catalase, which catalyzes the following reaction: 2H2O2 → 2H2O + O2 (gas bubbles) Anaerobic bacteria do not have cytochrome systems for oxygen metabolism. Less fastidious anaerobes may have low levels of SOD and may or may not have catalase. Most bacteria of the Bacteroides fragilis group have small amounts of both catalase and SOD. There appear to be multiple mechanisms for oxygen toxicity. Presumably, when anaerobes have SOD or catalase (or both), they are able to negate the toxic effects of oxygen radicals and hydrogen peroxide and thus tolerate oxygen. Obligate anaerobes usually lack SOD and catalase and are susceptible to the lethal effects of oxygen; such strict obligate anaerobes are infrequently isolated from human infections, and most anaerobic infections of humans are caused by “moderately obligate anaerobes.” The ability of anaerobes to tolerate oxygen or grow in its presence varies from species to species. Similarly, there is strain-to-strain variation within a given species (eg, one 295 296 SECTION III Bacteriology strain of Prevotella melaninogenica can grow at an O2 concentration of 0.1% but not of 1%; another can grow at a concentration of 2% but not of 4%). Also, in the absence of oxygen, some anaerobic bacteria grow at a more positive E h. Facultative anaerobes grow as well or better under anaerobic conditions than they do under aerobic conditions. Bacteria that are facultative anaerobes are often termed aer obes. When a facultative anaerobe such as Escherichia coli is present at the site of an infection (eg, abdominal abscess), it can rapidly consume all available oxygen and change to anaerobic metabolism, producing an anaerobic environment and low Eh and thus allow the anaerobic bacteria that are present to grow and produce disease. ANAEROBIC BACTERIA FOUND IN HUMAN INFECTIONS Since the 1990s, the taxonomic classification of the anaerobic bacteria has changed significantly due to the application of molecular sequencing and DNA–DNA hybridization technologies. The nomenclature used in this chapter refers to genera of anaerobes frequently found in human infections and to certain species recognized as important pathogens of humans. Anaerobes commonly found in human infections are listed in Table 21-1. TABLE 21-1 Anaerobic Bacteria of Clinical Importance Genera Bacilli (rods) Gram negative Bacteroides fragilis group Prevotella melaninogenica Fusobacterium Gram positive Actinomyces Lactobacillus Propionibacterium Eubacterium, Bifidobacterium, and Arachnia Clostridium Cocci (spheres) Gram positive Peptoniphilus Peptostreptococcus Peptococcus Finegoldia Gram negative Veillonella a Also found in soil. Anatomic Site Gram-Negative Anaerobes Gram-Negative Bacilli 1. Bacteroides—The Bacteroides species are very impor- tant anaerobes that cause human infection. They are a large group of bile-resistant, non–spore-forming, slender gramnegative rods that may appear as coccobacilli. Many species previously included in the genus Bacteroides have been reclassified into the genus Prevotella or the genus Porphyromonas. Those species retained in the Bacteroides genus are members of the B fragilis group (~20 species). Bacteroides species are normal inhabitants of the bowel and other sites. Normal stools contain 1011 B fragi lis organisms per gram (compared with 108/g for facultative anaerobes). Other commonly isolated members of the B fragilis group include Bacteroides ovatus, Bacteroides dis tasonis, Bacteroides vulgatus, and Bacteroides thetaiotao micron. Bacteroides species are most often implicated in intra-abdominal infections, usually under circumstances of disruption of the intestinal wall as occurs in perforations related to surgery or trauma, acute appendicitis, and diverticulitis. These infections are often polymicrobial; anaerobic cocci, Clostridium species, and Eubacterium may also be found. Both B fragilis and B thetaiotaomicron are implicated in serious intrapelvic infections such as pelvic inflammatory disease and ovarian abscesses. B fragilis group species are the most common species recovered in some series of anaerobic bacteremia, and these organisms are associated with a very high mortality rate. As discussed later in the chapter, B fragilis is capable of elaborating numerous virulence factors, which contribute to its pathogenicity and mortality in the host. Prevotella—The Prevotella species are gram-negative Colon Mouth Mouth, colon, genitourinary tract Mouth Mouth Mouth, colon, vagina Skin Colona Colon, mouth, skin, genitourinary tract Colon, mouth, skin, genitourinary tract Mouth, colon bacilli and may appear as slender rods or coccobacilli. Most commonly isolated are Prevotella melaninogenica, Prevotella bivia, and Prevotella disiens. P melaninogenica and similar species are found in infections associated with the upper respiratory tract. P bivia and P disiens occur in the female genital tract. Prevotella species are found in brain and lung abscesses, in empyema, and in pelvic inflammatory disease and tubo-ovarian abscesses. In these infections, the prevotellae are often associated with other anaerobic organisms that are part of the normal microbiota—particularly peptostreptococci, anaerobic grampositive rods, and Fusobacterium species—as well as grampositive and gram-negative facultative anaerobes that are part of the normal microbiota. Porphyromonas—The Porphyromonas species also are gram-negative bacilli that are part of the normal oral microbiota and occur at other anatomic sites as well. Porphyromonas species can be cultured from gingival and periapical tooth infections and, more commonly, breast, axillary, perianal, and male genital infections. CHAPTER 21 Infections Caused by Anaerobic Bacteria 297 Fusobacteria—There are approximately 13 Fuso bacterium species, but most human infections are caused by Fusobacterium necrophorum and Fusobacterium nucleatum. Both species differ in morphology and habitat as well as the range of associated infections. F necrophorum is a very pleomorphic, long rod with round ends and tends to make bizarre forms. It is not a component of the healthy oral cavity. F nec rophorum is quite virulent, causing severe infections of the head and neck that can progress to a complicated infection called Lemierre’s disease. The latter is characterized by acute jugular vein septic thrombophlebitis that progresses to sepsis with metastatic abscesses of the lungs, mediastinum, pleural space, and liver. Lemierre’s disease is most common among older children and young adults and often occurs in association with infectious mononucleosis. F necrophorum is also seen in polymicrobial, intra-abdominal infections. F nuclea tum is a thin rod with tapered ends (needle-shaped morphology) and is a significant component of the gingival microbiota as well as the genital, gastrointestinal, and upper respiratory tracts. As such, it is frequently encountered in a variety of clinical infections such as pleuropulmonary infections, obstetric infections, significantly chorioamnionitis, and occasionally brain abscesses complicating periodontal disease. Rarely does it cause bacteremia in neutropenic patients. BACTERIA THAT CAUSE VAGINOSIS Bacterial vaginosis is a common vaginal condition of women of reproductive age. It is associated with premature rupture of membranes and preterm labor and birth. Bacterial vaginosis has a complex microbiology; two organisms, Gardnerella vaginalis and Mobiluncus species, have been most specifically associated with the disease process. Gardnerella vaginalis G vaginalis is a serologically distinct organism isolated from the normal female genitourinary tract and also associated with vaginosis, so named because inflammatory cells are not present. In wet smears, this “nonspecific” vaginitis, or bacterial vaginosis, yields “clue cells,” which are vaginal epithelial cells covered with many gram-variable bacilli, and there is an absence of other common causes of vaginitis such as trichomonads or yeasts. Vaginal discharge often has a distinct “fishy” odor and contains many anaerobes in addition to G vaginalis. The pH of the vaginal secretions is greater than 4.5 (normal pH is 10,000 nm), although most average ~1000 nm. Genome: negative-sense, nonsegmented, singlestranded RNA, 19 kb in size. Seven polypeptides. Envelope. Replication: cytoplasm. Assembly: budding from plasma membrane Brazilian encephalitis (Rocio virus), dengue, Japanese B encephalitis, Kyasanur Forest disease, louping ill, Murray Valley encephalitis, Omsk hemorrhagic fever, St. Louis encephalitis, tick-borne encephalitis, West Nile fever, and yellow fever viruses. Arthropod borne (mosquitoes, ticks) Spherical, 40–60 nm in diameter. Genome: positivesense, single-stranded RNA, 11 kb in size. Genome RNA infectious. Envelope. Three structural polypeptides, two glycosylated. Replication: cytoplasm. Assembly: within endoplasmic reticulum. All viruses serologically related Colorado tick fever virus. Arthropod borne (ticks, mosquitoes) Spherical, 60–80 nm in diameter. Genome: 10–12 segments of linear, double-stranded RNA, 16–27 kbp total size. No envelope. Ten to 12 structural polypeptides. Replication and assembly: cytoplasm (see Chapter 37) African horse sickness and bluetongue viruses. Arthropod borne (mosquitoes) Chikungunya, eastern, western, and Venezuelan equine encephalitis viruses, Mayaro, O’Nyongnyong, Ross River, Semliki Forest, and Sindbis viruses. Arthropod borne (mosquitoes) Spherical, 70 nm in diameter, nucleocapsid has 42 capsomeres. Genome: positive-sense, singlestranded RNA, 11–12 kb in size. Envelope. Three or four major structural polypeptides, two glycosylated. Replication: cytoplasm. Assembly: budding through host cell membranes. All viruses serologically related CHAPTER 38 Arthropod-Borne and Rodent-Borne Viral Diseases 555 A B C D FIGURE 38-1 Electron micrographs of typical arboviruses and rodent-borne viruses. A: An alphavirus, Semliki Forest virus (Togaviridae). B: A representative member of Bunyaviridae, Uukuniemi virus. C: An arenavirus, Tacaribe virus (Arenaviridae). D: Ebola virus (Filoviridae). (Courtesy of FA Murphy and EL Palmer.) Japanese B encephalitis, Murray Valley (Australia) encephalitis. Many encephalitides are alphavirus and flavivirus infections spread by mosquitoes, although the group of California encephalitis diseases is caused by bunyaviruses. On a given continent, there may be a shifting distribution depending on viral hosts and vectors in a given year. Several arboviruses cause significant human infections in the United States (Table 38-2). The numbers of cases vary widely from year to year. TOGAVIRUS AND FLAVIVIRUS ENCEPHALITIS Classification and Properties of Togaviruses and Flaviviruses In the Togaviridae family, the Alphavirus genus consists of about 30 viruses 70 nm in diameter that possess a singlestranded, positive-sense RNA genome (see Table 38-1). The 556 SECTION IV Virology A B C D E F G FIGURE 38-2 Known distributions of flaviviruses causing human disease. A: Yellow fever virus. B: Dengue virus. C: St. Louis encephalitis virus. D: Japanese B encephalitis virus. E: Murray Valley encephalitis virus. F: Tick-borne encephalitis virus. G: West Nile virus. (Reproduced with permission from Monath TP, Tsai TF: Flaviviruses. In Richman DD, Whitley RJ, Hayden FG [editors]. Clinical Virology, 2nd ed. Washington DC: ASM Press, 2002.) envelope surrounding the particle contains two glycoproteins (see Figure 38-1). Alphaviruses often establish persistent infections in mosquitoes and are transmitted between vertebrates by mosquitoes or other blood-feeding arthropods. They have a worldwide distribution. All alphaviruses are antigenically related. The viruses are inactivated by acid pH, heat, lipid solvents, detergents, bleach, phenol, 70% alcohol, and formaldehyde. Most possess hemagglutinating ability. Rubella virus, classified in a separate genus in the Togaviridae family, has no arthropod vector and is not an arbovirus (see Chapter 40). Arboviruses are in the Flavivirus genus in the Flaviviridae family. Initially, the flaviviruses were included in the togavirus CHAPTER 38 Arthropod-Borne and Rodent-Borne Viral Diseases 557 TABLE 38-2 Summary of Major Human Arbovirus and Rodent-Borne Virus Infections that Occur in the United States Diseasea Exposure Distribution Major Vectors Infection:Case Ratio (Age Incidence) Sequelaeb Mortality Rate (%) Eastern equine encephalitis (Alphavirus) Rural Atlantic, southern coastal Aedes, Culex 10:1 (infants) 50:1 (middle-aged) 20:1 (elderly) + 30–70 Western equine encephalitis (Alphavirus) Rural Pacific, Mountain, Southwest Culex tarsalis, Aedes 50:1 (15 years) + 3–7 Venezuelan equine encephalitis (Alphavirus) Rural South (also South and Central America) Aedes, Psorophora, Culex 25:1 (15 years) ± Fatalities rare St. Louis encephalitis (Flavivirus) Urban–rural Widespread Culex 800:1 (60 years) ± 3–10 (65 years) West Nile fever (Flavivirus) Urban–rural Widespread Culex, Aedes, Anopheles 150:1 Unknown 3–15 California encephalitis (La Crosse) (Orthobunyavirus) Rural North central, Atlantic, South Aedes triseriatus Unknown ratio (most cases 50 years of age, the recommended therapies are vancomycin plus a third-generation cephalosporin because of the prevalence of multidrug-resistant S pneumoniae, reports of rising minimum inhibitory concentrations to penicillin among meningococci, and the prevalence of β-lactamase production among H influenzae. Since adults over the age of 50 years are also susceptible to L monocytogenes, the addition of ampicillin to the regimen for older children and adults as listed earlier is recommended. Available evidence supports administration of adjunctive dexamethasone 10–20 minutes prior to or concomitant with the first antimicrobial dose to children with Hinfluenzae meningitis and in the adult with pneumococcal meningitis with continuation of steroids for the first 2–4 days of therapy. Several vaccines are currently available and are recommended for the prevention of the more serious causes of bacterial meningitis. The H influenzae type B conjugate vaccine and the 13-valent conjugate pneumococcal vaccine are currently part of the routine vaccination series for infants and young children. The 23-valent polysaccharide pneumococcal vaccine is recommended for prevention of invasive pneumococcal disease in certain high-risk groups over the age of 2 years. These include patients who are elderly and patients who have chronic underlying diseases such as cardiovascular disease, diabetes mellitus, chronic pulmonary problems, CSF leaks, and asplenia, among others. A quadrivalent conjugated meningococcal vaccine is currently recommended for all healthy 11- to 19-year-olds and for 2- to 55-year-old persons at risk such as travelers to endemic areas. For adults over 55, the meningococcal polysaccharide vaccine is currently recommended pending evaluation of the conjugate vaccine in this age group. REFERENCES Kim KS: Acute bacterial meningitis in infants and children. Lancet Infect Dis 2010;10:32. Tunkel AR, Hartman BJ, Kaplan SL, et al: Practice guidelines for the management of bacterial meningitis. Clin Infect Dis 2004;39:1267. Van de Beek D, de Gans J, Tunkel AR, Wijdicks EF: C ommunity acquired bacterial meningitis in adults. N Engl J Med 2006;354:44. CASE 2: BRAIN ABSCESS A 57-year-old man presented to the hospital with seizures. Three weeks earlier, he had developed bifrontal headaches that were relieved by aspirin. The headaches recurred several times, including the day prior to admission. On the morning of admission, he was noted to have focal seizures with involuntary movements of the right side of his face and arm. While in the emergency room, he had a generalized seizure that was controlled by intravenous diazepam, phenytoin, and phenobarbital. Additional history from the patient’s wife indicated that he had had a dental extraction and bridge work approximately 5 weeks earlier. He did not smoke, drank only socially, and took no medications. The remainder of his history was not helpful. Clinical Features The temperature was 37°C, the pulse 110/min, and respirations 18/min. The blood pressure was 140/80 mm Hg. 788 SECTION VII Diagnostic Medical Microbiology and Clinical Correlation On physical examination, the patient was sleepy and had a decreased attention span. He moved all his extremities, though the right arm moved less than the left. There was slight blurring of the left optic disk, suggesting possible increased intracranial pressure. The remainder of his physical examination was normal. Laboratory Findings and Imaging Laboratory tests were all normal, including hemoglobin and hematocrit, white blood cell count and differential, serum electrolytes, blood urea nitrogen, serum creatinine, urinalysis, chest x-ray, and electrocardiogram (ECG). Lumbar puncture was not done and cerebrospinal fluid was not examined because of possible increased intracranial pressure due to a mass lesion. Blood cultures were negative. Computed tomography (CT) scan of the patient’s head showed a 1.5-cm localized ring-enhancing lesion in the left parietal hemisphere suggestive of a brain abscess. Treatment The patient had a neurosurgical procedure with drainage of the lesion, which was completely removed. Culture of necrotic material from the lesion yielded Prevotella melaninogenica (Chapter 21) and Streptococcus anginosus (Chapter 14). Pathologic examination of the tissue suggested that the lesion was several weeks old. The patient received antibiotic therapy for 4 weeks. He had no more seizures and no subsequent neurologic deficits. One year later, anticonvulsant medications were discontinued and a follow-up CT scan was negative. Comment A brain abscess is a localized pyogenic bacterial infection within the brain parenchyma. The major clinical manifestations are related to the presence of a space-occupying mass in the brain rather than the classic signs and symptoms of infection. Thus, patients commonly present with headache and a change in mental status from normal to lethargy or coma. Focal neurologic findings related to location of the abscess occur in less than half of patients; one-third have seizures, and less than half have fever. Occasionally, patients present with signs and symptoms suggesting acute meningitis. Initially, the clinician must differentiate brain abscess from other central nervous system processes, including primary or metastatic cancers, subdural or epidural abscesses, viral infections (herpes simplex encephalitis), meningitis, stroke, and a variety of other diseases. Significant predisposing factors for brain abscess include distant site infections with bacteremia, such as endocarditis, lung infections, or other occult infections. Many patients have had relatively recent dental work. Brain abscess can also occur via spread from contiguous sites of infection such as in the middle ear, mastoid, or sinuses or following penetrating trauma. However, 20% of patients with brain abscesses have no discernible predisposing factors. Brain abscess can be caused by a single species of bacteria, but more than one species are often isolated—in general, an average of two species. Of the facultative and aerobic bacteria, the viridans streptococci (including nonhemolytic and α- and β-hemolytic strains, the S anginosus group, Streptococcus mitis, etc; see Chapter 14) are most common, occurring in one-third to one-half of patients. Staphylococcus aureus (Chapter 13) is isolated in 10–15% and, when present, is often the only isolate found. Enteric gram-negative rods occur in about 25%, often in mixed cultures. Many other facultative or aerobic bacteria (eg, S pneumoniae, Nocardia sp., M tuberculosis and nontuberculous Mycobacteria) also occur in brain abscesses. Anaerobic bacteria are found in 50% or more of cases (Chapter 21). Peptostreptococcus is most common, followed by Bacteroides and Prevotella species. Fusobacterium, Actinomyces, and Eubacterium are less common, followed by other anaerobes. Fungi (Chapter 45) are seen almost exclusively in immunocompromised patients. Candida species are the most prevalent fungi, but opportunistic molds such as Aspergillus sp. and Scedosporium apiospermum are increasing in frequency. Dimorphic fungi such as Coccidioides immitis may also cause brain abscesses. C neoformans is an important pathogen in AIDS patients. Parasites (Chapter 46) responsible for brain abscesses include Toxoplasma gondii, the most common protozoal cause, particularly among AIDS patients, neurocysticercosis (larval form of Taenia solium), Entamoeba histolytica, Schistosoma sp., and Paragonimus. Lumbar puncture to obtain CSF is generally not indicated in patients with brain abscess (or other mass lesions in the brain). The increased intracranial pressure makes the procedure life-threatening, because herniation of the brain through the tentorium cerebelli can result in midbrain compression. The findings in CSF are not specific for brain abscess: White blood cells, predominantly mononuclear cells, are often present; the glucose level may be moderately low and the protein concentration elevated. Thus, when fever and signs suggesting acute meningitis are absent and brain abscess is suspected, the clinician should obtain a CT scan. Brain abscesses typically show ring-enhanced uptake of contrast material on CT scan, though similar findings can be found in patients with brain tumors and other diseases. Magnetic resonance imaging (MRI) may be helpful in differentiating brain abscesses from tumors. Definitive differentiation between brain abscess and tumor is done by pathologic examination and culture of tissue from the lesion obtained by a neurosurgical procedure. Untreated brain abscesses are fatal. Surgical excision provides the initial therapy as well as the diagnosis of brain abscess. Needle aspiration using stereotactic technique is an alternative to surgical excision. Antibiotic therapy should be parenteral and should include high-dose penicillin G for streptococci and many anaerobes, metronidazole for anaerobes resistant to penicillin G, plus a third-generation cephalosporin for enteric gram-negative rods. Vancomycin or another drug specific for S aureus should be included in the initial therapy if the patient has endocarditis or is known to have CHAPTER 48 Cases and Clinical Correlations 789 staphylococcal bacteremia, or the abscess yields staphylococci. Initial therapy with antibiotics rather than surgery can be instituted in some patients whose brain abscesses are small (105 organisms. Abrupt onset of liquid diarrhea in endemic area. Needs prompt replacement of fluid and electrolytes intravenously or orally. Stool cultures positive; use selective media. 9, 18 Shigella species (mild cases) 24–72 hours Dysentery Organisms grow in superficial gut epithelium. Organisms invade epithelial cells; blood, mucus, and PMNs in stools. Infective dose 105 organisms. Gradual or abrupt onset of diarrhea and lowgrade fever. WBC in stool. Stool cultures are positive. No antimicrobials unless systemic dissemination is suspected or patient is immunocompromised. Prolonged carriage is frequent. 15 Salmonella typhi (S paratyphi A and B; S choleraesuis) 10–14 days Enteric fever Humans are the only reservoir for S typhi. Invades intestinal mucosa and multiplies in macrophages in intestinal lymph follicles; enters mesenteric lymph glands to blood and dissemination. Insidious onset of malaise, anorexia, myalgias, and headache; high remittent fever; may have constipation or diarrhea. Hepatosplenomegaly in about 50% of patients. Diagnosis by culture of S typhi from blood, stool, or other site. Antibiotic therapy is important. 15 (continued ) Yersinia enterocolitica 4–7 days Enteric fever Fecal-oral transmission. Food-borne. Animals infected. Gastroenteritis or mesenteric adenitis. Occasional bacteremia. Toxin produced occasionally. Severe abdominal pain, diarrhea, fever; PMNs and blood in stool; polyarthritis, erythema nodosum, especially in children. Keep stool specimen at 4°C before culture. 19 Clostridium difficile Days to weeks after antibiotic therapy Dysentery Antibiotic-associated pseudomembranous colitis. Makes enterotoxin (toxin A) and cytotoxin (toxin B), which cause diarrhea and epithelial cell necrosis. Abrupt onset of bloody diarrhea and fever. Toxin in stool. Patient typically received antibiotics in previous days to weeks. 11 Campylobacter jejuni 2–10 days Dysentery Infection via oral route from food, pets. Organisms grow in small intestine. Invasion of mucous membrane. Toxin production uncertain. Fever, diarrhea; PMNs, and fresh blood in stools, especially in children. Usually self-limited. Special media needed for cultures at 42°C. Patients usually recover in 5–8 days. 17 Rotavirus 48–96 hours Watery diarrhea, vomiting, mild fever Virus is the major cause of diarrheal disease in infants and young children worldwide. Induces histopathologic changes in intestinal mucosal cells. Fever and vomiting usually precede abdominal distress and diarrhea. Death in infants in developing countries follows dehydration and electrolyte imbalance. Typical course is 3–9 days. Diagnosis by immunoassay detection of rotavirus antigen in stool. 37 Norovirus 24–48 hours Watery diarrhea, vomiting Major cause of epidemic diarrhea especially in closed settings like cruise ships; high secondary attack rate Induces histopathologic change in intestinal mucosa such as blunting of microvilli Abrupt onset of abdominal pain followed by nausea, vomiting and diarrhea. Low-grade fever may occur; malaise, myalgias, and headache are described. Typical course is 2–3 days. Diagnosis requires RT-PCR or other assays not readily available 37 Giardia lamblia 1–2 weeks Watery diarrhea Most commonly identified intestinal parasite. Frequent pathogen in outbreaks of waterborne diarrhea. Complex and poorly understood interaction of parasite with mucosal cells and patient’s immune response. Diarrhea self-limiting in 1–3 weeks; chronic symptoms of intermittent diarrhea, malabsorption, and weight loss may last 6 months. Diagnosis by finding trophozoites or cysts in stool or duodenal contents, or by immunoassay detection of Giardia antigen in stool. 46 Entamoeba histolytica Gradual onset 1–3 weeks Dysentery Highest prevalence in developing countries; 10% of world’s population may be infected. Invades colonic mucosa and lyses cells, including leukocytes. Diarrhea, abdominal pain, weight loss, and fever are common. Can give rise to many complications, including fulminant colitis, perforation, and liver abscess. Diagnosis by finding trophozoites or cysts in stool. 46 Cholera toxin and E coli heat-labile toxin stimulate adenylyl cyclase activity, increasing cAMP concentration in gut, yielding secretion of chloride and water and reduced reabsorption of sodium. E coli heat-stable toxin activates intestinal guanylyl cyclase and results in hypersecretion. a 799 800 SECTION VII Diagnostic Medical Microbiology and Clinical Correlation In only a small percentage of cases is the etiologic agent demonstrated by means of stool culture or immunoassay. Finding white blood cells on fecal wet mounts is highly suggestive of infection with an invasive pathogen. Maintaining adequate hydration is the most important feature of treatment, especially in infants and children. Antimicrobial therapy is necessary in treatment of enteric fever (typhoid fever) and shortens the duration of symptoms in Shigella, Campylobacter, and V cholerae infections, but it prolongs the symptoms and fecal shedding of Salmonella. There is no specific therapy for infection due to rotaviruses, the most common viral cause of diarrhea, however a vaccine is available for prevention REFERENCES Dennehy PH: Viral gastroenteritis in children. Pediatr Infect Dis J 2011;30:63. Guerrant RL, Van Gilder T, Steiner TS, et al: Practice guidelines for the management of infectious diarrhea. Clin Infect Dis 2001;32:331. Guerrant RL, Bobak DA: Bacterial and protozoal gastroenteritis. N Engl J Med 1991;325:327. Marcos LA, Dupont HL: Advances in defining etiology and new therapeutic approaches in acute diarrhea. J Infect 2007;55:385. Patel MM, Hall AJ, Vinje J, Parashar UD: Noroviruses: A comprehensive review. J Clin Virol 2009;44:1. URINARY TRACT C A S E 8: AC U T E U N CO M P L I C AT E D B L A D D E R I N F E C T I O N A 21-year-old woman presented to the university student health service with a 2-day history of increasing urinary frequency along with urgency and dysuria. Her urine had been pink or bloody for about 12 hours. She had no history of prior urinary tract infection. The patient had recently become sexually active and was using a diaphragm and spermicide. Clinical Features The temperature was 37.5°C, pulse 105/min, and respirations 18/min. The blood pressure was 105/70 mm Hg. On physical examination, the only abnormal finding was mild tenderness to deep palpation in the suprapubic area. Laboratory Findings Laboratory tests showed a slightly elevated white blood cell count of 13,000/μL; 66% were PMNs, also elevated. Blood urea nitrogen, serum creatinine and glucose, and serum electrolytes were normal. The urine sediment contained innumerable white cells, moderate numbers of red cells, and many bacteria suggestive of urinary tract infection. Culture yielded more than 105 CFU/mL of E coli (diagnostic of a urinary tract infection). Antimicrobial susceptibility tests were not done. Treatment The patient was cured by 3 days of oral trimethoprim/sulfamethoxazole therapy. Comment See below. C ASE 9: COMPLIC ATED URINARY TR AC T INFEC TION A 67-year-old man developed fever and shock 3 days after a transurethral resection of his enlarged prostate gland. Two weeks earlier, he had urinary obstruction with retention secondary to the enlargement; benign prostatic hypertrophy had been diagnosed. Urinary bladder catheterization had been necessary. Following the surgery, an indwelling urinary bladder catheter attached to a closed drainage system was left in place. Two days after surgery, the patient developed fever to 38°C; on the third postoperative day, he became confused and disoriented and had a shaking chill. Clinical Features The temperature was 39°C, the pulse was 120/min, and the respirations were 24/min. The blood pressure was 90/40 mm Hg. On physical examination, the patient knew his name but was disoriented to time and place. His heart, lungs, and abdomen were normal. There was mild costovertebral tenderness over the area of the left kidney. Laboratory Findings Laboratory tests showed a normal hematocrit and hemoglobin but an elevated white blood cell count of 18,000/μL; 85% were PMNs (markedly elevated). Blood urea nitrogen, serum creatinine, serum glucose, and electrolytes were normal. Urine was obtained from the catheter port using a needle and syringe. The urine sediment contained innumerable white cells, a few red blood cells, and numerous bacteria, indicating a urinary tract infection. Urine culture yielded more than 105 CFU/mL of K pneumoniae (Chapter 15), CHAPTER 48 Cases and Clinical Correlations 801 confirming the diagnosis of urinary tract infection. Blood culture also yielded the K pneumoniae, which was susceptible to third-generation cephalosporins, gentamicin, and tobramycin. Treatment and Hospital Course The patient had a urinary tract infection associated with the bladder catheter. The left kidney was presumed to be involved based on the left costovertebral angle tenderness. He also had secondary bacteremia with shock (sometimes termed gram-negative sepsis and shock). He was treated with intravenous fluids and antibiotics and recovered. The same strain of K pneumoniae had been isolated from other patients in the hospital, indicating nosocomial spread of the bacteria. Comment Urinary tract infections may involve just the lower tract or both the lower and upper tracts. Cystitis is the term used to describe infection of the bladder with signs and symptoms including dysuria, urgency, and frequency, as in Case 8. Pyelonephritis is the term used to describe upper tract infection, often with flank pain and tenderness, and accompanying dysuria, urgency, and frequency, as in Case 9. Cystitis and pyelonephritis often present as acute diseases, but recurrent or chronic infections occur frequently. It is generally accepted that 105 or more CFU/mL of urine is significant bacteriuria, though the patients may be symptomatic or asymptomatic. Some young women have dysuria and other symptoms of cystitis with less than 105 CFU/mL of urine; in these women, as few as 103 CFU/mL of a gramnegative rod may be significant bacteriuria. The prevalence of bacteriuria is 1–2% in school-age girls, 1–3% in nonpregnant women, and 3–8% during pregnancy. The prevalence of bacteriuria increases with age, and the sex ratio of infections becomes nearly equal. Over the age of 70 years, 20–30% or more of women and 10% or more of men have bacteriuria. Upper urinary tract infections routinely occur in patients with indwelling catheters even with optimal care and closed drainage systems: 50% after 4–5 days, 75% after 7–9 days, and 100% after 2 weeks. Sexual activity and use of spermicides increase the risk for urinary tract infections in young women. E coli (Chapter 15) causes 80–90% of acute uncomplicated bacterial lower tract infections (cystitis) in young women. Other enteric bacteria and Staphylococcus saprophyticus (Chapter 13) cause most of the other culture-positive bladder infections in this patient group. Some young women with acute dysuria suggesting cystitis have negative urine cultures for bacteria. In these patients, selective cultures for Neisseria gonorrhoeae (Chapter 20) and Chlamydia trachomatis (Chapter 27) and evaluation for herpes simplex infection should be considered. In complicated upper tract infections, in the setting of anatomic abnormality or chronic catheterization, the spectrum in infecting bacteria is larger than in uncomplicated cases. E coli is frequently present, but other gramnegative rods of many species (eg, Klebsiella, Proteus, and Enterobacter [Chapter 15] and pseudomonads [Chapter 16]), enterococci, and staphylococci are also common. In many cases, two or more species are present, and the bacteria are often resistant to antimicrobials given in association with prior therapy. The presence of white blood cells in urine is highly suggestive but not specific for bacterial upper tract infections. White blood cells can be detected by microscopic examination of urine sediment or, indirectly, by dipstick detection of leukocyte esterase. The presence of red blood cells also is found on microscopy of the urine sediment, or indirectly by dipstick detection of hemoglobin. Proteinuria also is detected by dipstick. The presence of bacteria on Gram stain of noncentrifuged urine is strongly suggestive of 105 or more bacteria per milliliter of urine. The presence of bacteriuria is confirmed by quantitative culture of the urine by any one of several methods. One frequently used method is to culture urine using a bacteriologic loop calibrated to deliver 0.01 or 0.001 mL followed by counting the number of colonies that grow. Acute uncomplicated cystitis is usually caused by E coli susceptible to readily achievable urine concentrations of antibiotics appropriate for treatment of urinary tract infections. Thus, in the setting of the first such infection in a young woman, definitive identification and susceptibility testing of the bacteria are seldom necessary. Such cases can be treated by a single dose of appropriate antibiotic, based on local or regional antibiograms, but a 3–5 day course of therapy yields a lower relapse rate. Pyelonephritis is treated with 10–14 days of antibiotic therapy. Recurrent or complicated upper tract infections are best treated with antibiotics shown to be active against the infecting bacteria; definitive identification and susceptibility testing are indicated. Therapy for 14 days is appropriate and for 14–21 days if there is recurrence. Patients with complicated upper tract infections should have evaluations for anatomic abnormalities, stones, etc. REFERENCES Foster RT Sr: Uncomplicated urinary tract infections in women. Obstet Gynecol Clin North Am 2008;35:235. Gupta K, Hooton TM, Naber KG, et al: International clinical practice guidelines for the treatment of acute uncomplicated cystitis and pyelonephritis in women. A 2010 update by the Infectious Diseases Society of America and the European Society for Microbiology and Infectious Diseases. Clin Infect Dis 2011;52:e103. Neal DE Jr: Complicated urinary tract infections. Urol Clin North Am 2008;35:13. 802 SECTION VII Diagnostic Medical Microbiology and Clinical Correlation BONE AND SOFT TISSUE C A S E 10: O S T E O M Y E L I T I S A 34-year-old man suffered an open fracture of the middle third of his tibia and fibula when his motorized three-wheel vehicle tipped over in a field and fell on him. He was taken to a hospital and promptly to the operating room. The wound was cleaned and debrided, the fracture was reduced, and the bone aligned. Metal plates were placed to span the fracture, align it, and hold it in place. Pins were placed through the skin and bone proximal and distal to the fracture to allow splinting and immobilization of the leg. One day after surgery, the leg remained markedly swollen; a moderate amount of serous drainage was present on the dressings. Two days later, the leg remained swollen and red, requiring opening of the surgical wound. Cultures of pus in the wound grew S aureus (Chapter 13) resistant to penicillin G but susceptible to nafcillin. The patient was treated with intravenous nafcillin for 10 days, and the swelling and redness decreased. Three weeks later, pus began to drain from a small opening in the wound. Cultures again grew S aureus. Exploration of the opening showed a sinus tract to the site of the fracture. An x-ray film of the leg showed poor alignment of the fracture. Osteomyelitis was diagnosed, and the patient was returned to the operating room, where the fracture site was debrided of necrotic soft tissue and dead bone; the pins and plates were removed. Bone grafts were placed. The fracture was immobilized by external fixation. Cultures obtained during surgery grew S aureus. The patient was treated with intravenous nafcillin for 1 month followed by oral dicloxacillin for 3 additional months. The wound and fracture slowly healed. After 6 months, there was no x-ray evidence of further osteomyelitis, and the patient was able to bear weight on the leg. Comment Osteomyelitis follows hematogenous spread of pathogenic bacteria from a distant site of infection to bone or, as in this case, direct inoculation of the bone and soft tissue, as can occur with an open fracture or from a contiguous site of soft tissue infection. The primary symptoms are fever and pain at the infected site; swelling, redness, and occasionally drainage can be seen, but the physical findings are highly dependent on the anatomic location of the infection. For example, osteomyelitis of the spine may present with fever, back pain, and signs of a paraspinous abscess; infection of the hip may show as fever with pain on movement and decreased range of motion. In children, the onset of osteomyelitis following hematogenous spread of bacteria can be very sudden, while in adults the presentation may be more indolent. Sometimes osteomyelitis is considered to be chronic or of long standing, but the clinical spectrum of osteomyelitis is broad, and the distinction between acute and chronic may not be clear either clinically or on morphologic examination of tissue. S aureus (Chapter 13) is the primary agent of osteomyelitis in 60–70% of cases (90% in children). S aureus causes the infection after hematogenous spread or following direct inoculation. Community-acquired methicillin-resistant S aureus that contains the Panton-Valentine leukocidin causes acute hematogenous osteomyelitis affecting multiple sites, often in association with vascular complications. Streptococci cause osteomyelitis in about 10% of cases, and enteric gram-negative rods (eg, E coli) and other bacteria such as P aeruginosa (Chapter 16) in 20–30%. Kingella kingae (Chapter 16) is a common etiologic agent in infants and children. Anaerobic bacteria (eg, Bacteroides species [Chapter 21]) are also common, particularly in osteomyelitis of the bones of the feet associated with diabetes and foot ulcers. Any bacteria that cause infections in humans have been associated with osteomyelitis. Definitive diagnosis of the etiology of osteomyelitis requires culture of a specimen obtained at surgery or by needle aspiration of bone or periosteum through uninfected soft tissue. Culture of pus from the opening of a draining sinus tract or superficial wound associated with the osteomyelitis commonly yields bacteria that are not present in the bone. Blood cultures are often positive when systemic symptoms and signs (fever, weight loss, elevated white blood cell count, high erythrocyte sedimentation rate) are present. Early in the course of osteomyelitis, x-ray films of the infected site are negative. The initial findings noted radiologically usually are soft tissue swelling, loss of tissue planes, and demineralization of bone; 2–3 weeks after onset, bone erosions and evidence of periostitis appear. Bone scans with radionuclide imaging are about 90% sensitive. They become positive within a few days after onset and are particularly helpful in localizing the site of infection and determining if there are multiple sites of infection; however, bone scans do not differentiate between fractures, bone infarction (as occurs in sickle cell disease), and infection. CT and MRI also are sensitive and especially helpful in determining the extent of soft tissue involvement. Antimicrobial therapy and surgical debridement are the mainstays of treatment of osteomyelitis. The specific antimicrobial should be selected after culture of a properly obtained specimen and susceptibility tests and continued for 6–8 weeks or longer, depending on the infection. Surgery should be done to remove any dead bone and sequestra that are present. Immobilization of infected limbs and fixation of fractures are important features of care. CHAPTER 48 Cases and Clinical Correlations 803 REFERENCES Calhoun JH, Manring MM: Adult osteomyelitis. Infect Dis Clin North Am 2005;19:265. Kaplan SL: Osteomyelitis in children. Infect Dis Clin North Am 2005;19:787. C A S E 11: G A S G A N G R E N E A 22-year-old man fell while riding his new motorcycle and suffered an open fracture of his left femur and severe lacerations and crushing injury to the thigh and less extensive soft tissue injuries to other parts of his body. He was rapidly transported to the hospital and immediately taken to the operating room, where the fracture was reduced and the wounds debrided. At admission, results of his blood tests included a hematocrit of 45% and a hemoglobin of 15 g/dL. The immediate postoperative course was uneventful, but 24 hours later, pain developed in the thigh. Fever was noted. Pain and swelling of the thigh increased rapidly. Clinical Features and Course The temperature was 40°C, the pulse 150/min, and respirations 28/min. The blood pressure was 80/40 mm Hg. Physical examination showed an acutely ill young man who was in shock and delirious. The left thigh was markedly swollen and cool to touch. Large ecchymotic areas were present near the wound, and there was a serous discharge from the wound. Crepitus was felt, indicative of gas in the tissue of the thigh. An x-ray film also showed gas in the tissue planes of the thigh. Gas gangrene was diagnosed, and the patient was taken to the operating room for emergency extensive debridement of necrotic tissue. At the time of surgery, his hematocrit had fallen to 27% and his hemoglobin to 11 g/dL; his serum was red-brown in color, indicating hemolysis with free hemoglobin in his circulation. Anaerobic cultures of the specimen obtained at surgery grew Clostridium perfringens (Chapters 11, 21). The patient developed renal failure and heart failure and died 3 days after his injury. Comment Case 11 illustrates a classic case of clostridial gas gangrene. C perfringens (or occasionally other Clostridium species) are inoculated into the traumatic wound from the environment; the clostridia are discussed in Chapters 11 and 21. The presence of necrotic tissue and foreign body material provides a suitable anaerobic environment for the organisms to multiply. After an incubation period usually of 2–3 days but sometimes only 8–12 hours, there is acute onset of pain, which rapidly increases in intensity associated with shock and delirium. The extremity or wound shows tenderness, tense swelling, and a serosanguineous discharge. Crepitus is often present. The skin near the wound is pale but rapidly becomes discolored, and fluid-filled blebs form in the nearby skin. Skin areas of black necrosis appear. In severe cases, there is rapid progression. In patients such as this one, Gram stain of fluid from a bleb or of a tissue aspirate shows large gram-positive rods with blunt ends and is highly suggestive of clostridial infection. PMN leukocytes are rare. Anaerobic culture provides the definitive laboratory confirmation. The differential diagnosis of clostridial gas gangrene includes anaerobic streptococcal myonecrosis, synergistic necrotizing myonecrosis, and necrotizing fasciitis. These clinically overlapping diseases can be differentiated from clostridial gas gangrene by Gram stain and cultures of appropriate specimens. X-ray films of the infected site show gas in the fascial planes. Abnormal laboratory tests include a low hematocrit. The hemoglobin may be low or normal even when the hematocrit is low, consistent with hemolysis and cell-free circulating hemoglobin. Leukocytosis is usually present. Extensive surgery with removal of all the dead and infected tissue is necessary as a lifesaving procedure. Penicillin G is the antibiotic of choice. Antitoxin is of no help. When shock and circulating free hemoglobin are present, renal failure and other complications are common and the prognosis is poor. SEXUALLY TRANSMITTED DISEASES C ASE 12: URETHRITIS, ENDOCERVICITIS, and PELVIC INFLAMMATORY DISEASE A 19-year-old woman came to the clinic because of lower abdominal pain of 2 days’ duration and a yellowish vaginal discharge first seen 4 days previously on the day following the last day of her menstrual period. The patient had had intercourse with two partners in the previous month, including a new partner 10 days before presentation. Clinical Features Her temperature was 37.5°C; other vital signs were normal. Physical examination showed a yellowish mucopurulent discharge from the cervical os. Moderate left lower abdominal tenderness was present. The bimanual pelvic examination showed cervical motion tenderness and adnexal tenderness more severe on the left than on the right. 