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6500 | https://cdn.kutasoftware.com/Worksheets/PreAlg/Angle%20Sum%20of%20Triangles%20and%20Quadrilaterals.pdf | ©4 K2j051s2A MK4uqtRas cSyoWf9tdwRaRrdeY ULJLdCS.j a TAclwlq 2ruiFgBhRtjsE OrfeuskefrAvdesdW.j P JM1apdxef 9wmi6tshu II4n9f7iznDiotkeQ BPBrveJ-LA2lBgje4b2rEaD.A Worksheet by Kuta Software LLC Kuta Software - Infinite Pre-Algebra Name_____ Period_ Date___ Angle Sum of Triangles and Quadrilaterals Find the measure of angle b.
1) 115° 31° b 2) 33° 29° b 3) 119° 34° b 4) 123° 39° b 5) 75° 75° b 6) 26° 45° b 7) 72° 108° 81° b 8) 77° 103° 97° b 9) 105°75° b 10) 86° 109° 71° b -1-©a P200l132R jKMumtuaQ eSKoFf6tvwga9rbeh rLhLpCT.G 6 HAglHlu srAi7gBhctRs8 HrdedsSebrPvFeAds.8 H 8MXaUdUe5 GwRiUtnhA MIHnyfMi0nRiZtLej PPWrRe0-DAklbgieBb6rKaw.N Worksheet by Kuta Software LLC 11) 95° 121° 59° b 12) 86° 94° 72° b Find the value of x.
13) 77° 50° (7x - 3)° 14) 45° 45° (4x + 18)° 15) 31° 39° (11x + 11)° 16) 57° 89° (5x - 6)° 17) (2x + 3)° 119° 90° 90° 18) (5x + 16)° 102° 100° 92° 19) (6x + 22)° 85° (x + 37)° 69° 20) 115° 118° 81° (2x + 8)° -2-©q e2B0T1y28 pKNu8tzah XSnocfLtvwDaprJea 7L1LYCQ.u m XAvlZlb rroiVg9hrtMsE rrMeDsYeDrTvpeddm.4 F NMMaSdae5 8wIiDtChC BI7nyfNipn7irtuea vPQrNeY-rAml4gwecbLrhaA.q Worksheet by Kuta Software LLC Kuta Software - Infinite Pre-Algebra Name_____ Period Date__ Angle Sum of Triangles and Quadrilaterals Find the measure of angle b.
1) 115° 31° b 34° 2) 33° 29° b 118° 3) 119° 34° b 27° 4) 123° 39° b 18° 5) 75° 75° b 30° 6) 26° 45° b 109° 7) 72° 108° 81° b 99° 8) 77° 103° 97° b 83° 9) 105°75° b 90° 10) 86° 109° 71° b 94° -1-©N G2Q0R1T2B bKQukt3aC 1STo3fetzwHacr6eb iLjLnCk.c N 0AIlVlg QrXiPgbhHt0sk er7eMsOeUrWvqegdJ.3 V SMlaEdgep rwCimtOhj TIEnPf9i8nUi1tRea EPRrGeE-iA7l1gAeXbEryaq.k Worksheet by Kuta Software LLC 11) 95° 121° 59° b 85° 12) 86° 94° 72° b 108° Find the value of x.
13) 77° 50° (7x - 3)° 8 14) 45° 45° (4x + 18)° 18 15) 31° 39° (11x + 11)° 9 16) 57° 89° (5x - 6)° 8 17) (2x + 3)° 119° 90° 90° 29 18) (5x + 16)° 102° 100° 92° 10 19) (6x + 22)° 85° (x + 37)° 69° 21 20) 115° 118° 81° (2x + 8)° 19 -2-Create your own worksheets like this one with Infinite Pre-Algebra. Free trial available at KutaSoftware.com |
6501 | https://or.stackexchange.com/questions/5057/divisibility-constraint-in-integer-programming | optimization - Divisibility constraint in Integer programming - Operations Research Stack Exchange
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Divisibility constraint in Integer programming
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I have a simple question regarding the divisibility in integer programming
suppose the objective function is
max x 1+x 2 max x 1+x 2
where the constraint is that the sum of x 1 x 1 and x 2 x 2 are divisible by either 5, 7 or 9
I'm wondering how can I model the constraint of divisibility?