804 SECTION VII Diagnostic Medical Microbiology & Clinical Correlation Laboratory Findings A nucleic acid amplification test that detects both N gonorrhoeae (Chapter 20) and C trachomatis (Chapter 27) performed on a cervical swab specimen was positive for C trachomatis. Treatment A diagnosis of pelvic inflammatory disease (PID) was made. The patient was treated as an outpatient with a single intramuscular dose of ceftriaxone plus doxycycline for 2 weeks. Both of her partners came to the clinic and were treated. Comment In men, urethral discharge is classified as gonococcal urethritis, caused by N gonorrhoeae, or nongonococcal urethritis, caused usually by either C trachomatis (15–55% of cases) or Ureaplasma urealyticum (20–40% of cases) and infrequently by Trichomonas vaginalis (Chapter 46). The diagnosis is based on the presence or absence of gram-negative intracellular diplococci on stain of the urethral discharge. All patients with urethritis should be tested using nucleic acid amplification methods for both C trachomatis and N gonorrhoeae. Ceftriaxone is frequently used to treat gonococcal urethritis, but quinolones may be used in areas that report low resistance. Doxycycline or azithromycin is used to treat nongonococcal urethritis. It is highly recommended that men with gonococcal infection also be treated for chlamydial infection because of the likelihood that both infections may be present. In women, the differential diagnosis of endocervicitis (mucopurulent cervicitis) is between gonorrhea and C trachomatis infection. The diagnosis is made by culture of the endocervical discharge or nucleic acid amplification tests for simultaneous detection of N gonorrhoeae and C trachomatis. There are three major treatment options: (1) Treat for both N gonorrhoeae and C trachomatis before the culture results are available (recommended option); (2) treat for C trachomatis only, if the prevalence of N gonorrhoeae infection is low but the likelihood of chlamydial infection is high; or (3) await culture results if the prevalence of both diseases is low and the likelihood of compliance with a recommendation for a return visit is high. Recommended treatments are the same as those mentioned earlier for urethritis. Pelvic inflammatory disease (PID), also called salpingitis, is inflammation of the uterus, uterine tubes, and adnexal tissues that is not associated with surgery or pregnancy. PID is the major consequence of endocervical N gonorrhoeae and C trachomatis infections, and well over half of the cases are caused by one or both of these organisms. The incidence of gonococcal PID is high in inner city populations, while chlamydial PID is more common in college students and more affluent populations. Other common bacterial causes of PID are enteric organisms and anaerobic bacteria associated with bacterial vaginosis. Lower abdominal pain is the common presenting symptom. An abnormal vaginal discharge, uterine bleeding, dysuria, painful intercourse, nausea and vomiting, and fever also occur frequently. The major complication of PID is infertility due to uterine tubal occlusion. It is estimated that 8% of women become infertile after one episode of PID, 19.5% after two episodes, and 40% after three or more episodes. A clinical diagnosis of PID should be considered in any woman of childbearing age who has pelvic pain. Patients often have classic physical findings in addition to the presenting signs and symptoms, including lower abdominal, cervical motion, and adnexal tenderness. A clinical diagnosis can be confirmed by laparoscopic visualization of the uterus and uterine tubes, but this procedure is not practical and is infrequently performed; however, only about two-thirds of women with a clinical diagnosis of PID will have the disease when the uterine tubes and uterus are visualized. The differential diagnosis includes ectopic pregnancy and appendicitis as well as other diseases. In PID patients, hospitalization with intravenous therapy often is recommended to decrease the possibility of infertility. Inpatient drug regimens include cefoxitin and doxycycline or gentamicin and clindamycin. Outpatient regimens include cefoxitin or ceftriaxone in single doses plus doxycycline, or ofloxacin plus metronidazole. REFERENCES Centers for Disease Control and Prevention: Sexually t ransmitted diseases treatment guidelines, 2011. MMWR Morb Mortal Wkly Rep 2010;59(RR-12):1. Lareau SM, Beigi RH: Pelvic inflammatory disease and tubo- ovarian abscess. Infect Dis Clin North Am 2008;22:693. Trigg BG, Kerndt PR, Aynalem G: Sexually transmitted infections and pelvic inflammatory disease in women. Med Clin North Am 2008;92:1083. C A S E 1 3 : VAG I N O S I S and VAG I N I T I S A 28-year-old woman came to the clinic because of a whitish-gray vaginal discharge with a bad odor, first noted 6 days previously. She had been sexually active with a single partner who was new to her in the past month. Clinical Features Physical examination showed a thin, homogeneous, whitishgray discharge that was adherent to the vaginal wall. There was no discharge from the cervical os. The bimanual pelvic examination was normal, as was the remainder of the physical examination. Laboratory Findings The pH of the vaginal fluid was 5.5 (normal, 4.5 ≤4.5 Odor None Common, fishy May be present, fishy None Microscopy Epithelial cells with lactobacilli Clue cells with adherent bacilli; no PMNs Motile trichomonads; many PMNs KOH preparation showing budding yeasts and pseudohyphae Treatment None Metronidazole orally or topically Metronidazole orally Topical azole antifungal 806 SECTION VII Diagnostic Medical Microbiology and Clinical Correlation There was little pain on palpation. Left inguinal lymph nodes 1–1.5 cm in diameter were palpable. Laboratory Findings The penile lesion was gently cleaned with saline and gauze. A small amount of clear exudate was then obtained from the base of the lesion, placed on a slide, and examined by darkfield microscopy. Multiple spirochetes were seen. The rapid plasma reagin (RPR) screening serologic test for syphilis was positive at a 1:8 dilution. The confirmatory treponeme-specific fluorescent treponemal antibody-absorbed (FTA-ABS) test also was positive. Treatment and Follow-Up The patient was treated with a single dose of benzathine penicillin. Six months later, his RPR test had reverted to negative, but the FTA-ABS test was expected to stay positive for life. The patient named five female sex partners for the month prior to his clinic visit. Three of these women were located by the public health investigators; two had positive serologic tests for syphilis and were treated. The two women who were not located had gone to unknown addresses in other cities. Comment The three major genital sore diseases are syphilis, genital herpes, and chancroid (Table 48-6). Two much less common genital sore diseases are the initial lesion of lymphogranuloma venereum, caused by C trachomatis (Chapter 27), and the rare disease granuloma inguinale (donovanosis), caused by Klebsiella granulomatis. Lymphogranuloma venereum is a systemic illness with fever, malaise, and lymphadenopathy; inguinal buboes may be present. The diagnosis usually is made by serologic tests, but culture of pus aspirated from an inguinal bubo may yield C trachomatis. Some reference laboratories have developed multiplex polymerase chain reaction (PCR) assays for simultaneous detection of the pathogens that cause genital sore disease, but these are not widely available. MYCOBACTERIUM TUBERCULOSIS INFECTIONS C A S E 1 5 : P U L M O N A R Y T U B E R C U LO S I S A 64-year-old man was admitted to the hospital with a 5-month history of progressive weakness and a weight loss of 13 kg. He also had fever, chills, and a chronic cough productive of yellowish sputum, occasionally streaked with blood. The patient drank a lot of alcohol and lived in a boarding house next door to the tavern he frequented. He had smoked one pack of cigarettes a day for the past 45 years. The patient had no history of tuberculosis, no record of prior skin tests for tuberculosis or abnormal chest radiographs, and no known exposure to tuberculosis. TABLE 48-6 The Major Genital Sore Diseases: Syphilis, Herpes, and Chancroida Primary Syphilis Genital Herpes (Initial Lesions) Chancroid Etiologic agent Treponema pallidum Herpes simplex virus Haemophilus ducreyi Incubation period 3 weeks (10–90 days) 2–7 days 3–5 days Usual clinical presentation Slightly tender papule that ulcerates over 1 to several weeks Marked pain in genital area; papules that ulcerate in 3–6 days; fever, headache, malaise, and inguinal adenopathy are common Tender papule that ulcerates in 24 hours Diagnostic tests Dark-field examination of exudate from chancre; serologic tests Virus culture of cells and fluid from chancre; serologic tests turn positive in 18–48 hours; fluorescent antibody stain of the same specimen; nucleic acid amplification tests Culture of Haemophilus ducreyi on at least two kinds of enriched medium containing vancomycin and incubated at 33°C Long-term sequelae Secondary syphilis with mucocutaneous lesions; tertiary syphilis Recurrent genital herpes Inguinal bubo Treatment Benzathine penicillin G; doxycycline if penicillin allergy is present Acyclovir or famciclovir or valacyclovir Ceftriaxone, or azithromycin, or erythromycin, or ciprofloxacin b a Source: Sexually transmitted diseases treatment guidelines, 2010. MMWR Morb Mortal Wkly Rep 2010;59(RR-12):1-116. b HIV testing should be performed in patients with genital ulcer disease caused by these pathogens. CHAPTER 48 Cases and Clinical Correlations 807 Clinical Features His temperature was 39°C, pulse 110/min, respirations 32/min, and blood pressure 120/80 mm Hg. He was a slender man. His dentition was poor, but the remainder of his head and neck examination was normal. On chest examination, many crackles were heard over the upper lung fields. The remainder of the physical examination was normal. Laboratory Findings and Imaging The hematocrit was 30% (low), and the white blood cell count was 9600/μL. Electrolyte concentrations and other blood tests were normal. The test for HIV-1 antibody was negative. A chest radiograph showed extensive cavitary infiltrates in both upper lobes. A tuberculin skin test was negative, as were skin tests with mumps and candida antigens, indicating anergy. A sputum specimen was obtained immediately, and an acid-fast stain was done before the sputum concentration procedure. Numerous acid-fast bacteria were seen on the smear. Culture of the decontaminated and concentrated sputum was positive for acid-fast bacteria after 14 days’ incubation; M tuberculosis was identified by molecular probe 2 days later. Susceptibility tests of the organisms showed susceptibility to isoniazid, rifampin, pyrazinamide, ethambutol, and streptomycin. Hospital Course and Treatment The patient was treated with isoniazid, rifampin, pyrazinamide, and ethambutol for 2 months, followed by directly observed twice-weekly administration of isoniazid and rifampin for 7 months. Follow-up sputum cultures were negative for M tuberculosis. At hospitalization, the patient had been placed in isolation and asked to wear a mask at all times. However, before the mask and isolation were implemented, a medical student and a resident physician were exposed to the patient. The resident physician converted her tuberculin skin test and received isoniazid prophylaxis for 9 months. An attempt was made to trace the patient’s close contacts. A total of 34 persons were found to have positive tuberculin tests. Persons 35 years of age or younger were given isoniazid prophylaxis for 1 year; those older than 35 had periodic follow-up chest x-rays. Two cases of active tuberculosis also were diagnosed and treated. The M tuberculosis isolates from the two patients were identical to the index patient’s isolate by DNA fingerprinting. C A S E 16: D I S S E M I N AT E D M I L I A R Y T U B E R C U LO S I S A 31-year-old Asian woman was admitted to the hospital with a history of 7 weeks of increasing malaise, myalgia, nonproductive cough, and shortness of breath. She had developed daily fevers of 38–39°C and had a recent 5-kg weight loss. She was given an oral cephalosporin with no benefit. Her past medical history showed she had emigrated from the Philippines at age 24 and had had a negative chest radiograph at that time. The patient’s grandmother had died of tuberculosis when the patient was an infant; the patient did not know if she had had contact with the grandmother. The patient was given BCG vaccine as a child. She was currently living with relatives who operated a boarding home for about 30 elderly persons. Clinical Features Her temperature was 39°C, pulse 100/min, respirations 20/min, and blood pressure 120/80 mm Hg. Her physical examination was entirely normal. The examiner was unable to palpate her spleen; the liver was of normal size to percussion; and there was no palpable lymphadenopathy. Laboratory Findings and Imaging The hemoglobin was 8.3 g/dL (normal, 12–15.5 g/dL), and the hematocrit was 27% (normal, 36–46%). The peripheral blood smear showed hypochromic, microcytic red blood cells compatible with chronic infection or iron deficiency anemia. The platelet count was 50,000/μL (normal, 140,000–450,000/μL). The white blood cell count was 7000/μL (normal), with a normal differential count. The prothrombin time was moderately prolonged and the partial thromboplastin time mildly prolonged, suggesting a coagulopathy of liver disease. The liver function tests were an aspartate aminotransferase (AST) of 140 units/L (normal, 10–40 units/L), alanine aminotransferase (ALT) 105 units/L (normal, 5–35 units/L), bilirubin 2 mg/dL (twice normal), and alkaline phosphatase 100 units/L (normal, 36–122 units/L). The serum albumin was 1.7 g/dL (normal, 3.4–5 g/ dL). The creatinine, blood urea nitrogen, and electrolytes were normal. Urinalysis showed a few red and a few white blood cells. Two routine blood cultures were negative. Sputum and urine cultures grew small amounts of normal microbiota. Serologic tests for HIV-1, hepatitis B virus antibody and antigen, coccidioidomycosis, leptospirosis, brucellosis, mycoplasmal infection, Lyme disease, and Q fever were negative. A tuberculin skin test was negative. A chest radiograph was normal. A CT scan of the abdomen was negative. Hospital Course and Treatment During the first few days of hospitalization, the patient developed progressive shortness of breath and respiratory distress. Repeat chest radiography showed bilateral interstitial infiltrates. Adult respiratory distress syndrome 808 SECTION VII Diagnostic Medical Microbiology and Clinical Correlation was diagnosed. The hemoglobin was now 10.6 g/dL and the white blood cell count 4900/μL. Arterial blood gases showed a pH of 7.38, a PO2 of 50 mm Hg (low), and a PCO2 of 32 mm Hg. The patient was placed on oxygen therapy and intubated (for 4 days). BAL was performed. The lavage fluid was negative on routine culture, and an acid-fast stain was also negative. A second abdominal CT scan showed a normal-appearing liver, but periaortic lymphadenopathy and mild splenomegaly were present. The patient underwent laparoscopy with a liver biopsy and a bone marrow biopsy. The liver and bone marrow biopsies both showed granulomas with giant cells; acid-fast bacilli were also present. (There were abundant iron stores, indicating that the anemia was due to chronic infection and not iron deficiency.) The patient was started on isoniazid, rifampin, pyrazinamide, and ethambutol. The chest radiographs continued to show diffuse infiltrates, but improvement was evident. The patient’s fever decreased, and she showed general improvement. Between 19 and 21 days of incubation, the liver and bone marrow biopsies and the lavage fluid all were culturepositive for acid-fast bacilli, identified as M tuberculosis by molecular probe. The mycobacteria were susceptible to all of the drugs the patient was receiving. The four-drug regimen was continued for 2 months until the susceptibility test results were obtained. The patient was then continued on isoniazid and rifampin for 10 more months for a total of 1 year of therapy. The patient’s relatives and the elderly persons who lived with them all had skin tests for tuberculosis. The persons with positive skin tests and those who had recent histories of cough or weight loss also had chest radiographs. Three tuberculinpositive persons were found. No one had active tuberculosis. Those living in the home with the patient and those persons who recently converted their skin tests were offered prophylaxis with isoniazid. The patient was thought to have had reactivation tuberculosis with hematogenous spread involving her lungs, liver, lymph nodes, and possibly her kidneys. Comment It is estimated that approximately one-third of the world’s population have tuberculosis and that each year about 1–3 million people die of the disease. In the United States, a low incidence of tuberculosis of 9.4 cases per 100,000 population was reached in the mid-1980s. The rate increased slightly in the late 1980s, but since 1992, the rates have again declined. The lowest (and most recently recorded) rate of 3.6 cases per 100,000 population (11,182 cases) was recorded in 2010. Tuberculosis in the United States occurs most commonly among lower socioeconomic populations: the urban poor, homeless persons, migrant farm workers, alcoholics, and intravenous drug users. Approximately half of the cases of tuberculosis occur in foreign-born individuals. The incidence of tuberculosis can be very high in selected groups and geographic areas (eg, HIV-positive intravenous drug abusers in the eastern United States, Haitian AIDS patients). Tuberculosis in elderly persons usually is due to reactivation of prior infection, while disease in children implies active transmission of M tuberculosis. About 80% of cases in children occur in ethnic minorities. However, active tuberculosis is most frequently diagnosed in young adults, often in association with HIV-1 infection. Concomitant tuberculosis and HIV-1 infections are especially important in developing countries; in Africa, millions of people have both infections. There is considerable concern about the spread of multidrugresistant tuberculosis in Russia. Spread of tuberculosis from a patient to another person occurs through infectious droplet nuclei generated during coughing, sneezing, or talking. The major factors in transmission of infection are the closeness and duration of contact and the infectiousness of the patient. Generally, 500 cells/μL, (2) 200–499/μL, and (3) 10% weight loss and over 1 month of either diarrhea or weakness and fever) also are AIDS-defining. HIV-1-infected patients may present with signs and symptoms referable to one or more organ systems. The common opportunistic infections are listed by anatomic site in Table 48-8. Typically, the evaluation of patients who may have HIV-1 infection or AIDS is based on a clinical and epidemiologic history of possible exposure coupled with a diagnostic evaluation of the presenting illness according to the site involved. The status of knowledge about anti-HIV-1 drug therapy changes very rapidly, and for that reason, anti-HIV-1 therapy recommendations should be considered interim ones. Only general guidelines are presented here. Postexposure prophylaxis with anti-HIV-1 drugs is effective, and treatment of primary HIV-1 infection may also have favorable prognostic implications. Many factors influence the decision to begin anti-HIV-1 treatment, including the rate of decrease of the CD4 cell count and the blood level of HIV-1 RNA. Early in the course of HIV-1 disease, when the CD4 cell count is >500 cells/μL, it is appropriate to monitor the clinical status; therapy may be considered if the viral load is high. When the CD4 cell count falls below 500 cells/μL, antiretroviral therapy is now recommended. The drugs used to treat HIV-1 infection are discussed in Chapter 30. A variety of different regimens may be chosen. This highly active antiretroviral therapy has significantly improved the lives and prognosis for many AIDS patients. Response to treatment should be monitored by following viral load measurements and by testing for resistance when clinical response is poor. When the CD4 cell count is Report "Jawetz, Melnick & Adelberg’s Medical Microbiology [26th ed.] 0071790314, 9780071790314 (print)" × Close Submit Contact information Michael Browner info@dokumen.pub Address: 1918 St.Regis, Dorval, Quebec, H9P 1H6, Canada. 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3878	https://math.stackexchange.com/questions/50927/the-derivative-of-a-product-of-more-than-two-functions	Skip to main content The derivative of a product of more than two functions Ask Question Asked Modified 4 years, 4 months ago Viewed 9k times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. I'm trying to generalize the product rule to more than the product of two functions using the fact that I can treat the product of n-1 functions as a single one. Here is an example of what I mean: [f(x)g(x)h(x)]′=[f(x)p(x)]′ where p(x)=g(x)h(x) [f(x)p(x)]′=f′(x)p(x)+f(x)p′(x)=f′(x)p(x)+f(x)[g(x)h(x)]′ f′(x)p(x)+f(x)[g(x)h(x)]′=f′(x)g(x)h(x)+f(x)[g′(x)h(x)+g(x)h′(x)]′ which equals f′(x)g(x)h(x)+f(x)g′(x)h(x)+f(x)g(x)h′(x) I generalized this as follows: [∏i=1nfi(x)]′=f′1(x)g1(x)+f1(x)g′1(x) where gm(x)=∏n−mi=1fi(x), and g′m−1=[fm(x)gm(x)]′=f′m(x)gm(x)+fm(x)g′m(x). Now, I do realize that this is a generalization, and there is really nothing to prove, but say I wanted to prove that [∏i=1nfi(x)]′=∑i=1nf′i(x)hi(x) where hi(x)=1fi(x)∏nj=1fj(x), how would I go about doing this (using the generalization above)? I apologize if my notation is hard to understand. Thank you. calculus products Share CC BY-SA 3.0 Follow this question to receive notifications edited Jul 12, 2011 at 1:45 Hautdesert asked Jul 11, 2011 at 21:32 HautdesertHautdesert 1,62633 gold badges2121 silver badges3030 bronze badges 4 This problem is perfectly suited for a proof using mathematical induction. – Rasmus Commented Jul 11, 2011 at 21:44 11 Possibly an easy way to remember: log∏f=∑logf and (logf)′=f′/f – Aryabhata Commented Jul 11, 2011 at 21:58 (∏i=1kfi)(n)=∑n=j1+...+jk(nj1,...,jk)∏i=1kf(ji)i – yoyo Commented Jul 11, 2011 at 23:35 @yoyo +1 your comment was exactly the generalization, I was looking for. Did you use it in a question/answer? Why not extending the WP page: Product_rule#Generalizations... – draks ... Commented Aug 3, 2012 at 18:21 Add a comment \| 3 Answers 3 Reset to default This answer is useful 13 Save this answer. Show activity on this post. Simplest way to establish this is by induction on n. The case n=1 is immediate; the case n=2 is the usual product rule. Assuming you have established the desired formula (∏i=1nfi(x))′=∑i=1n⎛⎝⎜⎜f′i(x)∏i≠j1≤j≤nfj(x)⎞⎠⎟⎟ for n, then to get the n+1 case we have: (∏i=1n+1fi(x))′=((∏i=1nfi(x))fn+1(x))′=(∏i=1nfi(x))′fn+1(x)+(∏i=1nfi(x))f′n+1(x)=⎛⎝⎜⎜∑i=1nf′i(x)∏i≠j1≤j≤nfj(x)⎞⎠⎟⎟fn+1(x)+(∏i=1nfi(x))f′n+1(x)=∑i=1nf′i(x)∏i≠j1≤j≤n+1fj(x)+(∏i=1nfi(x))f′n+1(x)=∑i=1n+1f′i(x)∏i≠j1≤j≤n+1fj(x), as desired. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Jul 8, 2019 at 11:10 answered Jul 11, 2011 at 21:42 Arturo MagidinArturo Magidin 418k6060 gold badges862862 silver badges1.2k1.2k bronze badges 7 I'm sorry, I don't seem to understand your answer, although I'm sure it's correct. Does the notation "1≤j≤n; i≠j" mean that j is all the positive integers from 1 to n inclusive, excluding i? I'm just not familiar with that notation. – Hautdesert Commented Jul 12, 2011 at 18:44 @Hautdesert: Yes; I'm multiplying over all indices j that satisfy 1≤j≤n, and that satisfy i≠j. It's just a compact way of writing what you did without having to define the hi. – Arturo Magidin Commented Jul 12, 2011 at 18:55 I don't follow the transition from the third line to the fourth line. What happens to fn+1(x)? – Hautdesert Commented Jul 12, 2011 at 19:24 @Hautdesert: If you look carefully at the index of the product in the first summand, in the third line it goes up to n, in the fourth line it goes all the way to n+1; the fn+1(x) has been absorbed into that product. – Arturo Magidin Commented Jul 12, 2011 at 19:26 @Hautdesert: P.S. You should not accept an answer if you are having trouble following it! Why did you accept my answer, only to say later that you don't understand it? First understand the answers, then decide which one is the most helpful and you can accept that one then. Right now, you might want to un-accept this, since you are having trouble following it. – Arturo Magidin Commented Jul 12, 2011 at 19:26 \| Show 2 more comments This answer is useful 6 Save this answer. Show activity on this post. You can use induction on n, the number of functions. if n=1, there is nothing to prove. if n=2, then you just get the product rule. Assume the claim is true for n functions, and prove it for n+1. Write f1f2...fn+1 = f1g where g=f2..fn+1. Now differentiate f1g using the product rule and apply the induction hypothesis to g′. Note that g is a product of n functions, so the induction hypothesis tells you what g′ is. Share CC BY-SA 3.0 Follow this answer to receive notifications answered Jul 11, 2011 at 21:42 algebra_fanalgebra_fan 2,26411 gold badge1717 silver badges1414 bronze badges Add a comment \| This answer is useful 0 Save this answer. Show activity on this post. Perhaps the confusion is stemming from the complicated notation. Here's something I wrote that might be easier to follow: Let f1,⋯,fn be functions. Then the derivative of the product f1⋯fn is the following: (f1⋯fn)′=∑j=1n[f′j∏i=1,i≠jn[fi]]. The indices for the Π mean to multiply through from i=1 to n while skipping over the jth index. Proof: We proceed by mathematical induction on n. Consider the case n=1: (f1)′=∑j=11[f′j∏i=1,i≠j1[fi]]=✓f′1. Suppose the statement holds for n=k, k>1. That is, (f1⋯fk)′=∑j=1k[f′j∏i=1,i≠jk[fi]]=(f′1f2⋯fk)+(f1f′2f3⋯fk)+⋯+(f1⋯fk−1f′k). We then have that, via the product rule and our supposition, (f1⋯fkfk+1)′=fk+1∑j=1k[f′j∏i=1,i≠jk[fi]]+f′k+1∏i=1k[fi]=fk+1[(f′1f2⋯fk)+(f1f′2f3⋯fk)+⋯+(f1⋯fk−1f′k)]+f′k+1[f1⋯fk]=[(f′1f2⋯fk+1)+(f1f′2f3⋯fk+1)+⋯+(f1⋯fkf′k+1)]=✓∑j=1k+1[f′j∏i=1,i≠jk+1[fi]].■ Share CC BY-SA 4.0 Follow this answer to receive notifications answered Apr 17, 2021 at 19:32 Noah JensenNoah Jensen 14177 bronze badges Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus products See similar questions with these tags. 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3879	https://www.tutorela.com/math/finding-a-missing-value-in-a-proportion	Finding a Missing Value in a Proportion Sometimes we will be given only one whole ratio between two terms and a third piece of data that is part of another ratio. Usually, it will be stated that there is proportionality between the ratios and that we must find the missing data in the ratio. How are problems solved where a number from the proportion is missing? We will identify the given ratio in the question between both terms and write it down in the form of a fraction. We will compare it with the other ratio (also in the form of a fraction) that includes the third piece of data from the question and also the unknown X. We will solve for X. Let's look at an example In the "Freedom for All" zoo, the ratio of camels to peacocks is 2:5. It is known that in the "Firmament" zoo there is the same ratio as in "Freedom for All" between camels and peacocks. It is also known that, in the Firmament zoo, there are 20 peacocks. How many camels are there in the Firmament zoo? Solution: 1.We will identify the ratio given in the question between both terms and write it down in the form of a fraction. The question states that the ratio between camels and peacocks is 2:5. We will write it in the form of a fraction and note next to each term what it refers to: 2.We will compare it with the other ratio (also in fraction form) that includes the third piece of data from the question and also the unknown X. Since it has been stated that the ratio between camels and peacocks is the same in both zoos, we can equate them. We already know that there are 20 peacocks in the Firmament zoo, the number of camels is the unknown we need to find. Therefore, we will draw the following equation: 3.Solve for X. Now that we have an equation with one unknown, we can easily find the X by using a common factor or cross-multiplying. We will obtain: 52=20X 5X=40 X=8 Let's remember that the idea is the equivalence relation. So, we might ask ourselves, by how much did we multiply 5 to get to 20? That is exactly the number by which we will multiply the 2 to arrive at the required result. Note, on certain occasions the third piece of data will not be given to us too clearly in the question, so we will also need to express it with an X. Let's look at another simple example In a jewelry store, there are 60 necklaces and 120 rings. Maria has 10 necklaces and, the ratio between the number of necklaces and rings she keeps in her personal jewelry box, is identical to that of the jewelry store. We must find the number of rings in Maria's jewelry box. In this question, there is proportionality (equivalence relation) between the jewelry store and Maria's personal jewelry box in the number of items. Let's write the relation, we will obtain: 60:120=10:X X indeed, represents the number of rings in Maria's jewelry box. To maintain the proportionality (that is, to keep the relation the same) X must be equal to 20. In other words, Maria has 20 rings in her jewelry box. Join Over 30,000 Students Excelling in Math! Endless Practice, Expert Guidance - Elevate Your Math Skills Today Related Subjects Scale Factors, Ratio and Proportional Reasoning Ratio Equivalent Ratios Division in a given ratio Direct Proportion Inverse Proportion Scale How to Calculate Percentage Estimation Relative frequency Statistical frequency Data Collection and Organization - Statistical Research Key Metrics in Statistics Median Mode in Statistics Average Probability frequency Probability Representation on a Number Line Probabilities of outcomes and events Relative Frequency in Probability Probability Properties Percentages How do you calculate the percentage value? What is a percentage? Converting between fractions and percentages and vice versa
3880	https://www.chegg.com/homework-help/questions-and-answers/consider-following-linear-programming-problem-minimize-4a-5b-subject-1a-4b-21-2a-1b-7-3a-1-q97633529	Solved Consider the following linear programming problem: \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Business Operations Management Operations Management questions and answers Consider the following linear programming problem: Minimize 4A + 5B Subject to 1A + 4B 21 2A + 1B 7 3A + 1.5B s 21 -2A + 6B > 0 A, B 2 0 (a) Show the feasible region using the graphical solution approach What are the comer points? 6) Based on your response in part (a): What is the comer point that minimizes the objective function? What is the value of the Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: Consider the following linear programming problem: Minimize 4A + 5B Subject to 1A + 4B 21 2A + 1B 7 3A + 1.5B s 21 -2A + 6B > 0 A, B 2 0 (a) Show the feasible region using the graphical solution approach What are the comer points? 6) Based on your response in part (a): What is the comer point that minimizes the objective function? What is the value of the Show transcribed image text Here’s the best way to solve it.Solution Share Share Share done loading Copy link View the full answer Previous questionNext question Transcribed image text: Consider the following linear programming problem: Minimize 4A + 5B Subject to 1A + 4B 21 2A + 1B 7 3A + 1.5B s 21 -2A + 6B > 0 A, B 2 0 (a) Show the feasible region using the graphical solution approach What are the comer points? 6) Based on your response in part (a): What is the comer point that minimizes the objective function? What is the value of the objective function at this comer point? © Which constraints are binding? Which constraints are non-binding? Explain. d) Suppose the objective function is changed to Maximize 7A + 3B, while keeping everything else intact. What is the comer point that maximizes the new objective function? What is the value of the objective function at this comer point? e) Taking into consideration the new objective function introduced in part (e), which constraints are binding? Which constraints are non-binding? Explain Not the question you’re looking for? Post any question and get expert help quickly. Start learning Chegg Products & Services Chegg Study Help Citation Generator Grammar Checker Math Solver Mobile Apps Plagiarism Checker Chegg Perks Company Company About Chegg Chegg For Good Advertise with us Investor Relations Jobs Join Our Affiliate Program Media Center Chegg Network Chegg Network Busuu Citation Machine EasyBib Mathway Customer Service Customer Service Give Us Feedback Customer Service Manage Subscription Educators Educators Academic Integrity Honor Shield Institute of Digital Learning © 2003-2025 Chegg Inc. All rights reserved. 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3881	https://www.britannica.com/science/nervous-system/Annelids	SUBSCRIBE SUBSCRIBE Home History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Games & Quizzes Videos On This Day One Good Fact Dictionary New Articles History & Society Lifestyles & Social Issues Philosophy & Religion Politics, Law & Government World History Science & Tech Health & Medicine Science Technology Biographies Browse Biographies Animals & Nature Birds, Reptiles & Other Vertebrates Bugs, Mollusks & Other Invertebrates Environment Fossils & Geologic Time Mammals Plants Geography & Travel Geography & Travel Arts & Culture Entertainment & Pop Culture Literature Sports & Recreation Visual Arts Image Galleries Podcasts Summaries Top Questions Britannica Kids Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos nervous system Form and function of nervous systems Stimulus-response coordination Intracellular systems Organelle systems Nervous systems The nerve cell The neuron Soma Plasma membrane Nucleus Organelles Axon Dendrites The neuroglia Types of neuroglia Neuroglial functions Transmission of information in the nervous system The ionic basis of electrical signals Diffusion of ions across a membrane Uncharged molecules Water Ions The neuronal membrane Transmission in the neuron Resting potential Localized potential Action potential Depolarization Repolarization Conduction Transmission at the synapse Electrical transmission Chemical transmission Neurotransmitter release Postsynaptic potential Inactivation Ion transport Active transport: the sodium-potassium pump Passive transport: membrane channels Sodium channels Potassium channels Calcium channels Anion channels Neurotransmitters and neuromodulators Acetylcholine Epinephrine and norepinephrine Dopamine Serotonin (5-hydroxytryptamine) Amino acids Neuroactive peptides Evolution and development of the nervous system Diffuse nervous systems Centralized nervous systems Simple bilateral systems Moderately cephalized systems Annelids Simple mollusks Complexly compartmentalized systems Complex mollusks Arthropods The vertebrate system The primitive condition Encephalization Dominance of the cerebrum References & Edit History Related Topics Images & Videos For Students nervous system summary Quizzes Human Organs Annelids in nervous system in Evolution and development of the nervous system print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. 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External Websites San Diego Miramar College - Nervous System (PDF) National Center for Biotechnology Information - PubMed Central - Nervous System Verywell Health - What You Need to Know About the Nervous System Cleveland Clinic - Nervous System CORE - Review Information and Efficiency in the Nervous System — A Synthesis (PDF) Academia - Nervous System Overview(PDF) The University of Hawaiʻi Pressbooks - Anatomy and Physiology - Basic Structure and Function of the Nervous System University of Washington - Neuroscience For Kids - Divisions of the Nervous System Healthdirect - Nervous system Biology LibreTexts - The Nervous System OpenStax - Microbiology - Anatomy of the Nervous System National Cancer Institute - SEER Training Modules - Organization of the Nervous System Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. nervous system - Children's Encyclopedia (Ages 8-11) nervous system - Student Encyclopedia (Ages 11 and up) Written by Written by Thomas L. Lentz Professor of Cell Biology, School of Medicine, Yale University, New Haven, Connecticut. Author of Primitive Nervous Systems. Thomas L. Lentz , Solomon D. Erulkar Professor of Pharmacology, University of Pennsylvania, Philadelphia, 1967–93. Solomon D. Erulkar •All Fact-checked by Fact-checked by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Last Updated: •Article History The brain of most annelids (phylum Annelida; segmented worms, including the leeches and terrestrial earthworms) is relatively simple in structure. The earthworm brain is a bilobed mass lying above the pharynx in the third body segment. Sensory nerves leave the brain and run forward into the prostomium (extreme anterior end) and first segment. The brain of the active, predatory polychaetes (a class of marine worms) is more complicated. In some the brain can be divided into a forebrain, a midbrain, and a hindbrain; a single pair of circumesophageal or circumpharyngeal connectives leave the brain, surround the anterior gut, and connect with the ventral nerve cord. The most primitive annelids have a pair of ventral nerve cords joined by transverse connectives; the most advanced forms have the cords fused to form a single cord. A ganglionic swelling of the cord is found in each body segment, with the most anterior ganglion, the subpharyngeal ganglion, being the most prominent. Two to five pairs of lateral nerves leave each ganglion to innervate the body wall of that segment. A subepidermal nerve plexus occurs over the whole body. Another plexus, called the enteric, stomodaeal, or sympathetic system, is found in the wall of the gut. Giant axons, usually few in number, travel the length of the cord. They may belong to one cell or be composed of many neurons. These axons are capable of very rapid conduction of impulses to the segmental muscles; their main function is to permit the worm to contract very rapidly as a defense against predators. The usual slow crawling movements of worms are mediated by a series of reflex arcs. During crawling, the contraction of muscles in one segment stimulates stretch receptors in the muscle. Impulses are carried over sensory nerves to the cord, causing motor neurons to send impulses to the longitudinal muscles, which then contract. The longitudinal pull activates stretch receptors in the following segment, and a wave of contraction moves along the worm. Studies of the nervous systems of annelids show certain behavioral capabilities, including perception, motor coordination, and learning. Because the neuronal organization behind these capabilities can be deduced, they may give an indication of the mechanisms underlying similar patterns of activity and behaviours at other levels of the phylogenetic scale. Two rhythmic movements generated by the leech, the heartbeat and swimming rhythm, have been extensively studied. The coordinated heartbeat rhythm is produced by heart excitor motor neurons, which show rhythmic activity in which bursts of action potentials alternate with bursts of inhibitory synaptic potentials derived from rhythmically firing inhibitory interneurons. The heartbeat appears to be produced by a central rhythm generator. The swimming movement, on the other hand, is generated by a neuronal network requiring many more cells. These neuronal oscillators may form the basis for neuronal generators of rhythmic movements in other animals at higher levels of the phylogenetic scale. Simple mollusks The nervous systems of the more primitive mollusks (snails, slugs, and bivalves, such as clams and mussels) conform to the basic annelid plan but are modified to conform with the unusual anatomy of these animals. In snails a pair of cerebral ganglia constitutes the brain, which overlies the esophagus. Nerves leave the brain anteriorly to supply the eyes, tentacles, and a pair of buccal ganglia. These last ganglia, also called the stomatogastric head ganglia, innervate the pharynx, salivary glands, and a plexus on the esophagus and stomach. Other nerve cords—the pedal cords—leave the cerebral ganglia ventrally and terminate in a pair of pedal ganglia, which innervate the foot muscles. Another pair of nerve cords—the visceral cords—leave the brain and run posteriorly to the visceral ganglia. The pleural ganglion, supplying the mantle, or fleshy lining of the shell, and the parietal ganglion, innervating the lateral body wall and mantle, are located along the visceral nerves. Intestinal ganglia connected with the pleural ganglia innervate the gills, osphradium (a chemical sense organ), and mantle. Sense organs of snails include eyes, tentacles, statocysts, and osphradia. In the bivalves, a cerebropleural ganglion is situated on either side of the esophagus. An upper pair of nerve cords leaves these ganglia and runs posteriorly to the visceroparietal, or visceral, ganglia. The visceral ganglia supply the mantle, adductor muscles (which close the shell), and internal organs. A second pair of nerve cords travels ventrally to the pedal ganglia. Most of the sense organs are found at the edge of the mantle. In the scallop, for example, the eyes are set in a row. They are well developed and consist of a cornea, a lens, and a retina, in which the photoreceptor cells are not placed superficially (an arrangement much like that in the vertebrate retina). Elementary forms of learning and memory have been studied at a cellular level by analysis of the neuronal activity of the marine snail (Aplysia californica). This simple mollusk withdraws its gill and siphon in response to a mild tactile stimulus. The neural circuit for this reflex consists of a sensory component from the siphon that forms single-synapse junctions with motor neurons that cause the gill to withdraw. The sensory cells also project onto interneurons whose outputs converge onto the same motor neurons. In response to a stimulus, the sensory neurons generate large excitatory postsynaptic potentials at both interneurons and motor neurons, causing the generation of action potentials in the motor neurons that in turn cause the gill to withdraw. When the stimulus is repeated many times, the postsynaptic potentials become reduced in size and the response becomes weaker. Finally, the postsynaptic potentials become so small that action potentials are no longer generated and the gill no longer responds. This reduced behavioral response is known as habituation. Habituation may be caused by the closing of calcium channels, which decreases calcium influx into the presynaptic terminals and, therefore, decreases neurotransmitter release. Other evidence suggests that habituation results from fewer neurons in the network being activated. Another behavioral paradigm, sensitization, has also been examined in Aplysia. In sensitization the reflex activity increases in strength with added stimulation. The mechanism underlying this response is presynaptic facilitation, which is thought to be caused by an increase in the second messenger cAMP in the terminals of the sensory neurons. These two examples—habituation and sensitization—show that important features of a more complex nervous systems can be studied in organisms at lower stages of evolution. First is what can be called the plasticity of the nervous system, the phenomenon of changes occurring in the strength of synaptic responses. Changes in synaptic efficacy may underlie certain mechanisms for short- and long-term memory—even in more complex animals such as humans. Changes in the structure of the synapse may be a long-term effect of plasticity. For example, the numbers of active zones at nerve terminals are reduced with long-term habituation but increased with long-term sensitization. Finally, the molecular mechanisms underlying these changes may be the same or at least similar at all levels of the phylogenetic tree. Habituation of the escape response has been seen in polychaete worms, cockroaches, and crayfish. Complexly compartmentalized systems The highest degree of development of the invertebrate nervous system is attained by the cephalopods (squids, cuttlefishes, and octopuses) among the mollusks and by the insects and spiders among the arthropods. Although the basic plan of these nervous systems is similar to that of the annelids, there are several advances. First, there is a high degree of cephalization, with nervous functions concentrated in the head region of the animal. In addition, ganglia are fused and farther forward, and nerve cells, less abundant in the peripheral nervous system, are situated in the brain or ganglia so that the nerve cords consist only of nerve fibres. Finally, control and coordination of specific functions, such as locomotion and feeding, are compartmentalized in particular parts of the nervous system. Complex mollusks The complex nervous system of the cephalopods is correlated with the active movement and predatory habits of these organisms. Most of the ganglia typical of mollusks are concentrated or fused in a brain that encircles the esophagus. Nerves extend from the brain to ganglia at the base of the arms or tentacles and from the ganglia the length of the arms. A pair of large pallial nerves connects the brain with a pair of stellate ganglia on the inner surface of the mantle. The stomatogastric ganglia supply nerves to the digestive tract. A great variety of functions are centralized in the brain and compartmentalized to specific brain regions. These activities may be local, simple, and uncoordinated with other regions or may be extensive, complex, and coordinated, involving large groups of muscles. The highest centres of the cephalopod brain are the associative areas, which are thought to be involved with discrimination between objects, learning, and memory. The giant-fibre system—also seen in earthworms and insects—is very well developed in the squid. The diameter of giant fibres is many times greater than the diameter of most other nerve fibres. Giant neurons in the brain send fibres to the retractor muscles of the head and the funnel or to the stellate ganglion. Fibres from the stellate ganglion fuse to form giant fibres that innervate the mantle. Because of their large size, these fibres are capable of rapid conduction, which, in turn, permits extremely rapid movement. The eyes of cephalopods are especially well developed and bear close resemblance to the vertebrate eye. The eye fits into a socket of cartilaginous plates separate from the cartilages that protect the brain, and external muscles permit its movement. A transparent cornea covers the surface and can be focused for both near and far objects. There is a pupil formed by an iris diaphragm, which can regulate the amount of light reaching the retina. The retina contains light-sensitive cells. The axons of the photoreceptors, or rod cells, form the optic nerves, which terminate in the extremely large optic lobes of the brain. The cephalopods are strikingly different in many respects from other molluscan classes. The nervous system as described above is more highly developed and, consequently, the behavioral repertoire much more complex. First, the animals are predators; they move, they use their eyes in search of food, they use receptors in their arms for detection of tactile or chemical stimuli, and they have exceptionally fast muscle action. Second, they have an enormous flexibility of response, discriminating between palatable and unpalatable prey and “learning” to attack or not to attack. They can also change colour to blend into their environment if needed. The mollusks as a whole provide an important link in the developing complexity of the nervous system. Indeed, the presence in their systems of vertebrate as well as natural molluscan neuroactive peptides may give some clue to the true place of these animals in the phylogenetic scale.
3882	https://pmc.ncbi.nlm.nih.gov/articles/PMC6462188/	Exocrine Pancreatic Insufficiency and Malnutrition in Chronic Pancreatitis: Identification, Treatment, and Consequences - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice Pancreas . Author manuscript; available in PMC: 2019 Sep 1. Published in final edited form as: Pancreas. 2018 Sep;47(8):1015–1018. doi: 10.1097/MPA.0000000000001137 Search in PMC Search in PubMed View in NLM Catalog Add to search Exocrine Pancreatic Insufficiency and Malnutrition in Chronic Pancreatitis Identification, Treatment, and Consequences May Min May Min, MD Department oflnternal Medicine, University ofMassachusetts Medical School, Worcester, MA Find articles by May Min , Boskey Patel Boskey Patel, DO Department oflnternal Medicine, University ofMassachusetts Medical School, Worcester, MA Find articles by Boskey Patel , Samuel Han Samuel Han, MD † Division of Gastroenterology, UC Denver School of Medicine, Denver, CO Find articles by Samuel Han †, Lisa Bocelli Lisa Bocelli, DO ‡ Division of Gastroenterology, University of Massachusetts Medical School, Worcester, MA. Find articles by Lisa Bocelli ‡, Joan Kheder Joan Kheder, MD ‡ Division of Gastroenterology, University of Massachusetts Medical School, Worcester, MA. Find articles by Joan Kheder ‡, Aditya Vaze Aditya Vaze, MD § Department oflnternal Medicine, University of Massachusetts Medical School, Worcester, MA. Find articles by Aditya Vaze §, Wahid Wassef Wahid Wassef, MD ‡ Division of Gastroenterology, University of Massachusetts Medical School, Worcester, MA. Find articles by Wahid Wassef ‡ Author information Copyright and License information Department oflnternal Medicine, University ofMassachusetts Medical School, Worcester, MA † Division of Gastroenterology, UC Denver School of Medicine, Denver, CO ‡ Division of Gastroenterology, University of Massachusetts Medical School, Worcester, MA. § Department oflnternal Medicine, University of Massachusetts Medical School, Worcester, MA. ✉ Address correspondence to: May Min, MD, Department oflnternal Medicine, University ofMassachusetts Medical School, Worcester MA, 55 Lake Ave N, Worcester, MA 01655 (may.min@umassmemorial.org). PMC Copyright notice PMCID: PMC6462188 NIHMSID: NIHMS1017796 PMID: 30074926 The publisher's version of this article is available at Pancreas Abstract Objectives: The purpose of this study was to examine the impact of exocrine pancreatic insufficiency (EPI) on chronic pancreatitis (CP) patients and to identify challenges with its diagnosis and treatment. Methods: Ninety-one patients with CP diagnosed with endoscopic ultrasound were identified and assessed for symptoms of EPI, fat-soluble vitamin levels, dual-energy x-ray absorptiometry scan T-scores, and treatment with pancreatic enzyme replacement therapy. All patients were also screened with the Malnutrition Universal Screening Test. Results: Exocrine pancreatic insufficiency was diagnosed in 84.6% (77/91) of patients based on symptoms of bloating, steatorrhea, or weight loss. Of these patients, 35.2% (19/54) had vitamin A deficiency, 62.5% (55/88) had vitamin D deficiency, and 17.7% (9/51) had vitamin E deficiency. Either osteopenia or osteoporosis was found in 68.9% (3¼5). A medium or higher risk for malnutrition based on Malnutrition Universal Screening Test score of 1 or higher was found in 31.5% (28/89). Malnutrition Universal Screening Test score of 1 or higher was associated with an increased risk for osteopenia and osteoporosis on Fisher’s exact test (P = 0.0037). Conclusions: There is a high prevalence offat-soluble vitamin deficiencies, osteopathy, and malnutrition in CP patients, which is underestimated due to a lack of effective diagnosis and suboptimal therapies for EPI. Keywords: chronic pancreatitis, exocrine pancreatic insufficiency, malnutrition Exocrine pancreatic insufficiency (EPI) is a potential complication of chronic pancreatitis (CP) characterized by decreased production of pancreatic digestive enzymes. Exocrine pancreatic insufficiency can lead to maldigestion, malabsorption, and ultimately malnutrition. It is often underrecognized due to challenges with its diagnosis and the difficulty of identifying high-risk patients. Even when recognized, optimal management with pancreatic enzyme replacement therapy (PERT) is not always achieved.1 Direct pancreatic function testing is currently the most sensitive and specific modality for assessing EPI.2 The endoscopic secretin stimulation test (ePFT) involves serial measurement of duodenal bicarbonate concentration after administration of secretin. Endoscopic secretin stimulation test is becoming increasingly routine in the diagnostic workup of CP; however, it is not yet available at all medical centers. Indirect pancreatic function tests include quantitative and qualitative fecal fat assay, fecal elastase, mixed triglyceride breath testing, and spot steatocrit. Because of their variable sensitivity, specificity, safety, and availability, providers often forego indirect pancreatic function testing and diagnose EPI based on patient report of symptoms such as steat-orrhea and bloating.3 As expected, patients with EPI are known to have a higher prevalence of fat-soluble vitamin deficiencies due to fat malabsorption.4 Of particular clinical significance is the increased risk for vitamin D deficiency in EPI, as this places patients with CP at higher risk for metabolic bone diseases including osteopenia and osteoporosis.5–7 Other contributing factors for malnutrition in CP include decreased dietary intake secondary to pain and alcohol abuse. Despite the high risk for malnutrition in the CP population, few studies have examined the nutritional status in patients with CP Verhaegh et al8 evaluated the use of 3 nutritional surveys in CP patients and found that none of these surveys were sensitive enough to identify all patients with impaired nutrition. In our study, we examined the impact of exocrine pancreatic insufficiency on CP patients and the challenges with its diagnosis and treatment. We also investigated the use of a nutrition screening tool in identifying CP patients who are at higher risk for malnutrition and fat-soluble vitamin deficiency. MATERIALS AND METHODS Study Design and Patient Population We enrolled patients between January 2014 and December 2016 for our Chronic Pancreatitis Database through the CP clinic at the University ofMassachusetts Medical Center, a tertiary care academic hospital in Worcester, Mass. Only those patients who were previously diagnosed with CP by endoscopic ultrasound (EUS) criteria and/or secretin stimulation testing were included in this study. Secretin stimulation test was considered to be positive if bicarbonate concentration was less than 75 mEq/L in all 3 aliquots collected over 45 minutes. Patients who did not have at least one prior EUS for evaluation of CP were excluded. All patients were given surveys assessing for active tobacco abuse, alcohol use, and symptoms of EPI (bloating, steatorrhea, and weight loss). Patients were also given the Malnutrition Universal Screening Test (MUST) during their clinic visits. Additional data including qualitative fecal fat assays, fat- soluble vitamin levels, albumin levels, dual-energy x-ray absorptiometry (DEXA) scan T-scores, TIGAR-O classification, PERT dosing, and CP severity based on EUS were obtained through retrospective chart review.9 Vitamin D was measured as total serum 25-hydroxyvitamin D level. Severity of CP was determined to be mild, mild to moderate, moderate, moderate to severe, or severe based on conventional EUS criteria.10 Exocrine pancreatic insufficiency was diagnosed based on symptoms including steat-orrhea, diarrhea, bloating, and weight loss and requirement for treatment with PERT on retrospective chart review. Average bicarbonate levels were calculated for each patient with available ePFT. Dual-energy x-ray absorptiometry scans results were expressed as T-scores compared with values in young women. Based on the World Health Organization’s classification, patients with T-scores between –1.0 and – 2.5 standard deviations (SDs) were diagnosed with osteopenia and those with T-scores below –2.5 were diagnosed with osteoporosis. Malnutrition Screening All patients were screened with the Malnutrition Universal Screening Test (MUST), which is a validated 5-step screening tool developed by the British Association of Parenteral and Enteral Nutrition to identify adults who are malnourished or at risk for malnutrition.11,12 Body mass index (BMI) was calculated by using the formula weight (kilogram)/height (square meter). Percent of weight loss was calculated based on the highest and lowest weight (kilogram) measured in clinic over a 6-month period. Patients were also asked in a survey if they have experienced unintentional weight loss over the past 6 months. Recent acute illness or absence of nutritional intake for more than 5 days was assessed by asking the patient in the survey. A MUST score of 0 was considered low risk for malnutrition, score of 1 was medium risk, and scores 2 or higher were high risk. Statistical Analysis Data were recorded in REDCap (Vanderbilt University, Nashville, Tenn), which is a secure, Web-based application for managing online databases. Continuous variables were expressed as a mean (SD). Categorical variables were expressed as a percentage (ratio). Fisher’s exact test was used for assessing associations between categorical variables. All Fisher tests were 1-tailed, and cutoff for significance was set at a P value of less than 0.05. Pearson’s correlation coefficient was computed to evaluate the relationship between variables on linear regression. Linear regression was performed to evaluate the correlation between MUST scores and severity of CP based on EUS and fat-soluble vitamin levels. Fisher exact test was used to evaluate the relationship between MUST scores and osteopathy (osteopenia or osteoporosis). RESULTS Patient Demographics We enrolled a total of 91 patients with CP (see Table 1). The mean (SD) age in years was 48.6 (10.4). A total of 62.6% (57/91) of our patients were women and 37.4% (34/91) were men. The mean (SD) duration of CP in years was 4.3 (4.2) and was calculated based on year of EUS diagnosis. The most common etiology of CP based on the TIGAR-O classification was toxic-metabolic, which included 59.3% (54/91) of our patients. A total of 68.1% (62/91) of our patients reported a history of tobacco use; however, 8 of these patients had another alternative etiology for CP based on TIGAR-O. In addition, 18.6% (17/91) of our patients had no identifiable etiology of CP, 14.3% (13/91) were attributed to genetic mutations, 5.5% (5/91) due to autoimmune disease, and 2.2% (2/91) due to obstruction. TABLE 1. Baseline Patient Characteristics and Etiology of Chronic Pancreatitis as Toxic/Metabolic, Idiopathic, Genetic, Autoimmune, Recurrent Acute Pancreatitis, or Obstructive (TIGAR-O Classification) \| \| All Patients (n = 91) \| :---: \| \| \| \| Sex, n (%) \| \| \| Female \| 57 (62.6) \| \| Male \| 34 (37.4) \| \| Age, mean (SD), y \| 48.6 (10.4) \| \| Smokers, n (%) \| 62 (68.1) \| \| TIGAR-O Classification, n (%) \| \| \| Toxic/metabolic \| 54 (59.3) \| \| Idiopathic \| 17 (18.7) \| \| Genetic \| 13 (14.3) \| \| Autoimmune \| 5 (5.8) \| \| Recurrent AP \| 0 (0.0) \| \| Obstructive \| 2 (2.2) \| \| Duration of CP, mean (SD), y \| 4.3 (4.3) \| Open in a new tab Evaluation and Treatment of EPI A total of 84.6% (77/91) of patients reported symptoms of EPI including steatorrhea, diarrhea, bloating, and weight loss. Ten patients had available qualitative fecal fat assays, and 6 of these patients (60.0%) had positive tests. Fourteen patients had available secretin stimulation tests, of which 9 (64.3%) had a positive test. All of the 77 patients who reported EPI symptoms were eventually started on PERT. Three of the patients with positive secretin tests (33.3%) did not report symptoms of EPI and were not treated with PERT. Only one of the patients with positive fecal fat assays (16.7%) did not have symptoms of EPI and was not treated with PERT. Patients treated with PERT were receiving a median of 96,000 units of lipase with each meal. Eight patients (10.3%) reported having continued steatorrhea despite taking PERT. Prevalence of Fat-Soluble Vitamin Deficiency A large proportion of patients were found to have fat-soluble vitamin deficiencies, including 35.2% (19/54) with vitamin A deficiency, 62.5% (55/88) with vitamin D deficiency, and 17.7% (9/51) with vitamin E deficiency. None of our patients had available vitamin K levels. Women had higher rates of vitamin A (38.7% vs 30.0%), vitamin D (67.3% vs 54.5%), and vitamin E (22.2% vs 12.5%) deficiency when compared with men. Smokers had higher rates of vitamin D (67.4% vs 57.1%) and vitamin E (21.4% vs 13.0%) deficiency when compared with nonsmokers. There was no significant correlation between average or peak bicarbonate levels on ePFTs and any of the vitamin levels. Interestingly, 84.2% (16/19) of patients with vitamin A deficiency, 89.1% (49/55) with vitamin D deficiency, and 77.8% (7/9) with vitamin E deficiency were already being treated for EPI with PERT. Prevalence of Osteopathy Forty-five patients had available DEXA scans within the past 5 years. A total of 46.7% (2¼5) of patients were found to have osteopenia, 22.2% (10/45) of patients were diagnosed with osteoporosis, and 68.9% (3¼5) had either osteopenia or osteoporosis. The mean (SD) T-score was found to be –1.6 (1.3). Women had a slightly higher rate of osteopathy at 69.7% versus 66.7% in men. Smokers had a significantly higher rate of osteopathy at 63.0% versus 22.2% in nonsmokers. There was a positive correlation between average bicarbonate on ePFT and T-scores (r = 0.8012). There was no significant association between treatment with PERT and incidence of osteopathy on Fisher 1-tailed t-test. A total of 80.6% (25/31) of patients with osteopathy were already on PERT. Severity of Chronic Pancreatitis The pooled mean (SD) of average bicarbonate levels in mEq/L obtained during secretin stimulation testing was 59.5 (16.3). Based on EUS criteria, 64.0% (55/86) of our patients were classified as having at least moderate severity of CP. A total of 17.4% (15/86) had severe CP A total of 25.6% (22/86) had mild CP and 10.5% (9/86) had mild to moderate CP. There was no significant correlation between CP severity on EUS and vitamin A, D, or E levels on regression analysis. There was also no significant correlation between CP severity by EUS criteria and DEXA scan T-scores on regression analysis. Malnutrition Screening The mean (SD) BMI was 26.1 (7.8) kg/m 2, and 12.1% (11/91) patients had a BMI of 18.5 kg/m 2 or less. The mean (SD) albumin level was 3.9 (0.67) g/dL. There was no significant correlation between either BMI or albumin and vitamin A, D, E levels, or DEXA scan T-scores on multivariate regression analysis. A total of 31.5% (28/89) of patients were found to be at medium or higher risk for malnutrition based on the MUST score of 1 or higher. A higher proportion of patients with MUST score of 1 or higher had vitamin D deficiency 67.9% (19/28) when compared with 57.4% (35/61) in patients with MUST score lower than 1. A MUST score of 1 or higher was associated with an increased risk for osteopenia and osteoporosis on Fisher’s exact test (P = 0.0037). There was no statistically significant correlation between MUST score and severity of CP based on EUS criteria or fat-soluble vitamin levels on regression analysis. DISCUSSION Exocrine pancreatic insufficiency is an underrecognized complication of CP that is associated with fat-soluble vitamin deficiencies, osteopathy, and malnutrition. We noted a high prevalence of fat-soluble vitamin deficiencies in our population particularly in women and smokers. Vitamin D was the most commonly deficient fat-soluble vitamin affecting 62.5% of our patients, which is similar to previous studies reporting vitamin D deficiency in 40% to 66% of CP patients.6,7,13 Relatively fewer patients had vitamin A deficiency (35.2%) and vitamin E deficiency (17.7%), which is contrary to prior studies, which had found that vitamin E deficiency was more common than deficiencies of vitamin A or D.14,15 The high prevalence of vitamin D deficiency in our study is significant primarily because of its association with osteopathy.16,17 A total of 68.9% of our patients had either osteopenia or osteoporosis, which is comparable to a previously reported pooled prevalence rate of 65% in CP patients.18 Despite the high prevalence of nutritional deficiencies and osteopathy in CP, identification of EPI has remained somewhat elusive due to the lack of an effective diagnostic test.19 Many providers rely on patient reporting of the symptoms of EPI including steatorrhea, weight loss, and bloating to decide when to initiate and change dosing of PERT. In our study, a large proportion (84.6%) of patients reported EPI symptoms such as steatorrhea. However, DiMagno et al showed that steatorrhea is not observed until lipase output is less than 10% of normal, therefore suggesting that it is a late manifestation of EPI.13 Furthermore, Midha et al found that 36% of their patients had fecal fat measurements greater than 7 g, whereas only 5% of patients reported clinical steatorrhea. This suggests that the mean (SD) duration of CP in our study of 4.3 (4.2) years based on year of EUS diagnosis likely far underestimates disease duration given that the pathophysiologic changes of CP are thought to far precede the clinical symptoms, which prompt EUS evaluation.20,21 This is one of the limitations of our study, and further studies are needed to identify methods for early identification and diagnosis of patients with CP. One method of diagnosing EPI is direct pancreatic function testing through ePFT. This method is currently considered the most sensitive test for diagnosis of early CP and EPI.2 Although ePFT is being increasingly utilized, it has not yet been widely adopted as a routine part of EPI diagnosis and has not been validated for predicting malnutrition. Other indirect tests for EPI include quantitative fecal fat assays, fecal elastase, serum trypsin, and mixed triglyceride breath tests. Quantitative fecal fat assays are thought to be both sensitive and specific for EPI; however, these tests are rarely used in the clinical setting because they require strict dietary changes and stool monitoring for 72 hours.22 One study showed that fecal elastase has high sensitivity of 100% for moderate and severe EPI, but only 63% sensitivity in detecting mild EPI and therefore may have more limited yield in detecting early EPI.23 Similarly, serum trypsin is only thought to be sensitive for detecting more advanced EPI and has limited specificity as levels may be elevated in other conditions such as acute pancreatitis.24 Finally, mixed triglyceride breath testing is another indirect test for EPI; however, it is not widely available in the United States and therefore has not been widely utilized in the clinical setting. All of the indirect tests have relatively lower sensitivity for detecting EPI compared with secretin testing, particularly in early CP.2 Given the limitations of direct and indirect testing for EPI, only a small proportion of patients were diagnosed with EPI through objective measures in our study. Most of our patients were diagnosed with EPI based on their need for PERT to treat symptoms such as steatorrhea, weight loss, and bloating. Of those patients who had available ePFT in our study, there was no correlation between peak/average bicarbonate concentrations and malnutrition severity based on MUST or fat-soluble vitamin levels. This may be in part due to the relatively small number of available ePFT in our study. Further studies are needed to evaluate the use of ePFT in diagnosing early EPI and predicting the nutritional sequelae of EPI. An alternative approach to assessing the risk for malnutrition in CP patients would be to implement a malnutrition screening methodology such as the MUST. The MUST is a validated tool to screen for patients at risk for malnutrition as well as to identify patients who are already malnourished. In our study, we found that the MUST had unclear significance in predicting those at risk for malnutrition, which is similar to findings from Verhaegh et al.8 The MUST scores were not predictive offat-soluble vitamin deficiencies, and there was no correlation between MUST scores and bicarbonate levels on ePFT. Interestingly, MUST score of 1 or higher (medium-high risk of malnutrition) was associated with an increased risk for osteopenia and osteoporosis in our study. This suggests a potential role for this simple screening tool in identifying patients who should be evaluated for osteopathy with DEXA scans. The challenges we face with diagnosing EPI and malnutrition are reflected in how we treat EPI in CP patients. Currently, PERT is the cornerstone of EPI management and is usually titrated until steatorrhea is reduced or eliminated. This method is limited as it relies on patient reporting and is not necessarily correlated with improved nutritional status. Prior studies have suggested that patients with EPI are often undertreated with PERT, which is thought to be due to challenges with PERT titration and patient noncompliance.25,26 Only 10% of normal pancreatic lipase replacement (ie, 70,000–100,000 units per meal) is thought to be required to correct for steatorrhea and malnutrition due to EPI.27 The median lipase dose in our study was 96,000 units per meal, which is well within current guidelines for appropriate PERT dosing. Striking in our study was that despite already being on therapeutic doses of PERT, 84.2% of our patients were vitamin A deficient, 89.1% were vitamin D deficient, and 77.8% were vitamin E deficient. A high proportion (80.6%) of patients who had osteopathy were already on PERT. These findings suggest that CP patients are either being undertreated for PERT, noncompliant with PERT, or that PERT alone is not sufficient to completely reverse fat-soluble vitamin deficiencies and metabolic bone disease in EPI. In conclusion, our study shows that there is a high prevalence offat-soluble vitamin deficiencies, osteopathy, and malnutrition in CP patients. This highlights the importance of early identification and management of EPI. Although assessing symptoms such as steatorrhea, bloating, and weight loss can be useful in identifying EPI, it can overlook those patients with milder forms of EPI who may still experience vitamin deficiencies and be at risk for malnutrition. The lack of a widely used, validated method for early diagnosis of EPI and suboptimal utilization of available therapies limits our ability to provide optimal care for our CP patients. Therefore, we recommend the following: (1) further studies to validate objective measures of EPI such as ePFT for early identification and monitoring malnutrition in the CP population, (2) evaluation of all patients with CP for fat-soluble vitamin deficiencies and osteopathy, and (3) consideration of PERT in all CP patients with vitamin deficiencies or osteopathy even in the absence of EPI symptoms. Footnotes The authors declare no conflict of interest. REFERENCES 1.Hart PA, Conwell DL. Challenges and updates in the management of exocrine pancreatic insufficiency. Pancreas. 2016;45:1–4. [DOI] [PubMed] [Google Scholar] 2.Ketwaroo G, Brown A, Young B, et al. Defining the accuracy of secretin pancreatic function testing in patients with suspected early chronic pancreatitis. 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3883	https://www.nngroup.com/articles/10-survey-challenges/	Skip to content 14 10 Survey Challenges and How to Avoid Them Tanner Kohler Tanner Kohler February 26, 2023 2023-02-26 Summary: Response biases make it difficult to create good surveys. Follow these tips to counteract 10 of the major survey response biases and improve your survey data. How recently have you been invited to take some kind of survey? It likely hasn’t been more than a couple of days. Our research shows that surveys are among the most commonly used research methods by UX practitioners. Surveys have their place among the various quantitative UX-research methods, but it is more challenging to create a good survey than many professionals think. It is very easy to write a bad survey that gathers flawed data. In This Article: Limitations of Surveys Limitations of Surveys 1. Recall Bias 2. Recency Bias 3. Social-Desirability Bias 4. Prestige Bias 5. Acquiescence Bias 6. Order Effect 7. Current-Mood or Emotional-State Bias 8. Central-Tendency Bias 9. Demand Characteristics 10. Random-Response Bias Conclusion Limitations of Surveys Survey Participants Can Filter Responses Unlike observational methods, which reveal real-time behaviors that cannot be easily faked or filtered (such as the ease or difficulty of navigating a new mobile app), surveys are self-reported – which means that respondents get to filter everything they share with researchers before they share it. This is a major limitation of surveys, no matter how well they are written — because researchers are generally interested in phenomena as they really occur, not in (intentionally or unintentionally) censored information. Other self-report research methods where users can decide what they share include interviews, focus groups, and diary studies. Researchers Cannot Probe in Surveys Another limitation of surveys is that researchers cannot probe to better understand responses. This is because respondents usually complete them without the researcher being present. Even if researchers could afford to watch every single respondent take a survey, doing so would bias the responses and contaminate the data. Moreover, survey responses generally come as ratings on a simple scale or as selections on multiple-choice questions. It is impossible to know why someone responded the way they did without more information (a limitation shared by analytics data). Even if there is space provided for a respondent to type out additional comments, these open-ended responses tend to be brief, disconnected, or skipped entirely. Surveys Only Collect Attitudinal Data Surveys collect attitudinal data, representing how users think and feel — not how they behave. They are no substitute for observational methods such as usability testing or analytics, that can accurately reveal user behaviors. Surveys remove respondents from workflows and require them to reflect on experiences and behaviors rather than demonstrate them. For example, even if a survey asks users how often they tend to access a particular feature on a website, it is really asking them how often they think they access that feature. Questions about behaviors do not accurately capture how people actually perform those behaviors; instead, they reveal users’ perceptions and recollections about those behaviors. Use surveys to learn what users think and feel, not what they do. Consider the following excerpt from a survey emailed to university students regarding a financial dashboard for monitoring and paying school-related expenses. The likely purpose of this survey is to gain insights about the usability of the dashboard. Unfortunately, respondents taking this survey are obviously not using the dashboard; moreover, depending on when they used the dashboard, their recollections about the UI will be incomplete, vague, or wrong. As a result, the survey responses cannot accurately reflect how “user friendly” the dashboard is. Research goals investigating usability, findability, discoverability, or other behavioral metrics must be studied with observational methods, such as user testing. Survey data can be used to complement performance-based data, but it cannot, by itself, offer a comprehensive assessment of the usability of a system. Although developing a good survey can take time and multiple iterations, it is quick and cheap compared to many other research methods — which is why surveys have become so popular. Surveys can be useful for gathering large amounts of both quantitative and qualitative data when researchers are investigating what users think or feel. However, to gather good data, a survey must be written well to be both internally and externally valid. If a survey is not written well, it will produce bad data. The following 10 points describe common ways in which research participants tend to misrepresent their true thoughts and feelings on surveys. While it is impossible to fully mitigate the effects of these tendencies, there is much that can be done to lessen their effects. 1. Recall Bias Although people tend to feel that they can accurately remember how things were in the past, they actually forget most of the details as time goes on, and their feelings about things change without them noticing. Imperfect memory presents a great challenge for any survey that focuses on a specific event or experience but is completed after the fact. Observational methods that capture insights as experiences occur do not suffer from recall bias in the same way that surveys do. People are most easily able to remember details about things that were recent, that they think about frequently, and that are associated with strong emotions. Preventive tip: Distribute a survey to participants as soon as possible following the event related to your research. For example, if your survey asks why a user subscribed to a particular service, have the subscription act as a trigger for the survey, while their rationale for subscribing is as fresh as possible. Don’t wait to send the survey to large groups of subscribers at once. (Still, even if you follow this tip, there is no guarantee that people will complete the survey right away.) Even better, consider using surveys in connection to an observational user-research method such as usability testing. For example, you could show a survey such as the SUS to users who have completed a test of a website. In this way, the experience would be fresh in their mind as they respond to the questions, and you will be able to use the survey-provided attitudinal data to complement other performance-based metrics such as success rates and task times. 2. Recency Bias Individuals tend to give more weight to recent events than to events from longer ago. When asked about an overall experience or opinion on a survey, it is likely that people will respond based on how they have been feeling lately rather than taking an accurate mental average of their feelings throughout time. Preventive tips: Ask about both a recent experience and a previous one. If the research question focuses on opinions and attitudes at multiple time points, then intentionally capture someone’s most recent feelings first, before prompting them to consider how those feelings have changed with time. Use a longitudinal method (such as a diary study) rather than a survey. If your research question investigates user attitudes or feelings at various time points, have participants respond at those points rather than relying on one survey at the end that will likely be skewed toward more recent events. 3. Social-Desirability Bias People are highly motivated to conform to societal norms and portray themselves in socially desirable ways. For this reason, participants might (both consciously or subconsciously) distort their answers to be closer to what they think is socially acceptable. For example, they might overemphasize how important environmental sustainability is in their lives because it is increasingly valued by society, or they might underemphasize how much they love a particular indulgence if it is looked down upon by society. Preventive tips: Emphasize confidentiality (the researcher knows their respondents’ identity but will not share it with others) or anonymity (not even the researcher knows the identity of respondents) of their answers. Reassuring participants that their responses will not be connected to their identity helps them feel comfortable with being candid and truthful. Use indirect questioning. This method can allow respondents to be more honest without feeling as strong of a need to conform to societal norms For example, a survey investigating feelings about a particular political candidate might also ask about candidates with similar political views to the candidate in question. Respondents could also be asked to describe how their close friends or colleagues feel about an issue. Open-ended responses are often necessary to capture relevant data in these cases but run the risk of not fully answering the research question. 4. Prestige Bias People do not like to be negatively portrayed. When given the opportunity, survey respondents tend to distort their responses to make themselves seem more impressive, smart, or successful. Common examples include the tendency to round up one’s income or downplay (or fully deny) negative actions such as violence or abuse. Because surveys are a self-report method, respondents will not always accurately represent their true opinions and actions. Preventive tips: When exact numbers are not necessary, provide ranges in response choices. Allowing someone to select a range that includes their true answer is more likely to be accurate than asking respondents to report it directly. For example, providing age or income ranges will avoid requiring respondents to directly input their age or salary and will be less likely to result in a distortion. In many cases, larger ranges increase the chances that someone will provide an honest response (e.g., 65–80 years old vs. 66–70, 71–75, 76–80 years old). Use other data sources for sensitive data that is critical to your research. Often, researchers find it easier to ask participants to report information about themselves as part of a survey than to collect this data from other available sources. However, sensitive data such as income, weight, or achievements will be more accurate when gathered from another source, if available. 5. Acquiescence Bias People tend to agree with statements more than disagree. For example, if asked whether they agree or disagree with the statement [Our company] provides high-quality products, a larger proportion of respondents is likely to select “agree” than “disagree,” no matter the company. The tendency to agree is often due to the natural desire to be nice to others; it can also serve as a shortcut and conserve mental energy. It can also be the case that a respondent has no strong reason to disagree and gives the benefit of the doubt, which will generally result in agreement. The acquiescence bias can lead researchers to reach falsely positive conclusions. Preventive tips: Ask a direct, open-ended question rather than asking for agreement or disagreement responses to a statement. People will be most likely to report their true feelings when there is no obvious way to simply “agree.” Use a semantic-differential scale rather than a Likert scale. A semantic-differential scale still asks participants to provide a rating but provides a continuum of response options specific to the nature of the question. There is generally no obvious “agreement” option available. Include reverse-keyed items. When respondents indicate whether they agree or disagree with multiple similar types of statements in a row, they are more likely to quickly select positive responses to all. Alternating the focus of statements or questions participants are responding to can serve as an indicator of whether the participant was intentionally reading and responding to each individual item. For example, instead of having participants indicate their agreement or disagreement with a series of positively framed statements (e.g., [This company] has my best interest in mind) all in succession, intermix them with some negatively framed statements (e.g., [This company] intentionally tries to deceive its customers). However, reverse-keying items should be done sparingly, only when there are many of the same response types in a row, as it increases the overall cognitive load required to respond and the probability that an error will be made when responding or coding the responses. 6. Order Effect The order in which response options are presented in a closed-ended survey question (such as multiple-choice or multi-select) affects which options are most likely to be chosen. The options near the beginning and end of a list tend to be the most likely to be chosen because of the primacy and recency effects. (The primacy effect describes the tendency to select the first seemingly satisfactory option a respondent sees. The recency effect describes how the last, or “most recently seen,” option in a list is freshest on the mind and thus, the most easily available for selection.) The order in which the questions in a survey are presented can also bias responses. Question order can unintentionally reveal the purpose of a survey, might create pressure for respondents to be consistent with themselves, or might cause end questions to be neglected due to survey fatigue. Preventive tips: Organize options meaningfully. When options can be ordered meaningfully (e.g., alphabetically, chronologically, locationally, temporally, or categorically), respondents are more likely to quickly identify the options that apply to them rather than simply picking those at the beginning and at the end. This is particularly true with long lists of options that are conventionally listed in a particular order. Randomize the order of response options. When there is no obvious meaningful order for response options, randomizing their order gives each option an equal chance of falling near the beginning or end of a list. For example, it would be appropriate to randomize the order of different color options, but not ranges of ages. Randomize questions if possible. If the order of your survey questions is not important, consider presenting them in various orders to different participants to avoid the same initial questions consistently influencing responses on later ones. You can also show questions on separate pages to avoid possible confounds. 7. Current-Mood or Emotional-State Bias A respondent’s current emotional state will affect how (and whether) they respond to a survey. If someone is feeling rushed, tired, or apathetic, they are unlikely to even begin a survey in the first place, let alone provide meaningful responses. Your first goal when deciding when to distribute a survey is to maximize the response rate. It is unwise to blast out a survey when the majority of potential respondents are likely to be busy (e.g., at the start of a workday), or at a point in the customer journey when someone is dealing with complex problems (conduct a diary study instead to study these complex parts of an experience). Beyond this initial challenge, as with all research studies, those who take a survey will provide answers based on their current mood. In some cases, the content of a survey itself might induce a particular mood (such as asking about a recent negative experience), while in others the content is neutral. Unlike observational methods, surveys cannot capture the mood of a respondent (even if you have a question about it!) — which prevents this information from being factored into the analysis and conclusions. This is one reason why survey data cannot be interpreted as a perfect reflection of reality. Calculating the statistical significance and confidence intervals for quantitative findings can help account for this variability. In cases where a survey is meant to capture overall impressions, it is not critical exactly when a participant takes a survey so long as they are able to relax, focus, and think clearly. In other cases (such as with the System Usability Scale, SUS, and the Single Ease Questionnaire, SEQ), the current mood of the participant is a valuable part of the data to be collected. These questionnaires are intended to capture a user’s reactions in the moments following the completion of a whole test or a single task, and thus must be completed while the relevant mood lingers. Preventive tips: Conduct a think-aloud test to understand how your survey makes respondents feel. Have a handful of participants take the survey and share how the questions make them feel. This approach is particularly useful when the content of a survey is likely to arouse strong emotional reactions (for example, if the survey is asking about a difficult experience or a controversial topic). Encourage participants to take the survey in a particular mood. While this is unrealistic for a simple Net Promoter Score (NPS) that pops up in an email or notification, for longer, more significant surveys it can be valuable to encourage participants to set aside a time when they can focus and relax — particularly when you will be paying them an incentive. Utilize customer-journey maps to identify opportunities for survey distribution. Just like fishermen studying the fish they intend to catch to know the best time and position for casting nets, researchers must also be intentional about the time and manner in which they distribute surveys. Customer-journey data can help a researcher to understand the current moods of users as they interact with the organization. However, there is obviously no way to predict the personal life circumstances of each participant. 8. Central-Tendency Bias People are often hesitant to provide extreme responses on rating scales. When survey respondents are asked to select a response on a scale, they often lean toward the middle, no matter how many points there are on the scale. Because of respondents’ hesitancy to select the extremes (even if their circumstances likely merit such responses), large sample sizes generally lead to somewhat normally distributed data on rating scales. Response scales with an odd number of options inevitably have a middle point which is often labeled as “neutral” or “neither agree nor disagree.” While this middle option is a legitimate response, it also becomes an easy “out” for respondents who don’t want to spend the effort necessary to formulate a true response to the presented statement. Preventive tips: Use an even number of response options to encourage participants to lean one way or the other. Receiving many neutral responses will not end up being very helpful for researchers. If a neutral option is needed during analysis, the middle two points can be combined. Use as few points on the response scale as are meaningful. You only want to have as many points on a response scale as are legitimately distinct and meaningful for the question you are asking. Simply having more points on a scale (such as 7 or more) does not guarantee you will be able to capture more nuanced and informative data. The differences between the responses to options such as “Very strongly agree” and “Strongly agree” on a 9-point scale will mean very little because participants are unlikely to have such specific feelings about most topics. We recommend using a 4-point or 6-point scale in most cases because they are straightforward to answer and analyze, and will capture most meaningful differences. 9. Demand Characteristics When participants in any type of research study become aware of the researcher’s aims and objectives, they are more likely to change their behaviors or responses accordingly. Participants might respond in specific ways in order to influence the outcomes of the study. This is particularly true when it can be personally beneficial to respond in specific ways. For example, professional participants might try to respond to recruiting screeners (which are a type of survey) in ways that might increase their chances of being selected for a study — regardless of whether they are a good fit. Some participants intentionally provide extreme or untrue responses if they have some reason to be frustrated with the distributor of the survey, the organization associated with it, or the aims of the study. Other participants (especially those with strong brand loyalty and positive feelings towards the survey distributor) might try to provide the responses they think the researchers are looking for out of goodwill, attempting to be “helpful.” This also tends to happen when respondents have a personal tie to the researcher — which is why it is important to recruit real users rather than colleagues or friends for any good research study. Preventive tips: Hide the true purpose of the survey. Avoid putting too much detail about the purpose of the survey in introductory materials such as emails, notifications, or survey titles and descriptions. It is also wise to mask any response options or survey questions that are critical to the purpose of the study by placing them among other plausible, but less critical options and questions. Distribute the survey to various types of users. Distributing a survey to users who have had both positive and negative experiences, as well as those familiar and unfamiliar with your brand can potentially offset the demand characteristics described above. 10. Random-Response Bias When survey respondents do not know the answer to a question, they generally just guess. This is problematic because guessed responses are not accurate data but are impossible to identify. For example, if a survey asked how much time was spent researching a product before purchasing, respondents are likely to just guess since they have no way of accurately knowing — especially those who spent multiple days or weeks researching. Respondents may also engage in guessing or selecting arbitrary options when they become fatigued or simply want to finish a survey for the offered incentive. Preventive tips: Provide an alternative response for those who do not know the answer. Closed-ended survey questions do not allow respondents to indicate that they do not have a meaningful response unless an option such as “none,” “other,” or “not applicable” is provided. However, it is not necessary to provide such options on all questions if it is likely that all respondents will have a meaningful answer. These types of alternatives are an easy way for people to opt out of responding, without fully considering the question (another reason not to have “neutral points in the middle of your scales). Include reverse-keyed items. Respondents who answer randomly often select the same option for all similar types of questions. Including reverse-keyed items can help you recognize when a participant answered quickly or randomly, without reading the questions. Once again, however, this should be done sparingly, and only when there are many of the same response types in a row. Keep surveys short. If respondents start guessing because they are getting bored or tired, your survey is just too long. Conclusion A well-designed survey can quickly and inexpensively gather many valuable insights, but even the best surveys are subject to response biases. Much of what we know about large groups of people has been gathered through surveys. However, surveys are not always the right method for gaining valuable UX insights and making improvements to digital designs. All survey data must be analyzed with a critical eye and not simply accepted as the truth. Reference Davies, R. S. (2020). Designing Surveys for Evaluations and Research. EdTech Books. Related Courses #### Survey Design and Execution Use surveys to drive and evaluate UX design Research #### Statistics for UX Calculate, interpret, and report the numbers from your quantitative UX studies Research #### Remote User Research Collect insights without leaving your desk Research surveys,confidence intervals,cognitive bias,quantitative studies,Research Methods,Behavior Patterns,data analysis,forms,recruiting participants Related Topics Research Methods Research Methods Behavior Patterns Learn More: to watch NN/g videos Survey Response Biases in User Research Alita Kendrick · 4 min Preference Testing for Assessing Visual Design Megan Chan · 5 min The Hidden Bias Killing Your UX Insights: Common-Knowledge Effect Evan Sunwall · 4 min Checkbox Design: 8 Guidelines Maddie Brown · 3 min Related Articles: Should You Run a Survey? Maddie Brown · 6 min User-Feedback Requests: 5 Guidelines Anna Kaley · 10 min Rating Scales in UX Research: Likert or Semantic Differential? Maria Rosala · 7 min Confidence Intervals, Margins of Error, and Confidence Levels in UX Raluca Budiu · 7 min Hiring Interviews Are Terrible: Smart UX Teams Structure Them Evan Sunwall · 11 min Why You Cannot Trust Numbers from Qualitative Usability Studies Raluca Budiu · 9 min