The only solution I can think of is like
```markdown
max
x1+x2+ 0x3
subject to
y1+y2+y3 >= 0
y1(x1+x2) = 5x3y1
y2(x1+x2) = 7x3y2
y3(x1+x2) = 9x3y3
x1>=0,x2>=0,x3>=0
```
Is it correct to solve divisibility constraints like this?
Thank you!
optimization
modeling
integer-programming
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edited Oct 19, 2020 at 3:13
RobPratt
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asked Oct 19, 2020 at 1:47
whtitefallwhtitefall
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Which of your variables are integer?RobPratt –RobPratt 2020-10-19 02:16:09 +00:00 Commented Oct 19, 2020 at 2:16
And do you have any upper bounds on x 1 x 1 and x 2 x 2?RobPratt –RobPratt 2020-10-19 02:56:02 +00:00 Commented Oct 19, 2020 at 2:56
@RobPratt x1 and x2 are integers and no upper bounds on x1 and x2, what could be a possible solution to this?whtitefall –whtitefall 2020-10-19 03:02:29 +00:00 Commented Oct 19, 2020 at 3:02
Do you have other constraints? Because otherwise the problem is unbounded.RobPratt –RobPratt 2020-10-19 03:03:27 +00:00 Commented Oct 19, 2020 at 3:03
@RobPratt yeah I know, the question is only to model the problem, not to solve the problem, so no other constraints...whtitefall –whtitefall 2020-10-19 03:05:40 +00:00 Commented Oct 19, 2020 at 3:05
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Suppose x 1+x 2 x 1+x 2 is bounded above by some M M; otherwise the problem is unbounded. Let D={5,7,9}D={5,7,9}, and for d∈D d∈D, introduce binary variable z d z d and nonnegative integer variable w d w d. You can enforce the desired behavior by imposing the following linear constraints:
x 1+x 2 d⋅w d∑d∈D z d=∑d∈D d⋅w d≤M⋅z d=1 for d∈D x 1+x 2=∑d∈D d⋅w d d⋅w d≤M⋅z d for d∈D∑d∈D z d=1
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answered Oct 19, 2020 at 3:13
RobPrattRobPratt
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6502 | https://www.youtube.com/watch?v=mCRMsn6tEAU | Systems of Linear Equations (Fractions & Decimals)
Mario's Math Tutoring
458000 subscribers
119 likes
Description
9671 views
Posted: 24 Oct 2018
Solve Systems of Linear Equations containing fractions or decimals in this video math tutorial by Mario's Math Tutoring. We discuss how to clear the denominators to make the systems easier to solve. In the case of decimals we discuss how to clear the decimals. We go through 2 examples using the elimination method to solve the two variable two equation systems.
Timestamps:
00:00 Intro
0:14 Example 1 How to Clear the Fractions
3:12 Solve System by Using the Elimination Method
4:25 Tip For Multiplying Whole Numbers and Fractions
5:32 Example 2 How to Clear the Decimals
6:30 Using Elimination Method to Solve the System
8:13 How to Double Check Your Solution
Related Videos:
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15 comments
Transcript:
Intro but all these fractions it's pretty difficult to work with so we're gonna talk about is how do we clear those fractions how do we get rid of those fractions so we just have integers and whole numbers to work with right well what you want to do is you want to look at all the denominators right so when you look at the diners we've got 4 2 & 2 okay so when we're Example 1 How to Clear the Fractions looking at this first equation what's the common denominator meaning like if we were going to get a common denominator use fractions what's that lowest common denominator now a lot of students will mistakenly say 2 because they're thinking oh 2 goes into 4 and 2 goes into 2 but what we're looking for is the smallest number that 2 would divide into and form and also divide into and the smallest number that that would be in this case would be 4 so what we're going to do is we're gonna multiply the whole equation by 4 now when you multiply this whole equation by 4 what you're gonna do is you're gonna distribute that 4 to every term okay left and right sides because you want to keep that equation balanced now another way to do this and I think a lot of students might find this a little bit easier is when you're multiplying by 4 you're really multiplying by 4 over 1 because anything divided by 1 is itself right so what I'm gonna do is right next to each one of these fractions I'm going to write 4 over 1 4 over 1 and 4 over 1 okay so here with me so far the reason I did that is because you can see we can do some cross reducing numerator and denominator you see those fours are cancelling out one another so you're really left with 1 over 1 which is just 1 and that just gives us X okay over here you can see 4 & 2 we can cross reduce 2 goes in here once 2 goes into 4 twice 2 times 1 is 2 1 times 1 is 1 2 divided by 1 is 2 so we have 2y and over here we can also cross reduce okay and we get negative 1 times 2 is negative 2 1 times 1 is 1 negative 2 over 1 is negative 2 so you can see we have effectively cleared the denominators we've gotten rid of those fractions we just have integers now which are easier to work with right now in the second equation here what can we multiply by to clear the denominators what do you think well you can see this has a denominator of 3 these ones you can think of as having a denominator of 1 okay because anything divided by 1 is itself right so what's that lowest common denominator meaning what's the smallest number that one divides into and 3 also divides into okay if you said 3 you're absolutely right so what we're gonna do is we're gonna multiply everything by three and remember three is like three over one so what I'm gonna do is I'm just gonna again write three over one three over one next to each one of those terms remember the terms are separated by minus or plus and you want to make sure you're multiplying everything by three not just the left side not just the right side everything every turn right okay so now notice what happens 3x divided by one is just 3x over here you can see that 3 is our cross reducing we're left with negative 5 over 1 which is negative 5y and over here we have 21 okay perfect so now we're in business right we can see we've got just integers and we want to try to solve the system so what we're gonna do here is we're gonna try to do the elimination method now what's the elimination method well elimination method is you try to multiply one of the equations by a constant such that when you add it to the other equation that either the X's cancel out or the Y's cancel out so I'll have links to other videos about solving systems of equations if you want more practice but just to show you here let's multiply everything in this equation by Solve System by Using the Elimination Method negative 3 because if we do that I'll write it right down here we're gonna get negative 3x negative 6y and negative 3 times negative 2 is positive 6 now when we add see how those X's are cancelling one another out because one's positive ones negative and over here we've got 27 okay so now let's see so ok so this is coming out to a little bit of a strange answer but that's okay so we're gonna divide everything by negative 11 so that's 27 over negative 11 okay that's our Y value now if you want to solve for X what do we do well we can put it back into any one of the equations but pick an easy one I'm going to pick this first one here because you can see the numbers are quite small so we have X plus 2 times 27 over negative 11 equals negative 2 and I'm just going to distribute the 2 so that's gonna be 54 over negative 11 okay so 2 is like 2 over 1 sometimes students have trouble with these fractions but just turn a whole number into a fraction by putting it over 1 so the numerators line up and the denominators line up and you just multiply horizontally right and so then this equals 2 over 1 we're gonna subtract actually this is a negative 54 11 so we do the ops we're going to add 54 over 11 to both sides right so plus 54 over a lab Tip For Multiplying Whole Numbers and Fractions we have to get common denominators the common denominator is 11 so I'm going to multiply this by 11 over 11 so we have negative 22 plus 54 all over 11 so let's see that comes out to what is that that is 32 elevenths okay that's what x equals and so now you can see our final answer we want to write it like a coordinate X comma Y we have 32 over 11 27 over negative 11 that's the point or the coordinate where the two lines would cross if you were to graph both of these lines now you can turn these into mixed numbers if you like you can turn them into decimals if you want to get an approximation in round but we'll just leave it like that for right now so that's how you would work with fractions how about example number two where you're working with some decimals well in this case what we want to do is we want to clear the decimals so what can we multiply by to get rid of those decimals what do you think well and the first one see wouldn't be great if this decimal point was right here so to move it to here we actually have to multiply it by a hundred because when you multiply by a hundred you move that decimal point two places to the right so Example 2 How to Clear the Decimals let's do that we're gonna multiply everything by 100 remember you're distributing it to each term so that's going to actually move this two places two places we're going to put a placeholder here a zero and then this one also two places so what we have is we have 5x minus 10 y equals negative 45 right this one here we only really need to multiply by 10 because that'll move this decimal place one to the right one to the right one to the right here we need a placeholder right so in this case we just multiply everything by 10 so 28 X plus 12 y equals 20 okay so now we're in business so again let's use the elimination method and what we're going to do is we're going to try to either eliminate the XS or the Y's so when we add the two equations together either the XS or the Y's cancel out this was a little bit more challenging because let's look at the Y's what's the smallest