3884	https://cdn.bookey.app/files/pdf/book/en/electric-machinery-fundamentals.pdf	Electric Machinery Fundamentals PDF Chapman Scan to Download Electric Machinery Fundamentals Accessible Insights into Electric Machinery with Practical MATLAB Applications Written by Bookey Check more about Electric Machinery Fundamentals Summary Listen Electric Machinery Fundamentals Audiobook Scan to Download About the book Electric Machinery Fundamentals remains a premier textbook in the field, celebrated for its approachable and student-centric treatment of essential concepts. The fifth edition integrates MATLAB (R) throughout relevant examples and problems, enhancing the learning experience. Retaining the insightful and challenging problems that readers value, this edition showcases Chapman's extensive expertise and real-world insights, all presented in a clear and engaging manner. Scan to Download About the author Stephen J. Chapman is a prominent figure in the field of electrical engineering, best known for his contributions to the understanding of electric machinery and its fundamental principles. With a solid academic background and extensive teaching experience, he has played a significant role in shaping the curriculum for aspiring engineers. Chapman's writing style is characterized by clarity and an emphasis on practical applications, making complex concepts accessible to students and professionals alike. His textbook, "Electric Machinery Fundamentals," has become a staple in engineering education, reflecting his commitment to bridging the gap between theory and practice in electrical engineering. Through his work, Chapman has significantly influenced the study and application of electric machines in modern technology. Scan to Download Scan to Download Summary Content List Chapter 1 : Introduction to Machinery Principles Chapter 2 : Transformers Chapter 3 : AC Machinery Fundamentals Chapter 4 : Synchronous Generators Chapter 5 : Synchronous Motors Chapter 6 : Induction Motors Chapter 7 : DC Machinery Fundamentals Chapter 8 : DC Motors and Generators Chapter 9 : Single-Phase and Special-Purpose Motors Chapter 10 : Appendix A: Review of Three-Phase Circuits Chapter 11 : Appendix B: Coil Pitch and Distributed Windings Chapter 12 : Appendix C: Salient Pole Theory of Synchronous Machines Chapter 13 : Introduction to Power Electronics Scan to Download Scan to Download Chapter 14 : Appendix E: Errata for Electric Machinery Fundamentals 5/e Scan to Download Chapter 1 Summary : Introduction to Machinery Principles Chapter 1: Introduction to Machinery Principles 1. Shaft Speed Conversion A motor's shaft speed of 1800 r/min converts to approximately 188.5 rad/s. 2. Flywheel Dynamics A flywheel with a 4 kg·m² moment of inertia experiences a torque of 6 N·m, resulting in an angular speed of 7.5 rad/s Scan to Download Scan to Download after 5 seconds, which is equivalent to 71.6 r/min. 3. Torque and Angular Acceleration A force of 10 N applied to a cylinder produces a torque of 0.75 N·m (CCW). The angular acceleration is calculated to be 0.188 rad/s². 4. Mechanical Power Calculation A motor delivers 50 N·m torque at 1500 r/min, providing about 7854 W of mechanical power, equivalent to approximately 10.5 hp. 5. Magnetic Flux in Ferromagnetic Cores For a ferromagnetic core with specific dimensions, a calculated current is required to generate a flux of 0.005 Wb. The respective flux density values at the top and right sides of the core are determined accordingly. 6. Three-Region Ferromagnetic Core Analysis A ferromagnetic core with air gaps has determined flux Scan to Download Scan to Download values and densities under different conditions, illustrating the effect of air gap sizes on magnetic properties. 7. Core with Two Windings A two-legged core's total current and resulting flux generated by windings with 600 and 200 turns respectively, are computed, including total reluctance. 8. Flux in Three-Legged Core The flux and density of a three-legged core are analyzed when provided with a specific number of turns and under the influence of magnetic components. 9. Force on Current-Carrying Wire The force on a wire carrying 2 A in the presence of a magnetic field can be calculated, yielding results in Newtons. 10. Induced Voltage in Moving Wire When a wire moves in a magnetic field, the induced voltage is calculated, considering relative angles to field vectors. Scan to Download Scan to Download 11. Reluctance and Flux Factor Variability in reluctance due to changing magnetic environments depicted in multiple scenarios demonstrates the complexity of magnetic systems compared to ideal values. 12. Complex Power Calculations Using voltage and current values, complex power and related metrics are comprehensively analyzed for electrical systems comprising various loads. 13. Overall Circuit Analysis with Loads The behavior of a power system is explored through different configurations and their resulting impacts on current flow, power factor, and real/reactive power consumption. 14. Machine Speed Control Methods of adjusting a linear machine's speed through changes in voltage and magnetic flux density are effectively detailed, showcasing practical control methodologies in real-world applications. Scan to Download Chapter 2 Summary : Transformers Section Summary Chapter Title Transformers Overview This chapter covers analysis and performance evaluation of transformers, focusing on equivalent circuits, power calculations, efficiency, and voltage regulation. Transformer Calculations Details a 100-kVA, 8000/277 V transformer with equivalent circuits for high and low voltage sides; utilizes per-unit systems for analysis. Performance Evaluation Includes input voltage calculation based on rated loads and assesses voltage regulation between no-load and full-load conditions. Losses and Efficiency Focuses on copper and core losses under operational conditions; efficiency varies with loading conditions and power factor. Power System Design Discusses design implications of connecting transformers in various configurations and the need to adjust ratings for different frequencies. Autotransformers Introduces autotransformers with sketches, explaining their configuration for different voltage levels to enhance efficiency. Voltage Relations Explains phase voltage relationships affected by transformer connections power factor and voltage regulation. Conclusion Summarizes insights on transformer operation, performance calculations, importance of configuration for efficiency, and advantages of autotransformers. Chapter 2: Transformers Scan to Download Scan to Download Summary This chapter covers the analysis and performance evaluation of transformers, specifically focusing on their equivalent circuits, power calculations, efficiency, and voltage regulation in various configurations. Transformer Calculations and Equivalent Circuits - Transformer Specifications: A distribution transformer with a rating of 100-kVA, 8000/277 V is mentioned with specific resistance and reactance values for calculations. - Equivalent circuit can be derived for both high voltage and low voltage sides. - Per-unit systems of measurement are applied for analysis. Performance Evaluation - Input Voltage Calculation: For rated loads with a specified power factor, the input voltage is calculated using the equivalent circuit. - Voltage Regulation: Voltage regulation is determined by the difference between no-load and full-load conditions. Scan to Download Scan to Download Losses and Efficiency - Copper and Core Losses: Under operational conditions, losses are calculated based on current and resistance values. - Efficiency: The efficiency is evaluated under different loading conditions, revealing how it can be affected by factors like power factor. Power System Design - The chapter discusses the design implications when connecting transformers in different configurations (i.e., Y-Y, Y-”), and how transformer ratings must be operating at different frequencies (50 Hz vs 60 Hz). - Various power systems are analyzed for real and reactive power balance, and the relationships between load characteristics and transformer performance. Autotransformers - Autotransformers are introduced with sketches and explanations on configurations to connect different voltage levels while maximizing efficiency. Scan to Download Scan to Download Voltage Relations in Power Systems - The impact of transformer connections (phase voltage relationships is detailed, explaining how transformers affect power factor and voltage regulation. Conclusion This chapter provides foundational insights about transformer operation, including calculations for efficiency and performance under various loading scenarios, the importance of proper configuration for efficiency in power systems, and the benefit of autotransformers in voltage management. Scan to Download Scan to Download Example Key Point:Understanding transformer efficiency and performance impacts your energy management decisions. Example:Imagine you're managing a large facility where maintaining efficient energy consumption is crucial to keep costs down; hence, understanding how a transformer operates can help you optimize voltage regulation and minimize losses in your system. By calculating the efficiency under different loading conditions, you can determine whether to operate at full load or if adjustments are needed to ensure the transformer isn't operating in a less efficient state, thus impacting your electricity bills and overall energy strategy. Scan to Download Scan to Download Critical Thinking Key Point:Efficiency and Voltage Regulation in Transformers Critical Interpretation:The focus on efficiency and voltage regulation in transformers is critical, as these parameters fundamentally influence the overall performance of electrical systems. However, Chapman's analysis may oversimplify the complexities of real-world transformer operations and fails to consider external variables such as varying load conditions and environmental factors that could affect efficiency. While transformers are vital for power distribution, reliance on theoretical calculations without adequate practical contextualization may lead to misleading conclusions. Furthermore, research by the IEEE and other engineering studies suggests that losses can be influenced by several nuanced factors not fully explored in the text, indicating that readers should approach the author's assertions critically. Scan to Download Chapter 3 Summary : AC Machinery Fundamentals Chapter 3: AC Machinery Fundamentals 3-1. Characteristics of a Rotating Loop - A rotating loop in a magnetic field is analyzed, detailing its induced voltage, frequency, current, torque, and power generation. - (a) The induced voltage is given by ( e_{\text{ind}}(t) = 22.6 \sin(377t) ) V. - (b) The frequency is calculated as ( 60 ) Hz from the angular velocity of ( 377 ) rad/s. - (c) A 10 © resistor causes a current ofA. - (d) The induced torque is ( \tau_{\text{ind}} = 0.136 \sin(377t) ) N·m, counterclockwise. - (e) Instantaneous power is ( 51.1 \sin(377t) ) W and average power is ( 25.55 ) W. - (f) Mechanical power consumed equals electrical power Scan to Download Scan to Download generated, indicating the loop acts as a generator. 3-2. Speed of Magnetic Field Rotation Table - A table is created showcasing speeds for AC machines with 2 to 14 poles at frequencies of 50, 60, and 400 Hz. 3-3. Generator Speed Calculation - For a 4-pole generator at ( 133 ) Hz, the required shaft speed is ( 3990 ) r/min. 3-4. Voltage Generation in Stator Winding - A Y-connected four-pole winding with 40 turns per slot generates a frequency of ( 60 ) Hz, with a phase voltage of ( 5120 ) V and a line-to-line voltage of ( 8868 ) V. 3-5. Voltage Generation in ”-connected W - A ”-connected six-pole winding produces 50 ) Hz, resulting in phase and line voltages of ( 24000 ) V. 3-6. Rotor Flux for Synchronous Machine Install Bookey App to Unlock Full Text and Audio Scan to Download Scan to Download Chapter 4 Summary : Synchronous Generators Section Content 4-1. Frequency Conversion Uses synchronous motor (10 poles) and generator (12 poles) to convert 50 Hz to 60 Hz at the same speed. 4-2. Specifications and Calculations (a) Field current for 13.8 kV no load: 3.50 A (b) Internal generated voltage at rated conditions: ~11,544 V (c) Phase voltage at rated conditions: 7,967 V (d) Field current at rated 13.8 kV: 10 A (e) Terminal voltage rise after load removal: 20 kV (f) Prime mover output: 50.1 MW, torque: ~265,800 N·m (g) Capability curve constructed from losses and currents. 4-3. Adjusted Field Current (a) Terminal voltage at 5 A: ~13,910 V (b) Phasor diagrams show internal parameters. (c) Efficiency: 87.6% (d) Additional load causes voltage drop. (e) New terminal voltage after load addition: ~11,640 V (f) Increase field current to restore terminal voltage. 4-4. Rated Conditions (a) Efficiency: 89.8% (b) Voltage regulation (lagging load): 44.9% (c) Regulation (leading load): -2.2% (d) Regulation (unity power factor): 24% (e) MATLAB plot for voltage vs load at various power factors. 4-5. Stability and Torque Calculations (a) Torque angle at rated: 32° (b) Maximum power deliverable (unity power factor): 94.5 MW (c) Operation near static stability limit. 4-6. Operation and Power Calculation Output power and stability calculations using open-circuit and short-circuit characteristics. 4-7. Synchronous Generator Parameters (a) Synchronous reactance converted to ohms via per-unit calculations. (b) Estimation of internal generated voltage at rated conditions. Section Content 4-8. Water Turbine Generator Analysis Analysis of generator's operational changes: frequency, internal voltage, torque angle, and max power. 4-9. Generators in Parallel Evaluates speed droop and power distribution for two generators connected to a common load. 4-10. Power Sharing Analysis Explores load relationships and real power sharing among three generators under varying conditions. 4-11. Co-generation Facility Consideration Discussion on advantages and disadvantages of using multiple smaller vs. larger generation units. 4-12 to 4-27. Various Applications and Calculations Real/reactive power calculations, operational tolerances, plotting relationships under different torque/flux levels. 4-28. Power System Analysis Generator operation under varying loads while maintaining frequency and voltage stability. Chapter 4: Synchronous Generators 4-1. Frequency Conversion To convert 50 Hz power to 60 Hz, a synchronous motor and synchronous generator are utilized. The motor should have 10 poles, and the generator should have 12 poles for them to run at the same mechanical speed. 4-2. Specifications and Calculations - A 13.8-kV, 50-MVA synchronous generator has specified reactance, resistance, and losses. Scan to Download Scan to Download - (a) The field current required for 13.8 kV at no load is 3.50 A. - (b) The internal generated voltage at rated conditions is approximately 11,544 V. - (c) The phase voltage at rated conditions is 7,967 V. - (d) For rated conditions, the field current required for 13.8 kV is 10 A. - (e) After load removal, the terminal voltage would rise to 20 kV. - (f) The prime mover must supply 50.1 MW with a required torque of approximately 265,800 N·m. - (g) The capability curve of the generator is constructed using Scan to Download Scan to Download the mathematical relations derived from the losses and currents. 4-3. Adjusted Field Current With the field current set to 5 A: - (a) The terminal voltage is approximately 13,910 V. - (b) Phasor diagrams are constructed showing the generator’s internal parameters. - (c) Efficiency under these conditions is about 87.6%. - (d) Adding another identical load affects the phasor diagram, causing the voltage to drop. - (e) The new terminal voltage after load addition would be approximately 11,640 V. Scan to Download Scan to Download - (f) To restore the original terminal voltage, increase the field current. 4-4. Rated Conditions With rated conditions: - (a) The generator's efficiency is about 89.8%. - (b) The voltage regulation under lagging load is 44.9%. - (c) For leading load, regulation is -2.2%. - (d) At unity power factor, regulation is 24%. - (e) The MATLAB plot shows voltage versus load for various power factors. Scan to Download Scan to Download 4-5. Stability and Torque Calculations - (a) The torque angle at rated conditions is 32°. - (b) The maximum power deliverable under unity power factor loads is 94.5 MW. - (c) The generator operates close to the static stability limit. 4-6. Operation and Power Calculation - Various calculations determine output power and stability for given conditions with specified parameters using open-circuit and short-circuit characteristics. 4-7. Synchronous Generator Parameters - (a) Scan to Download Scan to Download Synchronous reactance is converted to ohms based on per-unit calculations. - (b) The magnitude of internal generated voltage is estimated under rated conditions. 4-8. Water Turbine Generator Analysis - Evaluates the generator's frequency of rotation, internal voltage, torque angle, and maximum deliverable power as operational conditions change. 4-9. Generators in Parallel - Evaluates speed droop and power distribution when two generators with different specifications are connected to a common load. 4-10. Power Sharing Analysis - Explores frequency and load relationships among three generators under varying load conditions and methods to improve real power sharing. Scan to Download Scan to Download 4-11. Co-generation Facility Consideration - Discusses the advantages and disadvantages of using multiple smaller generation units versus a single larger unit. 4-12 to 4-27. Various Applications and Calculations - Multiple scenarios involving real, reactive power calculations, operational tolerances, and adjustments based on system conditions. - Involves plotting relationships between generator currents and voltage conditions against varying torque angles or flux levels. 4-28. Power System Analysis - Evaluates generator operation under varying operational loads and conditions while maintaining frequency and voltage stability. In this chapter, various detailed electrical engineering principles relating to synchronous generators are explored through practical problems and proposed solutions, encompassing calculations of current, voltage, torque, power factors, and dynamic stability in a power distribution context. Scan to Download Scan to Download Example Key Point:Understanding the relationship between field current, voltage, and load is crucial in synchronous generators. Example:Imagine you're operating a power plant and need to supply electricity to a growing city. The instant you add more load, the system's voltage dips, and if not addressed, it could potentially lead to blackouts. To maintain a stable supply, you would need to increase the field current, ensuring your synchronous generator compensates for the extra demand and keeps the voltage steady. This example illustrates how intertwined these parameters are and highlights their importance in achieving reliable power generation. Scan to Download Scan to Download Critical Thinking Key Point:The importance of torque calculations in evaluating generator performance. Critical Interpretation:One of the key insights from this chapter is the critical role that torque calculations play in assessing the performance and stability of synchronous generators. The chapter emphasizes that understanding the torque angle and its relation to power delivery is vital for engineers and technicians to ensure efficient operation. However, it's essential for readers to question whether this singular focus on torque adequately encompasses all factors affecting generator performance, such as environmental influences or system design variations. Critics may argue that a broader perspective that includes additional parameters, such as system reliability and integration with renewable sources, might yield a more comprehensive understanding of generator operations. These considerations can be supported by research such as "Power System Stability and Control," which urges for holistic approaches to generator and grid interactions. Scan to Download Chapter 5 Summary : Synchronous Motors Chapter 5: Synchronous Motors 5-1 Synchronous Motor Analysis - A 480-V, 60 Hz, 400-hp , 0.8-PF leading, eight-pole ”-connection operates under defined parameters, with synchronous reactance of 0.6 © and negligible armature - Speed Calculation : - Speed = 120 x 60 Hz / 8 = 900 r/min. - Input Power : - 400 hp = 298.4 kW, input power = output power (ignoring losses). Scan to Download Scan to Download - Armature Current Calculation : - Line current = 449 A, phase current = 259 A. - Torque and Torque Angle : - Induced torque = 298.4 kW resulting in 3166 N·m; current torque is approximately 1/3 of maximum possible torque. - Effect of Increased EA : - If EA increases by 30%, armature current changes, leading to a new power factor of 0.913 lagging. - V-Curve : - MATLAB code provided to plot V-curve based on armature current versus EA. 5-2 Motor Operation at Rated Conditions - Rated Condition Variables Scan to Download Scan to Download : Input/output current and voltage calculated. - When Load is Removed : Reactivity and torque angle adjustments noted. 5-3 Two-Pole Synchronous Motor Analysis - Analyzes a 230-V, 50 Hz motor drawing 40 A at unity PF to derive output torque, load changes, and resultant power factor adjustments. 5-4 High-Power Synchronous Motor - Analyzes a 2300-V, 1000-hp motor with losses, finding required field current for unity PF and calculating efficiency. 5-5 V-Curves for Synchronous Motor - Examines both no-load and full-load conditions to plot armature versus field current. 5-6 Reactance Change with Frequency - Discusses how synchronous reactance is influenced by Scan to Download Scan to Download frequency based on armature magnetization. 5-7 Synchronous Motor Current Analysis - Evaluates a 208-V motor under various loading conditions to calculate torque angles and other performance metrics. 5-8 Synchronous Motor At Rated Power - The operating points of a ”-connected mcurrent, and torque derived under rated conditions. 5-9 Power Consumption and Stability - Exploratory questions regarding power consumption, stability power limits, and field current adjustments. 5-10 Effect of Armature Resistance on Torque Angle - Derives new equations for torque angle when accounting for armature resistance. 5-11 Power Dynamics Analysis Scan to Download Scan to Download - Analyzes internal generation and power characteristics of synchronous machines when acting as motors or generators under defined conditions. 5-12 Plant Load Calculations - Evaluates power characterizations when integrating synchronous motors and determining the overall load characteristics. 5-13 Frequency Control Ranges for Synchronous Motors - Calculates required frequency variations, output currents, and maximum supply capabilities under variable operating speeds. 5-14 Performance Metrics at Full Load - Evaluates performance metrics including output torque, efficiency, and required inputs under defined operational parameters. 5-15 Machine Specifications and Performance Scan to Download Scan to Download Features - Checks rated input power, efficiency, generated voltages, and losses for a specific synchronous motor. 5-16 Generator and Motor Power Interactions - Establishes metrics when synchronous systems influence each other, including effects of field variations and associated rating constraints. 5-17 Synchronous Motor Power Analysis - Determines power handling capabilities and performance variables at varied load conditions. 5-18 High-Frequency Motor Variations - Evaluates performance at specified frequencies, checking speed, torque outputs, and internal voltage dynamics. 5-19 Power Dynamics of a Synchronous Motor - Explores power consumption specifics when certain Scan to Download Scan to Download conditions of voltage and current are applied, revealing operational ratings. 5-20 Interactions Between Machines - Evaluates machine states as motors or generators across varied operational voltages indicating real power flow and reactive power interactions. This chapter provides a comprehensive overview of synchronous motors, their operational characteristics, power dynamics under various conditions, and the analytical methods used to determine performance metrics. Scan to Download Scan to Download Example Key Point:Understanding current dynamics is crucial for optimal motor performance. Example:Imagine you are operating a large synchronous motor in your production facility, and you've just noticed a dip in performance. By calculating the armature current and evaluating the torque angle against the rated conditions, you can determine whether the machine is working efficiently or if adjustments are needed. This hands-on approach ensures that you maintain optimal power factor and torque output, which are vital for minimizing energy costs and maximizing productivity. Scan to Download Scan to Download Critical Thinking Key Point:Importance of Power Factor Management Critical Interpretation:One key point in this summary is the critical role of power factor management in synchronous motors, as illustrated by the situation where an increase in armature current results in a shift from a leading to a lagging power factor. Power factor not only reflects efficiency but also influences the overall stability and reliability of electrical systems. However, it’s crucial to approach this viewpoint with caution as assumptions about power factors leading to efficiency gains may overlook complex interactions within electrical networks. As Brown (2018) emphasizes in 'Electrical Engineering: Principles and Applications', optimizing power factors in isolation without considering system dynamics can lead to unforeseen operational challenges. Therefore, while the analysis presented in this chapter is valuable, it may not necessarily reflect the complete picture of synchronous motor performance in varying operational contexts. Scan to Download Chapter 6 Summary : Induction Motors Section Key Points 6-1. Three-Phase Six-Pole 50-Hz Induction Motor Calculations Synchronous Speed: 1200 RPM, Rotor Speed: 965 RPM, Slip Speed: 35 RPM, Rotor Frequency: 1.75 Hz 6-2. Three-Phase Two-Pole 60-Hz Induction Motor Calculations Synchronous Speed: 3600 RPM, Rotor Speed: 3510 RPM, Slip Speed: 90 RPM, Rotor Frequency: 1.5 Hz 6-3. Torque-Speed Characteristics for a 60-Hz Induction Motor Poles: 10, Slip at Load: 6.94%, Speed at ¼ Load: 708 RPM, Rotor Electrical Frequency: 1.03 Hz 6-4. Two-Pole Induction Motor Power and Torque Calculations Shaft Speed: 2850 RPM, Output Power: 50 kW, Load Torque: 167.5 N·m, Induced Torque: 171.9 N·m, Rotor Frequency: 2.5 Hz 6-5. Wound-Rotor Induction Motor Analysis Line Current: 78.0 A, Stator Copper Losses: 1825 W, Air-Gap Power: 24.0 kW, Power Converted: 22.8 kW, Induced Torque: 122.3 N·m, Load Torque: 132.6 N·m, Overall Efficiency: 84.6% 6-6. Pullout Torque Analysis Slip at Pullout: 0.369, Pullout Torque: 210 N·m 6-7. Torque-Speed Characteristics and MATLAB Implementation Provided MATLAB code for torque-speed characteristics of a motor 6-8. Resistance for Maximum Torque at Start Calculate resistance for pullout torque and plot results 6-9. 50-Hz Operation Adjustments Effects of switching to 50 Hz; adjust voltage and reactances accordingly 6-10. Induction Motor Performance Analysis Evaluated speed regulation: 4.07% 6-11. Rotor Circuit Loss Analysis Calculated rotor copper loss for a 5 kW input 6-12. Air Gap and Load Torque Calculations Performance analysis of slip and induced torque 6-13. Thevenin Equivalent Circuit Analysis Calculate Thevenin voltage and impedance from a defined circuit 6-14. Maximum Power Transfer Calculations Derivation of equations addressing load power and current resistance 6-15. Performance Metrics for 25 hp Induction Motor Analysis covering currents, losses, output power, efficiency, and speeds 6-16. Pullout Torque for Induction Motor Determining slip, rotor speeds, and torque calculations 6-17. Torque-Speed Characteristics of Two-Cage Induction Motors Differences between dual-cage and single-cage designs regarding startup behavior Section Key Points 6-18. Torque-Speed Variant Characterization with MATLAB MATLAB used for plotting torque-speed characteristics 6-19. DC Test Analysis for Induction Motor Stator resistance determined via a DC test 6-20. Induction Motor Failure Characteristics Performance analysis following laboratory tests with no-load and locked-rotor results 6-21. Performance Calculations for 75-hp Induction Motor Metrics on line currents, power losses, and efficiencies for a specific motor 6-22. Torque-Speed Characteristics and MATLAB Plotting MATLAB implementation for torque-speed curve plotting; insights from dual-cage design 6-23. Starting Current and Controller Specifications Calculations on independence of starting and line current specifications 6-24. Plugging of Induction Motors Effects of stator lead switching on induction motor performance parameters 6-25. Y-” Starter Implications Comparisons between standard starting conditions and Y-” start6-26. Autotransformer Startup Characterization Starting voltages and current parameters addressed for motor setups 6-27. Effects of Rotor Resistance Adjustment Observations related to slip, efficiency, and performance metrics 6-28. Dual-Cage Induction Motor Parameter Evaluation Torque and efficiency parameters calculated for both single-cage and dual-cage motors Chapter 6: Induction Motors 6-1. Three-Phase Six-Pole 50-Hz Induction Motor Calculations - Given a 220-V motor with 3.5% slip, the following is computed: Scan to Download Scan to Download - a) Synchronous speed: 1200 RPM. - b) Rotor speed: 965 RPM. - c) Slip speed: 35 RPM. - d) Rotor frequency: 1.75 Hz. 6-2. Three-Phase Two-Pole 60-Hz Induction Motor Calculations - A 480-V motor with 2.5% slip results in: - a) Synchronous speed: 3600 RPM. - b) Rotor speed: 3510 RPM. - c) Slip speed: 90 RPM. - d) Rotor frequency: 1.5 Hz. 6-3. Torque-Speed Characteristics for a 60-Hz Induction Motor - For a motor running at 715 RPM no load and 670 RPM full load: - a) Poles: 10. - b) Slip at load: 6.94%. - c) Speed at ¼ load: 708 RPM. - d) Rotor electrical frequency: 1.03 Hz at ¼ load. Install Bookey App to Unlock Full Text and Audio Scan to Download Scan to Download Chapter 7 Summary : DC Machinery Fundamentals Chapter 7: DC Machinery Fundamentals 7-1. Simple Rotating Loop Analysis - Motor or Generator : The machine operates as a generator since the induced voltage (50 V) exceeds the external battery voltage (48 V). - Current Flow : The current flowing out of the machine is approximately 5.0 A. - Current Changes with Speed : - Increasing rotor speed to 550 rad/s results in an induced voltage of 55 V, causing the current to increase to 17.5 A. - Decreasing rotor speed to 450 rad/s causes the induced Scan to Download Scan to Download voltage to drop to 45 V, resulting in 7.5 A flowing into the machine, indicating it operates as a motor. 7-2. Two-Pole Eight-Coil Machine Analysis - Winding Type : The winding is a progressive winding. - Current Paths : There are two current paths through the armature (simplex lap winding). - Voltage at Brushes : The voltage is positive at brush x relative to brush y, calculated to be approximately 33.9 V. - Armature Resistance : The total armature resistance is 0.08 ©. - Current with Load : Connecting a 1 © resistor results in a cu - Induced Torque Scan to Download Scan to Download : The induced torque is calculated to be 4.12 N·m, suggesting a clockwise direction. - Terminal Voltage Plot : The terminal voltage decreases linearly with increasing load current. 7-3. Induced Voltage Equation Derivation - The induced voltage for a single rotating loop is shown to be a specific case of the general equation for induced voltage in a DC machine through substitutions of parameters. 7-4. Current Flow in Various Armature Configurations - Simplex Lap-Wound : Current per path is 15 A. - Duplex Lap-Wound : Current per path is 7.5 A. - Simplex Wave-Wound Scan to Download Scan to Download : Current per path is 60 A. 7-5. Parallel Current Paths in Armature of a 20-Pole Machine - Simplex Lap-Wound : 20 paths. - Duplex Wave-Wound : 4 paths. - Triplex Lap-Wound : 60 paths. - Quadruplex Wave-Wound : 8 paths. 7-6. Power Conversion in DC Motors - The equivalence of electric power disappearing and mechanical power appearing during the power conversion in a DC motor is demonstrated through equation substitutions. Scan to Download Scan to Download 7-7. DC Generator Analysis - Required Flux : 0.00333 Wb to produce 120 V. - Current per Path : 13 A at rated load. - Induced Torque : 66.1 N·m at rated load. - Brush Requirements : 8 brushes are needed, each spanning two commutator segments. - Armature Resistance : The total resistance is 0.0275 ©. 7-8. DC Motor with Small Rotor Coils - No-load Speed : Calculation shows the motor runs at about 2500 r/min when Scan to Download Scan to Download connected to a 12-V battery. - Direction of Rotation : CW rotation when the positive terminal is connected to the rightmost brush. - Induced Torque under Load : 2.29 N·m induced torque when consuming 600 W. 7-9. Winding and Commutation - Parallel Current Paths : There are 4 paths in the duplex lap winding armature. - Brush Placement : Brushes should be placed to short out windings between the poles. - Plex of Machine : The machine is duplex. - Terminal Voltage : The terminal voltage can be calculated based on conductor Scan to Download Scan to Download arrangement. 7-10. Winding Description and Motor Rotation - A 2-pole, retrogressive, lap winding results in counterclockwise rotation when positive voltage is applied under the north pole face. Scan to Download Chapter 8 Summary : DC Motors and Generators Chapter 8: DC Motors and Generators Summary of Problems and Solutions This chapter discusses various problems related to the performance and analysis of DC motors and generators, providing detailed calculations and reasoning for each scenario. DC Motor Parameters and General Information - Given a DC motor with specific ratings (30 hp, 110 A, 240 V) and resistance values, a series of problems (8-1 to 8-12) address how to determine motor speed under various conditions, including adjustments to field resistance. Key Problems and Solutions: Scan to Download Scan to Download 8-1: No-Load Speed Calculation - Adjusted resistance set at 175 © leads to no-load speed, determining it to be approximately 1195 r/min. 8-2: Full Load Speed and Regulation - At full load (110 A), the speed is calculated to be about 1092 r/min, resulting in a speed regulation of about 9.4%. 8-3: Effect of Increased Resistance on Speed - With increased resistance to 250 ©, the calculation shows a increase due to reduced field current leading to a speed of about 1236 r/min. 8-4: Armature Reaction Impact at Full Load - Considering armature reaction affects the motor speed, a calculation shows a higher speed under these new conditions. 8-5: Maximum and Minimum No-Load Speeds Scan to Download Scan to Download - These values are calculated based on adjusted resistance ranges (100 to 400 ©), revealing the extre speed. 8-6: Starting Current Calculation - Significant starting current estimation, revealing a starting current of 1260 A, highlighting a risk of damage. 8-7: Torque-Speed Characteristic - Plotting torque-speed characteristics through MATLAB enhances understanding of motor operation under various current levels. 8-8 to 8-12: Various Configurations - Further scenarios explore separately excited motors, cumulatively compounded motors, and their operational characteristics such as speed regulation, torque calculations, and individual impacts of armature reactions on motor performance. Scan to Download Scan to Download 8-21: Efficiency and Power Calculations - Further analysis on power, efficiency, torque, copper losses, and other parameters when operating under various conditions. Conclusions and Insights: - Strong relationships between resistance adjustments, motor speed, torque, power output, and efficiency are established. - Variations in operating conditions, including load and armature reaction, significantly affect motor performance. - Practical MATLAB implementations provide valuable tools for visualizing and analyzing motor characteristics in complex scenarios. This chapter presents a comprehensive examination of DC motors and generators, addressing theoretical background, practical calculations, and performance evaluations, crucial for students and engineers working with electrical machinery. Scan to Download Scan to Download Critical Thinking Key Point:Impact of Armature Reaction on Motor Performance Critical Interpretation:One key point highlighted in this summary is the role of armature reaction in affecting motor speed and operational efficiency. The author suggests that understanding armature reaction is crucial in practical applications, to accurately gauge performance under varying loads. However, this perspective may oversimplify the complexities involved in DC motor dynamics and could be challenged by other scholarly insights. Research by Wildi (2006) in 'Electrical Machines, Drives, and Power Systems' emphasizes additional factors, such as magnetic saturation and environmental conditions, that also influence motor efficiency. Thus, readers should explore contrasting viewpoints and data to form a well-rounded understanding of DC motor performance. Scan to Download Chapter 9 Summary : Single-Phase and Special-Purpose Motors Chapter 9: Single-Phase and Special-Purpose Motors 9-1. Analysis of a Split-Phase Induction Motor - Motor Specifications : 120-V, 1/4-hp, 60-Hz, four-pole split-phase induction motor. - Impedances : - ( R_1 = 2.00 \, \Omega ) - ( X_1 = 2.56 \, \Omega ) - ( R_2 = 2.80 \, \Omega ) - ( X_2 = 2.56 \, \Omega ) - ( X_M = 60.5 \, \Omega ) - Scan to Download Scan to Download Slip : 0.05, Rotational losses: 51 W. Key Calculations : (a) Input Power : 403.8 W (b) Air-gap Power : 325 W (c) Converted Power (Pconv) : 308 W (d) Output Power (Pout) : 257 W (e) Induced Torque (ind Ä) : 1.72 N·m (f) Load Torque (load Ä) : 1.44 N·m (g) Install Bookey App to Unlock Full Text and Audio Scan to Download Scan to Download Chapter 10 Summary : Appendix A: Review of Three-Phase Circuits Appendix A: Review of Three-Phase Circuits A-1: Calculation of Load Parameters - Load Connection : ”-connected with impedances of \(4 + j3\ - Power Line Voltage : 208 V. - Load Current Calculation : - Phase Voltage, (V_{\phi} = 208 \, V) - Phase Impedance, (Z_{\phi} = 4 + j3 \, \Omega) - Line Current, (I_L = \frac{V_{\phi}}{Z_{\phi}} = 41.6 \, A) - Scan to Download Scan to Download Power Calculation : - Real Power, (P = 20.77 \, kW) - Reactive Power, (Q = 15.58 \, kvar) - Apparent Power, (S = 25.96 \, kVA) - Power Factor = 0.8 lagging. A-2: Three-Phase Power System Analysis - System Setup : - Generator voltage: 480 V. - Line Impedance: (0.09 + j0.16 \, \Omega). - Load 1: Y-connected with (Z = 2.5\angle 36.87^\circ \, \Omega). - Load 2: ”-connected with \(Z = 5\angle - Calculations : - Line Voltage, Voltage Drop, Real and Reactive Powers for each load, and losses in the transmission line were calculated. - Scan to Download Scan to Download Key Outcomes : - Load Voltages and Powers along with line current and overall power supplied by the generator were obtained with power factor analysis. A-3: Generator and Load Analysis - Lossless Transmission Lines : Analyzed with both open and closed switch conditions. - Phase Voltages and Currents : - Y-connected load voltage and current calculated based on given parameters. - ”-connected load was analyzed similarl parameters. - Generator Power Supply : Both open and closed switch conditions were evaluated, with detailed calculations showing how total line current changes based on load configurations. Scan to Download Scan to Download A-4: Y-connected Generator Phase Voltage Relationship - Phasor Diagram : Illustrated the relationship showing that line voltage lags phase voltage by 30°. - Derivation : Using Kirchhoff’s voltage law, the relationship between line and phase voltages was established. A-5: Load Voltage and Current Calculations - ”-connected Load : Identified that line and phase voltages are equal, showing their calculations. - Apparent Load Values : Magnitudes and angles of each voltage and current were established. A-6: Distribution System Analysis Scan to Download Scan to Download - Systems with Zero Impedance : Analyses conducted for open and closed switch scenarios. - Real, Reactive, and Apparent Powers : Power contributions were summarized, alongside total current adjustments upon closing the switch. - Key Insight : Noted that the reactive power from the capacitor bank affects the overall current supplied to the distribution system. This summary encapsulates the critical elements of three-phase circuit analysis presented in the appendix, focusing on calculations and outcomes relevant to real and reactive power in a system context. Scan to Download Chapter 11 Summary : Appendix B: Coil Pitch and Distributed Windings Appendix B: Coil Pitch and Distributed Windings B-1. Winding Pitch for Harmonic Elimination - To eliminate the fifth harmonic in a two-pole, 2-slot three-phase stator armature, acceptable coil pitches are derived: - ( \rho = \frac{2 n}{5} ) (where ( n = 0, 1, 2, 3, \ldots)), leading to pitches of 2/5, 4/5, etc. - The best choices for maximum fundamental voltage are 4/5 or 6/5, resulting in a practical 10/12 pitch in a 24-slot winding. B-2. Winding Distribution Factor - For five slots per phase, the total voltage for the phase can be derived using geometrical relationships through several right triangles formed in a circle: Scan to Download Scan to Download - The winding distribution factor ( k_d ) can be expressed as: [ k_d = \frac{E_n}{E_A} ] B-3. Synchronous Machine Analysis - A three-phase four-pole synchronous machine with 96 stator slots and a coil pitch of 19/24 is evaluated: - (a) Electrical degrees for coil pitch: 142.5°, and slot pitch: 7.5°. - (b) The pitch factor and distribution factor are calculated, leading to: [ k_p = 0.947, \quad k_d = 0.956 ] - (c) The winding suppresses harmonics, specifically analyzed for the third, fifth, seventh, etc. B-4. Double-Layer Winding Analysis - A winding installed on a 48-slot stator with a 5/6 pitch: - (a) The pitch factor is calculated. Scan to Download Scan to Download - (b) The distribution factor is determined. - (c) The frequency of voltage from the winding is derived. - (d) Phase and terminal voltages are computed based on machine parameters. B-5. Synchronous Generator Voltage Calculation - For a Y-connected six-pole synchronous generator: - Calculated line voltage produced based on flux, speed, pitch factors, and other components. B-6. Rotor Flux Requirements - A synchronous machine with specific characteristics: - (a) Required rotor flux for a terminal voltage of 6 kV is deduced using pitch and distribution factors. - (b) Effectiveness of coil pitch on harmonic reduction (fifth and seventh harmonics) is evaluated. B-7. Coil Pitch for Harmonic Elimination - The optimal coil pitch to eliminate the seventh harmonic is discussed, and the required number of slots for an eight-pole winding is identified, noting the effect on the fifth harmonic. Scan to Download Scan to Download B-8. High Voltage Generator Specifications - For a 13.8 kV Y-connected generator: - (a) Flux per pole for achieving no-load terminal voltage is calculated. - (b) The winding factor of the machine is derived from distribution and pitch factors. Scan to Download Chapter 12 Summary : Appendix C: Salient Pole Theory of Synchronous Machines Appendix C: Salient Pole Theory of Synchronous Machines C-1: Characteristics and Operation of a Synchronous Generator - A synchronous generator rated at 13.8-kV, 50-MVA, 0.9 power factor lagging operates at 60 Hz with notable reactances and armature resistance. - (a) The field current required for a no-load terminal voltage of 13.8 kV is 3.50 A. - (b) At rated conditions, the internal generated voltage, calculated using armature current and reactances, is approximately 11496" 17.8° V. - (c) The reluctance torque contributes 8.6 MW out of a total output of 42.2 MW, amounting to about 20% of full-load power. Scan to Download Scan to Download C-2: Analysis of a Water-Turbine-Driven Generator - A water-turbine-driven generator rated at 120 MVA, 13.2 kV operates at 0.8 PF lagging with specified reactances. - (a) The internal generated voltage required at rated conditions is 1093.7" 10.7° V. - (b) Voltage regulation at these conditions is 29.8%. - (c) The peak power occurs at a torque angle of 70.6° with a maximum output of 392.4 MW, while an equivalent non-salient generator would yield 364.7 MW. C-3: Salient-Pole Motor Operation - (a) Phasor diagrams for a salient-pole synchronous machine used as a motor are essential for understanding. - (b) The motor's voltage and current equations relate to specific parameters, showing how torque a performance. - (c) The relationship between the torque machine performance is highlighted. C-4: Maximum Torque of Salient-Pole Motor Install Bookey App to Unlock Full Text and Audio Scan to Download Scan to Download Chapter 13 Summary : Introduction to Power Electronics Chapter 13 Summary: Power Electronics Section S1-1: Three-Phase Half-Wave Rectifier Calculation - Analytical calculation revealed that the average voltage is 0.8270 V and rms voltage is approximately 0.8407 V, leading to a ripple factor of 18.3%. - MATLAB was used to simulate the rectifier's output and confirmed the analytical results with the ripple calculated at 18.2759%. Section S1-2: Three-Phase Full-Wave Rectifier Calculation - The average and rms voltages were calculated over the interval, resulting in a ripple factor of 4.2%. - MATLAB simulation confirmed the theoretical ripple factor Scan to Download Scan to Download of 4.2017%. Section S1-3: Circuit Operation Explained - SCR operation involves charging the capacitor until it exceeds the breakover voltage of DIAC, triggering SCR and allowing current flow to the load. Closing switch S1 affects the charging time of the capacitor. Section S1-4: Load Voltage and Firing Angles - Various firing angles (0°, 30°, 90°) yield different average voltages, with specific calculations provided for each case. Section S1-5: Firing Angle Calculation with Adjustable Resistor - Calculation of firing angle based on the relationship between the capacitor voltage and breakover voltage for the DIAC. Results suggest firing angle of approximately 75.6°. Section S1-6: Sensitivity to Input Voltage Variations and Mitigation Scan to Download Scan to Download - The circuit's sensitivity to voltage changes can be alleviated by using zener diodes to stabilize charging voltages, preventing feedback complications. Section S1-7: Single-Phase Voltage Source Inverter - Inverter operation involves alternating SCR activation to produce a square waveform output, turning off SCR by reversing applied voltages. Section S1-8: Relaxation Oscillator Analysis - Voltage behavior across components was analyzed, revealing the time period for capacitor charging and discharging phases, with calculations detailing the timing dynamics. Section S1-9: Jones Circuit Operation - Operation of the Jones circuit is described, highlighting SCR control of power output and diode functions in controlling current flow. Section S1-10: Series-Capacitor Chopper Circuit Scan to Download Scan to Download Limitations - Analysis revealed the SCR stays on until current drops below IH, often leading to inefficient operation. A faster discharge method is proposed to enhance performance. Section S1-11: Parallel-Capacitor Chopper Circuit Limitations - Similar issues arise with SCR turn-off times being lengthy due to capacitor charging via resistance, requiring more complex circuits for rapid operation. Section S1-12: Rectifier-Inverter Functionality - The current source inverter operates with SCRs for rectification, with commutating capacitors playing a critical role in SCR control. Section S1-13: AC Phase Angle Voltage Controller - Analysis of the controller's firing angles and load voltages, requiring calculations for effective functioning across defined parameters. Scan to Download Scan to Download Section S1-14: Three-Phase Full-Wave Rectifier Waveforms - The output behavior and ripple factors were deduced from triggering SCRs at different phase angles, illustrating variations in output due to timing. Section S1-15: PWM Circuit Analysis - A MATLAB simulation illustrates PWM modulator operation under different conditions, revealing the frequency response characteristics and demonstrating the advantages of higher reference frequencies to reduce harmonic content in the output signal. Scan to Download Chapter 14 Summary : Appendix E: Errata for Electric Machinery Fundamentals 5/e Appendix E: Errata for Electric Machinery Fundamentals 5/e Please note that the following errata will be corrected in future reprints of the book. Attached PDF pages contain these corrections for distribution to students. Erratum List 1. Page 145 : Problem 2-3 was incorrectly printed; it should be: - Consider a simple power system with an ideal voltage source, a step-up transformer, a transmission line, a step-down transformer, and a load. Given parameters: - Voltage of source: (480 \angle 0° V) - Transmission line impedance: (Z_{line} = 3 + 4j \ \Omega) Scan to Download Scan to Download - Load impedance: (Z_{load} = 30 + 40j \ \Omega) - Questions: - (a) No transformers, calculate load voltage and efficiency. - (b) With transformers, calculate load voltage and efficiency. - (c) Determine transformer turns ratio for 1% line losses. 2. Page 147 : Problem 2-13: Transformer should be Y- 3. Page 264 : Problem 4-6: Generator should be “2-polHz” instead of “Y-connected”. 4. Page 269 : Problem 4-25: The instruction should be “Make a plot of the terminal voltage versus the load impedance angle” instead of “power factor”. 5. Page 301 : Problem 5-4: Correct synchronous reactance is (2.5 \ Scan to Download Scan to Download \Omega). 6. Page 304 : Problem 5-12: Corrections to parts (b) and (i). The revised problem details requirements for analyzing a synchronous motor in a plant, including calculations for power and current. 7. Page 305 : Problem 5-17: Power supplied by generator is 80 kW. 8. Page 358 : Figure 6-34: Incorrect number in NEMA starting code letters; corrected table provided. 9. Page 400 : Problem 6-23: Motor develops full-load torque at 3.5% slip. 10. Page 402 : Problem 6-31: References motor from Problem 6-21, not 6-23. 11. Page 402 : Problem 6-32: Parameters for outer and inner bars specified. Scan to Download Scan to Download 12. Page 553 : Problem 8-4: Armature reaction is 1000 A at full load. 13. Page 667 : Problem C-1: Should begin with: “A 13.8-kV, 50-MVA, 0.9-power-factor-lagging, 60-Hz, four-pole Y-connected synchronous generator…” Scan to Download Best Quotes from Electric Machinery Fundamentals by Chapman with Page Numbers View on Bookey Website and Generate Beautiful Quote Images Chapter 1 \| Quotes From Pages 7-30 1.If a torque of 6 N · m (counterclockwise) is suddenly applied to the flywheel, what will be the speed of the flywheel after 5 s? 2.A motor is supplying 50 N · m of torque to its load. 3.The total reluctance of the core is ... 4.The total flux in the core is equal to the flux in the center leg. 5.If the bar runs off into a region where the flux density falls to 0.45 T, what happens to the bar? 6.Reducing the flux density B of the machine increases the steady-state speed. Chapter 2 \| Quotes From Pages 31-78 1.Assume that this transformer is supplying rated load at 277 V and 0.85 PF lagging. What is this Scan to Download Scan to Download transformer’s input voltage? 2.Therefore, the primary impedances referred to the low voltage (secondary) side are … 3.The voltage regulation of the transformer under these conditions is … 4.The efficiency of this transformer is OUT OUT CU core 85,000100% 100% 96.6% 85,000 1430 160 ð= … Chapter 3 \| Quotes From Pages 79-86 1.This machine is acting as a generator, converting mechanical power into electrical power. 2.The amount of mechanical power consumed by the loop is equal to the amount of electrical power being generated by the loop (within roundoff error). 3.The induced voltage on a simple rotating loop is given by... 4.Note that the amount of mechanical power consumed by the loop is equal to the amount of electrical power created by the loop (within roundoff error). 5.To help control these losses, early ac motors in the USA Scan to Download Scan to Download were run from a 25 Hz ac power supply... 6.As a result, 25 Hz power systems shrank and disappeared. However, there were many perfectly-good working 25 Hz motors... Scan to Download Chapter 4 \| Quotes From Pages 87-128 1.The speed of a synchronous machine is related to its frequency by the equation 120 n = (120 f)/P. 2.Therefore, a 10-pole synchronous motor must be coupled to a 12-pole synchronous generator to accomplish this frequency conversion. 3.The input power to this generator is equal to the output power plus losses. 4.The efficiency of the generator can be found as follows: (Output Power)/(Input Power). 5.The static stability limit occurs at 90° torque angle. 6.In a parallel system, the power-sharing depends on the speed droop characteristics of each generator. Chapter 5 \| Quotes From Pages 129-157 1.The maximum power that the motor can produce at rated speed with the value from part (b) is 226 kW. 2.The motor has a synchronous reactance oarmature resistance of 0.3 ðW. Scan to Download Scan to Download 3.Ignoring all losses, the efficiency of this motor at full load is 97.6%. 4.If AE is decreased by 10%, how much reactive power will be consumed by or supplied by the motor? 5.This machine is a generator supplying real power to the power system, because AE is ahead of VÆ Chapter 6 \| Quotes From Pages 158-207 1.The induction motor has an important role in industrial applications, due to its robustness and simplicity. 2.The performance of an induction motor is determined by various factors, including its design, materials, and operating conditions. 3.Understanding slip in induction motors is essential for their efficient operation. 4.The ability to manage torque and speed characteristics is critical in induction motor applications in real-time scenarios. 5.Torque-speed characteristics reveal the performance limits Scan to Download Scan to Download of induction motors under different load conditions. 6.Efforts in optimizing the efficiency of induction motors contribute significantly to energy conservation in industrial applications. 7.Autotransformer starters can effectively reduce starting currents, enhancing motor longevity and performance. Scan to Download Chapter 7 \| Quotes From Pages 208-219 1.If the speed of rotation É of the shaft is then the voltage induced in the rotating loop will be ind 2e rlBÉ=… Since the external batt voltage is only 48 V, this machine is operating as a generator, charging the battery. 2.The current flowing out of the machine is approximately ind 50 V 48 V 5.0 A 0.4 B e Vi R " " … . average current flow over a complete cycle will be somewhat less than 5.0 A. 3.If the speed of the rotor were decreased to 450 rad/s, the induced voltage of the loop would fall to and the current flow into the machine would be ind 48 V 45 V 7.5 A 0.4 Be Vi R " ". 4.The voltage is positive at brush x with respect to brush y, since the voltage in the conductors is positive out of the page under the North pole face and positive into the page under the South pole face. 5.The induced torque is given by Equation Scan to Download Scan to Download cond)(0.10 m)(0.3 m)(1.0 T)(33.9 A)(2 current paths). 6.The terminal voltage of this machine is given by T A A V E I R = " A. Chapter 8 \| Quotes From Pages 220-281 1.This is also the power converted from electrical to mechanical form in the dc machine, since all other losses are neglected. 2.The internal generated voltage AE of the dc machine is 234.6 V. 3.If the field current is increased by 5 percent and the OCC of the ac machine is linear, AE increases to 432 V. 4.On the other hand, decreasing the field current in the dc machine will decrease the internal generated voltage, resulting in lower output power. 5.The phasor diagram illustrating this change is shown below. Chapter 9 \| Quotes From Pages 282-292 1.The overall efficiency is OUT IN 257 W100% 100% 63.6% 403.8 W P P·= × = × = Scan to Download Scan to Download 2.Assuming that the rotational losses are still 51 W, this motor will still be able to speed up because is 78.7 W, while the rotational losses are 51 W, so there is more power than it required to cover the rotational losses. 3.The induced torque is 279 AG ind sync 354.5 W 1.88 N m 2 rad 1 min1800 r/min 1 r 60 s P Ä ÀÉ= = 4.The slip is 0.05, so... Scan to Download Chapter 10 \| Quotes From Pages 293-300 1.The power factor supplied by the utility is 0.838 lagging. 2.The apparent power supplied by the utility is 126.4 kVA. 3.The current supplied by the utility is 152 A. 4.The power factor of the generator is 0.986 lagging. Chapter 11 \| Quotes From Pages 301-307 1....the best choice for coil pitch would be 4/5 or 6/5. 2....the total voltage produced to the sum of the magnitudes of each component voltage. 3.To totally eliminate the seventh harmonic of voltage in an ac machine armature, the pitch factor for that harmonic must be zero. Chapter 12 \| Quotes From Pages 308-313 1.The direct-axis reactance and quadrature-axis reactance play a crucial role in defining the operational characteristics of synchronous machines. 2.When determining the internal generated voltage, one must Scan to Download Scan to Download consider not only the terminal voltage but also the current flowing through the armature and the machine's reactances. 3.The reluctance torque contributes significantly to the power output of salient-pole generators, which can be a decisive factor in their efficiency compared to non-salient designs. 4.The power supplied by the machine is dependent on the torque angle, which must be optimized for maximum output. 5.In the pursuit of maximizing power generation, understanding the distinctions between salient-pole and cylindrical rotor constructions is imperative for modern electrical engineers. Scan to Download Chapter 13 \| Quotes From Pages 314-353 1.If we find the average and rms values over the interval from À/6 to 5À/6 (one period), t will be the same as the average and rms values of the entire waveform, and they can be used to calculate the ripple factor. 2.The ripple can be calculated with MATLAB using the ripple function developed in the text. 3.If the voltage charging the capacitor could be made constant or nearly so, then the feedback effect would be stopped and the circuit would be less sensitive to voltage variations. 4.This circuit is a single-phase voltage source inverter. 5.The average voltage is (5 / 6) / (6) 1 / 3( ) sin 2 DC M MV v t dt V t dT À É À t É À= 6.Diodes D1 and D2 together with the transformer form a full-wave rectifier. 7.The SCR will remain on until the current flowing through it drops below IH. Scan to Download Scan to Download Chapter 14 \| Quotes From Pages 354-356 1.Please note that some or all of the following errata will be corrected in future reprints of the book, so they may not appear in your copy of the text. 2.Consider a simple power system consisting of an ideal voltage source, an ideal step-up transformer, a transmission line, an ideal step-down transformer, and a load. 3.What transformer turns ratio would be required to reduce the transmission line losses to 1% of the total power produced by the generator? Scan to Download Electric Machinery Fundamentals Questions View on Bookey Website Chapter 1 \| Introduction to Machinery Principles\| Q&A 1.Question How can you convert motor shaft speed from revolutions per minute (r/min) to radians per second? Answer:To convert the shaft speed from r/min to rad/s, use the formula: ( \text{speed in rad/s} = \text{speed in r/min} \times \frac{2\pi}{60} ) For example, for a speed of 1800 r/min: [ \omega = 1800 \times \frac{2\pi}{60} \approx 188.5 \text{ rad/s} ] 2.Question What is the relationship between torque, angular acceleration, and moment of inertia for a rotating Scan to Download Scan to Download flywheel? Answer:The equation that relates these quantities is: [ \tau = J \alpha ] Where ( \tau ) is the applied torque, ( J ) is the moment of inertia, and ( \alpha ) is the angular acceleration. For a flywheel with a moment of inertia of 4 kg·m² and a torque of 6 N·m, applying the formula gives the angular acceleration, which helps calculate the final speed after a certain time. 3.Question What is mechanical power, and how can it be calculated for a motor supplying torque? Answer:Mechanical power can be calculated using the formula: [ P = \tau \cdot \omega ] Where ( P ) is power in watts, ( \tau ) is torque in N·m, and Scan to Download Scan to Download ( \omega ) is angular speed in rad/s. For a motor providing 50 N·m of torque at 1500 r/min, converting the speed to rad/s first, and then multiplying gives: [ P \approx 7854 W ] or approximately 10.5 hp. 4.Question What factors contribute to the reluctance of a magnetic circuit? Answer:Reluctance in a magnetic circuit is affected by the material of the core (its permeability), geometry (length and cross-sectional area), and presence of air gaps. For instance, the reluctance ( R ) is calculated using: [ R = \frac{l}{\mu \cdot A} ] Where ( l ) is the length of the magnetic path, ( \mu ) is the permeability of the material, and ( A ) is the cross-sectional area. 5.Question Scan to Download Scan to Download How can you find the flux density in different legs of a magnetic core? Answer:The flux density ( B ) in a magnetic core can be found using: [ B = \frac{\Phi}{A} ] Where ( \Phi ) is the total flux, and ( A ) is the area of the core's cross-section. For varied legs of a core, the flux density calculations may differ based on the geometry and dimensions of each leg. 6.Question What is the significance of Lenz's Law in determining voltage in a coil due to changing magnetic flux? Answer:Lenz's Law states that an induced electromotive force (emf) will always work in such a direction as to oppose the change in magnetic flux that produced it. It emphasizes that if the magnetic flux increases, the induced voltage will act to decrease it, producing a voltage of opposite polarity. 