number that 10 goes into and 12 goes into 60 right so what I'm going to do is I'm gonna multiply this top one by Using Elimination Method to Solve the System six to make this negative 60 the bottom equation by five to make this positive 60 this way when we add the Y's are going to cancel each other out so let's go ahead and do that to the top equation we're distributing the six that gives us 30 X - 60 y equals let's see what does that come out to that's negative 240 plus 3 so negative 270 right and then here we're distributing the 5 to each term each term in the equation right so 5 times 28 is how much 140 X we're getting some large numbers here plus 60 y equals 100 okay but notice what happens when we add the Y's are canceling right because one's negative ones positive this gives us 170 x equals negative 170 divided by 170 you can see X's coming out to negative 1 that worked out pretty nice right all we have to do to solve for y is put this x equals negative 1 back into any one of the equations doesn't matter which one and solve for y let's go ahead and use this one here because the numbers are a little bit smaller so I'm gonna say 5 instead of X I'm going to put in negative 1 okay and then what we're going to do is this is a negative 5 so we do the OPP's I'm going to add 5 to both sides so that comes out to negative 10y equals negative 40 divide both sides by negative 10 that's going to give us y equals positive 4 so our answer is negative 1 comma 4 now if you want |
6503 | https://tutoring.k12.com/articles/rounding-decimals-tips-and-tricks-for-precision/ | Rounding Decimals: Tips and Tricks for Precision
Whether you’re calculating grocery bills, estimating travel distances, or solving problems in your math class, rounding decimals ensures your calculations are clean, understandable, and efficient. This article will break down what rounding decimals means, how it works, and provide useful tips to help you ensure precision. Let’s simplify the world of numbers, one decimal at a time!
What is Rounding Decimals?
At its core, rounding decimals is part of number theory—a branch of mathematics that deals with the properties and relationships of numbers. More specifically, rounding decimals helps to simplify numbers by adjusting them to a value that’s easier to work with, while staying as close as possible to the original number.
For example:
The key is to use rounding rules, which make the process systematic and easy to apply.
The Key Rule of Rounding
The primary rule for rounding is straightforward:
For instance:
Why is Rounding Decimals Useful?
Rounding decimals comes in handy for many real-life situations where precision is important but exact numbers aren’t necessary. Here are a few practical examples:
1. Calculating Grocery Bills
When you’re estimating your total at the grocery store, rounding decimal prices (e.g., $3.75 to $4.00) can simplify mental math and help you stay within budget.
2. Measuring Ingredients for Recipes
Ever adjusted a recipe? If the recipe calls for 1.333 tablespoons of sugar, you’ll likely round that to 1.33 or 1.5 tablespoons to make measuring easier.
3. Planning Road Trips
When calculating travel distances, you might round your map readings (like 152.47 miles to 152.5 miles) for quicker, rough estimates of fuel and time.
4. Determining Sales Tax
Sales tax calculations often involve decimals. For example, a 5% tax on $47.75 equals $2.3875. Rounding this to $2.39 makes it simpler to understand.
5. Currency Conversions
If you’re figuring out how much 100 euros will cost in dollars and get $106.357, rounding to $106.36 keeps things tidy without losing accuracy.
Understanding how and when to round not only helps you make quick calculations but also ensures your results are practical for decision-making.
How to Round Decimals Step by Step
Here’s a simple step-by-step process to round decimals accurately:
1. Identify the Decimal Place to Round To
Decide which digit place (tenths, hundredths, thousandths, etc.) you want to round to.
2. Look at the Following Digit
Check the number immediately after the place value you’re rounding to.
3. Apply the Rounding Rule
4. Adjust and Rewrite the Number
Replace all digits after the rounded value with zeros (or drop them entirely, depending on your rules).
Example:
Round 12.3461 to two decimal places.
Tips and Tricks for Accuracy
For more interactive practice, check out Khan Academy or Mathway, which offer exercises and tools to master these concepts.
Want to Improve Your Math Skills?
Rounding decimals is just one step toward mastering math, but there’s so much more to explore! If you’re ready to take your learning further, consider brushing up on other essential skills like subtracting vectors. Check out Mathsisfun.com’s article on vector subtraction for a deeper understanding.
And if you’re looking for hands-on guidance, K12 Tutoring has expert pre-algebra tutors ready to help you reach your full potential. From decimals to algebra, they’ve got you covered.