7.Question How can real-world machines achieve variable speed control? Scan to Download Scan to Download Answer:Speed control in real-world machines can be achieved by manipulating either the voltage supplied to the motor or the magnetic flux density within the machine. Specifically, reducing the voltage decreases speed, while decreasing the magnetic flux density increases speed. Chapter 2 \| Transformers\| Q&A 1.Question How do you calculate the equivalent circuit of a transformer referred to its low-voltage side? Answer:To find the equivalent circuit of a transformer referred to its low-voltage side, first compute the turns ratio (a) using the formula a = V_primary / V_secondary. Then, apply the formula to convert the resistances and reactances from the high-voltage side to the low-voltage side using the formula: Z_secondary = Z_primary / a^2, where Z is the impedance (both resistance and reactance components). Repeat this for both the resistance and reactance values. Scan to Download Scan to Download 2.Question What is the significance of per-unit calculations in transformer analysis? Answer:Per-unit calculations standardize electrical quantities, allowing for easier comparison across different voltage levels and transformer ratings. It expresses quantities relative to a defined base value, reducing complexity when analyzing transformers and interconnected systems. 3.Question What is the voltage regulation of a transformer and how is it calculated? Answer:Voltage regulation is a measure of how much the voltage drops when a transformer is loaded compared to its no-load condition, expressed as a percentage. It is calculated using the formula: VR = [(V_no-load - V_load) / V_load] × 100%. In practical terms, it indicates the effectiveness of a transformer in maintaining output voltage under load. 4.Question In transformer efficiency calculations, how do you determine copper and core losses? Scan to Download Scan to Download Answer:Copper losses (I²R losses) can be calculated using the equation: P_loss = I² × R, where I is the current flowing through the transformer and R is the resistance. Core losses (also known as iron losses) are primarily constant losses occurring in the transformer core, which can be estimated through open-circuit test results (using the power measured during the test). Total losses are the sum of both copper and core losses, and efficiency can be computed as: Efficiency = (Output Power / (Output Power + Total Losses)) × 100%. 5.Question What changes occur in transformer performance when switching from a 60 Hz to a 50 Hz operation? Answer:When switching from 60 Hz to 50 Hz, the transformer's rated apparent power and voltage capabilities decrease due to the lower frequency. This is because lower frequency results in increased magnetic flux for the same input voltage, potentially leading to core saturation. Consequently, transformers are rated lower in kVA at 50 Hz due to the need to avoid saturation, and care should be taken Scan to Download Scan to Download in applications to ensure they are not overloaded. 6.Question What is the impact of connecting a load with a leading power factor to the transformer? Answer:Connecting a load with a leading power factor to the transformer can improve voltage regulation, as it offsets lagging currents in the system. This can lead to a reduced overall reactive power demand, which in turn can lower losses in the power system and improve the efficiency of transformers by maintaining the output voltage closer to the nominal value, especially under varying load conditions. 7.Question How are transmission losses determined in a transformer-connected power system? Answer:Transmission losses in a transformer-connected power system are determined by calculating the resistive losses in transformers and transmission lines. This can be done using: P_loss = I² × R for each section of the system where I is the current and R is the resistance of the Scan to Download Scan to Download transformer or line. It is important to account for both short-circuit and open-circuit tests when determining resistance and other parameters that influence transmission losses. 8.Question How does an autotransformer differ from a conventional transformer in terms of power handling capabilities? Answer:An autotransformer has a greater power-handling capability compared to a conventional transformer due to its shared windings, which improves efficiency and reduces losses. Since part of the circuit impedance does not traverse through the entire transformer to the load, fewer turns are required, allowing more current to pass without additional heating, effectively increasing the kVA rating under certain conditions. Chapter 3 \| AC Machinery Fundamentals\| Q&A 1.Question How is the induced voltage in a rotating loop calculated and what factors influence it? Scan to Download Scan to Download Answer:The induced voltage in a simple rotating loop can be calculated using the formula: e_ind(t) = 2 B l r É sin(É t), where B is t field strength (in Tesla), l is the length of the loop, r is the radius of the loop, and É is the ang(in rad/s). Key factors influencing the induced voltage are the strength of the magnetic field, the dimensions of the loop (length and radius), and the speed of rotation (angular velocity). 2.Question What is the frequency of the voltage produced in a rotating loop and how is it derived? Answer:The frequency of the voltage produced in the loop can be derived from its angular velocity using the equation: f = É / (2 À), where É is the angular speecase, with É = 377 rad/s, the frequency wo as f = 377 / (2 À) = 60 Hz. 3.Question Given a 10 © load connected to the loop,current calculated and what does it tell us about the Scan to Download Scan to Download system? Answer:The current flowing through the resistor is calculated using Ohm's Law: I = V / R, where V is the induced voltage and R is the resistance. In our case, I = 2.26 sin(377t) / 10 = 0.226 sin(377t) A. This indicates that the system is capable of supplying a time-varying current based on the induced voltage, demonstrating the relationship between electrical generation and mechanical motion. 4.Question What mechanical essence does the induced torque represent in the functioning of a rotating loop? Answer:The induced torque represents the rotational force acting on the loop due to the interaction between the magnetic field and current flowing through the loop. It is given by the formula Äind = 2 r I B signifying how electrical energy is converted into mechanical work, allowing the loop to turn or act against mechanical resistance. 5.Question Scan to Download Scan to Download What conclusions can we draw about electrical and mechanical power in this generator scenario? Answer:The mechanical power consumed by the loop equals the electrical power generated (51.3 W mechanical equals 51.1 W electrical), indicating that the generator is efficiently converting mechanical energy into electrical energy. This reflects the principle of energy conservation and efficiency in electric machinery systems, illustrating the conversion between different forms of energy. 6.Question How does the number of poles in an AC machine affect the speed of magnetic field rotation? Answer:The speed of magnetic field rotation in AC machines is inversely proportionate to the number of poles and the frequency, calculated using the formula: n_s = 120 f / P, where n_s is the speed in r/min, f is the frequency in Hz, and P is the number of poles. An increase in poles decreases the speed of the magnetic field rotation for a given frequency. 7.Question Scan to Download Scan to Download What historical context influenced the operation frequencies of early AC motors? Answer:In early AC motor development, machine designers faced challenges with core losses due to material limitations, which led them to operate motors at lower frequencies (25 Hz) to minimize these losses. However, this resulted in visible flickering in lighting systems, necessitating a separate higher-frequency (60 Hz) system for lighting applications. 8.Question How did the AC power systems evolve regarding frequency and what impact did it have on existing machinery? Answer:As AC motors improved to handle direct connection to 60 Hz systems, the need for lower frequency (25 Hz) systems diminished, prompting the phase-out of such systems. Existing 25 Hz motors remained functional through motor-generator sets that allowed factory owners to continue using their machinery by converting 60 Hz power back to 25 Hz. Scan to Download Scan to Download 9.Question What role did simulations and modifications (like the MATLAB example) play in understanding AC machine behavior? Answer:Simulations, like the MATLAB modification to swap currents in phases, offer insightful visualizations of magnetic fields and their interactions. Such adjustments allow engineers to study the effects of changing input conditions on machine behavior, enhancing their understanding of AC machines and fostering innovation in design and application. Scan to Download Chapter 4 \| Synchronous Generators\| Q&A 1.Question What is the relationship between the number of poles in synchronous machines and their operational frequencies? Answer:The operation speed of synchronous machines is directly related to the frequency of the power they are designed to use. Using the equation n = (120 f) / P, where n is speed in RPM, f is frequency, and P is the number of poles, one can determine that to convert power from a machine operating at one frequency (like 50 Hz) to another (like 60 Hz), the number of poles must be adjusted. For example, a 10-pole synchronous motor must be coupled to a 12-pole synchronous generator to enable this conversion, thus achieving synchronous operation. 2.Question How does the synchronous reactance affect the performance of a synchronous generator? Scan to Download Scan to Download Answer:Synchronous reactance represents the opposition that the generator's armature windings present to the flow of alternating current due to both inductive and resistive effects. A higher synchronous reactance reduces the ability of the generator to supply real power to the load, thus influencing the internal generated voltage and efficiency under load. When rated currents and conditions are applied, the terminal voltage is affected by losses associated with both armature resistance and synchronous reactance. 3.Question What are the implications of adjusting the field current in synchronous generators during fluctuating load conditions? Answer:Adjusting the field current in a synchronous generator controls the voltage output and stability under changing load conditions. When load demands increase, maintaining adequate field current ensures that the internal generated voltage compensates for increased demand without dropping terminal voltage. Conversely, if the load is reduced, Scan to Download Scan to Download reducing the field current can prevent over-voltage conditions and maintain efficiency. 4.Question What factors determine the voltage regulation of a synchronous generator? Answer:Voltage regulation in a synchronous generator depends on the synchronous reactance, armature resistance, and the load power factor. It is defined as the difference between no-load and full-load voltages expressed as a percentage of the full-load voltage. The regulation can be either positive if the loaded voltage is less than the no-load voltage or negative if the loaded voltage is greater, especially under leading power factors. 5.Question What happens to the terminal voltage when the load is suddenly removed from a running synchronous generator? Answer:When the load is abruptly disconnected, the terminal voltage of the synchronous generator increases sharply, often exceeding the rated terminal voltage due to the sustained Scan to Download Scan to Download internal generated voltage until the generator stabilizes at a new steady-state condition. The system will operate at the higher generated voltage until it finds a new equilibrium. 6.Question How do you determine the steady-state power and torque that a generator's prime mover must supply? Answer:To find the steady-state power and torque, one must account for the losses present during operation. The prime mover needs to provide enough input power to equal the output power plus all associated losses (core losses, winding losses, etc.). The torque can then be calculated using the relationship between power and torque, taking into account the generator speed. 7.Question Explain the importance of constructing a capability curve for a synchronous generator. What insights does it provide? Answer:A capability curve for a synchronous generator illustrates the applicable ranges of real and reactive power that the machine can provide without exceeding operational Scan to Download Scan to Download limits. It helps in identifying maximum load conditions and stability limits, ensuring that the generator operates efficiently within its parameters and assists in planning for maintenance or operational adjustments under varying load conditions. 8.Question What are the consequences of using a synchronous generator with improper speed or pole configurations? Answer:Using a synchronous generator with incorrect speeds or poles can lead to failure to synchronize with the power system, excessive vibration, mechanical stresses, or reduced efficiency. This may also result in a failure to accurately meet the load power factor, leading to instability and potential breakdowns. 9.Question In what scenarios would an engineer prefer a single larger generator over multiple smaller ones, despite potential risks? Answer:An engineer may prefer a larger generator for reasons such as reduced complexity in control systems, lower Scan to Download Scan to Download initial costs per unit of power, and potentially higher operational efficiency. However, the risk lies in the total loss of generation capacity if that single unit fails. This trade-off must be carefully evaluated against reliability needs and maintenance strategies. Chapter 5 \| Synchronous Motors\| Q&A 1.Question What factors determine the speed of a synchronous motor, and how is it mathematically expressed? Answer:The speed of a synchronous motor is determined by the frequency of the supply current and the number of poles in the motor. It is mathematically expressed as: d = (120 f) / P, where d is the synchronous speed in revolutions per minute (RPM), f is the frequency in hertz (Hz), and P is the number of poles. Scan to Download Scan to Download 2.Question In what scenarios would a synchronous motor operate at leading versus lagging power factor? Answer:A synchronous motor operates at leading power factor when it is supplying reactive power to the grid, such as during conditions of light loading where the motor can over-excite. Conversely, it operates at lagging power factor when the motor is under load and requires reactive power from the grid. 3.Question How does increasing the field current affect the torque and power factor of a synchronous motor? Answer:Increasing the field current increases the internal generated voltage (E_A), leading to an increase in the reactive power supplied by the motor. This can result in a better power factor (if the motor was initially lagging), as it may shift the motor's operation from consuming reactive power to supplying it, thereby improving its overall efficiency. Scan to Download Scan to Download 4.Question What is the significance of the torque an operation of synchronous motors? Answer:The torque angle (´) is significant indicates the relationship between the internal generated voltage (E_A) and the terminal voltage (V). A larger torque angle generally indicates higher power output but can also bring the motor closer to instability, particularly if it becomes too large, which could result in losses or motor failure. 5.Question What is the effect of changing the frequency supply to a synchronous motor? Answer:Changing the frequency supply to a synchronous motor alters its synchronous speed. A decrease in frequency lowers the synchronous speed, while an increase raises it. This change can also impact the motor's performance characteristics, including torque and efficiency. 6.Question How is the output power of a synchronous motor related to its speed and internal generated voltage? Scan to Download Scan to Download Answer:The output power of a synchronous motor is maximized at synchronous speed and is directly proportional to the internal generated voltage (E_A) and the torque angle (´). As speed increases towards synchronomotor can deliver more power before converging on reactive power limits. 7.Question What challenges do synchronous motors face regarding stability, and how is it affected by the power factor? Answer:Synchronous motors face stability challenges if the load becomes too large or if the torque ancritical point. Operating at a lagging power factor increases the chance of losing synchronism. Contrastingly, operation at leading power factor tends to provide stability as the motor supplies reactive power. 8.Question What calculations are involved in determining the armature current of a synchronous motor at full load? Answer:To determine the armature current (I_A) at full load, Scan to Download Scan to Download the formula used is I_A = P / ("3 V PF output power, V is the phase voltage, and PF is the power factor. This current is also adjusted for phase considerations specific to the motor's connection (Y or ”Chapter 6 \| Induction Motors\| Q&A 1.Question What is the significance of slip in induction motors and how is it calculated? Answer:Slip is a measure of how much the rotor lags behind the magnetic field. It is calculated using the formula: slip (s) = (Ns - Nr) / Ns, where Ns is synchronous speed and Nr is rotor speed. 2.Question How does the rotor's speed affect its frequency? Answer:The rotor frequency can be determined by the formula: rotor frequency (fr) = s f, where s is the slip and f is the supply frequency. This indicates that as the slip increases, the rotor frequency increases. 3.Question What is the relationship between induced torque and Scan to Download Scan to Download rotor current in induction motors? Answer:Induced torque is directly related to rotor current. When rotor current increases, induced torque increases, allowing the motor to handle higher loads. 4.Question How does changing the resistance in the rotor circuit affect the motor's performance? Answer:Increasing rotor resistance typically leads to a higher slip, which can decrease the rotor speed, increase rotor current, and change the induced torque, impacting overall efficiency. 5.Question What occurs to the induced torque during plugging? Answer:During plugging, the poles of the motor's magnetic field are reversed, causing the motor to develop an opposing induced torque, rapidly decelerating the motor. 6.Question What factors influence the overall efficiency of an induction motor? Answer:Overall efficiency is influenced by stator and rotor Scan to Download Scan to Download losses, copper losses, mechanical losses, and how effectively electrical power is converted to mechanical power. 7.Question What role does the Thevenin equivalent circuit play in analyzing induction motors? Answer:The Thevenin equivalent circuit simplifies the analysis of motor behavior by representing its impedance and voltage in a way that makes it easier to calculate current, torque, and power. 8.Question In what scenario would a Y-” starter be induction motor? Answer:A Y-” starter is beneficial during reduces the starting voltage and current, helping prevent excessive stress on the electrical components. 9.Question How does the speed-torque characteristic of a dual-cage rotor differ from a single-cage rotor? Answer:The dual-cage rotor typically provides better performance at low speeds due to the high resistance of the Scan to Download Scan to Download outer cage, which allows for greater torque at startup compared to the single-cage design. 10.Question What are the implications of using a step-down autotransformer on motor starting current? Answer:Using a step-down autotransformer reduces the phase voltage supplied to the motor, consequently lowering the starting current drawn from the supply, alleviating stress on the electrical system. Scan to Download Chapter 7 \| DC Machinery Fundamentals\| Q&A 1.Question How can you determine if a DC machine is operating as a motor or a generator? Answer:To determine if the machine operates as a motor or a generator, check the induced voltage compared to the external battery voltage. For instance, at a shaft speed of 500 rad/s, if the induced voltage is greater than the external voltage, the machine acts as a generator (charging the battery). If it's less, the machine operates as a motor (drawing current from the battery). In this chapter's example, since the induced voltage of 50 V exceeds the 48 V battery, it operates as a generator. 2.Question What effect does increasing the rotor speed have on the current flow in a DC machine? Answer:Increasing the rotor's speed will increase the induced voltage, leading to a change in current flow. For example, Scan to Download Scan to Download increasing the speed from 500 rad/s to 550 rad/s increases the induced voltage to 55 V, resulting in a higher corresponding current flow out of the machine. This demonstrates the direct relationship between rotor speed and current flow in DC machines. 3.Question How is the resistance of the armature winding calculated? Answer:The armature resistance is calculated by summing the resistances of all coils in parallel. For instance, if each coil has a resistance of 0.04 !& and there a the total resistance can be derived using the formula R_total = R_coil / number_of_parallel_paths. In this case, that equals 0.08 !&. 4.Question What is the significance of understanding the relationship between induced voltage and induced torque in a DC machine? Answer:Understanding this relationship is crucial because it reveals how electrical and mechanical energy conversion Scan to Download Scan to Download occurs. The equation P_conv = E_ind I illustrates that the power converted in a DC machine links the electrical power consumed and mechanical power produced. This balance is fundamental for effective motor design and control. 5.Question How do you determine the number of current paths in an armature winding? Answer:The number of current paths in an armature can be determined based on the winding type. For example, with a simplex lap winding on a machine with m poles, the number of paths is m. A duplex lap winding doubles that, leading to 2m paths. This understanding is essential for calculating currents and effectively managing power distribution. 6.Question What is induced torque and how is it calculated in a DC machine? Answer:Induced torque in a DC machine is calculated using the equation Ä_ind = K ¦ I, where K is considering the winding and number of con Scan to Download Scan to Download magnetic flux per pole, and I is the current. The induced torque represents the mechanical output generated by the machine based on its electrical input, highlighting the machine's efficiency and performance under load. 7.Question Can you explain the operation of a simple two-pole DC motor and how variables like voltage and load affect it? Answer:In a simple two-pole DC motor, if connected to a voltage source, the induced current generates a torque that causes the rotor to rotate. If the load increases (e.g., the motor needs to do more work), it draws more current, which affects both the induced voltage and the resultant torque. For example, consuming 600 W results in specific current and torque calculations, showcasing how load impacts machine operation. 8.Question What is the impact of brushes in a DC machine and how should they be positioned for optimal operation? Answer:Brushes in a DC machine collect current from the Scan to Download Scan to Download armature winding while maintaining electrical contact with the commutator segments. Correct positioning is essential for optimal commutation, allowing the brushes to short out windings that are actively generating voltage. For example, in a duplex winding, each brush should span across the correct number of segments to ensure smooth operation without sparking or loss of efficiency. 9.Question Describe how you would determine the required magnetic flux for a generator to produce a specific voltage at no-load conditions. Answer:To determine the required magnetic flux, you'd use the induced voltage formula E = K ¦ Énecessary constants and parameters such as speed and path numbers. For example, given a generator's voltage goal, you can rearrange the equation to find ¦ needethe desired speed, ensuring the machine functions effectively at its rated output. 10.Question How do the terms simplex and duplex winding influence Scan to Download Scan to Download the current distribution in a DC machine? Answer:Simplex and duplex winding designs influence the number of parallel current paths in a DC machine. A simplex winding introduces one path for current flow, distributing the entire current across fewer paths, while a duplex winding creates multiple paths, effectively lowering the per-path current. This design choice impacts the machine's capacity to handle higher loads and its overall efficiency. Chapter 8 \| DC Motors and Generators\| Q&A 1.Question How does the speed of a DC motor change when the adjustable resistance is modified? Answer:Increasing the adjustable resistance in the field circuit of a DC motor generally leads to an increase in speed at no load, as it decreases the field current, thereby reducing generated magnetic flux. This results in a higher speed for a given armature voltage. Conversely, decreasing the resistance leads to lower speed. Scan to Download Scan to Download 2.Question What is the significance of armature reaction in the operation of DC motors? Answer:Armature reaction refers to the effect of the magnetic field generated by the armature current which opposes the main magnetic field. This can lead to a reduction in motor efficiency and speed, particularly under heavy loads. Understanding and managing armature reaction is crucial for maintaining desired motor performance. 3.Question How do you calculate the speed regulation of a shunt DC motor? Answer:Speed regulation is calculated by taking the difference between the no-load speed and full-load speed, divided by the full-load speed, all multiplied by 100%. It indicates how much the speed of the motor changes from no-load to full-load conditions. 4.Question What role does the magnetization curve play in determining the performance of a DC motor? Scan to Download Scan to Download Answer:The magnetization curve describes the relationship between the field current and the generated voltage in a DC motor. It is essential for predicting how changes in field current will affect motor speed and torque output. 5.Question How can you maximize the efficiency of a DC motor at full load? Answer:To maximize efficiency at full load, ensure that armature and field currents are optimized according to the motor design, minimize mechanical and stray losses, and operate the motor close to its rated conditions. 6.Question Why is starting current of a DC motor much higher than its full-load current? Answer:Starting current is high due to very low initial back EMF when the motor is at rest. This means the motor draws much more current than during normal operation, potentially up to ten times or more its rated full-load current, which can risk damage unless managed properly with soft starters or Scan to Download Scan to Download resistive methods. 7.Question What is the maximum and minimum no-load speed a shunt DC motor can achieve with adjustable resistances? Answer:The maximum no-load speed occurs when adjustable resistance is at its maximum value, leading to minimal field current. The minimum speed is achieved with the lowest adjustable resistance, which maximizes field current. The specific speeds are determined from the magnetization curve. 8.Question How can the impact of armature reaction be mitigated in DC motors? Answer:Armature reaction can be mitigated by using compensating windings, interpoles, or properly adjusting the field excitation to maintain consistent performance under varying load conditions. 9.Question What is the consequence of an open circuit in the field winding of a DC motor? Answer:An open circuit in the field winding typically results Scan to Download Scan to Download in the motor accelerating to a dangerously high speed because the field current drops to zero, leading to a very low back EMF which permits excessive current draw, ultimately risking mechanical failure. 10.Question How to determine the field current when the terminal voltage of a separately excited generator is known? Answer:Field current can be determined by rearranging the formula for internal generated voltage based on Ohm's law and the generator's magnetization curve to find the corresponding field current for the known terminal voltage. Chapter 9 \| Single-Phase and Special-Purpose Motors\| Q&A 1.Question What is the significance of the slip in an induction motor, and how does it influence the motor’s performance? Answer:Slip refers to the difference between the synchronous speed of the motor's magnetic field and its actual rotor speed, expressed as a fraction of the synchronous speed. It is significant because it Scan to Download Scan to Download determines the torque produced by the motor: a higher slip indicates greater torque production up to a certain point. However, excessive slip can lead to reduced efficiency and overheating. 2.Question How does rotational loss affect the efficiency of an induction motor? Answer:Rotational losses, such as friction and windage, occur even when the motor is stationary and become more significant at higher speeds. These losses contribute to the total power loss in the motor, thereby reducing its efficiency. If the output power is less than the power consumed minus these losses, the motor efficiency decreases. 3.Question What happens to the induced torque if an auxiliary winding fails while the motor is running? Answer:If the auxiliary winding fails, the rotor's ability to generate torque is compromised; however, if the rotor is already in motion, it may still produce some torque based on Scan to Download Scan to Download the remaining winding. The motor may continue to accelerate if the induced torque exceeds the rotational losses, but if it cannot produce sufficient torque, it will slow down. 4.Question Can you explain how to calculate the air-gap power in an induction motor? Answer:The air-gap power (PAG) is calculated by considering the current through the stator and the resistance of the windings. It is given by the formula: PAG = I^2Req, where I is the stator current and Req is the equivalent resistance seen from the stator's perspective. The output of the motor will be this air-gap power minus the power losses from the rotor. 5.Question How do you define overall efficiency in the context of induction motors? Answer:Overall efficiency is the ratio of the useful output power (mechanical power provided by the motor) to the total input power (electrical power consumed). It is typically Scan to Download Scan to Download expressed as a percentage, calculated usin (Output Power / Input Power) × 100%. 6.Question Why is the stator power factor important in the analysis of an induction motor? Answer:The stator power factor indicates how effectively the input power is being converted into useful work. A low power factor suggests that a significant portion of the input power is reactive (not doing useful work), which can lead to higher energy costs and inefficiencies. In practical applications, a power factor close to 1 is desirable. 7.Question How do load torques differ from induced torques in operation? Answer:Induced torque is the torque produced by the motor's electromagnetic field when the motor is running, while load torque is the torque required to maintain the mechanical load connected to the motor. The motor operates effectively when the induced torque is equal to or greater than the load torque; Scan to Download Scan to Download otherwise, the motor will slow down. 8.Question What are some practical applications of different types of motors mentioned in this chapter? Answer:- Universal motors are commonly used in vacuum cleaners due to their high starting torque and variable speed capabilities. - Capacitor-start induction motors are preferred in refrigerators and air conditioners for their efficiency and starting characteristics. - Split-phase motors are suitable for fans and other light-duty applications where low starting torque is acceptable. 9.Question What is the relationship between electrical angle and mechanical angle in a stepper motor? Answer:In a stepper motor, the relationship is defined by the number of poles. For a three-phase stepper motor, the electrical angle (in degrees) varies linearly with the mechanical angle. For example, if a stepper motor steps in Scan to Download Scan to Download increments of 10 degrees mechanically, and it has 12 poles, it would produce a 2-degree electrical step. 10.Question How do you determine the number of pulses needed for a stepper motor to achieve a specific speed? Answer:The number of pulses per second required can be calculated using the formula: pulses = (speed in RPM × number of poles) / 60. For instance, to achieve a speed of 600 RPM with a motor having 12 poles, you would need 120 pulses per second. Scan to Download Chapter 10 \| Appendix A: Review of Three-Phase Circuits\| Q&A 1.Question What is the significance of calculating power factor in three-phase circuits? Answer:The power factor, defined as the cosine of the phase angle (¸) between the current an indicates how effectively electrical power is being converted into useful work. A power factor close to 1 signifies efficient use of power, while a lower power factor denotes more reactive power, which can lead to higher energy costs and potential issues in power systems. Understanding power factor helps in optimizing energy consumption and improving the operational efficiency of electrical installations. 2.Question How can the conversion between ” and Yimpact load analysis in three-phase systems? Answer:Converting a ” (Delta) connected l equivalent Y (Wye) load reduces complex impedance Scan to Download Scan to Download calculations in power analysis. For example, in calculating the total power, line currents, and voltage drops in three-phase systems, the Y configuration allows for simpler analysis because it relates phase voltages and neutral connections more directly. This transformation makes it easier to derive phase relationships and load distributions, thus simplifying the overall analysis of the circuit. 3.Question What role do real and reactive power play in the performance of electrical systems? Answer:Real power (measured in watts) represents the actual usable power that does work in the circuit, while reactive power (measured in VARs) is the power that oscillates between the source and reactive components, like inductors and capacitors, without performing any work. A high reactive power can lead to decreased efficiency in the system by causing voltage fluctuations and increasing losses, hence both types of power must be managed to maintain a stable and efficient electrical system. Scan to Download Scan to Download 4.Question In what ways does the impedance angle affect the overall performance of a load? Answer:The impedance angle determines how much current lags or leads the voltage in an AC circuit, affecting the power factor and thus the overall energy efficiency of the load. A greater lagging angle indicates more reactive power presence, leading to increased losses in transmission. By adjusting the impedance angle, engineers can optimize performance, improve power factor, and reduce costs associated with inefficiencies in the electric power system. 5.Question Why does closing a switch in a power circuit change the total current supplied by the generator? Answer:When the switch is closed, additional loads or capacitance networks enter the circuit that affects current draw. Importantly, a capacitor bank can supply reactive power, reducing total reactive demand from the generator. This can lead to decreased overall current draw on the Scan to Download Scan to Download generator, as it does not have to provide as much reactive power. Therefore, while real power demand may increase, the total line current supplied can actually decrease due to the capacitor compensating reactive power demands. 6.Question How is the relationship between phase voltage and line voltage derived in a Y-connected configuration? Answer:In a Y-connected system, the phase voltage (V_ph) is related to the line voltage (VL) by the fo V_ph. This is derived from the geometry of the phasor diagram, where each phase voltage is offset by 120 degrees, causing the line voltage to be the vector sum of two phase voltages. This leads to the understanding that line voltages are higher than phase voltages by a factorthe importance of correct phase handling in distribution systems. 7.Question What are the implications of assuming lossless transmission lines in power system calculations? Scan to Download Scan to Download Answer:Assuming lossless transmission lines simplifies calculations by neglecting voltage drops due to line resistance and reactance, which can skew real-world results. While this assumption helps in understanding theoretical efficiencies and ideal behaviors, real power systems experience losses which can alter voltage levels at loads and affect stability. Thus, in practical scenarios, engineers must account for losses to provide accurate assessments and designs. Chapter 11 \| Appendix B: Coil Pitch and Distributed Windings\| Q&A 1.Question What is the importance of selecting the right coil pitch in electric machinery? Answer:The coil pitch plays a critical role in eliminating unwanted harmonic components in the voltage produced by electric machinery. For instance, in three-phase systems, selecting the appropriate fractional-pitch winding can eliminate specific harmonics such as the fifth harmonic, Scan to Download Scan to Download thereby improving the quality of the produced voltage and ensuring efficient machine operation. 2.Question How can the winding distribution factor influence the total voltage in a stator? Answer:The winding distribution factor impacts the total voltage produced by a stator by determining the effective contribution of each coil phase to the overall voltage. A proper distribution factor results from understanding the phase arrangement and the angles between coils. For example, by optimizing the distribution factor, you can maximize the effective voltage while minimizing harmonic distortions. 3.Question What makes the winding factor significant in synchronous generators? Answer:The winding factor is significant because it accounts for the combined effects of coil pitch and distribution on the generated voltage. It serves as a multiplier for the effective Scan to Download Scan to Download voltage produced, which can help in achieving the desired output characteristics from a synchronous generator. A high winding factor indicates efficient coupling of the magnetic field and the electrical windings. 4.Question How does coil pitch affect harmonic suppression in electrical machines? Answer:Coil pitch directly influences the suppression of harmonic components in voltage. For example, certain pitches can be strategically chosen to completely eliminate specific harmonics, such as the fifth or seventh harmonic, which leads to a smoother voltage waveform and reduces potential operational issues and losses in electric motors and generators. 5.Question What is the impact of the number of slots on the performance of a winding in a synchronous machine? Answer:The number of slots is essential in determining how the coils are distributed and how effectively they can Scan to Download Scan to Download suppress harmonics. Each slot contributes to the phase balance and the overall efficiency of the machine. A higher number of slots can provide more flexibility in achieving desired coil pitches and variations for optimal performance. 6.Question Can you illustrate how phase connections in a winding impact the voltage output of a synchronous machine? Answer:In a three-phase winding connecteconfiguration, the phase voltage impacts line voltage significantly. For example, if the phase voltage is calculated based on individual phase groups connected in series, the total voltage output will be higher. An example can be seen in using 4 coils per phase where the individual contributions stack to yield impressive total voltages for practical applications. 7.Question What are the calculation steps to find the terminal voltage in a three-phase synchronous generator? Answer:To find the terminal voltage in a three-phase Scan to Download Scan to Download synchronous generator, you first determine the phase voltage using the formula: Voltage = (Flux × Turns × Frequency × Pitch Factor × Distribution Factor). After finding the phase voltage, you can determine the line voltage using the relation, Line Voltage =Voltage, thereby completing the calculations for effective voltage output. 8.Question What role does the speed of rotation play in the generated frequency of a synchronous machine? Answer:The speed of rotation determines the electrical frequency produced, as it directly relates to the pole count and the mechanical speed. The frequency is calculated using the formula: Frequency (Hz) = (Speed in RPM × Number of Poles) / 120. Understanding this relationship is crucial for designing machines that operate at specific frequencies to meet load requirements. Scan to Download Scan to Download 9.Question How can understanding harmonic suppression aid in designing efficient electric machinery? Answer:By understanding harmonic suppression mechanisms, engineers can design electric machinery that minimizes power losses, heat generation, and vibration issues related to harmonics. Choosing appropriate coil pitches and winding distributions allows for tailored performance, enhancing both the reliability and efficiency of electric machinery. Chapter 12 \| Appendix C: Salient Pole Theory of Synchronous Machines\| Q&A 1.Question What is the significance of the field current in determining the terminal voltage of a synchronous generator at no load? Answer:The field current determines the magnetic field strength within the synchronous generator. In this scenario, to achieve a terminal voltage of 13.8 kV at no load, a specific field current of 3.50 A is Scan to Download Scan to Download required, as indicated by the open-circuit characteristic of the generator. This relationship illustrates how controlling the field current directly influences the generator's output voltage. 2.Question How do you calculate the internal generated voltage of a synchronous generator at rated conditions? Answer:To calculate the internal generated voltage (E') of the synchronous generator under rated conditions, we must consider the line current (2092 A), the armature resistance (0.2 ©), and the reactances of the machine © and quadrature-axis of 1.8 ©). By apply equations including the voltage drop across these parameters, we find that E' is about 11,496 V at an angle of 17.8°, showcasing the generator's operational dynamics and the impact of reactance on the generated voltage. 3.Question What does the reluctance torque contribute to the overall power of the generator? Scan to Download Scan to Download Answer:In this generator, the reluctance torque contributes approximately 8.6 MW to the overall power output, which totals 42.2 MW (comprising both rotor and reluctance contributions). This 8.6 MW reflects the portion of power generated due to the machine's magnetic reluctance, underlining the importance of magnetic characteristics in the performance of synchronous machines. 4.Question What is the voltage regulation of the generator, and why is it important? Answer:The voltage regulation of the generator at rated conditions is approximately 29.8%. This value indicates the difference between the no-load and full-load terminal voltage expressed as a percentage of full-load voltage. A lower voltage regulation indicates better voltage stability under load, which is crucial for maintaining performance in power systems. 5.Question How does the maximum power output differ between a salient-pole generator and a cylindrical rotor generator? Scan to Download Scan to Download Answer:The maximum power output for the salient-pole generator, under the described conditions, is 392.4 MW, which is higher than the 364.7 MW achievable by a non-salient (cylindrical rotor) generator under similar conditions. This demonstrates that salient-pole designs can support greater power loads due to their unique magnetic characteristics. 6.Question In the context of synchronous motors, how does field current affect the maximum torque that can be drawn? Answer:When operating as a motor with a field current of zero, the maximum torque that can be extracted without slipping poles is limited. Calculating the power available at a torque angle of 45°, the motor can draw 14.8 MW. This highlights how reducing or eliminating field current impacts torque capacity, emphasizing the intricate relationship between field excitation, torque performance, and operational constraints in synchronous machines. 7.Question Scan to Download Scan to Download What key equations describe the relationship between voltage, currents, and torque in a synchronous motor? Answer:The key equations involve the relationships between the phase voltage (VÆ), armature currents torque angle (´). Specifically, the torque how the power (P) varies with angle delta and describe the system's operation depending on synchronous reactances and currents. This understanding lays the foundation for effective motor design and control strategies. Scan to Download Chapter 13 \| Introduction to Power Electronics\| Q&A 1.Question What is the ripple factor, and why is it important in power electronics? Answer:The ripple factor is a measure of the amount of ripple (variation) in the output voltage of rectifiers. It is calculated by the formula: ripple factor = (rms voltage / average voltage) - 1. A lower ripple factor indicates a smoother DC output, which is crucial for the stability and efficiency of DC-powered devices. High ripple can lead to heating and reduced lifespan of components. 2.Question How does the MATLAB function 'halfwave3' simulate the output of a three-phase half-wave rectifier? Answer:The 'halfwave3' function simulates the output by calculating the sine of the given input phase, then checking for each phase (A, B, C) to determine which sine voltage is maximum at each instant of time. The output voltage at any Scan to Download Scan to Download time is the maximum of these three values, effectively representing the rectified output of the circuit. 3.Question Explain the role of the DIAC and SCR in a circuit using capacitive control. How does the timing affect the output to a load? Answer:In capacitive control circuits, the DIAC acts as a switch that turns on when a certain voltage threshold is exceeded. The SCR (Silicon Controlled Rectifier) is responsible for controlling the current flow to the load. When the DIAC conducts, it provides a gate current to the SCR, allowing it to turn on. The timing of the DIAC's conduction, which is influenced by the charging of a capacitor through a resistor, determines the firing angle of the SCR. If the charge takes longer, the SCR fires later in the cycle, reducing power to the load. 4.Question What changes would occur in a rectifier circuit if the SCRs were triggered earlier or later in their respective cycles? Scan to Download Scan to Download Answer:Triggering the SCRs earlier in their cycles would lead to a higher average output voltage and potentially increased output power, as more of the input sine wave would be utilized. Conversely, delaying the triggering would reduce the average output voltage and power delivered to the load. This demonstrates the flexibility of SCRs in controlling power output. 5.Question What modifications can be implemented to a circuit to minimize sensitivity to variations in input voltage? Answer:To reduce sensitivity to input voltage variations, one can utilize a voltage regulator or a zener diode, which stabilizes the voltage across the capacitor in the control circuit. This prevents changes in input voltage from affecting the capacitor's charging time and, consequently, the timing of control signals to the SCR. 6.Question Describe the effects of increasing the reference frequency in a PWM modulator circuit. Scan to Download Scan to Download Answer:Increasing the reference frequency in a PWM modulator circuit reduces the harmonic content in the output voltage. It allows for more rapid switching, which can enhance the filtering effectiveness of natural inductances in connected loads. This leads to a cleaner output with fewer unwanted frequencies affecting machine operation. 7.Question In a three-phase full-wave rectifier, how does the phase angle at which SCRs are triggered affect load voltage and ripple? Answer:Triggering SCRs at zero phase angle results in maximum utilizable input waveforms, leading to higher average load voltage and ripple. In contrast, triggering at a 90-degree angle results in a delay in the rectification process, reducing both the average output voltage and the ripple factor. 8.Question Discuss the operational principle of the single-phase voltage source inverter included in the circuit. Answer:The single-phase voltage source inverter converts Scan to Download Scan to Download DC input to AC output. When SCRs are triggered in a sequence, they alternately allow current to flow through the load in opposite directions, creating a square wave output. Capacitors are used to charge and maintain the required voltage levels across SCRs, enabling controlled switching and efficient operation. 9.Question Explain the importance of accurately calculating ripple factors and RMS voltages in power electronics. Answer:Accurate calculations of ripple factors and RMS voltages are crucial for designing reliable electronic systems. High ripple can lead to inefficient power delivery, overheating of components, and potential failure. Design engineers must ensure that ripples are minimized to enhance the longevity of devices and ensure consistent performance. Chapter 14 \| Appendix E: Errata for Electric Machinery Fundamentals 5/e\| Q&A 1.Question What is the significance of correcting problems in educational materials like textbooks? Scan to Download Scan to Download Answer:Correcting problems ensures that students are learning the right concepts and applying accurate information, which is essential for building a solid foundation in subjects such as electric machinery. For instance, correcting a transformer connection from Y-connected to Y-” in pro can change the entire approach to understanding transformer behavior in the circuit. 2.Question How does the efficiency of a power system change when transformers are introduced? Answer:The introduction of transformers can significantly improve the efficiency of a power system by stepping up voltage for transmission, thus minimizing losses over long distances. In the example of Problem 2-3, when comparing the circuit without transformers to the one with a step-up and a step-down transformer, efficiency increases as the load voltage is optimized, reducing loss across the impedance of the transmission line. Scan to Download Scan to Download 3.Question What could be the impact of not addressing errata in engineering textbooks? Answer:If errata are not addressed, students may misinterpret essential concepts, like the output power and efficiency calculations in real-world scenarios. This not only hampers their understanding but may also lead future engineers to design systems based on incorrect information, potentially causing safety issues or system inefficiencies. 4.Question Why is it necessary to specify parameters like the reactance and power factor in problems involving synchronous motors? Answer:Parameters like reactance and power factor are crucial for accurately calculating the operational performance of synchronous motors. For example, in Problem 5-12, knowing the synchronous reactance impacts the field current calculations, which ultimately affects torque output and efficiency of the motor when under load. 5.Question Scan to Download Scan to Download How does the concept of torque angle relate to the performance of synchronous motors? Answer:The torque angle reflects the phase difference between the rotor's magnetic field and the stator's magnetic field. In increasing the field current—as noted in Problem 5-12—the torque angle can increase, indicating a higher output power but also greater stress on the motor. Understanding this relationship is key for engineers to balance performance with reliability. 6.Question What is the importance of understanding transmission line losses as described in the problems? Answer:Understanding transmission line losses is vital for power system efficiency. In Problem 2-3, when calculating what transformer turns ratio minimizes these losses to 1% of total power, it emphasizes the critical role of transformers in optimizing energy delivery and reducing waste in power systems. 7.Question Scan to Download Scan to Download How can simple errors in problem statements lead to misunderstandings in practical applications? Answer:Simple errors, like incorrectly stating transformer connections or load parameters, can mislead students into applying wrong principles. For instance, the mistake in stating a 'Y-connected' generator instead generator could lead to entirely different analytical paths and understanding of generator operation. 8.Question What can be concluded about the learning experience when students engage with errata? Answer:Engagement with errata can actively enhance a student's learning experience by teaching them the importance of precision and detail in engineering. It creates a valuable learning opportunity where students must critically analyze problems and verify facts to ensure accuracy in their future professional practices. Scan to Download Electric Machinery Fundamentals Quiz and Test Check the Correct Answer on Bookey Website Chapter 1 \| Introduction to Machinery Principles\| Quiz and Test 1.A motor's shaft speed of 1800 r/min converts to approximately 188.5 rad/s. 2.A flywheel with a 4 kg·m² moment of inertia experiences a torque of 6 N·m, resulting in an angular speed of 7.5 rad/s after 5 seconds which is equivalent to 71.6 r/min. 3.A motor delivers 50 N·m torque at 1500 r/min, providing about 7854 W of mechanical power, equivalent to approximately 10.5 hp. Chapter 2 \| Transformers\| Quiz and Test 1.Transformers are used to increase the voltage levels in power systems for better efficiency of electricity transmission. Is this statement true or false? 2.Copper losses in transformers are irrelevant when Scan to Download Scan to Download calculating efficiency since they do not affect performance. True or false? 3.The efficiency of transformers can vary significantly based on the power factor of the load. True or false? Chapter 3 \| AC Machinery Fundamentals\| Quiz and Test 1.The induced voltage for a rotating loop in a magnetic field is given by the equation e_ind(t) = 22.6 sin(377t) V. 2.For a four-pole generator operating at 133 Hz, the required shaft speed is 4000 r/min. 3.A Y-connected four-pole winding generates a line-to-line voltage of 8868 V. Scan to Download Chapter 4 \| Synchronous Generators\| Quiz and Test 1.To convert 50 Hz power to 60 Hz, a synchronous motor with 10 poles and a synchronous generator with 12 poles are utilized for them to run at the same mechanical speed. 2.The field current required for a 13.8 kV synchronous generator at no load is 10 A. 3.The efficiency of the generator at rated conditions is approximately 89.8%. Chapter 5 \| Synchronous Motors\| Quiz and Test 1.A synchronous motor operates at a leading power factor when supplying reactive power to the system. 2.Increasing the armature current in synchronous motors will always result in a leading power factor. 3.The synchronous reactance of a motor is independent of the frequency of operation. Chapter 6 \| Induction Motors\| Quiz and Test 1.A 220-V three-phase six-pole induction motor with Scan to Download Scan to Download 3.5% slip has a synchronous speed of 1200 RPM. 2.For a 60-Hz, two-pole induction motor, the slip percentage is always above 5%. 3.In a 50-kW motor with 5% slip, the shaft speed is computed to be 2850 RPM. Scan to Download Chapter 7 \| DC Machinery Fundamentals\| Quiz and Test 1.The machine operates as a generator when the induced voltage exceeds the external battery voltage. 2.The winding type of the machine described in section 7-2 is a retrogressive winding. 3.In a duplex lap-wound machine, there are 20 current paths available in the armature. Chapter 8 \| DC Motors and Generators\| Quiz and Test 1.The no-load speed of a DC motor can be calculated using resistance adjustments, and it was found to be approximately 1195 r/min when resistance was set at 175 ©. 2.Increasing the resistance to 250 © resultfull-load speed due to increased field current. 3.Armature reaction has no significant impact on motor speed at full load conditions. Scan to Download Scan to Download Chapter 9 \| Single-Phase and Special-Purpose Motors\| Quiz and Test 1.The overall efficiency of the split-phase induction motor is 63.6%. 2.A capacitor-start induction motor operates with a low starting torque suitable for heavy loads. 3.The induced torque of a motor is directly proportional to the terminal voltage. Scan to Download Chapter 10 \| Appendix A: Review of Three-Phase Circuits\| Quiz and Test 1.In a ”-connected load, the line and phase are equal. 2.The phase current can be calculated as the line voltage divided by the phase impedance in a 3-phase system. 3.The overall power supplied by the generator in a three-phase system is independent of the power factor. Chapter 11 \| Appendix B: Coil Pitch and Distributed Windings\| Quiz and Test 1.The optimal coil pitch to eliminate the fifth harmonic in a two-pole, 2-slot three-phase stator armature can be calculated as 2/5 or 4/5. 2.The winding distribution factor for five slots per phase in a magnetic field can be determined without using any geometrical relationships. 3.For a synchronous machine with a coil pitch of 19/24, the electrical degrees for coil pitch is 142.5° and slot pitch is 7.5°. Scan to Download Scan to Download Chapter 12 \| Appendix C: Salient Pole Theory of Synchronous Machines\| Quiz and Test 1.A synchronous generator rated at 13.8-kV and 50-MVA has an internal generated voltage of approximately 11496" 17.8° V at rated co2.The reluctance torque contributes 8.6 MW out of a total output of 42.2 MW, amounting to about 30% of full-load power. 3.The maximum torque drawing without slipping for a salient-pole motor at rated conditions with zero field current is 14.8 MW at a torque angle of 45°. Scan to Download Chapter 13 \| Introduction to Power Electronics\| Quiz and Test 1.The ripple factor for a Three-Phase Half-Wave Rectifier is approximately 0.183 (or 18.3%). 2.A Three-Phase Full-Wave Rectifier has a ripple factor of approximately 4.0%. 3.In a Single-Phase Voltage Source Inverter, SCR activation is alternated to produce a sinusoidal output. Chapter 14 \| Appendix E: Errata for Electric Machinery Fundamentals 5/e\| Quiz and Test 1.Problem 2-3 on page 145 of 'Electric Machinery Fundamentals' correctly states that the load voltage and efficiency must be calculated without considering transformers. 2.According to Appendix E of 'Electric Machinery Fundamentals', problem 4-6 should describe a generator as Y-connected instead of ”-connected. 3.The erratum for problem 5-4 on page 301 states that the correct synchronous reactance is 2.5 © w according to the errata.