Final Thoughts
Rounding decimals is a crucial skill in both math and everyday life—it simplifies calculations without compromising accuracy. From estimating grocery costs to planning your next vacation, it’s an essential tool that makes math accessible and practical. By following the tips and tricks outlined in this guide, you’ll be rounding decimals like a pro in no time.
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α-Synuclein colocalizes with AP180 and affects the size of clathrin lattices
Karina J Vargas
Karina J Vargas
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
2 Marine Biological Laboratory, Woods Hole, Massachusetts, USA
3 Department of Cell Biology, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
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1,2,3,‡, PL Colosi
PL Colosi
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
4 PREP Program, Yale University, New Haven, Connecticut, USA
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1,4,‡, Eric Girardi
Eric Girardi
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
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1,‡, Jae-Min Park
Jae-Min Park
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
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1, Leah E Harmon
Leah E Harmon
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
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1, Sreeganga S Chandra
Sreeganga S Chandra
1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
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1,∗
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1 Departments of Neurology and Neuroscience, Yale University, New Haven, Connecticut, USA
2 Marine Biological Laboratory, Woods Hole, Massachusetts, USA
3 Department of Cell Biology, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
4 PREP Program, Yale University, New Haven, Connecticut, USA
∗
For correspondence: Sreeganga S. Chandra sreeganga.chandra@yale.edu
‡
These authors contributed equally to this work.
Received 2022 Nov 16; Revised 2023 Jul 11; Collection date 2023 Sep.
© 2023 The Authors
This is an open access article under the CC BY license (
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PMCID: PMC10470054 PMID: 37516240
Abstract
α-Synuclein and family members β- and γ-synuclein are presynaptic proteins that sense and generate membrane curvature, properties important for synaptic vesicle (SV) cycling. αβγ-synuclein triple knockout neurons exhibit SV endocytosis deficits. Here, we investigated if α-synuclein affects clathrin assembly in vitro. Visualizing clathrin assembly on membranes using a lipid monolayer system revealed that α-synuclein increases clathrin lattices size and curvature. On cell membranes, we observe that α-synuclein is colocalized with clathrin and its adapter AP180 in a concentric ring pattern. Clathrin puncta that contain both α-synuclein and AP180 were significantly larger than clathrin puncta containing either protein alone. We determined that this effect occurs in part through colocalization of α-synuclein with the phospholipid PI(4,5)P2 in the membrane. Immuno-electron microscopy (EM) of synaptosomes uncovered that α-synuclein relocalizes from SVs to the presynaptic membrane upon stimulation, positioning α-synuclein to function on presynaptic membranes during or after stimulation. Additionally, we show that deletion of synucleins impacts brain-derived clathrin-coated vesicle size. Thus, α-synuclein affects the size and curvature of clathrin structures on membranes and functions as an endocytic accessory protein.
Keywords: electron microscopy; lipid monolayer assay; membrane curvature; synaptic vesicle endocytosis; PI(4,5)P2
α-Synuclein became a principal focus of neurodegeneration research when it was identified as the major constituent of Lewy Bodies, pathological protein aggregates found in the brains of Parkinson’s Disease (PD) patients (1). The importance of α-synuclein to PD pathogenesis was further underscored by the identification of families with Mendelian forms of PD that arise from point mutations and gene multiplications of SNCA, the α-synuclein gene (2, 3, 4, 5, 6, 7). Genome-wide association studies demonstrate that sequence variants in SNCA are associated with sporadic PD (8, 9). Additionally, Lewy body pathology is observed in several other neurodegenerative diseases, for instance, Lewy body dementia and multiple system atrophy, collectively known as synucleinopathies (10). Based on these observations, many current therapeutic strategies for PD, and more broadly synucleinopathies, are focused on eliminating or reducing α-synuclein levels in the brain (11, 12). Therefore, there is a growing interest in elucidating the underlying physiological functions of α-synuclein and how loss of α-synuclein impacts neuronal physiology.
α-Synuclein was initially discovered as a synaptic vesicle (SV)-associated protein in the electric organ of Torpedo (13), suggesting its physiological function(s) are linked to the SV cycle. Indeed, α-synuclein has been implicated to function in distinct steps of the SV cycle. We and others have demonstrated that α-synuclein regulates different stages of SV endocytosis (SVE) (14, 15, 16). In addition, there is evidence of α-synuclein interacting with the exocytic SNARE, synaptobrevin-2 (17, 18, 19), regulating fusion pore dynamics (20), and clustering of SVs, postfusion (21, 22, 23). The preponderance of functional studies supports synucleins acting postfusion, in SVE (14, 15, 16, 20, 24, 25).
Biochemical evidence in cell lines and neurons supports interactions between α-synuclein and the protein machinery of endocytosis (Hsc70, clathrin, AP2, endophilin) (24, 26). In previous studies, we showed, using pHluorin imaging and cholera toxin labeling, that αβγ-synuclein KO hippocampal neurons (27) exhibit slower SVE kinetics (14). Similar results have been obtained by monitoring transferrin uptake in α-synuclein KO neurons, which showed reduced endocytosis (24). In addition, we demonstrated that the interaction between α-synuclein and synaptobrevin-2 observed by other researchers (17, 23, 28) is part of a larger complex, of which AP180 is a major component (14). AP180 is a brain-specific adapter of synaptobrevin-2 (29) and along with AP2, acts to initiate clathrin-coated pit (CCP) formation by recruiting clathrin to synaptic membranes (19, 29, 30, 31, 32). With this in mind, we hypothesized that α-synuclein could be impacting SVE through a mechanism that involves lipids, AP180, and clathrin. Here, we aim to gain molecular insights regarding this mechanism by performing biochemical experiments with synthetic and physiological membranes as well as brain-derived clathrin-coated vesicles (CCVs).
Our experiments revealed that α-synuclein affects clathrin lattice size in vitro, and on physiological membranes, additionally, that AP180, α-synuclein, and clathrin colocalize and are arranged in a concentric pattern in larger clathrin puncta. Clathrin-α-synuclein colocalization is dependent on α-synuclein’s lipid-binding properties. We also show that the subsynaptic localization of α-synuclein is dynamic and can localize to the presynaptic plasma membrane, under conditions where bulk and SVE are promoted. Finally, we demonstrate that deletion of synucleins decreases brain-derived CCV perimeters. Collectively, these results indicate that α-synuclein may facilitate the growth of clathrin lattices on synaptic membranes.
Results
α-Synuclein affects the size of clathrin lattices on synthetic lipid monolayers
To explore the mechanisms by which α-synuclein regulates SVE, we investigated the effect of α-synuclein on AP180-dependent clathrin lattice formation on synthetic lipid membranes. We used a minimal component system for clathrin lattice assembly described by (32), in which a lipid monolayer containing PI(4,5)P2 is incubated, in a Teflon chamber, with different combinations of purified proteins in the presence of clathrin. In this lipid monolayer assay, AP180 alone is sufficient to recruit brain-purified clathrin (which contains small amounts of AP2) to the membrane and form clathrin lattices (32). We performed the lipid monolayer assay using the Teflon incubation chamber as shown in Figure 1A. We dropped the synthetic lipid mixture on top of the buffer in a well to form a monolayer, and after evaporation of the lipid diluent, chloroform, we placed a carbon-coated EM grid in the well face down onto the lipids (Fig.1A). We purified recombinant mouse α-synuclein (14 KDa), rat AP180 (190 KDa) from E.Coli, and clathrin from pig brain (130 KDa heavy and 30–35 KDa light chains) (Fig.1B). We injected these proteins in different combinations into the incubation chamber beneath the monolayer (Fig.1A). After incubation, the grids were lifted from the chamber and processed for negative staining and EM. Representative micrographs show that the addition of clathrin-only has no effect on membranes due to its inability to directly bind them (Fig.1C), while AP180 can bind the lipid monolayer and modestly deform it (Fig.1C). Injection of recombinant mouse α-synuclein alone or in simple combination with AP180 or clathrin-only resulted in membrane ruffles and deformations (Fig.1D) consistent with its ability to bend membranes (33, 34, 35, 36). Addition of both AP180 and clathrin led to the formation of discrete clathrin lattices (Fig.1E), confirming previous literature showing AP180 is sufficient for clathrin assembly (32). Inclusion of mouse α-synuclein (2 μM; endogenous concentration) in this protein mixture resulted in significantly larger clathrin lattices on the lipid monolayer (Fig.1, E–G) (AP180+clathrin= 8607± 413 nm 2, AP180+clathrin+α-synuclein= 15,569± 2131 nm 2; p< 0.01). This observation suggests that α-synuclein can impact the size of clathrin lattices under conditions where the formation of clathrin lattices is dependent on AP180 (Fig.1E).