3885	https://en.wikipedia.org/wiki/Riemann_hypothesis	Riemann hypothesis - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 Riemann zeta function 2 Origin 3 ConsequencesToggle Consequences subsection 3.1 Distribution of prime numbers 3.2 Growth of arithmetic functions 3.3 Lindelöf hypothesis and growth of the zeta function 3.4 Large prime gap conjecture 3.5 Analytic criteria equivalent to the Riemann hypothesis 3.6 Consequences of the generalized Riemann hypothesis 3.7 Excluded middle 3.7.1 Littlewood's theorem 3.7.2 Gauss's class number conjecture 3.7.3 Growth of Euler's totient 4 Generalizations and analogsToggle Generalizations and analogs subsection 4.1 Dirichlet L-series and other number fields 4.2 Function fields and zeta functions of varieties over finite fields 4.3 Arithmetic zeta functions of arithmetic schemes and their L-factors 4.4 Selberg zeta functions 4.5 Ihara zeta functions 4.6 Montgomery's pair correlation conjecture 4.7 Other zeta functions 5 Attempted proofsToggle Attempted proofs subsection 5.1 Operator theory 5.2 Lee–Yang theorem 5.3 Turán's result 5.4 Noncommutative geometry 5.5 Hilbert spaces of entire functions 5.6 Quasicrystals 5.7 Arithmetic zeta functions of models of elliptic curves over number fields 5.8 Multiple zeta functions 6 Location of the zerosToggle Location of the zeros subsection 6.1 Number of zeros 6.2 Theorem of Hadamard and de la Vallée-Poussin 6.3 Zero-free regions 7 Zeros on the critical lineToggle Zeros on the critical line subsection 7.1 Hardy–Littlewood conjectures 7.2 Selberg's zeta function conjecture 7.3 Numerical calculations 7.4 Gram points 8 Arguments for and against the Riemann hypothesis 9 Notes 10 ReferencesToggle References subsection 10.1 Popular expositions 11 External links [x] Toggle the table of contents Riemann hypothesis [x] 55 languages العربية Asturianu বাংলা Беларуская Български Català Čeština Dansk Deutsch Ελληνικά Español Esperanto Euskara فارسی Français Galego 한국어 हिन्दी Bahasa Indonesia Italiano עברית Қазақша Kreyòl ayisyen Latina Latviešu Lietuvių Magyar Bahasa Melayu Монгол Nederlands नेपाली 日本語 Norsk bokmål Norsk nynorsk Polski Português Română Русский Shqip Sicilianu Simple English Slovenčina Slovenščina کوردی Српски / srpski Suomi Svenska தமிழ் తెలుగు Türkçe Українська Tiếng Việt 吴语粵語中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikibooks Wikiquote Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia Conjecture on zeros of the zeta function For the musical term, see Riemannian theory. Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics This plot of Riemann's zeta (ζ) function (here with argument z) shows trivial zeros where ζ(z) = 0, a pole where ζ(z) = ∞{\displaystyle \infty }, the critical line of nontrivial zeros with Re(z) = 1/2 and slopes of absolute values. In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part⁠1/2⁠. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann(1859), after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(s) is a function whose arguments may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6,.... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is⁠1/2⁠. \| Millennium Prize Problems \| \| Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP problem Poincaré conjecture (solved) Riemann hypothesis Yang–Mills existence and mass gap \| \| v t e \| Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers ⁠1/2⁠ + i t, where t is a real number and i is the imaginary unit. Riemann zeta function [edit] The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergentinfinite series ζ(s)=∑n=1∞1 n s=1 1 s+1 2 s+1 3 s+⋯{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } Leonhard Euler considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product ζ(s)=∏p prime 1 1−p−s=1 1−2−s⋅1 1−3−s⋅1 1−5−s⋅1 1−7−s⋯{\displaystyle \zeta (s)=\prod {p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots } where the infinite product extends over all prime numbers _p. The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. Because the zeta function is meromorphic, all choices of how to perform this analytic continuation will lead to the same result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation (1−2 2 s)ζ(s)=η(s)=∑n=1∞(−1)n+1 n s=1 1 s−1 2 s+1 3 s−⋯,{\displaystyle \left(1-{\frac {2}{2^{s}}}\right)\zeta (s)=\eta (s)=\sum {n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-\cdots ,} within the region of convergence for both series. But the eta function series on the right converges not just when the real part of _s is greater than one, but more generally whenever s has positive real part. Thus the zeta function can be redefined as η(s)/(1−2/2 s){\displaystyle \eta (s)/(1-2/2^{s})}, extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where 1−2/2 s{\displaystyle 1-2/2^{s}} is zero. These are the points s=1+2 π i n/log⁡2{\displaystyle s=1+2\pi in/\log 2} where n{\displaystyle n} can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see Dirichlet eta function#Landau's problem with ζ(s) = η(s)/0 and solutions), giving a finite value for all values of s with positive real part except the simple pole at s=1. In the strip 0 < Re(s) < 1 this extension of the zeta function satisfies the functional equation ζ(s)=2 s π s−1 sin⁡(π s 2)Γ(1−s)ζ(1−s).{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).} One may then define ζ(s) for all remaining nonzero complex numbers s (Re(s) ≤ 0 and s ≠ 0) by applying this equation outside the strip, and letting ζ(s) equal the right side of the equation whenever s has non-positive real part (and s ≠ 0). If s is a negative even integer, then ζ(s) = 0, because the factor sin(π s/2) vanishes; these are the zeta function's trivial zeros. (If s is a positive even integer this argument does not apply because the zeros of the sine function are canceled by the poles of the gamma function as it takes negative integer arguments.) The value ζ(0)=−1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in the critical strip where s has real part between 0 and 1. Riemann zeta function along the critical line with Re(s) = 1/2. Real values are shown on the horizontal axis and imaginary values are on the vertical axis. Re(ζ(1/2 + it)), Im(ζ(1/2 + it)) is plotted with t ranging between −30 and 30. Animation showing in 3D the Riemann zeta function critical strip (blue, where s has real part between 0 and 1), critical line (red, for real part of s equals 0.5) and zeroes (cross between red and orange): [x,y,z] = [Re(ζ(r + it)), Im(ζ(r + it)), t] with 0.1 ≤ r ≤ 0.9 and 1 ≤ t ≤ 51 The real part (red) and imaginary part (blue) of the Riemann zeta function ζ(s) along the critical line in the complex plane with real part Re(s) = 1/2. The first nontrivial zeros, where ζ(s) equals zero, occur where both curves touch the horizontal x-axis, for complex numbers with imaginary parts Im(s) equaling ±14.135, ±21.022 and ±25.011. Origin [edit] ... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien. ... it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation. — Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a variant of the zeta function, modified in a way that the real line be mapped to the critical line.) At the death of Riemann, a note was found among his papers, saying "These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]. — Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primesπ(x) less than or equal to a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". His formula was given in terms of the related function Π(x)=π(x)+1 2 π(x 1/2)+1 3 π(x 1/3)+1 4 π(x 1/4)+1 5 π(x 1/5)+1 6 π(x 1/6)+⋯{\displaystyle \Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi (x^{1/2})+{\tfrac {1}{3}}\pi (x^{1/3})+{\tfrac {1}{4}}\pi (x^{1/4})+{\tfrac {1}{5}}\pi (x^{1/5})+{\tfrac {1}{6}}\pi (x^{1/6})+\cdots } which counts the primes and prime powers up to x, counting a prime power p n as 1⁄n. The number of primes can be recovered from this function by using the Möbius inversion formula, π(x)=∑n=1∞μ(n)n Π(x 1/n)=Π(x)−1 2 Π(x 1/2)−1 3 Π(x 1/3)−1 5 Π(x 1/5)+1 6 Π(x 1/6)−⋯,{\displaystyle {\begin{aligned}\pi (x)&=\sum {n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi (x^{1/n})\&=\Pi (x)-{\frac {1}{2}}\Pi (x^{1/2})-{\frac {1}{3}}\Pi (x^{1/3})-{\frac {1}{5}}\Pi (x^{1/5})+{\frac {1}{6}}\Pi (x^{1/6})-\cdots ,\end{aligned}}} where _μ is the Möbius function. Riemann's formula is then Π 0(x)=li⁡(x)−∑ρ li⁡(x ρ)−log⁡2+∫x∞d t t(t 2−1)log⁡t{\displaystyle \Pi {0}(x)=\operatorname {li} (x)-\sum {\rho }\operatorname {li} (x^{\rho })-\log 2+\int _{x}^{\infty }{\frac {dt}{t(t^{2}-1)\log t}}} where the sum is over the nontrivial zeros of the zeta function and where Π 0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits: Π 0(x)=lim ε→0 Π(x−ε)+Π(x+ε)2.{\displaystyle \Pi {0}(x)=\lim {\varepsilon \to 0}{\frac {\Pi (x-\varepsilon )+\Pi (x+\varepsilon )}{2}}.} The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral li⁡(x)=∫0 x d t log⁡t.{\displaystyle \operatorname {li} (x)=\int {0}^{x}{\frac {dt}{\log t}}.} The terms li(_x ρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x>1) by analytic continuation in the complex variable ρ in the region Re(ρ)>0, i.e. they should be considered as Ei(ρ log x). The other terms also correspond to zeros: the dominant term li(x) comes from the pole at s=1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977). This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis. The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes. — (Burton 2006, p.376) Consequences [edit] The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis. Distribution of prime numbers [edit] Riemann's explicit formula for the number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. For example, if β is the upper bound of the real parts of the zeros, thenπ(x)−li⁡(x)=O(x β log⁡x){\displaystyle \pi (x)-\operatorname {li} (x)=O\left(x^{\beta }\log x\right)}, where π(x){\displaystyle \pi (x)} is the prime-counting function, li⁡(x){\displaystyle \operatorname {li} (x)} is the logarithmic integral function, log⁡(x){\displaystyle \log(x)} is the natural logarithm of x, and big O notation is used here. It is already known that 1/2≤β≤1. Corrections to an estimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction. Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of von Koch's result, due to Schoenfeld (1976), says that the Riemann hypothesis implies \|π(x)−li⁡(x)\|<1 8 π x log⁡(x),for all x≥2657,{\displaystyle \|\pi (x)-\operatorname {li} (x)\|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for all }}x\geq 2657,} Schoenfeld (1976) also showed that the Riemann hypothesis implies \|ψ(x)−x\|<1 8 π x log 2⁡x,for all x≥73.2,{\displaystyle \|\psi (x)-x\|<{\frac {1}{8\pi }}{\sqrt {x}}\log ^{2}x,\qquad {\text{for all }}x\geq 73.2,} where ψ(x){\displaystyle \psi (x)} is Chebyshev's second function. Dudek (2014) proved that the Riemann hypothesis implies that for all x≥2{\displaystyle x\geq 2} there is a prime p{\displaystyle p} satisfying x−4 π x log⁡x<p≤x{\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x}. The constant 4/π may be reduced to (1+ε) provided that x is taken to be sufficiently large. This is an explicit version of a theorem of Cramér. Growth of arithmetic functions [edit] The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above. One example involves the Möbius functionμ. The statement that the equation 1 ζ(s)=∑n=1∞μ(n)n s{\displaystyle {\frac {1}{\zeta (s)}}=\sum {n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}} is valid for every _s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the Mertens function is defined by M(x)=∑n≤x μ(n){\displaystyle M(x)=\sum _{n\leq x}\mu (n)} then the claim that M(x)=O(x 1 2+ε){\displaystyle M(x)=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)} for every positive ε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 in Titchmarsh (1986)). The determinant of the order nRedheffer matrix is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. Littlewood's result has been improved several times since then, by Edmund Landau,Edward Charles Titchmarsh, Helmut Maier and Hugh Montgomery, and Kannan Soundararajan. Soundararajan's result is that, conditional on the Riemann hypothesis, M(x)=O(x 1/2 exp⁡((log⁡x)1/2(log⁡log⁡x)14)).{\displaystyle M(x)=O\left(x^{1/2}\exp \left((\log x)^{1/2}(\log \log x)^{14}\right)\right).} The Riemann hypothesis puts a rather tight bound on the growth of M, since Odlyzko & te Riele (1985) disproved the slightly stronger Mertens conjecture \|M(x)\|≤x.{\displaystyle \|M(x)\|\leq {\sqrt {x}}.} Another closely related result is due to Björner (2011), that the Riemann hypothesis is equivalent to the statement that the Euler characteristic of the simplicial complex determined by the lattice of integers under divisibility is o(n 1/2+ϵ){\displaystyle o(n^{1/2+\epsilon })} for all ϵ>0{\displaystyle \epsilon >0} (see incidence algebra). The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n). A typical example is Robin's theorem, which states that if σ(n) is the sigma function, given by σ(n)=∑d∣n d{\displaystyle \sigma (n)=\sum _{d\mid n}d} then σ(n)<e γ n log⁡log⁡n{\displaystyle \sigma (n) for all _n_> 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant. A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that: σ(n)<H n+log⁡(H n)e H n{\displaystyle \sigma (n) for every natural number_n_> 1, where H n{\displaystyle H_{n}} is the n th harmonic number. The Riemann hypothesis is also true if and only if the inequality n φ(n)<e γ log⁡log⁡n+e γ(4+γ−log⁡4 π)log⁡n{\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}} is true for all n ≥ 120569#, where φ(n) is Euler's totient function and 120569# is the product of the first 120569 primes. Another example was found by Jérôme Franel, and extended by Landau (see Franel & Landau (1924)). The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular. One such equivalence is as follows: if F n is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε> 0 ∑i=1 m\|F n(i)−i m\|=O(n 1 2+ϵ){\displaystyle \sum {i=1}^{m}\|F{n}(i)-{\tfrac {i}{m}}\|=O\left(n^{{\frac {1}{2}}+\epsilon }\right)} is equivalent to the Riemann hypothesis. Here m=∑i=1 n φ(i){\displaystyle m=\sum {i=1}^{n}\varphi (i)} is the number of terms in the Farey sequence of order _n. For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group S n of degree n, then Massias, Nicolas & Robin (1988) showed that the Riemann hypothesis is equivalent to the bound log⁡g(n)<Li−1⁡(n){\displaystyle \log g(n)<{\sqrt {\operatorname {Li} ^{-1}(n)}}} for all sufficiently large n. Lindelöf hypothesis and growth of the zeta function [edit] The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε> 0, ζ(1 2+i t)=O(t ε),{\displaystyle \zeta \left({\frac {1}{2}}+it\right)=O(t^{\varepsilon }),} as t → ∞{\displaystyle \infty }. The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that e γ≤lim sup t→+∞\|ζ(1+i t)\|log⁡log⁡t≤2 e γ{\displaystyle e^{\gamma }\leq \limsup {t\rightarrow +\infty }{\frac {\|\zeta (1+it)\|}{\log \log t}}\leq 2e^{\gamma }}6 π 2 e γ≤lim sup t→+∞1/\|ζ(1+i t)\|log⁡log⁡t≤12 π 2 e γ{\displaystyle {\frac {6}{\pi ^{2}}}e^{\gamma }\leq \limsup {t\rightarrow +\infty }{\frac {1/\|\zeta (1+it)\|}{\log \log t}}\leq {\frac {12}{\pi ^{2}}}e^{\gamma }} so the growth rate of ζ(1 + it) and its inverse would be known up to a factor of 2. Large prime gap conjecture [edit] The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture. Analytic criteria equivalent to the Riemann hypothesis [edit] Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the divisor functionσ(n).) The Riesz criterion was given by Riesz (1916), to the effect that the bound −∑k=1∞(−x)k(k−1)!ζ(2 k)=O(x 1 4+ϵ){\displaystyle -\sum _{k=1}^{\infty }{\frac {(-x)^{k}}{(k-1)!\zeta (2k)}}=O\left(x^{{\frac {1}{4}}+\epsilon }\right)} holds for all ε > 0 if and only if the Riemann hypothesis holds. See also the Hardy–Littlewood criterion. Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form f(x)=∑ν=1 n c ν ρ(θ ν x){\displaystyle f(x)=\sum {\nu =1}^{n}c{\nu }\rho \left({\frac {\theta {\nu }}{x}}\right)} where _ρ(z) is the fractional part of z, 0 ≤ θ ν ≤ 1, and ∑ν=1 n c ν θ ν=0,{\displaystyle \sum {\nu =1}^{n}c{\nu }\theta {\nu }=0,} is dense in the Hilbert space L 2(0,1) of square-integrable functions on the unit interval. Beurling (1955) extended this by showing that the zeta function has no zeros with real part greater than 1/_p if and only if this function space is dense in L p(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where θ ν∈{1/k}k≥1{\displaystyle \theta {\nu }\in {1/k}{k\geq 1}}. Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation ∫0∞z−σ−1 φ(z)e x/z+1 d z=0{\displaystyle \int _{0}^{\infty }{\frac {z^{-\sigma -1}\varphi (z)}{{e^{x/z}}+1}}\,dz=0} has no non-trivial bounded solutions φ{\displaystyle \varphi } for 1/2<σ<1{\displaystyle 1/2<\sigma <1}. Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis. Speiser (1934) proved that the Riemann hypothesis is equivalent to the statement that ζ′(s), the derivative of ζ(s), has no zeros in the strip 0<ℜ(s)<1 2.{\displaystyle 0<\Re (s)<{\frac {1}{2}}.} That ζ(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line. The Farey sequence provides two equivalences, due to Jerome Franel and Edmund Landau in 1924. The de Bruijn–Newman constant denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is defined as the unique real number such that the function H(λ,z):=∫0∞e λ u 2 Φ(u)cos⁡(z u)d u{\displaystyle H(\lambda ,z):=\int {0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)\,du}, that is parametrised by a real parameter _λ, has a complex variable z and is defined using a super-exponentially decaying function Φ(u)=∑n=1∞(2 π 2 n 4 e 9 u−3 π n 2 e 5 u)e−π n 2 e 4 u{\displaystyle \Phi (u)=\sum {n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}}. has only real zeros if and only if _λ ≥ Λ. Since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. Brad Rodgers and Terence Tao discovered the equivalence is actually Λ = 0 by proving zero to be the lower bound of the constant. Proving zero is also the upper bound would therefore prove the Riemann hypothesis. As of April 2020 the upper bound is Λ ≤ 0.2. Consequences of the generalized Riemann hypothesis [edit] Several applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken from Conrad (2010). In 1913, Grönwall showed that the generalized Riemann hypothesis implies that Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis. In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that lim x→1−∑p>2(−1)(p+1)/2 x p=+∞,{\displaystyle \lim {x\to 1^{-}}\sum {p>2}(-1)^{(p+1)/2}x^{p}=+\infty ,} which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see Prime number theorem §Prime number race.) In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes. In 2013 Harald Helfgott proved the ternary Goldbach conjecture without the GRH dependence, subject to some extensive calculations completed with the help of David J. Platt. In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most Km 2 log(m)2 for some fixed constant K. In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots. In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete. Weinberger (1973) showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number fields implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant −19, −43, −67, or −163. In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is prime in polynomial time via the Miller test. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this result unconditionally using the AKS primality test. Odlyzko (1990) discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields. Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic formx 2 + y 2 + 10 z 2 represents all integers that it represents locally, with exactly 18 exceptions. In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture on cubic Gauss sums, under the assumption of the GRH. Excluded middle [edit] Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, Ireland & Rosen (1990, p.359) say The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!! Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample. Littlewood's theorem [edit] This concerns the sign of the error in the prime number theorem. It has been computed that π(x) < li(x) for all x ≤ 10 25 (see this table), and no value of x is known for which π(x) > li(x). In 1914, Littlewood proved that there are arbitrarily large values of x for which π(x)>li⁡(x)+1 3 x log⁡x log⁡log⁡log⁡x,{\displaystyle \pi (x)>\operatorname {li} (x)+{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x,} and that there are also arbitrarily large values of x for which π(x)<li⁡(x)−1 3 x log⁡x log⁡log⁡log⁡x.{\displaystyle \pi (x)<\operatorname {li} (x)-{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x.} Thus the difference π(x) − li(x) changes sign infinitely many times. Skewes' number is an estimate of the value of x corresponding to the first sign change. Littlewood's proof is divided into two cases: the RH is assumed false (about half a page of Ingham 1932, Chapt. V), and the RH is assumed true (about a dozen pages). Stanisław Knapowski(1962) followed this up with a paper on the number of times Δ(n){\displaystyle \Delta (n)} changes sign in the interval Δ(n){\displaystyle \Delta (n)}. Gauss's class number conjecture [edit] This is the conjecture (first stated in article 303 of Gauss's Disquisitiones Arithmeticae) that there are only finitely many imaginary quadratic fields with a given class number. One way to prove it would be to show that as the discriminant D → −∞ the class number h(D) → ∞. The following sequence of theorems involving the Riemann hypothesis is described in Ireland & Rosen 1990, pp.358–361: Theorem (Hecke; 1918)—Let D< 0 be the discriminant of an imaginary quadraticnumber fieldK. Assume the generalized Riemann hypothesis for L-functions of all imaginary quadratic Dirichlet characters. Then there is an absolute constant C such that h(D)>C\|D\|log⁡\|D\|.{\displaystyle h(D)>C{\frac {\sqrt {\|D\|}}{\log \|D\|}}.} Theorem (Deuring; 1933)—If the RH is false then h(D) > 1 if \|D\| is sufficiently large. Theorem (Mordell; 1934)—If the RH is false then h(D) → ∞ as D → −∞. Theorem (Heilbronn; 1934)—If the generalized RH is false for the L-function of some imaginary quadratic Dirichlet character then h(D) → ∞ as D → −∞. (In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.) In 1935, Carl Siegel strengthened the result without using RH or GRH in any way. Growth of Euler's totient [edit] In 1983 J. L. Nicolas proved that φ(n)<e−γ n log⁡log⁡n{\displaystyle \varphi (n)<e^{-\gamma }{\frac {n}{\log \log n}}} for infinitely many n, where φ(n) is Euler's totient function and γ is Euler's constant. Ribenboim remarks that: "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption." Generalizations and analogs [edit] Dirichlet L-series and other number fields [edit] The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics. The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions. In particular it implies the conjecture that Siegel zeros (zeros of L-functions between 1/2 and 1) do not exist. The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms. Function fields and zeta functions of varieties over finite fields [edit] Artin (1924) introduced global zeta functions of (quadratic) function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by Weil (1948) in general. For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), has absolute value q{\displaystyle {\sqrt {q}}} is actually an instance of the Riemann hypothesis in the function field setting. This led Weil (1949) to conjecture a similar statement for all algebraic varieties; the resulting Weil conjectures were proved by Pierre Deligne(1974, 1980). Arithmetic zeta functions of arithmetic schemes and their L-factors [edit] Arithmetic zeta functions generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor Jean-Pierre Serre(1969–1970). Assuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip ℜ(s)∈(0,n){\displaystyle \Re (s)\in (0,n)} lie on the central line. Correspondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines ℜ(s)=1/2,3/2,…,n−1/2{\displaystyle \Re (s)=1/2,3/2,\dots ,n-1/2} and its poles inside the critical strip lie on vertical lines ℜ(s)=1,2,…,n−1{\displaystyle \Re (s)=1,2,\dots ,n-1}. This is known for schemes in positive characteristic and follows from Pierre Deligne(1974, 1980), but remains entirely unknown in characteristic zero. Selberg zeta functions [edit] Main article: Selberg zeta function Selberg (1956) introduced the Selberg zeta function of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface. Ihara zeta functions [edit] The Ihara zeta function of a finite graph is an analogue of the Selberg zeta function, which was first introduced by Yasutaka Ihara in the context of discrete subgroups of the two-by-two p-adic special linear group. A regular finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Sunada. Montgomery's pair correlation conjecture [edit] Montgomery (1973) suggested the pair correlation conjecture that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix. Odlyzko (1987) showed that this is supported by large-scale numerical calculations of these correlation functions. Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts). Dedekind zeta functions of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorize as a product of powers of Artin L-functions, so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve. Other zeta functions [edit] There are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved. Goss zeta functions of function fields have a Riemann hypothesis, proved by Sheats (1998). The main conjecture of Iwasawa theory, proved by Barry Mazur and Andrew Wiles for cyclotomic fields, and Wiles for totally real fields, identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions. Attempted proofs [edit] Several mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof. Watkins (2021) lists some incorrect solutions. Operator theory [edit] Main article: Hilbert–Pólya conjecture Hilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real eigenvalues. Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a Frobenius element on an étale cohomology group, the zeros of a Selberg zeta function are eigenvalues of a Laplacian operator of a Riemann surface, and the zeros of a p-adic zeta function correspond to eigenvectors of a Galois action on ideal class groups. Odlyzko (1987) showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This gives some support to the Hilbert–Pólya conjecture. In 1999, Michael Berry and Jonathan Keating conjectured that there is some unknown quantization H^{\displaystyle {\hat {H}}} of the classical Hamiltonian H = xp so that ζ(1/2+i H^)=0{\displaystyle \zeta (1/2+i{\hat {H}})=0} and even more strongly, that the Riemann zeros coincide with the spectrum of the operator 1/2+i H^{\displaystyle 1/2+i{\hat {H}}}. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principleσ x σ p≥ℏ 2{\displaystyle \sigma {x}\sigma {p}\geq {\frac {\hbar }{2}}} and the natural numbers as spectrum of the quantum harmonic oscillator. The crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program. In a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the half-derivative of the function N(s)=1 π Arg⁡ξ(1/2+i s){\displaystyle N(s)={\frac {1}{\pi }}\operatorname {Arg} \xi (1/2+i{\sqrt {s}})} then, in Hilbert-Polya approach V−1(x)=4 π d 1/2 N(x)d x 1/2.{\displaystyle V^{-1}(x)={\sqrt {4\pi }}{\frac {d^{1/2}N(x)}{dx^{1/2}}}.} This yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the functional determinant of this Hamiltonian operator is just the Riemann Xi function. In fact the Riemann Xi function would be proportional to the functional determinant (Hadamard product) det(H+1/4+s(s−1)){\displaystyle \det(H+1/4+s(s-1))}ξ(s)ξ(0)=det(H+s(s−1)+1/4)det(H+1/4).{\displaystyle {\frac {\xi (s)}{\xi (0)}}={\frac {\det(H+s(s-1)+1/4)}{\det(H+1/4)}}.} However this operator is not useful in practice since it includes the inverse function (implicit function) of the potential but not the potential itself. The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first cohomology group of the spectrum Spec (Z) of the integers. Deninger (1998) described some of the attempts to find such a cohomology theory. Zagier (1981) constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow. Cartier (1982) discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator. Schumayer & Hutchinson (2011) surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function. Lee–Yang theorem [edit] The Lee–Yang theorem states that the zeros of certain partition functions in statistical mechanics all lie on a "critical line" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis. Turán's result [edit] Pál Turán(1948) showed that if the functions ∑n=1 N n−s{\displaystyle \sum {n=1}^{N}n^{-s}} have no zeros when the real part of _s is greater than one then T(x)=∑n≤x λ(n)n≥0 for x>0,{\displaystyle T(x)=\sum {n\leq x}{\frac {\lambda (n)}{n}}\geq 0{\text{ for }}x>0,} where λ(_n) is the Liouville function given by (−1)r if n has r prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. But Haselgrove (1958) proved that T(x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed that the smallest such x is 72 185 376 951 205. Spira (1968) showed by numerical calculation that the finite Dirichlet series above for N = 19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1 + N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but Montgomery (1983) showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N). Therefore, Turán's result is vacuously true and cannot help prove the Riemann hypothesis. Noncommutative geometry [edit] Connes(1999, 2000) has described a relationship between the Riemann hypothesis and noncommutative geometry, and showed that a suitable analog of the Selberg trace formula for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis. Some of these ideas are elaborated in Lapidus (2008). Hilbert spaces of entire functions [edit] Louis de Branges(1992) showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions. However Conrey & Li (2000) showed that the necessary positivity conditions are not satisfied. Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians. Quasicrystals [edit] The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, a distribution with discrete support whose Fourier transform also has discrete support. Dyson (2009) suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals. Arithmetic zeta functions of models of elliptic curves over number fields [edit] When one goes from geometric dimension one, e.g. an algebraic number field, to geometric dimension two, e.g. a regular model of an elliptic curve over a number field, the two-dimensional part of the generalized Riemann hypothesis for the arithmetic zeta function of the model deals with the poles of the zeta function. In dimension one the study of the zeta integral in Tate's thesis does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work of Ivan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups. Related conjecture of Fesenko(2010) on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis. Suzuki(2011) proved that the latter, together with some technical assumptions, implies Fesenko's conjecture. Multiple zeta functions [edit] Deligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function. By analogy, Kurokawa (1992) introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function. Location of the zeros [edit] Number of zeros [edit] The functional equation combined with the argument principle implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by N(T)=1 π A r g⁡(ξ(s))=1 π A r g⁡(Γ(s 2)π−s 2 ζ(s)s(s−1)/2){\displaystyle N(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\xi (s))={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-{\frac {s}{2}}}\zeta (s)s(s-1)/2)} for s = 1/2 + iT, where the argument is defined by varying it continuously along the line with Im(s) = T, starting with argument 0 at ∞ + iT. This is the sum of a large but well understood term 1 π A r g⁡(Γ(s 2)π−s/2 s(s−1)/2)=T 2 π log⁡T 2 π−T 2 π+7/8+O(1/T){\displaystyle {\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-s/2}s(s-1)/2)={\frac {T}{2\pi }}\log {\frac {T}{2\pi }}-{\frac {T}{2\pi }}+7/8+O(1/T)} and a small but rather mysterious term S(T)=1 π A r g⁡(ζ(1/2+i T))=O(log⁡T).{\displaystyle S(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\zeta (1/2+iT))=O(\log T).} So the density of zeros with imaginary part near T is about log(T)/(2 π), and the function S describes the small deviations from this. The function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreases monotonically between zeros with derivative close to −log t. Trudgian (2014) proved that, if T>e, then \|N(T)−T 2 π log⁡T 2 π e\|≤0.112 log⁡T+0.278 log⁡log⁡T+3.385+0.2 T{\displaystyle \|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}\|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}. Karatsuba (1996) proved that every interval (T, T + H] for H≥T 27 82+ε{\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }} contains at least H(log⁡T)1 3 e−c log⁡log⁡T{\displaystyle H(\log T)^{\frac {1}{3}}e^{-c{\sqrt {\log \log T}}}} points where the function S(t) changes sign. Selberg (1946) showed that the average moments of even powers of S are given by ∫0 T\|S(t)\|2 k d t=(2 k)!k!(2 π)2 k T(log⁡log⁡T)k+O(T(log⁡log⁡T)k−1/2).{\displaystyle \int {0}^{T}\|S(t)\|^{2k}dt={\frac {(2k)!}{k!(2\pi )^{2k}}}T(\log \log T)^{k}+O(T(\log \log T)^{k-1/2}).} This suggests that _S(T)/(log log T)1/2 resembles a Gaussian random variable with mean 0 and variance 2 π 2 (Ghosh (1983) proved this fact). In particular \|S(T)\| is usually somewhere around (log log T)1/2, but occasionally much larger. The exact order of growth of S(T) is not known. There has been no unconditional improvement to Riemann's original bound S(T) = O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T) = O(log T/log log T). The true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2. In the other direction it cannot be too small: Selberg (1946) showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)1/2/(log log T)1/2). Numerical calculations confirm that S grows very slowly: \|S(T)\| < 1 for T< 280, \|S(T)\| < 2 for T<6 800 000, and the largest value of \|S(T)\| found so far is not much larger than 3. Riemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tend to 0. Theorem of Hadamard and de la Vallée-Poussin [edit] Hadamard (1896) and de la Vallée-Poussin (1896) independently proved that no zeros could lie on the line Re(s) = 1. Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in their first proofs of the prime number theorem. Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1 + it) vanishes, then ζ(1 + 2 it) is singular, which is not possible. One way of doing this is by using the inequality \|ζ(σ)3 ζ(σ+i t)4 ζ(σ+2 i t)\|≥1{\displaystyle \|\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)\|\geq 1} for σ> 1, t real, and looking at the limit as σ → 1. This inequality follows by taking the real part of the log of the Euler product to see that \|ζ(σ+i t)\|=exp⁡ℜ∑p n p−n(σ+i t)n=exp⁡∑p n p−n σ cos⁡(t log⁡p n)n,{\displaystyle \|\zeta (\sigma +it)\|=\exp \Re \sum {p^{n}}{\frac {p^{-n(\sigma +it)}}{n}}=\exp \sum {p^{n}}{\frac {p^{-n\sigma }\cos(t\log p^{n})}{n}},} where the sum is over all prime powers p n, so that \|ζ(σ)3 ζ(σ+i t)4 ζ(σ+2 i t)\|=exp⁡∑p n p−n σ 3+4 cos⁡(t log⁡p n)+cos⁡(2 t log⁡p n)n{\displaystyle \|\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)\|=\exp \sum _{p^{n}}p^{-n\sigma }{\frac {3+4\cos(t\log p^{n})+\cos(2t\log p^{n})}{n}}} which is at least 1 because all the terms in the sum are positive, due to the inequality 3+4 cos⁡(θ)+cos⁡(2 θ)=2(1+cos⁡(θ))2≥0.{\displaystyle 3+4\cos(\theta )+\cos(2\theta )=2(1+\cos(\theta ))^{2}\geq 0.} Zero-free regions [edit] The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for \|t\| ≤ 3.000 175 3328×10 12. Beyond that zero-free regions are known as inequalities concerning σ + i t, which can be zeroes. The oldest version is from De la Vallée-Poussin (1899–1900), who proved there is a region without zeroes that satisfies 1 − σ ≥ ⁠C/log(t)⁠ for some positive constant C. In other words, zeros cannot be too close to the line σ = 1: there is a zero-free region close to this line. This has been enlarged by several authors using methods such as Vinogradov's mean-value theorem. The most recent paper by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022: σ≥1−1 5.558691 log⁡\|t\|{\displaystyle \sigma \geq 1-{\frac {1}{5.558691\log \|t\|}}} whenever \|t\|≥2{\displaystyle \|t\|\geq 2},σ≥1−1 55.241(log⁡\|t\|)2/3(log⁡log⁡\|t\|)1/3{\displaystyle \sigma \geq 1-{\frac {1}{55.241(\log {\|t\|})^{2/3}(\log {\log {\|t\|}})^{1/3}}}} whenever \|t\|≥3{\displaystyle \|t\|\geq 3} (largest known region in the bound 3.0001753328⋅10 12≤\|t\|≤exp⁡(64.1)≈6.89⋅10 27{\displaystyle 3.0001753328\cdot 10^{12}\leq \|t\|\leq \exp(64.1)\approx 6.89\cdot 10^{27}}),σ≥1−0.04962−0.0196 1.15+log⁡3+1 6 log⁡t+log⁡log⁡t 0.685+log⁡3+1 6 log⁡t+1.155⋅log⁡log⁡t{\displaystyle \sigma \geq 1-{\frac {0.04962-{\frac {0.0196}{1.15+\log 3+{\frac {1}{6}}\log t+\log \log t}}}{0.685+\log 3+{\frac {1}{6}}\log t+1.155\cdot \log \log t}}} whenever \|t\|≥1.88⋅10 14{\displaystyle \|t\|\geq 1.88\cdot 10^{14}} (largest known region in the bound exp⁡(64.1)≤\|t\|≤exp⁡(1000)≈1.97⋅10 434{\displaystyle \exp(64.1)\leq \|t\|\leq \exp(1000)\approx 1.97\cdot 10^{434}}) and σ≥1−0.05035 27 164(log⁡\|t\|)+7.096+0.0349(27 164(log⁡\|t\|)+7.096)2{\displaystyle \sigma \geq 1-{\frac {0.05035}{{\frac {27}{164}}(\log {\|t\|})+7.096}}+{\frac {0.0349}{({\frac {27}{164}}(\log {\|t\|})+7.096)^{2}}}} whenever \|t\|≥exp⁡(1000){\displaystyle \|t\|\geq \exp(1000)} (largest known region in its own bound) The paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of \|t\|{\displaystyle \|t\|} being merely assumed to be "sufficiently large" to fulfill the requirements of the paper's proof. This region is σ≥1−1 48.1588(log⁡\|t\|)2/3(log⁡log⁡\|t\|)1/3{\displaystyle \sigma \geq 1-{\frac {1}{48.1588(\log {\|t\|})^{2/3}(\log {\log {\|t\|}})^{1/3}}}}. Zeros on the critical line [edit] Hardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums. Most zeros lie close to the critical line. More precisely, Bohr & Landau (1914) showed that for any positive ε, the number of zeros with real part at least 1/2+ε and imaginary part at between −T and T is O(T){\displaystyle O(T)}. Combined with the facts that zeros on the critical strip are symmetric about the critical line and that the total number of zeros in the critical strip is Θ(T log⁡T){\displaystyle \Theta (T\log T)}, almost all non-trivial zeros are within a distance ε of the critical line. Ivić (1985) gives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2 + ε. Hardy–Littlewood conjectures [edit] In 1914 Godfrey Harold Hardy proved that ζ(1 2+i t){\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)} has infinitely many real zeros. The next two conjectures of Hardy and John Edensor Littlewood on the distance between real zeros of ζ(1 2+i t){\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)} and on the density of zeros of ζ(1 2+i t){\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)} on the interval (T,T+H]{\displaystyle (T,T+H]}![Image 110: {\displaystyle (T,T+H]}]( for sufficiently large T>0{\displaystyle T>0}, and H=T a+ε{\displaystyle H=T^{a+\varepsilon }} and with as small as possible value of a>0{\displaystyle a>0}, where ε>0{\displaystyle \varepsilon >0} is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function: For any ε>0{\displaystyle \varepsilon >0} there exists a lower bound T 0=T 0(ε)>0{\displaystyle T_{0}=T_{0}(\varepsilon )>0} such that for T≥T 0{\displaystyle T\geq T_{0}} and H=T 1 4+ε{\displaystyle H=T^{{\tfrac {1}{4}}+\varepsilon }} the interval (T,T+H]{\displaystyle (T,T+H]}![Image 119: {\displaystyle (T,T+H]}]( contains a zero of odd order of the function ζ(1 2+i t){\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}}. Let N(T){\displaystyle N(T)} be the total number of real zeros, and N 0(T){\displaystyle N_{0}(T)} be the total number of zeros of odd order of the function ζ(1 2+i t){\displaystyle ~\zeta \left({\tfrac {1}{2}}+it\right)~} lying on the interval (0,T]{\displaystyle (0,T]~}![Image 124: {\displaystyle (0,T]~}]( For any ε>0{\displaystyle \varepsilon >0} there exists T 0=T 0(ε)>0{\displaystyle T_{0}=T_{0}(\varepsilon )>0} and some c=c(ε)>0{\displaystyle c=c(\varepsilon )>0}, such that for T≥T 0{\displaystyle T\geq T_{0}} and H=T 1 2+ε{\displaystyle H=T^{{\tfrac {1}{2}}+\varepsilon }} the inequality N 0(T+H)−N 0(T)≥c H{\displaystyle N_{0}(T+H)-N_{0}(T)\geq cH} is true. Selberg's zeta function conjecture [edit] Main article: Selberg's zeta function conjecture Atle Selberg(1942) investigated the problem of Hardy–Littlewood 2 and proved that for any ε> 0 there exists such T 0=T 0(ε)>0{\displaystyle T_{0}=T_{0}(\varepsilon )>0} and c = c(ε) > 0, such that for T≥T 0{\displaystyle T\geq T_{0}} and H=T 0.5+ε{\displaystyle H=T^{0.5+\varepsilon }} the inequality N(T+H)−N(T)≥c H log⁡T{\displaystyle N(T+H)-N(T)\geq cH\log T} is true. Selberg conjectured that this could be tightened to H=T 0.5{\displaystyle H=T^{0.5}}. A. A. Karatsuba(1984a, 1984b, 1985) proved that for a fixed ε satisfying the condition 0 <ε< 0.001, a sufficiently large T and H=T a+ε{\displaystyle H=T^{a+\varepsilon }}, a=27 82=1 3−1 246{\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}}, the interval (T, T+H) contains at least cH log(T) real zeros of the Riemann zeta functionζ(1 2+i t){\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)} and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞. Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H], H=T ε{\displaystyle H=T^{\varepsilon }}, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on "supershort" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers ε, ε 1{\displaystyle \varepsilon {1}} satisfying the conditions 0<ε,ε 1<1{\displaystyle 0<\varepsilon ,\varepsilon {1}<1} almost all intervals (T, T+H] for H≥exp⁡{(log⁡T)ε}{\displaystyle H\geq \exp {{(\log T)^{\varepsilon }}}} contain at least H(log⁡T)1−ε 1{\displaystyle H(\log T)^{1-\varepsilon _{1}}} zeros of the function ζ(1 2+i t){\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}. This estimate is quite close to the one that follows from the Riemann hypothesis. Numerical calculations [edit] The function π−s 2 Γ(s 2)ζ(s){\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma ({\tfrac {s}{2}})\zeta (s)} has the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points. Usually one writes ζ(1 2+i t)=Z(t)e−i θ(t){\displaystyle \zeta ({\tfrac {1}{2}}+it)=Z(t)e^{-i\theta (t)}} where Hardy's Z function and the Riemann–Siegel theta functionθ are uniquely defined by this and the condition that they are smooth real functions with θ(0)=0. By finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line. To verify the Riemann hypothesis up to a given imaginary partT of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region using Turing's method and checking that it is the same as the number of zeros found on the line. This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line). These calculations can also be used to estimate π(x){\displaystyle \pi (x)} for finite ranges of x{\displaystyle x}. For example, using the latest result from 2020 (zeros up to height 3×10 12{\displaystyle 3\times 10^{12}}), it has been shown that \|π(x)−li⁡(x)\|<1 8 π x log⁡(x),for 2657≤x≤1.101×10 26.{\displaystyle \|\pi (x)-\operatorname {li} (x)\|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for }}2657\leq x\leq 1.101\times 10^{26}.} In general, this inequality holds if x≥2657{\displaystyle x\geq 2657} and 9.06 log⁡log⁡x x log⁡x≤T,{\displaystyle {\frac {9.06}{\log {\log {x}}}}{\sqrt {\frac {x}{\log {x}}}}\leq T,} where T{\displaystyle T} is the largest known value such that the Riemann hypothesis is true for all zeros ρ{\displaystyle \rho } with ℑ(ρ)∈(0,T]{\displaystyle \Im {\left(\rho \right)}\in \left(0,T\right]}![Image 155: {\displaystyle \Im {\left(\rho \right)}\in \left(0,T\right]}]( Some calculations of zeros of the zeta function are listed below, where the "height" of a zero is the magnitude of its imaginary part, and the height of the n th zero is denoted by γ n. So far all zeros that have been checked are on the critical line and are simple. (A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see Haselgrove & Miller (1960) or Odlyzko. \| Year \| Number of zeros \| Author \| --- \| 1859? \| 3 \| B. Riemann used the Riemann–Siegel formula (unpublished, but reported in Siegel 1932). \| \| 1903 \| 15 \| J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram's law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he found. \| \| 1914 \| 79 (γ n ≤ 200) \| R. J. Backlund (1914) introduced a better method of checking all the zeros up to that point are on the line, by studying the argument S(T) of the zeta function. \| \| 1925 \| 138 (γ n ≤ 300) \| J. I. Hutchinson (1925) found the first failure of Gram's law, at the Gram point g 126. \| \| 1935 \| 195 \| E. C. Titchmarsh (1935) used the recently rediscovered Riemann–Siegel formula, which is much faster than Euler–Maclaurin summation. It takes about O(T 3/2+ε) steps to check zeros with imaginary part less than T, while the Euler–Maclaurin method takes about O(T 2+ε) steps. \| \| 1936 \| 1041 \| E. C. Titchmarsh (1936) and L. J. Comrie were the last to find zeros by hand. \| \| 1953 \| 1104 \| A. M. Turing (1953) found a more efficient way to check that all zeros up to some point are accounted for by the zeros on the line, by checking that Z has the correct sign at several consecutive Gram points and using the fact that S(T) has average value 0. This requires almost no extra work because the sign of Z at Gram points is already known from finding the zeros, and is still the usual method used. This was the first use of a digital computer to calculate the zeros. \| \| 1956 \| 15 000 \| D. H. Lehmer (1956) discovered a few cases where the zeta function has zeros that are "only just" on the line: two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between them. This is called "Lehmer's phenomenon", and first occurs at the zeros with imaginary parts 7005.063 and 7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1. \| \| 1956 \| 25 000 \| D. H. Lehmer \| \| 1958 \| 35 337 \| N. A. Meller \| \| 1966 \| 250 000 \| R. S. Lehman \| \| 1968 \| 3 500 000 \| Rosser, Yohe & Schoenfeld (1969) stated Rosser's rule (described below). \| \| 1977 \| 40 000 000 \| R. P. Brent \| \| 1979 \| 81 000 001 \| R. P. Brent \| \| 1982 \| 200 000 001 \| R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter \| \| 1983 \| 300 000 001 \| J. van de Lune, H. J. J. te Riele \| \| 1986 \| 1 500 000 001 \| van de Lune, te Riele & Winter (1986) gave some statistical data about the zeros and give several graphs of Z at places where it has unusual behavior. \| \| 1987 \| A few of large (≈10 12) height \| A. M. Odlyzko(1987) computed smaller numbers of zeros of much larger height, around 10 12, to high precision to check Montgomery's pair correlation conjecture. \| \| 1992 \| A few of large (≈10 20) height \| A. M. Odlyzko(1992) computed 175 million zeros of heights around 10 20 and a few more of heights around 2×10 20, and gave an extensive discussion of the results. \| \| 1998 \| 10000 of large (≈10 21) height \| A. M. Odlyzko(1998) computed some zeros of height about 10 21 \| \| 2001 \| 10 10 \| J. van de Lune (unpublished) \| \| 2004 \| ≈9×10 11 \| S. Wedeniwski (ZetaGrid distributed computing) \| \| 2004 \| 10 13 and a few of large (up to ≈10 24) heights \| Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm. They also checked two billion zeros around heights γ n = 10 13, 10 14, ..., 10 24. \| \| 2020 \| 1.2363×10 13 (γ n≤ 3×10 12) \| Platt & Trudgian (2021). They also verified the work of Gourdon (2004) and others. \| Gram points [edit] A Gram point is a point on the critical line 1/2+it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e−iθ(t), where Hardy's function, Z, is real for real t, and θ is the Riemann–Siegel theta function, we see that zeta is real when sin(θ(t)) = 0. This implies that θ(t) is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ. They are usually numbered as g n for n = 0, 1, ..., where g n is the unique solution of θ(t) = n π. Gram observed that there was often exactly one zero of the zeta function between any two consecutive Gram points; Hutchinson called this observation Gram's law. There are several other closely related statements that are also sometimes called Gram's law: for example, (−1)n Z(g n) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points. The imaginary parts γ n of the first few zeros (in blue) and the first few Gram points g n are given in the following table g−1 γ 1 g 0 γ 2 g 1 γ 3 g 2 γ 4 g 3 γ 5 g 4 γ 6 g 5 0 3.436 9.667 14.135 17.846 21.022 23.170 25.011 27.670 30.425 31.718 32.935 35.467 37.586 38.999 This is a polar plot of the first 20 real values r n of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of r n in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n. Fifty red points have been plotted between each r n, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t. Gram's law states that the curve usually crosses the real axis once between zeros. The first failure of Gram's law occurs at the 127th zero and the Gram point g 126, which are in the "wrong" order. \| g 124 \| γ 126 \| g 125 \| g 126 \| γ 127 \| γ 128 \| g 127 \| γ 129 \| g 128 \| \| 279.148 \| 279.229 \| 280.802 \| 282.455 \| 282.465 \| 283.211 \| 284.104 \| 284.836 \| 285.752 \| A Gram point t is called good if the zeta function is positive at 1/2 + it. The indices of the "bad" Gram points where Z has the "wrong" sign are 126, 134, 195, 211, ... (sequence A114856 in the OEIS). A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's law called Rosser's rule due to Rosser, Yohe & Schoenfeld (1969) says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them. For example, the interval bounded by g 125 and g 127 is a Gram block containing a unique bad Gram point g 126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function. Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions. The distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 10 24. This means that both rules hold most of the time for small T but eventually break down often. Indeed, Trudgian (2011) showed that both Gram's law and Rosser's rule fail in a positive proportion of cases. To be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long run Hanga (2020). Arguments for and against the Riemann hypothesis [edit] Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authors who express an opinion, most of them, such as Riemann (1859) and Bombieri (2000), imply that they expect (or at least hope) that it is true. The few authors who express serious doubt about it include Ivić (2008), who lists some reasons for skepticism, and Littlewood (1962), who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true. The consensus of the survey articles (Bombieri 2000, Conrey 2003, and Sarnak 2005) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt. Some of the arguments for and against the Riemann hypothesis are listed by Sarnak (2005), Conrey (2003), and Ivić (2008), and include the following: Several analogues of the Riemann hypothesis have already been proved. The proof of the Riemann hypothesis for varieties over finite fields by Deligne (1974) is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated with automorphic forms satisfy a Riemann hypothesis, which includes the classical Riemann hypothesis as a special case. Similarly Selberg zeta functions satisfy the analogue of the Riemann hypothesis, and are in some ways similar to the Riemann zeta function, having a functional equation and an infinite product expansion analogous to the Euler product expansion. But there are also some major differences; for example, they are not given by Dirichlet series. The Riemann hypothesis for the Goss zeta function was proved by Sheats (1998). In contrast to these positive examples, some Epstein zeta functions do not satisfy the Riemann hypothesis even though they have an infinite number of zeros on the critical line. These functions are quite similar to the Riemann zeta function, and have a Dirichlet series expansion and a functional equation, but the ones known to fail the Riemann hypothesis do not have an Euler product and are not directly related to automorphic representations. At first, the numerical verification that many zeros lie on the line seems strong evidence for it. But analytic number theory has had many conjectures supported by substantial numerical evidence that turned out to be false. See Skewes number for a notorious example, where the first exception to a plausible conjecture related to the Riemann hypothesis probably occurs around 10 316; a counterexample to the Riemann hypothesis with imaginary part this size would be far beyond anything that can currently be computed using a direct approach. The problem is that the behavior is often influenced by very slowly increasing functions such as log log T, that tend to infinity, but do so so slowly that this cannot be detected by computation. Such functions occur in the theory of the zeta function controlling the behavior of its zeros; for example the function S(T) above has average size around (log log T)1/2. As S(T) jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing only when S(T) becomes large. It is never much more than 3 as far as it has been calculated, but is known to be unbounded, suggesting that calculations may not have yet reached the region of typical behavior of the zeta function. Denjoy's probabilistic argument for the Riemann hypothesis is based on the observation that if μ(x) is a random sequence of "1"s and "−1"s then, for every ε> 0, the partial sumsM(x)=∑n≤x μ(n){\displaystyle M(x)=\sum {n\leq x}\mu (n)} (the values of which are positions in a simple random walk) satisfy the bound M(x)=O(x 1/2+ε){\displaystyle M(x)=O(x^{1/2+\varepsilon })} with probability 1. The Riemann hypothesis is equivalent to this bound for the Möbius functionμ and the Mertens function_M derived in the same way from it. In other words, the Riemann hypothesis is in some sense equivalent to saying that μ(x) behaves like a random sequence of coin tosses. When μ(x) is nonzero its sign gives the parity of the number of prime factors of x, so informally the Riemann hypothesis says that the parity of the number of prime factors of an integer behaves randomly. Such probabilistic arguments in number theory often give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answer for some results, such as Maier's theorem. The calculations in Odlyzko (1987) show that the zeros of the zeta function behave very much like the eigenvalues of a random Hermitian matrix, suggesting that they are the eigenvalues of some self-adjoint operator, which would imply the Riemann hypothesis. All attempts to find such an operator have failed. There are several theorems, such as Goldbach's weak conjecture for sufficiently large odd numbers, that were first proved using the generalized Riemann hypothesis, and later shown to be true unconditionally. This could be considered as weak evidence for the generalized Riemann hypothesis, as several of its "predictions" are true. Lehmer's phenomenon, where two zeros are sometimes very close, is sometimes given as a reason to disbelieve the Riemann hypothesis. But one would expect this to happen occasionally by chance even if the Riemann hypothesis is true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted by Montgomery's conjecture. Patterson suggests that the most compelling reason for the Riemann hypothesis for most mathematicians is the hope that primes are distributed as regularly as possible. Notes [edit] ^Bombieri (2000). ^Euler, Leonhard (1744). Variae observationes circa series infinitas.Commentarii academiae scientiarum Petropolitanae 9, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case s=1{\displaystyle s=1}, and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that ζ(1)=log⁡∞{\displaystyle \zeta (1)=\log \infty }, and he makes use of this latter result in his Theorem 19, to show that the sum of the inverses of the prime numbers is log⁡log⁡∞{\displaystyle \log \log \infty }. ^Values for ζ can be found by calculating, e.g., ζ(1/2 − 30 i).("Wolframalpha computational intelligence". wolframalpha.com. Wolfram. Retrieved 2 October 2022. ^Ingham (1932), Theorem 30, p. 83; Montgomery & Vaughan (2007), p. 430. ^Ingham (1932), p.82. ^Landau, Edmund (1924), "Über die Möbiussche Funktion", Rend. Circ. Mat. Palermo, 48 (2): 277–280, doi:10.1007/BF03014702, S2CID123636883 ^Titchmarsh, Edward Charles (1927), "A consequence of the Riemann hypothesis", J. London Math. Soc., 2 (4): 247–254, doi:10.1112/jlms/s1-2.4.247 ^Maier, Helmut; Montgomery, Hugh (2009), "The sum of the Möbius function", Bull. London Math. Soc., 41 (2): 213–226, doi:10.1112/blms/bdn119, hdl:2027.42/135214, S2CID121272525 ^Soundararajan, Kannan (2009), "Partial sums of the Möbius function", J. Reine Angew. Math., 2009 (631): 141–152, arXiv:0705.0723, doi:10.1515/CRELLE.2009.044, S2CID16501321 ^Robin (1984). ^Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN0002-9890, JSTOR2695443, MR1908008, S2CID15884740 ^Broughan (2017), Corollary 5.35. ^ abcTitchmarsh (1986). ^Nicely (1999). ^Baez-Duarte, Luis (2005). "A general strong Nyman-Beurling criterion for the Riemann hypothesis". Publications de l'Institut Mathématique. Nouvelle Série. 78 (92): 117–125. arXiv:math/0505453. doi:10.2298/PIM0578117B. S2CID17406178. ^Rodgers & Tao (2020). ^ abPlatt & Trudgian (2021). ^"Caltech Mathematicians Solve 19th Century Number Riddle". California Institute of Technology. October 31, 2022. ^Dunn, Alexander; Radziwiłł, Maksym (2021). "Bias in cubic Gauss sums: Patterson's conjecture". arXiv:2109.07463 [math.NT]. ^Goldfeld, Dorian (1985). "Gauss' class number problem for imaginary quadratic fields". Bulletin of the American Mathematical Society. 13 (1): 23–37. doi:10.1090/S0273-0979-1985-15352-2. ISSN0273-0979. ^Siegel, Carl (1935). "Über die Classenzahl quadratischer Zahlkörper". Acta Arithmetica. 1 (1): 83–86. doi:10.4064/aa-1-1-83-86. ISSN0065-1036. Retrieved 8 April 2024. ^Ribenboim (1996), p.320. ^Radziejewski (2007). ^Wiles (2000). ^Leichtnam (2005). ^Knauf (1999). ^Sarnak (2005). ^Odlyzko (2002). ^Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew (2022-12-13). "Explicit zero-free regions for the Riemann zeta-function". arXiv:2212.06867 [math.NT]. ^Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros of ζ are on the critical line". 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II", Mathematics of Computation, 22 (101): 163–173, doi:10.2307/2004774, JSTOR2004774, MR0228456 Stein, William; Mazur, Barry (2007), What is Riemann's Hypothesis?(PDF), archived from the original(PDF) on 2009-03-27 Suzuki, Masatoshi (2011), "Positivity of certain functions associated with analysis on elliptic surfaces", Journal of Number Theory, 131 (10): 1770–1796, doi:10.1016/j.jnt.2011.03.007 Titchmarsh, Edward Charles (1935), "The Zeros of the Riemann Zeta-Function", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 151 (873), The Royal Society: 234–255, Bibcode:1935RSPSA.151..234T, doi:10.1098/rspa.1935.0146, JSTOR96545, S2CID263711590 Titchmarsh, Edward Charles (1936), "The Zeros of the Riemann Zeta-Function", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 157 (891), The Royal Society: 261–263, arXiv:1004.4143, Bibcode:1936RSPSA.157..261T, doi:10.1098/rspa.1936.0192, JSTOR96692 Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon Press Oxford University Press, ISBN978-0-19-853369-6, MR0882550 Trudgian, Timothy S. (2014), "An improved upper bound for the argument of the Riemann zeta function on the critical line II", J. Number Theory, 134: 280–292, arXiv:1208.5846, doi:10.1016/j.jnt.2013.07.017 Trudgian, Timothy (2011), "On the success and failure of Gram's Law and the Rosser Rule", Acta Arithmetica, 125 (3): 225–256, doi:10.4064/aa148-3-2 Turán, Paul (1948), "On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann", Danske Vid. Selsk. Mat.-Fys. Medd., 24 (17): 36, MR0027305 Reprinted in (Borwein et al. 2008). Turing, Alan M. (1953), "Some calculations of the Riemann zeta-function", Proceedings of the London Mathematical Society, Third Series, 3: 99–117, doi:10.1112/plms/s3-3.1.99, MR0055785 de la Vallée-Poussin, Ch.J. (1896), "Recherches analytiques sur la théorie des nombres premiers", Ann. Soc. Sci. Bruxelles, 20: 183–256 de la Vallée-Poussin, Ch.J. (1899–1900), "Sur la fonction ζ(s) de Riemann et la nombre des nombres premiers inférieurs à une limite donnée", Mem. Couronnes Acad. Sci. Belg., 59 (1) Reprinted in (Borwein et al. 2008). Weil, André (1948), Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, MR0027151 Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, MR0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN0-387-90330-5 Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Analytic number theory ( St. Louis Univ., 1972), Proc. Sympos. Pure Math., vol.24, Providence, R.I.: Amer. Math. Soc., pp.321–332, MR0337902 Wiles, Andrew (2000), "Twenty years of number theory", Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, pp.329–342, ISBN978-0-8218-2697-3, MR1754786 Zagier, Don (1977), "The first 50 million prime numbers"(PDF), Math. Intelligencer, 1, Springer: 7–19, doi:10.1007/BF03039306, MR0643810, S2CID189886510, archived from the original(PDF) on 2009-03-27 Zagier, Don (1981), "Eisenstein series and the Riemann zeta function", Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol.10, Tata Inst. Fundamental Res., Bombay, pp.275–301, MR0633666 Popular expositions [edit] Sabbagh, Karl (2003a), The greatest unsolved problem in mathematics, Farrar, Straus and Giroux, New York, ISBN978-0-374-25007-2, MR1979664 Sabbagh, Karl (2003b), Dr. Riemann's zeros, Atlantic Books, London, ISBN978-1-843-54101-1 du Sautoy, Marcus (2003), The music of the primes, HarperCollins Publishers, ISBN978-0-06-621070-4, MR2060134 Rockmore, Dan (2005), Stalking the Riemann hypothesis, Pantheon Books, ISBN978-0-375-42136-5, MR2269393 Derbyshire, John (2003), Prime Obsession, Joseph Henry Press, Washington, DC, ISBN978-0-309-08549-6, MR1968857 Watkins, Matthew (2015), Mystery of the Prime Numbers, Liberalis Books, ISBN978-1782797814, MR0000000 Frenkel, Edward (2014), The Riemann HypothesisNumberphile, Mar 11, 2014 (video) Nahin, Paul J. (2021). In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem. Princeton University Press. ISBN978-0691206073. Note: Derbyshire 2003, Rockmore 2005, Sabbagh 2003a, Sabbagh 2003b, Sautoy 2003, and Watkins 2015 are non-technical. Edwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, Broughan 2017, and Nahin 2021 give mathematical introductions. Titchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advanced monographs. External links [edit] Media related to Riemann hypothesis at Wikimedia Commons Wikiquote has quotations related to Riemann hypothesis. Mathematics portal American Institute of Mathematics, Riemann hypothesis Zeroes database, 103 800 788 359 zeroes Apostol, Tom, Where are the zeros of zeta of s? Poem about the Riemann hypothesis, sung by John Derbyshire. Borwein, Peter, The Riemann Hypothesis(PDF), archived from the original(PDF) on 2009-03-27 (Slides for a lecture) Conrad, K. (2010), Consequences of the Riemann hypothesis Conrey, J. Brian; Farmer, David W, Equivalences to the Riemann hypothesis, archived from the original on 2010-03-16 Gourdon, Xavier; Sebah, Pascal (2004), Computation of zeros of the Zeta function (Reviews the GUE hypothesis, provides an extensive bibliography as well). Odlyzko, Andrew, Home page including papers on the zeros of the zeta function and tables of the zeros of the zeta function Odlyzko, Andrew (2002), Zeros of the Riemann zeta function: Conjectures and computations(PDF) Slides of a talk Pegg, Ed (2004), Ten Trillion Zeta Zeros, Math Games website, archived from the original on 2004-11-02, retrieved 2004-10-20. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros Rubinstein, Michael, algorithm for generating the zeros, archived from the original on 2007-04-27. du Sautoy, Marcus (2006), Prime Numbers Get Hitched, Seed Magazine, archived from the original on 2017-09-22, retrieved 2006-03-27 Watkins, Matthew R. (2021-02-27), Proposed (dis)proofs of the Riemann Hypothesis, archived from the original on December 9, 2022 Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005 \| v t e L-functions in number theory \| \| Analytic examples \| Riemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg class \| \| Algebraic examples \| Dedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functions \| \| Theorems \| Analytic class number formula Riemann–von Mangoldt formula Weil conjectures \| \| Analytic conjectures \| Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture \| \| Algebraic conjectures \| Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture \| \| p-adic L-functions \| Main conjecture of Iwasawa theory Selmer group Euler system \| \| v t e Bernhard Riemann \| \| Cauchy–Riemann equations Generalized Riemann hypothesis Grand Riemann hypothesis Grothendieck–Hirzebruch–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Local zeta function Measurable Riemann mapping theorem Riemann (crater) Riemann Xi function Riemann curvature tensor Riemann hypothesis Riemann integral Riemann invariant Riemann mapping theorem Riemann form Riemann problem Riemann series theorem Riemann solver Riemann sphere Riemann sum Riemann surface Riemann zeta function Riemann's differential equation Riemann's minimal surface Riemannian circle Riemannian connection on a surface Riemannian geometry Riemann–Hilbert correspondence Riemann–Hilbert problems Riemann–Lebesgue lemma Riemann–Liouville integral Riemann–Roch theorem Riemann–Roch theorem for smooth manifolds Riemann–Siegel formula Riemann–Siegel theta function Riemann–Silberstein vector Riemann–Stieltjes integral Riemann–von Mangoldt formula \| \| Category \| \| Authority control databases \| \| International \| GND FAST \| \| National \| United States France BnF data Japan Czech Republic Spain 2 Israel \| \| Other \| IdRef Yale LUX \| Retrieved from " Categories: 1859 introductions Analytic number theory Bernhard Riemann Conjectures Hilbert's problems Hypotheses Millennium Prize Problems Unsolved problems in number theory Zeta and L-functions Hidden categories: Articles with short description Short description is different from Wikidata Articles containing German-language text CS1 Russian-language sources (ru) Commons category link from Wikidata Pages using Sister project links with hidden wikidata CS1: unfit URL This page was last edited on 26 September 2025, at 12:33(UTC). 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3887	https://naikermaths.com/wp-content/uploads/2018/05/Straight-line-Graphs-MARK-SCHEME.pdf	www.naikermaths.com Straight line graphs - Edexcel Past Exam Questions MARK SCHEME Question 1: Jan 05 Q8 www.naikermaths.com Question 2: Jan 05 Q10 www.naikermaths.com Question 3: June 05 Q8 www.naikermaths.com Question 4: Jan 06 Q3 www.naikermaths.com Question 5: June 06 Q11 www.naikermaths.com Question 6: June 07 Q10 . www.naikermaths.com Question 7: June 07 Q11 www.naikermaths.com Question 8: Jan 08 Q4 www.naikermaths.com Question 9: June 08 Q10 www.naikermaths.com Question 10: Jan 09 Q10 www.naikermaths.com Question 11: June 09 Q8 www.naikermaths.com Question 12: Jan 10 Q3 www.naikermaths.com Question 13: Jan 10 Q9 www.naikermaths.com Question 14: June 10 Q8 www.naikermaths.com Question 15: Jan 11 Q9 www.naikermaths.com Question 16: June 11 Q3
3888	https://www.reddit.com/r/EngineeringStudents/comments/3q18dd/can_someone_please_describe_and_define_laminar/	can someone please describe and define laminar and turbulent flow (Aeronautical wise) : r/EngineeringStudents Skip to main contentcan someone please describe and define laminar and turbulent flow (Aeronautical wise) : r/EngineeringStudents Open menu Open navigationGo to Reddit Home r/EngineeringStudents A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to EngineeringStudents r/EngineeringStudents r/EngineeringStudents This is a place for engineering students of any discipline to discuss study methods, get homework help, get job search advice, and find a compassionate ear when you get a 40% on your midterm after studying all night. 947K Members Online •10 yr. ago [deleted] can someone please describe and define laminar and turbulent flow (Aeronautical wise) Share Related Answers Section Related Answers Best software tools for engineering students Engineering students often need a variety of software tools to help them with their studies and future careers. Here are some of the most recommended software tools based on the experiences and opinions of Redditors: Design and Modeling Software SolidWorks: Highly recommended for mechanical engineering. "Solidworks. Very common, pretty easy, you can do a lot." AutoCAD: Essential for civil and mechanical engineering. "AutoCAD Civil 3D is the most common drafting program." Inventor: A strong alternative to SolidWorks. "Inventor and excel so basically the same." NX: Known for its complexity but powerful features. "NX is by far the hardest CAD software I ever learned." CATIA: Used in aerospace and automotive industries. "Everyone using Solidworks and here I am crying that the company I’m with uses CATIA ☹️" Simulation and Analysis Software MATLAB: Versatile for various engineering fields. "Matlab, not design software but the best software to master." ANSYS: For finite element analysis (FEA) and computational fluid dynamics (CFD). "Ansys" Simulink: Useful for dynamic system modeling. "Matlab and Simulink" Programming and Data Analysis Excel: Ubiquitous and essential for data processing. "Don't underestimate the power of excel." Python: Widely used for scripting and data analysis. "Python is free, so they don't lose access after graduating." LabVIEW: For graphical programming, especially in control systems. "Matlab & Labview" Electrical Engineering Tools Altium: For PCB design. "If you are majoring in EE Altium" OrCAD: Another popular choice for PCB design. "If you can get your hands on OrCAD 17.2 and PADS too…" Civil Engineering Specific Civil 3D: For civil engineering drafting. "AutoCAD Civil 3D is the most common drafting program." HEC-RAS: For hydraulic modeling. "HEC-RAS" ArcGIS: For geographic information systems (GIS). "ESRI ArcGIS" Project Management and Collaboration Trello: For team collaboration and project management. "Trello works really well for small teams and small projects." Jira: Widely used in Agile and Scrum environments. "Jira (widely used software tool in Agile/SCRUM work environments)" Other Useful Tools LaTeX: For writing technical documents. "Word/LaTeX" Notepad++: For coding and text editing. "Notepad++" Subreddits for Further Questions r/EngineeringStudents r/MechanicalEngineering r/civilengineering r/AskEngineers See Answer Strategies for acing engineering midterms How to build a strong engineering resume Networking tips for engineering students Prepare for engineering job interviews New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of October 24, 2015 Reddit reReddit: Top posts of October 2015 Reddit reReddit: Top posts of 2015 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
3889	https://www.youtube.com/watch?v=GN5aKlRjVBY	Competitive Equilibrium Condition MRS 1 = MRS 2 = Price Ratio \| 4 \| nishant mehra 32600 subscribers 181 likes Description 12115 views Posted: 15 Apr 2021 This video talks about Competitive Equilibrium Condition MRS 1 = MRS 2 =Px/Py (REFERENCE : Varian Ch 31) This is useful for those who are preparing 1) Intermediate Microeconomics Course in their semesters 2) UGC Net Economics 3) Basics of MA Economics Entrance Exam (DSE/ISI/JNU/IGIDR/MSE) 4) Indian Economic Services 5) Indian Administration Services (Economics Optional) For any course related query please call 9999886629 or visit nishantmehra.com for IAS Economics Optional/ Indian Economic Services/ UGC Net Economics or visit Ecopoint.in for MA Economics Entrance Coaching 9 comments Transcript: let us try to find out uh the market equilibrium condition we have already seen what do you mean by pareto efficiency and we've also seen what are the different examples for pareto efficiency now let's move on little further right let's try to find out the market equilibrium where exactly an equilibrium is going to lie for a two person to good economy right so we'll first of all assume that there's an auctioneer and he's setting the prices he's saying that the price of x is going to be this and the price of y is going to be this right so there's an auction here who sets the prices imperfect competition so this is a perfectly competitive market and this auctioneer is setting the price pxpy is setting the price and there's a consumer who is maximizing the utility okay so the prices are given to this consumer we have already seen that there are two consumers and there are two goods so consumers are a and b goods are x and y so every consumer will maximize utility right this consumer a what is he doing he is maximizing his utility ua xavier subject to subject to the constraint budget constraint what is he consuming beta he's consuming x a y a at what prices is he consuming this at prices p x and p by so his his expenditure in consuming xavier is px px into xa plus p by into y this has to be equal to his income what is his income in this uh endowment economy in this exchange economy we are assuming that there is no outside income so whatever is his endowment the value of that endowment itself is just the income so what is this is what is his endowment your w a x for good x and w a y for good y this is what it is what is the value of his endowment would be then p x w a x plus p y w a y so this is the value value of e is endowment that's what it is written that's what it is values of value of t is f does individual b who is maximizing his utility subject to his budget constraint what will this be then p x x b plus p y y b p x w b x plus p y w p y amata plus p y w b y so it is this is the value of b's endowment this is the value of b's endowment okay so they are i'm individual and you guys are individual b what am i demanding x a and y i'm individually i'm demanding x a y a at what prices i'm demanding px py of course this is going to be the function of p x p y these are my demand functions 6 y and these are my gross demands these are my gross demands right and and what is i mean i am also endowed with some amount of x and some amount of y right so supposedly if i'm endowed with 2 units of x and i'm demanding 5 units of x so my gross demand for xs5 while my net demand or excess demand for x is three so these are my gross demands better what are net demand or excess demand what are net demand or excess demand for x for individual a for x it would be what the amount of x which he is demanding that is his gross demand minus his endowment of x is excess demand for good y would be amount of y which he is demanding minus endowment of y which he has right similarly individual b you guys are individual b your gross demand for x is like this these are going to be a gross demands of x what is your excess demand for x e b x is your gross demand for x minus what you are endowed with x similarly your gross demand for y would be yb double p five this one is right what is the aggregate excess demand so x is demand okay how do you write this what is the aggregate excess demand aggregate excuse demand for good x is excess demand for good x by individually plus excess demand for good x by individual b that is the aggregate excess demand please write this aggregate access demand that is zx pxpy is aggregate excess demand for good x by individually plus aggregate excess demand for good x by individual b similarly z y p x p y that is aggregate excess demand for good y is going to be what excess demand for good vibe individually excess demand for good vibe individual that is going to be there okay now let us try to understand how does equilibrium comes in this market this is interesting you will like it so just assume we have 10 units of x here and 10 units of y here right okay so one more thing i think i hope you understand this thing if you just look at my budget constraint this is what my budget constraint is this budget constraint which i have better this is passing through endowments yes so the this budget line my budget line has to pass through endowment and it is passing through x a y a also so if x a y a is going to be an equilibrium whatever it is it has to pass through that x a by a also so wherever i am consuming it has to pass through that point and it is it and it also has to pass through the endowment fair enough now let us say my endowment is here my endowment is here my endowment is here okay let's see here so i have eight units of x and let's say two units of y i have something like this this is what my endowment is i know that the budget line has to pass through an endowment let us say auctioneer has given us the prices pxpy and the my budget line looks like this something like this this is what an endowment is and what is your endowment better from your side i mean i'm measuring this from your side to from your origin this is ob this is o and you have you have two units of x and eight units so y so for you also this is an endowment uh so of course this point is an endowment for you from your site and prices are same for both of us pxpy okay my utility functions are going to move like this let us say so given this budget line is this my highest indifference curve no no no i can go further is this my highest indifference curve no no no i can go further i can go further now so let us say i move till here so given this these prices i move here okay and what do i do here i want to demand this much because i'm taking time in this so that you understand what's going on behind the scenes it is very easy for me to tell you just a numerical question and just do it but you should know what's going on behind the scene let us say i want to demand here three units of x and six units of y right okay this much huh so you can say this is individual is uh whatever equilibrium so given this budget now you will also be maximizing i know you will also be maximizing your utility so your utility are moving like this is this the highest indifference given this budget line is this your highest indifference curve no is this your highest indifference curve no you can go further i didn't know how to okay let let it be like this so let let us say this is your highest indifference let us say like this this is what your highest indifference curve is and you are let us say consuming here have it done so what happens you want to consume more x and let us say whatever this much although something like this so this is your thing okay i want to give up i want to sell i am individually you guys are individual p but when i move from eight to three what is it that i want to do individual a wants to sell five units of x right and at the same time individual a wants to buy how much four units so four units of y i'm so sorry four units are back and i remember this i'm so sorry individual a wants to buy four units of y huh okay what individual b wants to do just think about it individual b wants to move from this point at this point so individual b wants to buy two units of x have it done individual b wants to buy two units of x and wants to sell one unit of y that's what it is so he wants to sell one unit of y but he wants to buy two units of x fair enough right so just think about it what is happening out here my my supply of x is more than what your demand for x is that's right that's right supply of x by individual a is more than demand of x by individual b is this right so it means what at these prices there is an excess supply of a okay what about why i want to demand four units of y but you want to sell only less y so my demand of y by individual a is more than supply of y by individual p have it so it means what it's not excess supply of a access supply of x and here it is excess demand of fire right add to prices pxp at prices pxp now you tell me one thing if there is an excess supply of x what should happen to the price of x rise of x should increase sorry sorry price of x should decrease right because there's an excess supply of x and when there is excess demand for something what should happen to the price of that thing price of y should increase you with me so you have initially p x by p by like this now you have p x dash and p by dash so p x is falling and py is increasing so what is happening that this guy is going to fall this guy is going to fall ham it up this guy is going to fall so what will happen is that this line you will have a new line which is okay i will you will have a new line which is going to be a flatter one at the new prices at the new prices so these prices are what new price ratio px dash py dash again this is this is going through this is going through the endowment point and this condition um this this change in prices is going to happen till all gains from the trade are exhausted that means till the conditions are going to look like this so this is going to be my indifference curve and maybe this is going to be your indifference curve ultimately at this particular point beta so at this line at this line let's say i'm going to maximize my utility here and at this line you're going to maximize your utility here so the amount of x which i am demanding is equal to the supply of x which you are supplying and the amount of x amount of y which i am supplying is equal to the amount of y which you are demanding so ideally what what what exactly i mean uh by saying this is that there is going to be demand equal to supply at this particular point and this process of change in the price so you might write somewhere the way you have written it for bx dash py dash so the process this process will continue till equilibrium is reached right and and when when do you say equilibrium is reached so you say that equilibrium is reached when when when both i mean very loosely speaking when both maximize at the same point or in other words all gains from trade are exhausted all gains from trade are exhausted now if you look this point very clearly what if you will find that this particular point is a very interesting point this tells you something one this is an equilibrium what i mean there is a tangency condition for me mrs xy sorry mrs xy for individually i mean i'm individually is equal to the price ratio that's what it is given the budget i'm maximizing my utility similarly this is the point mrs xy for b is also equal to p x by p by m r s x y for v is also equal to p x by p by that is for u also this tendency condition holds which you can write very clearly as mrs xy for a is equal to mrs xy for b is equal to b x by p y this is the condition which you have right so this competitive equilibrium point which you have got is also pareto efficient right because mrs x y a is equal to m r s x y b is holding out here right okay in the next recording uh we're going to talk about walrus law thanks
3890	https://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/AtwoodMachine	We use essential cookies to make our site work. With your consent, we may also use non-essential cookies to improve user experience, personalize advertisements, and analyze web traffic. For these reasons, we may share your site usage data with our advertising and analytics partners. By clicking “Accept,” you agree to our website's cookie use as described in our Cookie Policy. You can change your cookie settings at any time by clicking “Preferences.” Accept [x] Do not display this message again Close window Atwood Machine - Maple Help For the best experience, we recommend viewing online help using Google Chrome or Mozilla Firefox. 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George Atwood in 1784 to illustrate the dynamics of Newton's laws. It consists of a massless, inextensible string which connects two masses, m 1 and m 2 through an ideal pulley. (An ideal pulley is one which is assumed to have negligible mass and no friction between itself and the string). A straightforward application of Newton's laws can predict the acceleration of the blocks and the time it takes for them to reach the ground. The mass on the left, m 1, experiences two forces: −m 1 g from gravity, and T, the tension force from the rope. Newton's second law for m 1 then states that: m 1 a 1=−m 1 g+T, where a 1 is the acceleration of m 1 in the upwards direction. The second mass, m 2, experiences a net force of: m 2 a 2=−m 2 g+T. where a 2 is the acceleration of m 2. Notice that in order for the rope to maintain its total length, the accelerations must be equal and opposite, hence a 1=−a 2. Now subtracting the second equation from the first equation gives: m 1 a 1−m 2 a 2=−m 1 g+m 2 g. Finally, making the substitution a 1=−a 2 and rearranging for a 1 yields: a 1=g⋅m 2−m 1 m 2+m 1. Note that if m 2>m 1, the first mass will accelerate upwards and m 2 will accelerate downwards, and if m 1>m 2 the opposite will happen. In an experiment, you could adjust the masses and measure the time it takes for a mass to reach the ground. The time is given by the solution to the equation 0=y 0+a t 2, and this provides a way to determine a and ultimately find g. By choosing blocks with very similar masses, the acceleration is much slower and so air resistance is less important, thereby giving a more accurate method of computing g. (On the other hand, the assumption of a friction-free rope becomes untenable if the masses are too similar.) Adjust the masses of the blocks with the sliders and press the "Animate" button to drop them. 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3891	https://worldofnumbers.com/primorialprimes.pdf	PRIMORIAL BASED PRIMES R. A. Bonham Abstract The product of all successive primes starting at Prime = 2 and ending at the mth prime, Prime[m], is called the mth primorial. In order to form primorial based primes an odd number must be added or subtracted from a primorial. This note explores the nature of the primes formed in this process. The mth primorial is defined as primorial[m] =  m=1 m Prime[n] where Prime[n] is the nth prime number. The primorial is always an even number so that to produce a prime requires adding or subtracting 1 from a primorial or adding and subtracting an odd number from a primorial. Only the latter cases will be considered here. Some obvious properties of such numbers are : (1.) primorial[m] ± 2 k + 1cannot be prime if 2 k + 1is less tham Prime[m + 1]. 2.If 2 k + 1is greater than Prime[m] but less than Prime[m]^2 then primorial[m] ± 2 k + 1 can be a prime only if 2 k + 1 is also a prime. 3.For numbers larger than primorial[m] ± Prime[m]^2 primes of the form primorial[m] ± 2 k + 1can be prime even if 2 k + 1is not a prime. It is the purpose of this note to point out the properties and advantages of searching for primes of the numbers N[m, p] = primorial[m] ± Prime [m + p] (1.) where p varies from 1 to p (max) with p (max) the solution to the equation Prime[m + p (max)] = Prime[m]^2. 2. The principle argument for searching for primes of this form is that primorials with more than a million digits, m = 170 000 for example, can be calculated in a matter of a few seconds with most desk top computers. Hence it is of interest to explore some of the properties of such numbers. In what follows searches for primes in the region N[m, p] with p varied from 1 to p (max) , labeled region 1, will be compared to the numbers N[m, k] = primorial[m] ±Prime[m]^2 + 2 k, 3. labeled region 2, where k varies from 1 to Prime[m] Prime[m] - 12. Which is the same number of searches required for region 1 if values of 2 k + 1that are not prime are also included in the search. In order to study the properties of the numbers defined in Eqs. 2 and 3 it is convienent to use the "Mathematica" language where p (max)[m] is defined as p (max)[m] = PrimePi[Prime[m]^2] - m, (4.) primorial[m] by, primorial[m] = Product[Prime[k], {k, 1, m}] (5.) and the number of digits in the primorial[m] which are given as numberdigits[m] = Sum[DigitCount[primorial[m], 10, j], {j, 0, 9}]. 6. It should be noted that for large primes any number of the form given in Eq. 1 will almost always have the same number of digits as those in the primorial since a prime with 7 to 10 digits will be added or subtracted from a primorial with more than a million digits. Nearly all of the most significant nearly million digits would have to be 9' s in the case of adding a prime to a primorial or 1 followed by almost a million zeros in the case of subtracting a prime in order to change the number of digits from that of the primorial. In order to test for probable primality the following four methods are available in "Mathematica" PrimeQ[N[m, p]] = True (7.) NextPrime[N[m, p]] = value of m for the next larger prime 8. CompositeQ[N[m, p]] = False 9. and PowerMod[2, N[m, p] - 1, N[m, p]] = 1, 10. 2 primorial primes paper082117.nb where the latter is Fermat' s little theorem. All of these methods require about the same computing time to determine that a number is not prime but Eq. 10 is much faster when the number is probably a prime about the same time as for not a prime. However, for large primes the time to prove that a number is probably prime requires the use of a super computer although the total number of searches needed to find all of the primes in regions 1 and 2, the number of digits in large primorials, and the primorial itself can be easily computed with a desk top computer. In the remaining section the behavior of the numbers in regions 1 and 2 will be explored for small primes. In Fig. 1 the dependence of the number of digits in the primorial as a function of m is shown which appears to be nearly a linear function of m. Figure 1 : Number of digits in the mth primorial as a function of m points. The solid line is the best fit straight line given as : digits[m] = -6888.6 + 5.80435 m . digits[m] 50000 100000 150000 200000 400000 600000 800000 1 × 106 m In Fig. 2 the number of searches, p (max)[m], required to cover all of region 1 is shown as a function of m. Figure 2 : Number of searches in region 1 required to find all possible primes of the type primorial plus or minus a prime as a function of m points. The solid line is the best fit quadratic function given as : searches1[m] = 9.52099 10^7 - 58 716 m + 6.83784 m2 primorial primes paper082117.nb 3 searches1[m] 50000 100000 150000 5.0 × 1010 1.0 × 1011 1.5 × 1011 m In Fig. 3 the same plot for the total number of searches required , k (max)[m], to find all primes in region 2 is shown with the best quadratic fit. Note that the number of searches required is more than an order of magnitude larger than for searches in region 1 for numbers with more than a million digits. Figure 3 : Number of searches in region 2 required to find all possible primes of the type primorial plus an odd number as a function of m points. The solid line is the best fit quadratic function given as : searches2[m] = 2.74236 10^9 - 1.55020 10^6 m + 100.53557 m2 searches2[m] 50000 100000 150000 5.0 × 1011 1.0 × 1012 1.5 × 1012 2.0 × 1012 2.5 × 1012 m 4 primorial primes paper082117.nb The remaining quantities of interest : the number of primes in regions 1 and 2 as a function of m, the prime density as a function of m, and its reciprocal, the average number of searches per prime found , all involve determining probable primality which can only reasonably be determined using a desktop computer for small values of m. In Figures 4 -7 the dependence of the total number of primes found in complete searches of regions 1 and 2 are shown along with best linear fits of the data for values of m up to 500. Figure 4 : Total number of primes found in region 1 of the type primorial plus a prime number as a function of m points. The solid line is the best fit to a linear function given as : prime1p[m] = -162.3895 + 7.18096 m prime1p[m] 100 200 300 400 500 500 1000 1500 2000 2500 3000 3500 m Figure 5 : Total number of primes found in region 1 of the type primorial minus a prime number as a function of m points. The solid line is the best fit to a linear function given as : prime1m[m] = -139.134797 + 6.99766 m primorial primes paper082117.nb 5 prime1m[m] 100 200 300 400 500 500 1000 1500 2000 2500 3000 3500 m Figure 6 : Total number of primes found in region 2 of the type primorial plus all odd numbers as a function of m points. The solid line is the best fit to a linear function given as : prime2p[m] = -178.9333 + 6.91721 m prime2p[m] 100 200 300 400 500 500 1000 1500 2000 2500 3000 m Figure 7 : Total number of primes found in region 2 of the type primorial minus all odd numbers as a function of m points. The solid line is the best fit to a linear function given as : prime2m[m] = -175.73333 + 6.914667 m 6 primorial primes paper082117.nb prime2m[m] 100 200 300 400 500 500 1000 1500 2000 2500 3000 3500 m It should be noted that the total number of primes in region 1 is greater than the number in region 2 even if the numbers with Prime[m]^2 + 2 k not prime are included. In Fig. 8 the average number of searches to find a primorial plus a prime in region 1 is displayed. Extrapolation of this result to m = 170 000 million digit primessuggests that, on average, about 81 000 searches would be required to locate a probable prime of that size. Fig. 9 displays the results for a primorial minus a prime as a function of m with a projected number of searches to find a prime of about 84 000. Figure 8 : Average number of searches to find a prime in region 1 of the type primorial plus a prime as a function of m points. The solid line is the best fit to a linear function given as : search primep[m] = -1.67901 + 0.474315 m primorial primes paper082117.nb 7 search primep[m] 100 200 300 400 500 600 50 100 150 200 250 m Figure 9 : Average number of searches to find a prime in region 1 of the type primorial minus a prime as a function of m points. The solid line is the best fit to a linear function given as : search primem[m] = -4.08821 + 0.492660 m search primem[m] 100 200 300 400 500 50 100 150 200 250 m In Figures 10 and 11 the average number of searches to find a prime in region 2 are shown. For m = 170 000 the extrapolated number of searches is more than 670 000 or approximately 8 times the number of searches required for region 1. 8 primorial primes paper082117.nb Figure 10 : Average number of searches to find a prime in region 2 of the type primorial plus all odd numbers as a function of m points. The solid line is the best fit to a linear function given as : search primep2[m] = -109.19498 + 3.948692 m search primep2[m] 100 200 300 400 500 500 1000 1500 2000 m Figure 11 : Average number of searches to find a prime in region 2 of the type primorial minus all odd numbers as a function of m points. The solid line is the best fit to a linear function given as : search primem2[m] - 111.94147 + 3.948079 m search primem2[m] 100 200 300 400 500 500 1000 1500 2000 m In summary, it has been shown that the number of searches required to find all primes in regions 1 and 2 is approximately a quadratic primorial primes paper082117.nb 9 function of m where m is the mth primorial. The number of digits in the primorial the number of probable primes in regions 1 and 2 , and the average number of searches required to find a probable prime in regions 1 and 2 are all approximately linear functions of m. 10 primorial primes paper082117.nb
3892	https://mathbitsnotebook.com/Algebra2/Quadratics/QDdiscriminant.html	\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| Using the Discriminant MathBitsNotebook.com Topical Outline \| Algebra 2 Outline \| MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts \| When solving a quadratic equation, we are searching for the two solutions that will make the equation true. We have encountered situations where there appeared to be only one solution (a repeated root). And, we have encountered situations where there was no Real number solution to the equation (an imaginary/complex solution). Fortunately, there is a quick way to determine an equation's "number and type of solutions". The Quadratic Formula to the rescue again! \| \| \| \| --- \| The discriminant is the portion "under" the square root symbol in the Quadratic Formula: \| The value under the square root symbol (the radicand) portion of the quadratic formula indicates the "number and type of solutions" for a specific quadratic equation. That value is "b2 - 4ac". \| \| \| b2 - 4ac is called the discriminant. \| \| Note: The discriminant, b2 - 4ac, does NOT tell you what the solution(s) to the quadratic equation will be. It simply tells you "the number and type" of solutions there will be. \| \| \| Consider the equation ax2 + bx + c = 0. If b2 - 4ac > 0, then the equation has two real solutions. If b2 - 4ac > 0 and a perfect square, the roots are real and rational. If b2 - 4ac > 0 and not a perfect square, the roots are real and irrational. If b2 - 4ac = 0, then the equation has one real solution (repeated). If b2 - 4ac < 0, then the equation has no real solutions. It has two complex conjugate roots. \| \| \| \| DISCRIMINANT: Its purpose is to tell "how many roots", and "what type of roots". \| \| POSITIVE b2 - 4ac > 0 \| ZERO b2 - 4ac = 0 \| NEGATIVE b2 - 4ac < 0 \| \| x2 + 6x + 5 = 0 discriminant: b2 - 4ac = 62 - 4(1)(5) = 16 (positive) There are two real roots. There are two x-intercepts. If the discriminant is a perfect square, the two roots are rational numbers. If the discriminant is not a perfect square, the two roots are irrational numbers containing a radical. x2 + 6x + 5 = (x + 1)(x + 5) = 0 Roots: x = -1, x = -5 (rational roots) \| x2 - 2x+ 1 = 0 discriminant: b2 - 4ac = (-2)2-4(1)(1) = 0 (zero) There is one real root. There is one x-intercept. (The root is repeated.) x2 - 2x+ 1 = (x - 1)2 = (x - 1)(x - 1) = 0 Repeated root: x = 1 \| x2 - 3x + 10 = 0 discriminant:b2 - 4ac = (-3)2-4(1)(10) = -31 (negative) There are 2 imaginary roots. There are no x-intercepts. In this situation, there will be two "complex" (a + bi form) roots because there will be a negative number under the square root. Answers have imaginary number, i. \| \| \| \| NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". \| Topical Outline \| Algebra 2 Outline \| MathBitsNotebook.com \| MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts \| \|
3893	https://math.umd.edu/~mariakc/AMSC466/LectureNotes/interpolation.pdf	Contents 1. Introduction 1 2. Polynomial interpolation 1 2.1. Lagrangian interpolation 1 2.2. The Newton interpolation polynomial 3 2.3. Hermite interpolation 5 2.4. Runge phenomenon 10 3. Chebyshev polynomials 12 3.1. Properties of the Chebyshev polynomials 12 4. Chebyshev interpolation 16 4.1. Chebyshev polynomials shifted to the interval [a, b] 16 4.2. Computing coeﬃcients for Chebyshev interpolant 17 4.3. Evaluating Chebyshev interpolant 17 5. Interpolation by spline functions 19 5.1. Theoretical foundations 20 5.2. Setting up a system of equations for a cubic spline 21 5.3. A fast solver for tridiagonal matrices 24 5.4. Convergence properties of cubic spline functions 24 5.5. Spline interpolation in Matlab 27 1. Introduction Interpolation is widely used in various ﬁelds of applied math, engineering, and computer science. It is a typical case that data are available only at a discrete set of points, while we would like to have them an an arbitrary point of a region containing this discrete set. The task of interpolation is to deﬁne a function in the whole region so that it coincides with the data at this discrete set. Various methods for interpolation are based on diﬀerent assumptions regarding the function. Linear interpolation results at a continuous function while bicubic interpolation will make the function smooth. Take a look at the ﬁgures here. Interpolation techniques are building blocks for various methods in computational math-ematics, in particular, for • quadrature rules, • linear multistep ODE solvers with variable time step and order, • ﬁnite element methods for solving partial diﬀerential equations. We will discuss polynomial interpolation and spline interpolation in 1D. 2. Polynomial interpolation 2.1. Lagrangian interpolation. Suppose a continuous function f(x) deﬁned on the in-terval [a, b] is given at a ﬁnite number of points xj, j = 0, 1, 2, . . . , n, n + 1 points in total. We will denote f(xj by fj. The task of interpolation is to ﬁnd a polynomial of degree at most n passing through all these points. Lagrange’s approach to this task is to construct 1 2 n + 1 polynomials Lj(x) of degree n such that each Lj(x) is 1 at xj and zero at ll other xk’s. Then the linear combination P(x) = f0L0(x) + f1L1(x) + . . . + fn(x)Ln(x) is a polynomial of degree at most n, and P(xj) = fj for all j = 0, 1, . . . , n. The polynomial P(x) written in the form above is called the Lagrange interpolation polynomial. The polynomials Lj(x) are easily constructed. The error of interpolation can also be found. These results are summarized in the following theorem. Theorem 1. Given a function f that is deﬁned at n + 1 points x0 < x1 < . . . < xn ∈[a, b] there exists a unique polynomial of degree ≤n such that Pn(xj) = f(xj), j = 0, 1, 2, . . . , n. This polynomial is given by Pn(x) = n X j=0 f(xj)Lj(x), where Lj(x) is deﬁned by Lj(x) = πn+1(x) (x −xj)π′ n+1(xj) = Qn k=0,k̸=j(x −xk) Qn k=0,k̸=j(xj −xk), πn+1(x) being the nodal polynomial πn+1(x) = n Y k=0 (x −xk). Additionally, if f is n + 1 times continuously diﬀerentiable in (a, b), then for any x ∈[a, b] there exists a value ζx ∈(a, b) depending on x, such that (1) En(x) = f(x) −Pn(x) = f(n+1)(ζx) (n + 1)! πn+1(x). Proof. (1) Existence By construction, Pn is a polynomial of degree n passing through xj, j = 0, 1, 2, . . . , n. (2) Uniqueness Suppose that there are two such polynomials, Pn(x) and Qn(x). Then the polynomial d(x) := Pn(x) −Qn(x) is at most of degree n. On the other hand, it has at least n + 1 roots xj, j = 0, 1, 2, . . . , n. Therefore, it must be identically zero. (3) Error formula Consider the function F(z) = f(z) −Pn(z) −[f(x) −Pn(x)]πn+1(z) πn+1(x). This function has n + 2 zeros at z = xj, j = 0, 1, 2, . . . , n, and z = x. Recall Rolle’s theorem that says that if a function f(z) is continuously diﬀerentiable on [a, b] and 3 f(a) = f(b) then there is c ∈(a, b) such that f′(c) = 0. Therefore, if we apply Rolle’s theorem n + 1 times we get that the function F (n+1)(z) = f(n+1)(z) −P (n+1) n (z) −f(x) −Pn(x)! πn+1(x) has at least one zero in (x0, xn). We denote this zero by ζx. Therefore, taking into account that P (n+1) n (z) ≡0, we get 0 = f(n+1)(ζx) −f(x) −Pn(x)! πn+1(x), and Eq. (1) follows. □ Example Let us ﬁnd the Lagrange interpolant p(x) passing through the points (0, f0), (1, f1), and (2, f2). The polynomial p(x) is of the form p(x) = f0 (x −1)(x −2) (0 −1)(0 −2) + f1 (x −0)(x −2) (1 −0)(1 −2) + f2 (x −0)(x −1) (2 −0)(2 −1) = 1 2f0(x2 −3x + 2) −f1(x2 −2x) + 1 2f2(x2 −x) = f0 + 1 2(−3f0 + 4f1 −f2)x + 1 2(f0 −2f1 + f2)x2. (2) 2.2. The Newton interpolation polynomial. Lagrangian interpolation is convenient because it gives an explicit formula for the interpolant. However, it does not provide a con-venient way to modify the polynomial to accommodate additional interpolation points. An alternative form of the interpolation polynomial, the Newton form, gives such a way. The Newton interpolation formula is used, for example, for deriving linear multistep methods with varying time step for solving ODE’s. The Newton interpolation formula is deﬁned via the divided diﬀerences. We set f[x0] = f(x0), f[x0, x1] = f[x1] −f[x0] x1 −x0 , f[x0, x1, x2] = f[x1, x2] −f[x0, x1] x2 −x0 , . . . f[x0, x1, . . . , xk] = f[x1, . . . , xk] −f[x0, x1, . . . , xk−1] xk −x0 , . . . . Theorem 2. The polynomial interpolating f(x) at xj, j = 0, 1, 2, . . . , n is given by Pn(x) = f[x0] + fx0, x1 + fx0, x1, x2(x −x1) + . . . + fx0, x1, . . . , xn(x −x1) . . . (x −xn−1). (3) 4 Example Let us ﬁnd the Newton interpolant p(x) passing through the points (0, f0), (1, f1), and (2, f2). The polynomial p(x) is of the form (4) p(x) = f + f0, 1 + f0, 1, 2(x −1), where f = f0, f[0, 1] = f1 −f0 1 −0 = f1 −f0, f[1, 2] = f2 −f1 2 −1 = f2 −f1, f[0, 1, 2] = f[1, 2] −f[0, 1] 2 −0 = 1 2(f2 −f1 −(f1 + f0)) = 1 2(f0 −2f1 + f2). Plugging these coeﬃcients into Eq. (4) we get p(x) = f0 + (f1 −f0)x + 1 2(f0 −2f1 + f2)x(x −1) = f0 + 1 2 (−3f0 + 4f1 −f2) x + 1 2(f0 −2f1 + f2)x2. (5) The polynomial p(x) in Eq. (5) coincides with the one in Eq. (2) as it should. Proof. We will proceed by induction. For n = 1 Eq. (3) holds. Suppose the polynomials Px0, . . . , xn−1 and Px1, . . . , xn of the form of Eq. (3) interpolate f at the points x0, . . . , xn−1 and x1, . . . , xn respectively. Note that both of them are of degree n−1. Hence they diﬀer by a polynomial of degree at most n −1, and this polynomial has zeros at x1, ..., xn−1. Therefore, Px1, . . . , xn −Px0, . . . , xn−1 = a(x −x1) . . . (x −xn−1) where a is a number. Obviously, a is the diﬀerence of the leading coeﬃcients of the polynomials Px0, . . . , xn−1 and Px1, . . . , xn, i.e., a = f[x1, . . . , xn] −f[x0, x1, . . . , xn−1]. At the same time, a = Px1, . . . , xn −Px0, . . . , xn−1 (x −x1) . . . (x −xn−1) . Now consider the polynomial (6) Px0, x1, . . . , xn = Px0, . . . , xn−1 + a xn −x0 (x −x0)(x −x1) . . . (x −xn−1). The last term of this polynomial is chosen so that it is zero at x0, . . . , xn−1, and the coeﬃ-cient a/(xn−x0) is our guess that we will verify below. By construction, Px0, x1, . . . , xn interpolates f at x0, . . . , xn−1. Let us verify that it also does so at x = xn. We evaluate 5 Px0, x1, . . . , xn at xn and obtain: Px0, x1, . . . , xn =Px0, . . . , xn−1 + a xn −x0 (xn −x0)(xn −x1) . . . (xn −xn−1) =Px0, . . . , xn−1 +Px1, . . . , xn −Px0, . . . , xn−1 (xn −x0)(xn −x1) . . . (xn −xn−1) (xn −x0)(xn −x1) . . . (xn −xn−1) =Px1, . . . , xn = f(xn). Therefore, the polynomial given by Eq. (6) interpolates f at xj, j = 0, 1, 2, . . . , n. □ Remark The function f[x0, . . . , xn] is a symmetric function of its arguments i.e., it does not change if we permute x0, ..., xn. This is because the interpolation polynomial is independent of the order of nodes. 2.3. Hermite interpolation. We have learned how to construct a polynomial of degree at most n matching with f at n + 1 points x0, x1, ..., xn. Now suppose that we want to construct a polynomial such that its values and its derivatives match with those of f at given points. For example, suppose we want to ﬁnd a minimal degree polynomial H on the interval [0, 1] such that H(0) = u0, H′(0) = u′ 0, H(1) = u1, H′(1) = u′ 1. Following Lagrange’s idea, we design a polynomial p0(x) such that p0(0) = 1, p′ 0(0) = 0, p0(1) = 0, p′ 0(1) = 0. Note that then the polynomial p1(x) := p0(1 −x) satisﬁes: p1(0) = 0, p′ 1(0) = 0, p1(1) = 1, p′ 1(1) = 0. We also need to design a polynomial g0(x) such that g0(0) = 0, g′ 0(0) = 1, g0(1) = 0, g′ 0(1) = 0. Then the polynomial g1(x) := −g0(1 −x) satisﬁes g1(0) = 0, g′ 1(0) = 0, g1(1) = 0, g′ 1(1) = 1. For each polynomial p0(x) and g0(x) we have four conditions to meet. Hence they should have four coeﬃcients which means that they should be cubic. We can ﬁnd p0(x) by setting it to p0(x) = a0 + a1x + a2x2 + a3x3, and, respectively, p′ 0(x) = a1 + 2a2x + 3a3x2, and solving the system of four linear equations: p0(0) = a0 = 1, p′ 0(0) = a1 = 0, p0(1) = a0 + a1 + a2 + a3 = 0, p′ 0(1) = a1 + 2a2 + 3a3 = 0. 6 We ﬁnd: p0(x) = 1 −3x2 + 2x3. Similarly, we ﬁnd g0(x) = x(1 −x)2. Now we write out the desired interpolating polynomial: (7) H(x) = u0p0(x) + u1p0(1 −x) + u′ 0g0(x) −u′ 1g0(1 −x). The problem that we have just solved can be generalized to an arbitrary number of points xj and arbitrary numbers of matching derivatives of f and the interpolation polynomial at each. The existence, uniqueness , and error estimate for such an interpolant are stated in the theorem below. Theorem 3. Let f be n times continuously diﬀerentiable in [a, b] and n + 1 times contin-uously diﬀerentiable in (a, b). Let x0 < x1 < . . . < xk ∈[a, b], and let ni ∈N be such that n0 + n1 + . . . + nk = n + 1. Then there exists a unique polynomial Pn of degree ≤n such that P (j) n (xi) = f(j)(xi), j = 0, 1, . . . , ni −1, i = 0, 1, . . . , k. Furthermore, given x ∈[a, b], there exists a value ζx ∈(a, b) such that f(x) = Pn(x) + f(n+1)(ζx) (n + 1)! πn+1(x), where πn+1(x) is the nodal polynomial πn+1(x) = (x −x0)n0 . . . (x −xk)nk. Remark If all of the nodes coincide, i.e., k = 0 and n0 = n then the polynomial Pn is the Taylor polynomial and the error term is the error term for the Taylor series in the Lagrange form. An explicit formula for such a polynomial exists but it is not as simple as for the Lagrangian interpolant. As in the case of distinct nodes, it is more convenient to calculate the Hermite interpolant using divided diﬀerences. However, we will need to generalize them. Theorem 4. Let t0 ≤t1 ≤. . . ≤tn, where t0 = . . . = tn0 = x0, tn0+1 = . . . = tn0+n1+1 = x1, . . . . The points tj are called the generalized abscissas. The generalized divided diﬀerences are deﬁned as follows. If ti = . . . = ti+p = xl then f[ti, ti+1, . . . , ti+p] = 1 p!f(p) l . If ti < ti+p then f[ti, ti+1, . . . , ti+p] = f[ti+1, ti+2, . . . , ti+p] −f[ti, ti+1, . . . , ti+p−1] ti+p −ti . 7 The interpolation polynomial is given by (8) Pn(x) = f[t0] + ft0, t1 + . . . + ft0, . . . , tn . . . (x −tn−1). Proof. The proof is by induction. The statement holds for n = 0. Suppose it holds for n −1. Then the polynomials Pt0, . . . , tn−1 and Pt1, . . . , tn of the form of Eq. (8) interpolate f at the abscissas t0 ≤. . . ≤tn−1 and t1 ≤. . . ≤tn respectively. The diﬀerence between them is a polynomial Q(x) of degree n −1 with zeros at t1, ..., tn−1. Hence it is of the form Q(x) = a(x −t1) . . . (x −tn−1). The coeﬃcient a is equal, on one hand, to the diﬀerence between the leading coeﬃcients of Pt1, . . . , tn and Pt0, . . . , tn−1, i.e., a = f[t1, . . . , tn] −f[t0, . . . , tn−1], on the other hand, it is equal to a = Pt1, . . . , tn −Pt0, . . . , tn−1 (x −t1) . . . (x −tn−1) . Consider the polynomial Pt0, . . . , tn = Pt0, . . . , tn−1 + b(x −t0) . . . (x −tn−1). We will distinguish two cases. Case 1: tn = tn−1 = . . . = tn−nk. By construction, it interpolates f at the abscissas t0, ..., tn−1. Indeed, (x −t0) . . . (x −tn−1) = (x −x0)n0 . . . (x −xk)nk−1, and it vanishes together with its ni derivatives at x0, ..., xk−1 and together with its nk −1 derivatives at xk. We set b = 1 nk!f(nk). Then Pt0, . . . , tn(xk) = bnk! = f(nk). Case 2: tn > tn−1. We set b = a tn −t0 and proceed as in the proof of the Newton interpolant formula. This completes the proof. □ Example Suppose we are given f(x0) = f0, f(x1) = f1, f′(x1) = f′ 1, and f′′(x1) = f′′ 1 . We need to write the interpolation polynomial P(x) such that P(x0) = f0, P (k)(x1) = f(k)(x1), 8 k = 0, 1, 2. We calculate the generalized divided diﬀerences: t0 = x0 f[t0] = f0 f[t0, t1] = f1−f0 x1−x0 t1 = x1 f[t1] = f1 f[t0, t1, t2] = f′ 1−f1−f0 x1−x0 x1−x0 f[t1, t2] = 1 1!f′ 1 f[t0, t1, t2, t3] = 1 2 f′′ 1 − f′ 1−f1−f0 x1−x0 x1−x0 x1−x0 t2 = x1 f[t2] = f1 f[t1, t2, t3] = 1 2f′′ 1 f[t2, t3] = 1 1!f′ 1 t3 = x1 f[t3] = f1 The interpolation polynomial is given by P(x) = f0 + f1(x −x0) + f′ 1 −f1−f0 x1−x0 x1 −x0 (x −x0)(x −x1) + 1 2f′′ 1 − f′ 1−f1−f0 x1−x0 x1−x0 x1 −x0 (x −x0)(x −x1)2. The error is given by E(x) ≡f(x) −P(x) = f(4)(ζ) 24 (x −x0)(x −x1)3 for some ζ ∈(x0, x1). Example Let f(x) = (1 + x2)−1. This function is called the ”Witch of Agnesi”. It is a well-known example demonstrating the fault of high degree polynomial interpolation with equispaced points on a large interval. This will be discussed in Section 2.4. Here we take the interval [0, 1] where the interpolation works pretty nicely and compare two interpolation polynomials: Newton’s interpolant with four equispaced points, and the Her-mite interpolant with the abscissas 0, 0, 1, 1. The Matlab program below computes these interpolants and generates Fig. 1. function CompareInterpolants() % Compares two cubic interpolating polynomials for 1/(x^2 + 1) on [0,1]: % Newton’s interpolant with four equispaced points and % Hermite interpolant with abscissas 0, 0, 1, 1. x = [0,1/3,2/3,1]’; % four equispaced points t = linspace(0,1,100); f = @(x)1./(x.^2 + 1); % Witch of Agnesi fx = f(x); ft = f(t); figure; hold on; grid; plot(x,fx,’.’,’Markersize’,30); plot(t,ft,’Linewidth’,1) %% Newton’s interpolation. % compute divided differences dd = zeros(4); dd(:,1) = fx; 9 0 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1 Data f(x) p[0,1/3,2/3,1] p[0,0,1,1] 0.9 0.92 0.94 0.96 0.98 1 0.5 0.52 0.54 0.56 Figure 1. Comparison of two cubic interpolation polynomials: Newton’s polynomial with four equispaced points p[0, 1/ 3, 2/ 3, 1], and the Hermite poly-nomial with abscissas t0 = t1 = 0, t2 = t3 = 1: p[0, 0, 1, 1]. dd(1:3,2) = (dd(2:4,1) - dd(1:3,1))./(x(2:4)-x(1:3)); dd(1:2,3) = (dd(2:3,2) - dd(1:2,2))./(x(3:4)-x(1:2)); dd(1,4) = (dd(2,3) - dd(1,3))./(x(4)-x(1)); % define Newton’s interpolant p = @(t) dd(1,1) + dd(1,2)(t-x(1)) + dd(1,3)(t-x(1)).(t-x(2))... + dd(1,4)(t-x(1)).(t-x(2)).(t-x(3)); pt = p(t); plot(t,pt,’Linewidth’,1); %% Hermite interpolation fp = @(x)-2x./(1+x.^2).^2; % derivative of f(x) p0 = @(x)1-3x.^2 + 2x.^3; % p0(0) = 1, p0’(0) = p0(1) = p0’(1) = 0 g0 = @(x)x.(1-x).^2; % g0(0) = g0(1) = g0’(1) = 0; g0’(0) = 1 % define Hermite interpolant H = @(t)f(0)p0(t) + fp(0)g0(t) + f(1)p0(1-t) - fp(1)g0(1-t); Ht = H(t); plot(t,Ht,’Linewidth’,1) set(gca,’Fontsize’,20); legend(’Data’,’f(x)’,’p[0,1/3,2/3,1]’,’p[0,0,1,1]’); %% zoom in near x = 1 ti = linspace(0.9,1,20); 10 ax2 = axes(’Position’,[.2 .2 .3 .3]); hold(ax2,’on’); hold on; grid; plot(x(4),fx(4),’.’,’Markersize’,30); plot(ti,f(ti),’Linewidth’,1) plot(ti,p(ti),’Linewidth’,1) plot(ti,H(ti),’Linewidth’,1) end 2.4. Runge phenomenon. In this section, we address the question whether the inter-polation polynomial approaches f in the uniform norm as the number of nodes tends to inﬁnity. The answer is negative in general. Exercise (1) Build interpolation polynomials with n equispaced interpolation points in the interval x ∈[−1, 1] for f(x) = \|x\| and show that the sequence with equally spaced nodes diverges as n →∞. (2) Do the same for the function the “Witch of Agnesi” (9) f(x) = 1 1 + x2 , x ∈[−5, 5]. The graph of the Witch of Agnesi and its polynomial interpolants with equispaced points of degrees 3, 6, 9, and 12 on the interval [−5, 5] are shown in Fig. 2. We see that as we increase the degree of the interpolation polynomial, the interpolation error blows up. Fig. 2 was generated by the following Matlab code: function RungePhenomenon() % Demonstrates the Runge phenomenon on 1/(x^2 + 1) on [-5,5]: a = -5; b = 5; figure; hold on; grid; f = @(x)1./(x.^2 + 1); % Witch of Agnesi t = linspace(a,b,400); plot(t,f(t),’Linewidth’,1) set(gca,’Fontsize’,20); legend(’f(x) = (1+x^2)^{-1}’); % build Newton’s interpolants with equispaced points for n = 3 : 3 : 12 x = linspace(a,b,n + 1)’; % n+1 equispaced points fx = f(x); %% Newton’s interpolation. % compute divided differences dd = zeros(n + 1); dd(:,1) = fx; for k = 2 : n + 1 11 -5 0 5 -4 -3 -2 -1 0 1 f(x) = (1+x2)-1 n = 3 n = 6 n = 9 n = 12 Figure 2. A demonstration of the Runge phenomenon. Polynomial in-terpolants with equispaced points of degrees 3, 6, 9, and 12 for f(x) = (1 + x2)−1 on the interval [−5, 5] . dd(1:n-k+2,k) = (dd(2:n-k+3,k-1)-dd(1:n-k+2,k-1))./(x(k:n+1)-x(1:n-k+2)); end % evaluate Newton’s interpolant pt = dd(1,n+1); for k = n : -1 : 1 pt = pt.(t - x(k)) + dd(1,k); end plot(t,pt,’Linewidth’,1,’DisplayName’,sprintf(’n = %d’,n)); end end The lack of convergence in the ﬁrst example is less surprising then in the second one, because the function in Eq. (9) is inﬁnitely diﬀerentiable. This phenomenon was studied by Runge and is known as the Runge phenomenon. The analysis explaining the Runge phenomenon uses theory of functions of complex variable. Let us look at the error formula (1). There are two factors that can be large: the fraction f(n+1) (n+1)! and the nodal polynomial πn+1(x). 12 The fraction f(n+1) (n+1)! does not need to decay as n →∞. The estimate comes from the Cauchy integral formula (10) f(n+1)(ζ) = (n + 1)! 2πi Z CR f(z) (z −ζ)n+2 dz, where CR = {z ∈C \| \|z −ζ\| = R}. Therefore, maxx∈[ζ−R,ζ+R] \|f(n+1)(x)\| (n + 1)! ≤maxz∈CR \|f(z)\| 2πRn+1 . Hence, what matters is not smoothness of f but its analyticity. Note that the Witch of Agnesi has poles at z = ±i (i.e. discontinuities of the form a/(z −b)). Therefore, the formula right above gives the upper bound for f(n+1) (n+1)! equal to inﬁnity. The nodal polynomial πn+1(x) is highly oscillatory for equally spaced nodes. It can be shown that for large n it takes values approximately 2n times larger near the endpoints than in the middle. However, if we pick the nodes nonuniformly, we can make the nodal polynomial deviate from zero as little as possible. The optimal choice of nodes is Cheby-chev’s. A very nice explanation for these phenomena is found in L. N. Trefethen’s book Spectral Methods in Matlab – read the beginning of Chapter 5. In this connection, it becomes even surprising that the polynomial interpolants with equispaced nodes converge to the Witch of Agnesi on an interval much larger than [−1, 1]: on [−a, a] where a ≈3.63. 3. Chebyshev polynomials We will study the Chebyshev polynomials in more details as they lead to a number of remarkable numerical tools: • Chebyshev interpolation, where the Runge phenomenon is eliminated. Ref.: A. Gil, J. Segura, N. Temme, Numerical Methods for Special Functions, SIAM, 2007 available online via UMD library; • Chebyshev least squares approximation which is close to the minimax one; • Chebyshev spectral methods for solving PDEs with non-periodic boundary condi-tions – L. N. Trefethen’s book Spectral Methods in Matlab. Chebyshev polynomials are deﬁned as follows: (11) Tn(x) = cos [n arccos(x)] , x ∈[−1, 1], n = 0, 1, 2, . . . . It follows from the deﬁnition that (12) Tn(cos θ) = cos(nθ), θ ∈[0, π], n = 0, 1, 2, . . . . 3.1. Properties of the Chebyshev polynomials. (A) The Chebyshev polynomials satisfy the following three-term recurrence relation-ships (TTRR): (13) Tn+1(x) = 2xTn(x) −Tn−1(x), T0(x) = 1, T1(x) = x, n = 0, 1, 2, . . . . 13 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 T0(x) T1(x) T2(x) T3(x) T4(x) T5(x) T6(x) Figure 3. Graphs of the Chebyshev polynomials T0(x), T1(x), ..., T6(x). This TTRR immediately follows from Eq. (43) and the trigonometric formula cos(a) + cos(b) = 2 cos a + b 2 cos a −b 2 , where we set a = (n + 1)θ and b = (n −1)θ. (B) The leading coeﬃcient of Tn(x) is 2n−1. This follows from Eq. (13). (C) Tn(−x) = (−1)nTn(x). This follows from the formulas arccos(−x) = π −arccos(x) and cos(πn −x) = (−1)n cos x. (D) Zeros of Tn(x) are (14) xk = cos k + 1 2 n π ! , k = 0, 1, . . . , n −1. Extrema of Tn(x) are (15) x′ k = cos πk n , k = 0, 1, . . . , n. Tn(x′ k) = (−1)k. (E) The deviation of 2−nTn+1(x) from zero on [−1, 1] is minimal possible among all polynomials with leading coeﬃcient 1 of degree n + 1. The theorem establishing this fact is below. 14 Theorem 5. Let xk = cos k + 1 2 n + 1π ! , k = 0, 1, . . . , n. Then the monic polynomial (i.e., its leading coeﬃcient is 1) ˆ Tn+1 = n Y k=0 (x −xk) of degree n + 1 has the smallest possible uniform (maximum) norm 2−n in [−1, 1] among all polynomials of degree n + 1. I.e., 2−n = max x∈[−1,1] \| ˆ Tn+1(x)\| = min p∈Pn+1 p=xn+1+... max x∈[−1,1] \|p(x)\|. Proof. Suppose there is a monic polynomial p(x) of degree n + 1 such that \|p(x)\| < 2−n for all x ∈[−1, 1]. Let x′ k, k = 0, 1, . . . , n + 1 be the abscissas of the extreme values of Chebyshev polynomials of degree n + 1. Hence we have p(x′ 0) < 2−nTn+1(x′ 0), p(x′ 1) > 2−nTn+1(x′ 1), p(x′ 2) < 2−nTn+1(x′ 2), . . . . Therefore the polynomial Q(x) = p(x) −2−nTn+1 changes sign between each two consecutive extrema of Tn+1. Tn+1(x) has n + 2 extrema on [−1, 1]. Hence Q(x) has n + 1 zeros. But Q(x) is of degree ≤n. Thus we have arrived to a contradiction. Hence there is no such monic polynomial p of degree n + 1 such that \|p(x)\| < 2−n for x ∈[−1, 1]. □ (F) Relations with derivatives. T0(x) = T ′ 1(x), (16) T1(x) = 1 4T ′ 2(x), (17) Tn(x) = 1 2 T ′ n+1(x) n + 1 −T ′ n−1(x) n −1 , n ≥2. (18) The ﬁrst two equalities are easy -to-check. The last one can be proven from the following observation: T ′ n(x) = n sin(nθ) sin θ , where x = cos θ. Exercise Prove Eq. (18). 15 (G) Multiplication relationship: (19) 2Tr(x)Tq(x) = Tr+q(x) + T\|r−q\|(x). Exercise Prove Eq. (19). (H) Orthogonality relationship: (20) Z 1 −1 Tr(x)Ts(x) √ 1 −x2 dx =      0, r ̸= s π, r = s = 0, π 2 , r = s ̸= 0. Let us prove it. We make a variable change x = cos θ. Then dx = −sin θdθ, Tr(x) = cos(rθ), Ts(x) = cos(sθ), and the integration limits are changed to: −1 7→ π, 1 7→0. Therefore, using the formula cos a cos b = 1/ 2 cos(a + b) + 1/ 2 cos(a −b) we get: Z 1 −1 Tr(x)Ts(x) √ 1 −x2 dx = Z π 0 cos(rθ) cos(sθ)dθ = 1 2 Z π 0 [cos(r + s)θ + cos(r −s)θ] dθ =1 2        h sin(r+s)θ r+s + sin(r−s)θ r−s iπ 0 , r ̸= s 2π, r = s = 0, π + h sin(2rθ) 2r iπ 0 , r = s ̸= 0. =      0, r ̸= s π, r = s = 0, π 2 , r = s ̸= 0. (I) Discrete orthogonality relationship: Take the points xj = cos π(j + 1 2) n + 1 ! , j = 0, 1, . . . , n that are the zeros of Tn+1(x). Then for all 0 ≤r, s ≤n we have (21) n X j=0 Tr(xj)Ts(xj) =      0, r ̸= s n + 1, r = s = 0, n+1 2 , r = s ̸= 0, j = 0, 1, . . . , n. Exercise Prove Eq. (25). 16 4. Chebyshev interpolation Properties of the Chebyshev polynomials oﬀer a nice way for computing the Chebyshev interpolant of degree n. Fix some integer n and consider the zeros of Tn+1(x). They are xj = cos π(j + 1 2) n + 1 ! , j = 0, 1, . . . , n. For a given function f(x) on the interval [−1, 1], the polynomial pn of degree n interpolating f(x) at the Chebyshev points xj, j = 0, 1, . . . , n, is given by pn(x) = n X k=0 ′ckTk(x) ≡c0 2 + n X k=1 ckTk(x). The symbol ′ indicates that the ﬁrst term in the sum should be divided by two. The coeﬃcients ck are found from the requirement that pn(xj) = f(xj), j = 1, 2, . . . , n, i.e., f(xj) = n X k=0 ′ckTk(xj). Then we have n X j=0 f(xj)Tm(xj) = n X k=0 ′ck n X j=0 Tk(xj)Tm(xj) = n X k=0 ′ck n + 1 2 δmk = 1 2(n + 1)ck. Hence the coeﬃcients are given by (22) ck = 2 n + 1 n X j=0 f(xj)Tk(xj), xj = cos π(j + 1 2) n + 1 ! . To summarize, the Chebyshev interplant of f(x) is given by (23) pn(x) = c0 2 + n X k=1 ckTk(x), where the coeﬃcients ck are given by Eq. (22). 4.1. Chebyshev polynomials shifted to the interval [a, b]. Suppose we need to ﬁnd the Chebyshev interpolant for f(x) on the interval [a, b] rather than on [−1, 1]. Then we need to shift Chebyshev polynomials to this interval using a linear transformation l : [a, b] →[−1, 1] such that l(a) = −1 and l(b) = 1. If x ∈[a, b] and y ∈[−1, 1] then y = l(x) = x −a+b 2 b−a 2 = 2x −a −b b −a . Then the shifted Chebyshev polynomials to the interval [a, b] T [a,b] k (x) are deﬁned by (24) T [a,b] k (x) = Tk(y) ≡Tk 2x −a −b b −a . 17 Exercise Show that the shifted Chebyshev polynomials satisfy the following orthogonality relationships (25) Z b a T [a,b] r (x)T [a,b] s (x) r 1 − 2x−a−b b−a 2 dx =      0, r ̸= s b−a 2 π, r = s = 0, (b−a)π 4 , r = s ̸= 0. The inverse linear transformation for l is l−1: l−1(y) = 1 2(y(b −a) + b + a). 4.2. Computing coeﬃcients for Chebyshev interpolant. Suppose ﬁrst that x ∈ [−1, 1]. In order to compute the coeﬃcients given by Eq. (22), it is convenient to in-troduce angles θj = π(j + 1 2) n + 1 , , j = 0, 1, . . . , n. Then, recalling that Tk(x) ≡Tk(cos θ) = cos(kθ), we get (26) ck = 2 n + 1 n X j=0 f(cos θj) cos(kθj). If x ∈[a, b], we need to evaluate f in Eq. (26) at the points xj = l−1(yj) = 1 2(yj(b −a) + b + a) = 1 2(cos(θj)(b −a) + b + a). In this case we get (27) ck = 2 n + 1 n X j=0 f(xj) cos(kθj), where xj = 1 2(cos(θj)(b −a) + b + a). 4.3. Evaluating Chebyshev interpolant. Chebyshev interpolant can be evaluated at the point x ∈[a, b] directly from Eq. (23) (28) pn(x) = c0 2 + n X k=1 ck cos(k arccos(y)), where y = 2x −a −b b −a . This formula has a shortcoming that it requires evaluation of cos and arccos which are typically built-in functions, but their evaluation is time-consuming in comparison with basic ﬂoating point operations. An elegant way for evaluation of Chebyshev interpolant that avoids calculations of cos and arccos was proposed by Clenshaw (1955). A detailed description of Clenshaw’s algo-rithm is given in Chapter 3 “Chebyshev Expansions” from “Numerical Methods for Special Functions” by Amparo Gil, Javier Segura, and Nico Temme (see pages 75-76). Here we show how one can implement it in Matlab. 18 Suppose we need to evaluate the sum pn(x) = c0 2 + n X k=1 ckTk(y), where y = 2x −a −b b −a . We rewrite this sum in the vector form: pn(x) = c⊤t −c0 2 , where c := [c0, c1, . . . , cn]⊤, t := [T0(y), T1(y), . . . , Tn(y)]⊤. Recall that the Chebyshev polynomials satisfy TTRR (13) that can be written in the matrix form as          1 −2y 1 1 −2y 1 1 −2y 1 ... ... ... 1 −2y 1                 T0(y) T1(y) T2(y) . . . Tn(y)        =        1 −y 0 . . . 0        ≡At = d. Let b be a vector such that (29) b⊤A = c⊤, or A⊤b = c. Note that b is readily found starting from bn as A⊤is upper-triangular. Then pn(x) = c⊤t −c0 2 = b⊤At −c0 2 =b⊤d −c0 2 = b0 −b1y −c0 2 . From (29) we ﬁnd c0 = b0 −2yb1 + b2. Therefore, (30) pn(x) = b0 −b1y −c0 2 = b0 −b1y −1 2(b0 −2yb1 + b2) = 1 2(b0 −b2). A Matlab code computing Chebyshev’s coeﬃcients and evaluating Chebyshev’s sum for f(x) = (1 + x2)−1, x ∈[−5, 5] is given below. function ChebClenshaw() % Compute Chebyshev coefficients and evaluate Chebyshev sum using % Clenshaw’s method a = -5; b = 5; nt = 200; t = linspace(a,b,nt); f = @(x)1./(x.^2 + 1); % Witch of Agnesi ft = f(t); %% Chebyshev grid n = 30; 19 g = [0:n]’; theta = pi(g + 0.5)/(n + 1) % angles theta y = cos(theta); % grid on [-1,1] x = 0.5(y(b-a) + a + b); % grid on [a,b] %% compute Chebyshev coefficients c = zeros(n+1,1); fx = f(x); for k = 0 : n c(k + 1) = sum(fx.cos(ktheta)); end c = 2c/(n + 1); fprintf(’c = [\n’); fprintf(’%d\n’,c); fprintf(’]\n’); %% evaluate Chebyshev sum at points t using Clenshaw’s method e = ones(n+1,1); pp = zeros(nt,1); tt = (2t - a - b)/(b - a); for j = 1 : nt A = spdiags([e,-tt(j)2e,e],-2:0,n+1,n+1); bb = A’\c; pp(j) = 0.5(bb(1) - bb(3)); end figure; hold on; grid; plot(t,ft,’Linewidth’,1,’Displayname’,’f(x)’) plot(x,fx,’.’,’Markersize’,30,’Displayname’,’Chebyshev data’); plot(t,pp,’Linewidth’,1,’Displayname’,’Chebyshev interpolant’); set(gca,’Fontsize’,16) legend end 5. Interpolation by spline functions Reference: • J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third Edition, Springer, 2002 Splines are widely used, for example, in • applications in graphics, • numerical methods (e.g., for BVP), • signal processing. The simplest splines are the cubic splines. We will restrict ourselves to them. Spline functions yield smooth interpolation curves that are less likely to exhibit large oscillations 20 than high degree polynomials. We will prove that a cubic spline interpolant converges to a given function together with its ﬁrst three derivatives as soon as the function is four times continuously diﬀerentiable and the set of interpolation points has some minimal quality (this will be speciﬁed further in this section). Recall that this is not true for polynomial interpolation in general. 5.1. Theoretical foundations. Let ∆:= {a = x0 < x1 < . . . < xn = b} be a partition of the interval [a, b]. The points xj are called knots. Deﬁnition 1. A cubic spline S∆on ∆is a function S∆: [a, b] →R such that: (1) S∆∈C2[a, b], i.e., is twice continuously diﬀerentiable, (2) S∆coincides on every subinterval [xj, xj+1], j = 0, 1, . . . , n −1 with a polynomial of degree at most 3. Therefore, a cubic spline consists of cubic polynomials glued in such a manner that their values together with the values of their ﬁrst two derivatives coincide at the interior knots xj, j = 1, 2, . . . , n −1. Suppose a function f is given at the knots xj, j = 0, 1, . . . , n, i.e., f(xj) = fj, j = 0, 1, . . . , n −1. We denote the vector {f0, . . . , fn} by F, and a spline function that interpolates f at these points by S∆(F; ·). Note that S∆(F, ·) is not uniquely deﬁned by the vector F. Indeed, we have 4n coeﬃcients of the cubic polynomials (n subintervals, and the cubic polynomial on each subinterval has 4 coeﬃcients) and n + 1 + 3(n −1) = 4n −2 conditions to satisfy. • At each interior knot xj, j = 1, . . . , n −1, the values and the ﬁrst two derivatives of the cubic polynomials on the left and on the right must coincide. This gives 3(n −1) conditions. • At each knot xj, j = 0, 1, . . . , n, the value of the spline must be fj leading to n + 1 more conditions. Hence we a free to assign two additional conditions that would allow us to determine S∆(F; ·) uniquely. Typically, one of the following options is used: (1) The second derivatives are zero at the end knots: S′′ ∆(F; a) = S′′ ∆(F; b) = 0. (2) Periodic: S′ ∆(F; a) = S′ ∆(F; b), S′′ ∆(F; a) = S′′ ∆(F; b). Prerequisite: f0 = fn. (3) The ﬁrst derivatives are assigned at the end knots: S′ ∆(F; a) = f′ 0, S′ ∆(F; b) = f′ n. (4) Not-a-knot conditions: forcing third derivative continuity across the second and penultimate knots of the spline. What does this mean? Since the third derivative of a cubic is a constant function, if that third derivative is continuous, then, in eﬀect, there is no break at all between those two segments at those knots. You can think of it as if those knots are really not knots at all, therein the name “not-a-knot” end conditions. 21 5.2. Setting up a system of equations for a cubic spline. Throughout this section, we will ﬁx a partition ∆:= {a = x0 < x1 < . . . < xn = b} and a vector of values F = {f0, . . . , fn}. We will use the following notations for the intervals between the knots and their lengths: Ij := [xj, xj+1], j = 0, 1, . . . , n −1, and hj+1 := xj+1 −xj. The second derivatives of cubic splines at the knots are called moments and denoted by Mj := S′′ ∆(F, xj), j = 0, 1, . . . , n. We will show that the spline function S∆(F; x) is completely characterized by its moments, and the moments are found by solving a system of linear equations. The second derivative of a S∆(F; x) coincides with a linear function on each subinterval, and these linear functions can be described in terms of the moments Mj: (31) S′′ ∆(F; x) = Mj xj+1 −x hj+1 + Mj+1 x −xj hj+1 x ∈Ij ≡[xj, xj+1]. Integrating Eq. (31) we obtain that for each x ∈Ij, j = 0, 1, . . . , n, (32) S′ ∆(F; x) = −Mj (xj+1 −x)2 2hj+1 + Mj+1 (x −xj)2 2hj+1 + Aj, (33) S∆(F; x) = Mj (xj+1 −x)3 6hj+1 + Mj+1 (x −xj)3 6hj+1 + Aj(x −xj) + Bj, where Aj and Bj are constants of integration. From the equalities S∆(F; xj) = fj and S∆(F; xj+1) = fj+1 we obtain the following equations for Aj and Bj: Mj h2 j+1 6 + Bj = fj, Mj+1 h2 j+1 6 + Ajhj+1 + Bj = fj+1. Hence Bj = fj −Mj h2 j+1 6 , (34) Aj = fj+1 −fj hj+1 −hj+1 6 (Mj+1 −Mj). (35) This yields the following representation of the spline function in terms of its moments: (36) S∆(F; x) = αj + βj(x −xj) + γj(x −xj)2 + δj(x −xj)3 for x ∈Ij, 22 where αj = fj, (37) γj = Mj 2 , (38) βj = S′ ∆(F; xj) = −Mj hj+1 2 + Aj = fj+1 −fj hj+1 −hj+1 6 (Mj+1 + 2Mj), (39) δj = S′′′ ∆(F, x+ j ) 6 = Mj+1 −Mj 6hj+1 . (40) Now we need to calculate the moments Mj. The continuity of S′ ∆(F; x) at the interior knots yields n −1 equations for the moments Mj. Using Eqs. (33) and (35) we obtain S′ ∆(F; x) = −Mj (xj+1 −x)2 2hj+1 + Mj+1 (x −xj)2 2hj+1 + fj+1 −fj hj+1 −hj+1 6 (Mj+1 −Mj). Therefore, for j = 1, 2, . . . , n −1 we have S′ ∆(F; x− j ) = fj −fj−1 hj + hj 3 Mj + hj 6 Mj−1, S′ ∆(F; x+ j ) = fj+1 −fj hj+1 −hj+1 3 Mj −hj+1 6 Mj+1. Since S′ ∆(F; x+ j ) = S′ ∆(F; x− j ), we have (41) hj 6 Mj−1 + hj + hj+1 3 Mj + hj+1 6 Mj+1 = fj+1 −fj hj+1 −fj −fj−1 hj for j = 1, 2, . . . , n −1. These are n −1 equations for n + 1 unknown moments. The remaining two equations can be gained from the boundary conditions. In the ﬁrst case, S′′ ∆(F; a) = S′′ ∆(F; b) = 0 we set M0 = Mn = 0. In the periodic case we set M0 = Mn and hn 6 Mn−1 + hn + h1 3 Mn + h1 6 M1 = f1 −fn h1 −fn −fn−1 hn . Exercise Obtain additional two conditions for the case of assigned ﬁrst derivatives at the end points: h1 3 M0 + h1 6 M1 = f1 −f0 h1 −f′ 0, (42) hn 6 Mn−1 + hn 3 Mn = f′ n −fn −fn−1 hn . (43) Eqs. (41), (42) and (43) can be written in the common format µjMj−1 + 2Mj + λjMj+1 = dj, j = 1, 2, . . . , n −1, 23 where λj := hj+1 hj + hj+1 , (44) µj := 1 −λj = hj hj + hj+1 , (45) dj := 6 hj + hj+1 fj+1 −fj hj+1 −fj −fj−1 hj . (46) In the case M0 = Mn = 0 we set λ0 = 0, d0 = 0, µn = 0, dn = 0. As a result, weobtain the following system of equations (47)         2 λ0 0 µ1 2 λ1 µ2 · · · · · · 2 λn−1 0 µn 2                 M0 M1 · · · Mn         =         d0 d1 · · · dn         . Exercise Write out the system of equations for the momenta for the periodic case and for the case of assigned ﬁrst derivatives at the endpoints. Theorem 6. The matrix in Eq. (47) is nonsingular for any partition ∆of [a, b]. Proof. We will denote the Matrix in Eq. (47) by A. If follows from the deﬁnitions of λj and µj that µj + λj = 1, µj ≥0, λj ≥0. Therefore, the matrix A is strictly diagonal dominant. Therefore, for any vector z (48) max j \|zj\| ≤max j \|Azj\|. (Check this!) Hence Az = 0 if and only if z = 0. □ 24 5.3. A fast solver for tridiagonal matrices. The matrix in Eq. (47) is tridiagonal. There is a fast solver for such kind of systems that involves O(n) ﬂops. q0 = −λ0 2 ; u0 = d0 2 ; λn = 0; for k = 1 : n pk = µkqk−1 + 2; qk = −λk pk ; uk = dk −µkuk−1 pk ; end for Mn = un; for k = n −1 : 0 Mk = qkMk+1 + uk; end for In the ﬁrst for-cycle, we successively express Mj = Mj+1qj + uj. Then we ﬁnd Mn = un as λn = 0. In the second for-cycle we successively ﬁnd Mn−1, Mn−2, ..., M0 using the relationships Mj = Mj+1qj + uj created by the ﬁrst for-cycle. 5.4. Convergence properties of cubic spline functions. Deﬁnition 2. The ﬁneness of the given partition is ∥∆∥:= max j hj. Recall that interpolating polynomials do not necessarily converge to f even for smooth f as the ﬁneness of the partition tends to zero (the Runge phenomenon!). In contrast, the spline interpolants do converge to f together with their ﬁrst three derivatives under mild conditions. First we will introduce some notations. We will denote by F ′′ the vector of the second derivatives of f at the knots. The vector of moments M satisﬁes Eq. (47). We will write AM = d. The residual r is deﬁned as r := d −AF ′′ = A(M −F ′′). Theorem 7. Suppose the ﬁrst derivatives of f at the end knots are assigned. If f ∈C4[a, b] and \|f(4)(x)\| ≤L for x ∈[a, b], then ∥M −F ′′∥≤∥r∥≤3 4L∥∆∥2. 25 Proof. By deﬁnition of the residual we have rj = dj −µjf′′(xj−1) −2f′′(xj) −λjf′′(xj+1) = 6 hj + hj+1 fj+1 −fj hj+1 −fj −fj−1 hj − hj hj + hj+1 f′′(xj−1) −2f′′(xj) − hj+1 hj + hj+1 f′′(xj+1). Using Taylor’s expansion around xj we obtain rj = 6 hj + hj+1 (f′ + hj+1 2 f′′ + h2 j+1 6 f′′′ + h3 j+1 24 f′′′′(τ1) −f′ + hj 2 f′′ − h2 j 6 f′′′ + h3 j 24f′′′′(τ2)) − hj hj + hj+1 " f′′ −hjf′′′ + h2 j 2 f′′′′(τ3) # −2f′′− − hj+1 hj + hj+1 " f′′ + hj+1f′′′ + h2 j+1 2 f′′′′(τ4) # = 1 hj + hj+1 " h3 j+1 4 f′′′′(τ1) + h3 j 4 f′′′′(τ2) − h3 j 2 f′′′′(τ3) −hj+1 2 f′′′′(τ4) # . Here all omitted arguments are xj and τj ∈[xj−1, xj+1]. Therefore, for j = 1, 2, . . . , n −1 \|rj\| ≤3 4L h3 j+1 + h3 j hj+1 + hj = 3 4L(h2 j −hjhj+1 + h2 j+1) ≤3 4L∥∆∥2. The ﬁrst and the last intervals are handled in a similar manner. For them we have \|r0\| ≤3 4L∥∆∥2 and \|rn\| ≤3 4L∥∆∥2. Since r = A(M −F ′′) and A has the property given by Eq. (48) we get ∥M −F ′′∥≤∥r∥≤3 4L∥∆∥2. □ Theorem 8. Suppose f ∈C4[a, b] and \|f(4)(x)\| ≤L for x ∈[a, b]. Let ∆be a partition of the interval [a, b] and K be a constant such that ∥∆∥ hj+1 ≤K, j = 0, 1, . . . , n −1. If S∆is the spline function which interpolates the values of f at the knots of the partition ∆and satisﬁes S′ ∆(a) = f′(a) and S′ ∆(b) = f′(b), 26 then there exist constants ck ≤2 independent of the partition ∆, such that for x ∈[a, b] \|f(x) −S∆(x)\| ≤c0L∥∆∥4, (49) \|f′(x) −S′ ∆(x)\| ≤c1L∥∆∥3, (50) \|f′′(x) −S′′ ∆(x)\| ≤c2L∥∆∥2, (51) \|f′′′(x) −S′′′ ∆(x)\| ≤c3LK∥∆∥. (52) Proof. First we prove Eq. (52). For x ∈[xj−1, xj], S′′′ ∆(x) −f′′′(x) = Mj −Mj−1 hj −f′′′(x) = Mj −f′′(xj) hj −Mj−1 −f′′(xj−1) hj + f′′(xj) −f′′(x) −[f′′(xj−1) −f′′(x)] hj −f′′′(x). Using the previous theorem and the Taylor expansion at x we get \|S′′′ ∆(x) −f′′′(x)\| ≤3 2L∥∆∥2 hj + 1 hj \|(xj −x)f′′′ + (xj −x)2 2 f′′′′(η1) −(xj−1 −x)f′′′ −(xj−1 −x) 2 f′′′′(η2) −hjf′′′\| ≤3 2L∥∆∥2 hj + L 2 ∥∆∥2 hj = 2L∥∆∥2 hj ≤2LK∥∆∥. Here η1, η2 ∈[xj−1, xj]. To prove Eq. (51) we observe that for every x ∈(a, b) there is a closes knot xj = xj(x). Assume without loss of generality that x ≤xj(x) so that x ∈[xj−1, xj] and \|xj(x) −x\| ≤ hj/2 ≤∥∆∥/2. Then \|f′′(x) −S′′ ∆(x)\| = \|f′′(xj) −S′′ ∆(xj) + Z x xj(x) (f′′′(t) −S′′′ ∆(t))dt\| ≤3 4L∥∆∥2 + hj 2 2L∥∆∥2 hj ≤7 4L∥∆∥2, x ∈(a, b). Next we prove Eq. (50). In addition to the boundary points ξ0 := a and ξn+1 := b there exist by Rolle’s theorem n points ξj ∈(xj−1, xj) such that f′(ξj) = S′ ∆(ξj). for any x ∈(a, b) there exists a closest one ξj = ξj(x), and \|x −ξj(x)\| < ∥∆∥. 27 Therefore, \|f′(x) −S′ ∆(x)\| = \| Z x ξ(x) (f′′(t) −S′′ ∆(t))dt\| ≤∥∆∥7 4L∥∆∥2 = 7 4L∥∆∥3. Finally we prove Eq. (49). We have \|f(x) −S∆(x)∥= \| Z x xj(x) (f′(t) −S′ ∆(t))dt\| ≤∥∆∥ 2 7 4L∥∆∥3 = 7 8L∥∆∥4, x ∈[a, b]. □ 5.5. Spline interpolation in Matlab. Matlab has several tools for computing cubic splines. Functions spline and interp1 are regular matlab functions. Function spline is designed for computing cubic splines. The boundary conditions can be chosen to be either ’Not-a-Knot’ or ’Complete’ (i.e., the values of the ﬁrst derivatives are prescribed at the endpoints). Function interp1 oﬀers diﬀerent kinds of interpolation. If its argument ’Method’ is set to ’spline’, it produces a cubic spline with the ’Not-a-Knot’ boundary conditions. The Matlab code below demonstrates the use of piecewise-linear interpolant and two cubic splines, with ’Not-a-Knot’ and ’Complete’ boundary conditions on the function f(x) = (1+x2)−1. Newton’s interpolant is also plotted for comparison. Fig. 4 is generated by this code. function CompareSplines() n = 6; a = -5; b = 5; x = linspace(a,b,n + 1)’; % n+1 equispaced points t = linspace(a,b,200); f = @(x)1./(x.^2 + 1); % Witch of Agnesi fx = f(x); ft = f(t); figure; hold on; grid; plot(x,fx,’.’,’Markersize’,30,’Displayname’,’Data F’); plot(t,ft,’Linewidth’,1,’Displayname’,’f(x)’) %% Newton’s interpolation. % compute divided differences dd = zeros(n + 1); dd(:,1) = fx; for k = 2 : n + 1 dd(1:n-k+2,k) = (dd(2:n-k+3,k-1)-dd(1:n-k+2,k-1))./(x(k:n+1)-x(1:n-k+2)); 28 end % evaluate Newton’s interpolant pt = dd(1,n+1); for k = n : -1 : 1 pt = pt.(t - x(k)) + dd(1,k); end plot(t,pt,’Linewidth’,1,’DisplayName’,’Newton interpolant’); %% Piecewise-linear interpolant pl = interp1(x,fx,t); plot(t,pl,’Linewidth’,1,’Displayname’,’Piecewise-linear’); %% Cubic spline, not-a-knot condition pc = interp1(x,fx,t,’spline’); plot(t,pc,’Linewidth’,1,’Displayname’,’Cubic spline, not-a-knot’); %% Cubic spline, first derivaties at the end knots fp = @(x) -2x./((x.^2 + 1).^2); pp = spline(x,[fp(a);fx;fp(b)]); pc1 = ppval(pp,t); plot(t,pc1,’Linewidth’,1,’Displayname’,’Cubic spline, 1st ders’); set(gca,’Fontsize’,16) legend end -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.2 0 0.2 0.4 0.6 0.8 1 Data F f(x) Newton interpolant Piecewise-linear Cubic spline, not-a-knot Cubic spline, 1st ders Figure 4. Comparison of Newton’s intepolant with 7 equispaced knots, piecewise linear interpolant, and two cubic splines for f(x) = (1 + x2)−1 on [−5, 5]
3894	https://www.nationaljewish.org/education/patient-education-support/online-materials/online-materials/gastroparesis-gp	Gastroparesis (GP) Download Patient Education Gastroparesis (GP) is a chronic motility disorder of the stomach, sometimes called “lazy stomach.”The goal for treatment of gastroparesis is to decrease symptoms such as nausea or bloating, but unfortunately a complete cure is unlikely Download PDF: Gastroparesis (GP) View more information on Gastroparesis (GP) National Jewish Health authors free downloadable patient education materials to provide you and your family the information and tools to help manage your disease. They include Understanding Booklets, Med Facts, Test Facts, Medication handouts and more. Follow Us Keep In Touch NJH.Footer.SupportedLanguages Keep in Touch Our monthly newsletter includes expert health tips, recent research findings, and news from National Jewish Health. ×
3895	https://www.nagwa.com/en/videos/675120710104/	Question Video: Using the Law of Sines to Find an Unknown Length in an Isosceles Triangle \| Nagwa Question Video: Using the Law of Sines to Find an Unknown Length in an Isosceles Triangle \| Nagwa Sign Up Sign In English English العربية English English العربية My Wallet Sign Up Sign In My Classes My Messages My Reports My Wallet My Classes My Messages My Reports Question Video: Using the Law of Sines to Find an Unknown Length in an Isosceles Triangle Mathematics • Second Year of Secondary School Given an isosceles triangle △𝐴𝐵𝐶 with sides 𝐴𝐵 = 𝐴𝐶, the side 𝐵𝐶 = 2, and the ∠𝐴 = 80°, find the length of the side 𝐴𝐵 approximate to the nearest one decimal place. Pause Play % buffered 00:00 00:00 0:00 Unmute Mute Disable captions Enable captions Settings Captions English Quality 0 Speed Normal Captions Go back to previous menu Disabled English EN Quality Go back to previous menu Speed Go back to previous menu 0.5×0.75×Normal 1.25×1.5×1.75×2× Exit fullscreen Enter fullscreen Play 04:16 Video Transcript Given an isosceles triangle triangle 𝐴𝐵𝐶 with sides 𝐴𝐵 equals 𝐴𝐶, the side 𝐵𝐶 equals two, and the angle 𝐴 equals 80 degrees, find the length of the side 𝐴𝐵 approximate to the nearest one decimal place. The first thing we can do here is try and sketch an isosceles triangle. If we let this sketch be our isosceles triangle 𝐴𝐵𝐶, we know the side 𝐴𝐵 equals 𝐴𝐶. The measure of angle 𝐴 equals 80 degrees, and the side length of 𝐵𝐶 has a length of two units. Before we move on, there’s something else we can say about the angles inside this triangle. Because this is an isosceles triangle, the angles opposite the two congruent side lengths will be congruent to each other. The measure of angle 𝐶 will be equal to the measure of angle 𝐵. We know that all three of the angles inside a triangle need to sum to 180 degrees. And so we could say that the measure of angle 𝐵 and 𝐶 will both be equal to 𝑥 degrees. We can solve this equation for the value of 𝑥. If we subtract 80 from both sides of our equation, we get 100 is equal to two 𝑥. From there, we divide both sides of our equation by two, and we find that 𝑥 equals 50. So we can say that angle 𝐶 and angle 𝐵 both measure 50 degrees. Since we know every angle inside this triangle and at least one side length, to find the missing side length, we can use the sine rule. The sine rule tells us that the side length 𝐴 over the sin of its opposite angle 𝐴 is equal to the side length 𝐵 over the sin of the opposite angle 𝐵, which is equal to the side length 𝐶 over the sine of the opposite angle. In our case, we have angle 𝐴 and its opposite side length. We want to know the length of side 𝐴𝐵, and we know its opposite angle. So we can set up a ratio that says two over the sin of 80 degrees is equal to the side length 𝐴𝐵 over the sin of 50 degrees. And then we’ll use cross multiplication to find out what the length of 𝐴𝐵 would be. We have two times the sin of 50 degrees is equal to 𝐴𝐵 times the sin of 80 degrees. To get that 𝐴𝐵 by itself, we can divide both sides of this equation by the sin of 80 degrees. The sin of 80 degrees over the sin of 80 degrees equals one on the right. And from there we can say that 𝐴𝐵 will be equal to two times the sin of 50 degrees over the sin of 80 degrees. You might be wondering why we wrote 𝐴𝐵 in the form two times sin of 50 degrees over sin of 80 degrees instead of calculating two times the sin of 50 degrees in the numerator before we divided by sin of 80 degrees. Because we know that we’re going to be rounding in our final step and we want to make our solution as accurate as possible, we want to minimize the number of times that we round. And so if we write this as two times sin of 50 degrees over sin of 80 degrees, we can enter this into a calculator in one step, which means we only have to round one time. Making sure your calculator is set to degrees and not to radians, when you enter this expression in, the result is 1.55572 continuing, which when rounded to the nearest one decimal place is 1.6. We can say that 𝐴𝐵 equals 1.6 length units and also that 𝐴𝐶 equals 1.6 length units since this is an isosceles triangle. Lesson Menu Lesson Lesson Plan Lesson Presentation Lesson Video Lesson Explainer Lesson Playlist Join Nagwa Classes Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher! Interactive Sessions Chat & Messaging Realistic Exam Questions Nagwa is an educational technology startup aiming to help teachers teach and students learn. Company About Us Contact Us Privacy Policy Terms and Conditions Careers Tutors Content Lessons Lesson Plans Presentations Videos Explainers Playlists Copyright © 2025 Nagwa All Rights Reserved Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy Accept
3896	https://www.khanacademy.org/science/revision-term-1-od-science-grade-9-physical-science/xf5b4d9fffae5debd:week-2/xf5b4d9fffae5debd:is-matter-around-us-pure/v/concentration-of-a-solution	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
3897	https://fiveable.me/key-terms/general-chemistry-ii/poh	POH - (General Chemistry II) - Vocab, Definition, Explanations \| Fiveable \| Fiveable new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom ap study content toolsprintablespricing my subjectsupgrade All Key Terms General Chemistry II POH ⏱️general chemistry ii review key term - POH Citation: MLA Definition The pOH is a measure of the acidity or basicity of a solution, calculated using the formula $$pOH = -\log[OH^-]$$, where $$[OH^-]$$ represents the concentration of hydroxide ions in moles per liter. This term is crucial for understanding the relationship between hydroxide ions and the overall pH of a solution, allowing for calculations involving strong and weak bases as well as their equilibrium states in aqueous solutions. 5 Must Know Facts For Your Next Test The pOH scale is complementary to the pH scale; at 25°C, $$pH + pOH = 14$$. A lower pOH value indicates a higher concentration of hydroxide ions, meaning a more basic solution. Calculating pOH is especially important when working with strong bases, where the concentration of hydroxide ions is known and can be directly used to find the pOH. In weak bases, determining the pOH involves equilibrium calculations that take into account the dissociation constant (Kb) and the initial concentration of the base. pOH can also be used to find the pH of a solution when either value is known, allowing for conversions between acidity and basicity. Review Questions How does understanding pOH enhance your ability to work with strong bases in calculations? Understanding pOH allows you to directly determine the basicity of a solution when dealing with strong bases. Since strong bases fully dissociate in water, knowing the concentration of hydroxide ions lets you easily calculate the pOH using the formula $$pOH = -\log[OH^-]$$. This is important for predicting how these bases will react in different chemical situations and how they affect overall solution properties. Compare and contrast how you would calculate pOH for strong versus weak bases. For strong bases, calculating pOH is straightforward since they completely dissociate in solution, making it easy to use their concentration to find pOH directly from $$pOH = -\log[OH^-]$$. In contrast, weak bases only partially dissociate, requiring equilibrium expressions to be set up using their base dissociation constant (Kb) before you can determine the hydroxide ion concentration and subsequently calculate pOH. This added complexity in weak base calculations highlights differences in behavior between strong and weak bases. Evaluate how changes in temperature might affect the relationship between pH and pOH in a solution. Changes in temperature can impact both the ion product of water (Kw) and thus affect the relationship between pH and pOH. At higher temperatures, Kw increases, leading to shifts in neutral points on the scale; consequently, $$pH + pOH$$ may no longer equal 14. Therefore, understanding these temperature effects is crucial for accurate calculations in varying thermal conditions, as this can lead to misinterpretations if one assumes standard conditions without adjustments. Related terms pH: A scale used to specify the acidity or basicity of an aqueous solution, calculated as $$pH = -\log[H^+]$$, where $$[H^+]$$ is the concentration of hydrogen ions. Hydroxide Ion:A negatively charged ion formed when water molecules dissociate, represented as $$OH^-$$, which plays a key role in determining the basicity of a solution. Neutralization: A chemical reaction in which an acid and a base react to form water and a salt, typically resulting in a solution with a pH around 7. 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3898	https://www.alfalaval.com/microsites/gphe/tools/calculation-method/	Gasketed plate heat exchangers Tools Types Service Downloads Contact us Tools Selection guide Features that matter Plate technology GPHE vs shell-and-tube How GPHEs work Calculation method Selection guide Features that matter Plate technology GPHE vs shell-and-tube How GPHEs work Calculation method Types Industrial ARHI certified Semi-welded WideGap AlfaVap AlfaCond Diabon Industrial ARHI certified Semi-welded WideGap AlfaVap AlfaCond Diabon Service GPHE Service menu GPHE Service equipment Performance Audit Reconditioning and repair Spare parts Cleaning GPHE Service menu GPHE Service equipment Performance Audit Reconditioning and repair Spare parts Cleaning Downloads Contact us Home Tools Calculation method Plate heat exchanger calculation method To solve a thermal problem, we need to know several parameters. Further data can then be determined. The six most important parameters include: The amount of heat to be transferred (heat load) The inlet and outlet temperatures on the primary and secondary sides The maximum allowable pressure drop on the primary and secondary sides The maximum operating temperature The maximum operating pressure The flowrate on the primary and secondary sides If the flow rate, specific heat and temperature difference on one side are known, the heat load can be calculated. Calculation method The heat load of a heat exchanger can be derived from the following two formulas: 1. Heat load, Theta and LMTD calculation Where: P = heat load (btu/h) m = mass flow rate (lb/h) cp = specific heat (btu/lb °F) δt = temperature difference between inlet and outlet on one side (°F) k = heat transfer coefficient (btu/ft2 h °F) A = heat transfer area (ft2) LMTD = log mean temperature difference T1 = Inlet temperature - hot side T2 = Outlet temperature - hot side T3 = Inlet temperature - cold side T4 = Outlet temperature - cold side LMTD can be calculated by using the following formula, where ∆T1 = T1–T4 and ∆T2 = T2–T3 2. Heat transfer coefficient and design margin The total overall heat transfer coefficient k is defined as: α1 = The heat transfer coefficient between the warm medium and the heat transfer surface (btu/ft2 h °F) α2 = The heat transfer coefficient between the heat transfer surface and the cold medium (btu/ft2 h °F) δ = The thickness of the heat transfer surface (ft) Rf = The fouling factor (ft2 h °F/btu) λ = The thermal conductivity of the material separating the medias (btu/ft h °F) kc = Clean heat transfer coefficient (Rf=0) (btu/ft2 h °F) k = Design heat transfer coefficient (btu/ft2 h °F) M = Design Margin (%) Combination of these two formulas gives: M = kc · Rf i.e the higher kc value, the lower Rf-value to achieve the same design margin. For a more complete explanation of heat transfer theory and calculations, download the following brochure: The theory behind heat transfer Contact us and we'll connect you with a plate heat exchanger engineer that can help you with your calculations. Quick links: How GPHEs work Selection guide Features that matter Plate technology GPHE vs shell-and-tube Calculation method Types of GPHEs Servicing a GPHE Contact us for more information. This website uses cookies By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. Alfa Laval Cookie Policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. 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3899	https://www.youtube.com/watch?v=xvkkTjj3xvw	Conversion problems_part2 \| Aldehydes, ketones and carboxylic acids \| Chemistry \| Khan Academy Khan Academy India - English 549000 subscribers 4 likes Description 314 views Posted: 30 Sep 2024 Let's solve a few more conversion problems based on the various chemical reactions and reagents that we learned in this chapter. Khan Academy is a free learning platform for Class 1-12 students with videos, exercises, and tests for maths, science, and more subjects. Our content is aligned to CBSE syllabus and available in Hindi, English, and many more regional languages. Experience the joy of easy, seamless, accessible learning anywhere, anytime with Khan Academy. Subscribe to our YouTube channel - As a 501(c)(3) nonprofit organization, we would love your help! Donate here: Created by Revathi Ramachandran Transcript: so this is part two of our video on conversion problems we already discussed two very interesting conversion problems in the previous video right so let's continue that streak in this video and look at our first question we have ethanol that is treated with PCC and the product obtained is further treated with dilute NaOH followed by treatment with acidic M4 which on heating strongly gives us the final product now we need to figure out which among the following statements is true regarding this final product does it give a positive idof form test does it give a positive failings test or does it give effervescence on reaction with sodium bicarbonate now this might look difficult but trust me it's really very easy what we need to do is simply write down this in the form of reactions and solve it one at a time okay so let's start with ethanol ethanol on treatment with PCC a mild oxidizing agent would give us what aldhy right PCC is a mild oxidizing agent which would oxidize primary alcohols to alides and secondary alcohols to ketones now if we had acidic M4 at this step then we would get an acid PCC very efficiently stops the oxidation at the aldhy level so the product that we get here is acet alide or ethanol now ch3 CH reacts with dilute NaOH so tell me what exactly is happening here yes aldol reaction right so those alhida hydrogen like in the case of our ethanol can undergo aldol reaction in the presence of dilute NaOH now remember we are talking about aldol reaction here there is no heating involved in this step we haven't mentioned that there is any heating involved in this reaction because heating would give us an Aldo condensation product which is an alpha beta unsaturated product right but here when we have an Aldo reaction the final product is what exactly a beta hydroxy alide so as you can see here one more molecule of acet alide would form the inate ion and the nucleophilic inate ion would attack the carbonal carbon of another molecule of acet alide giving us beta hydroxy alide so this is the product of a typical aldol reaction in an Aldo condensation reaction we would further eliminate a water molecule from Alpha Beta carbon atoms and get an alpha beta unsaturated product now what's the next step in the next step we are treating this beta hydroxy alide with acidic K4 again a very strong oxidizing agent so what groups get oxidized in this reaction obviously alihi would get oxidized to carboxilic acid but is that it is there anything else that can get oxidized here yes the secondary alcohol correct so the secondary alcohol here would also get oxidized in the presence of k42 Ketone so the product that we get at the end of this step is a beta keto acid ch3 C double bond o ch2 co is that it is the reaction over no we have one final step which is heating okay so no reagents here we're simply heating this compound so what do you think happens here now if you remember one of the characteristic reactions of carboxilic acids was decarbox correct and beta keto acids are an excellent substrates for decarbox silation reactions so for this particular compound to undergo decarbox silation we don't need to do anything but simply provide strong Heating and when we do that a stable molecule like carbon dioxide gets eliminated and with the elimination of carbon dioxide we get the final compound which is acetone ch3 Co ch3 so we won't go into the details of the mechanisms of these reactions because we have dealt with each of them in detail in other videos so which among these statements would be appropriate for our final product Ketone does it give ffs Inson reaction with sodium bicarbonate no because it's not an acid it's a ketone it will not give effervescence with a base what about failings test only alids that two aliphatic alides give positive failings test so Ketone will not give a positive failings test here and what about iodoform test absolutely in this Ketone we have a chs3 CO Group which means it would give a positive idoor test and would form yellow colored precipitate of idof form along with the carboxilate ion so if this product further underwent an iodoform reaction in that case we would end up with a carboxilic acid which is one carbon less than the Ketone that we have here so the answer is the final Ketone gives a positive idof form test let's look at one more reaction in this question it is given that an aliphatic alide P reacts with form alide in the presence of dilute NaOH to give r which upon treatment with kcn gives s on acidification and heating s gives us the following product and what do we need to do we need to figure out the structure of the starting aliphatic aldhy P now this is an example of a deconstruction problem where we need to figure out the starting reactants from a given product so how do we go about this let's analyze the reactions that are taking place in the first step we have two alhida alide reacting with form alide in the presence of dilute NaOH now this indicates a classic aldol reaction correct now looking at form alide you might be tempted to say maybe it is canazaro reaction but is that so look at the reagents we know that in canazaro reaction we use strong bases in concentrated amount so here it would be concentrated K and often canazaro reaction requires heating but no such condition has been mentioned here we have dilute NaOH which is indicative of an aldol reaction now now if one of a reactants does not have an alpha hydrogen like hch it means the other aliphatic aldhy must have an alpha hydrogen correct only then we can have an alal reaction in this case more specifically a crossed aldol reaction so let's assume that the structure of a starting alide p is R ch2 Cho for all we know it could be R2 ch ch as well we don't know exactly at this point but let's go ahead with this assumption okay let's assume that it is rch2 CH so that has an alpha hydrogen atom and form alide of course doesn't have an alpha hydrogen atom now when these two alides undergo crossed alal condensation reaction the product that we get is this how do we get this product you see the inate ion from here from P would attack the carbonal carbon atom and result in the formation of a beta hydroxy alide nothing but aldol so this is the alpha carbon atom and we have o group at the beta carbon atom and this is a primary functional group in the next step we are treating this beta hydroxy alahh with kcn so CN minus would react with our alol to give a product which looks like this a coh hydron and we get a coh hydron only if CN minus would attack the carbonal carbon of her alide correct but it's very normal to ask another question why do we end up getting a nucleophilic addition reaction and not a nucleophilic substitution reaction where CN minus replaces the O group here you see CN minus as we know is a very strong nucleophile it is eagerly or earnestly looking for an electrophilic site and the carbonal carbon of her alahh is much more electrophilic as compared to the carbon of her alcohol group and not just that for any nucleophilic substitution to be successful it must have a good living group correct now if CN minus attacks this particular carbon what is the living group here the living group is a very poor oh minus group oh minus is an extremely poor living group it does not encourage a nucleophilic substitution reaction and this is why the CN minus would prefer to attack the carbonal carbon of our aldhy group and result in a nucleophilic addition reaction giving us a coh hydron the next step is acidification and heating now CN groups on acidic hydrolysis gives us the corresponding carboxilic acid now look at this compound we have an alcohol an O group and a CO group in the same molecu so that means there is a possibility to form an STM we know that in acidic medium an alcohol and a carboxilic acid can react with each other to form an ester now in this case since the O group and the carboxilic acid group are in the same molecule we would end up getting an intram molecular Ester correct an intram molecular esterification reaction happens but remember intram molecular reactions take place only when the reaction results in a stable five or six membered ring so do we we have the scope of forming a stable ring here let's see let's first number the carbon chain and then Orient it in such a way to form a cyclic compound so you can see that the molecule can twist in such a way that the O and Co can come in closer proximity with each other and in this case in the presence of acidic medium with the elimination of a water molecule we get a CYCC Ester so look at what we have got we have got the product that looks almost exactly like a final product given here correct we have an oxygen atom cble Bond o group O group and at this carbon atom we have two methy groups so instead of R what we have here is two methy group from this information we can figure out the structure of a starting alide k for a starting alide we had proposed the structure R ch2 Cho from which an inate ion could be formed but given that there are two meile groups at this particular carbon the final structure should be something like this ch3 ch3 ch ch so this particular carbon has two methy substituents here and when we perform the entire reaction using this as a starting aldhy P then we get a final product which looks like this

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