Figure 1.
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Reconstitution of clathrin assembly using recombinant proteins.A, schematic of Teflon block and experimental setup used in monolayer reconstitutions. Wells (W) and side-injection ports (SIP) labeled. B, coomassie stain showing proteins used in reconstitution conditions. C–E, electron micrographs of constituent proteins added to lipid monolayers. C, electron micrographs of negative controls. From left to right: Lipids only, lipid monolayer with no added protein; AP180, recombinant AP180 added to lipid monolayer; Clathrin, brain-purified clathrin added to lipid monolayer. D, from left to right: all proteins without a lipid monolayer; α-Synuclein, recombinant WT mouse α-synuclein added to lipid monolayer; α-Syn+AP180, recombinant mouse AP180 and α-synuclein added to lipid monolayer; α-Syn+Clathrin, α-Synuclein and isolated clathrin added to lipid monolayer. E, electron micrographs of clathrin and AP180 added to lipid monolayer. Characteristic fullerene-shaped clathrin cages form. Inset: Zoomed image outlined in green. F, electron micrographs of α-synuclein, clathrin, and AP180 added to lipid monolayer. Inset: Zoomed image outlined in blue. Scale bar represents 100 nm. G, quantification of area of clathrin puncta obtained in conditions (E) and (F) includes expansion of data from 0 to 4× 10 4. Addition of α-synuclein increases the area of clathrin puncta. N= 3 independent experiments with at least ten micrographs per experiment. ∗p< 0.05; ∗∗p< 0.01. Scale bar represents 100 nm.
Next, we evaluated clathrin lattice assembly with mouse brain cytosol which contains the entire repertoire of soluble brain proteins and compared the effects of cytosol lacking α-synuclein. For this, we adapted the lipid monolayer assay for incubation with brain cytosol (Fig.1A). (32). WT and an αβγ-synuclein triple knockout (TKO) brain cytosol were used to evaluate clathrin assembly into lattices. TKO cytosol was chosen as we previously showed that all synucleins are functionally redundant to regulate SVE (14). In these experiments, the lipid monolayer was incubated with brain cytosol (WT and TKO) in the absence (0 min) or presence of ATP (15 min) to monitor basal and active clathrin assembly, respectively. In the absence of ATP, we observed the formation of minute clathrin lattices, representing the baseline lattice size (Fig.2, A and B, 0 min). The baseline assembly of clathrin structures, albeit very small, was significantly smaller when the membranes were incubated with TKO cytosol and was not changed by the addition of mouse α-synuclein (Fig.2, A and B, t= 0, WT= 10,298± 1370 nm 2 and TKO= 6644± 527 nm 2, p< 0.05, TKO+ α-syn= 6229± 942 nm 2, p< 0.05).
Figure 2.
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Lipid monolayer assay to visualize clathrin recruitment from brain cytosol.A, electron micrographs of clathrin assembly into flat lattices and CCPs on membrane monolayers over time (0 and 15 min). Membrane monolayers were incubated with cytosol from WT, αβγ-synuclein KO (TKO), and mouse α-synuclein added to TKO cytosol (rescue). Clathrin puncta are highlighted with red outlines. N= 3 independent experiments with ten images per experiment. Scale bar represents 300 nm. B, comparison of clathrin patch areas in the three conditions mentioned above at 0, 15 min. C, examples of pentagon, hexagon, and heptagon clathrin lattices. Scale bar represents 25 nm. D, ratio of nonhexagons (pentagons and heptagons) to hexagon in clathrin lattices for each of the three conditions, t= 15 min. E, histogram of nonhexagon to hexagon ratio for the three conditions. The sum of all weights was used to normalize the data. For all graphs: ∗p< 0.05 and ∗∗∗p< 0.001. Welsh’s t test. TKO, triple knockout.
Next, we tested clathrin assembly in the presence of an ATP reconstitution system for 15 min. We observed that incubation of the lipid monolayer with WT cytosol led to robust assembly of clathrin lattices (Fig.2, A and B, WT, 15 min), in agreement with previous publications (32, 37, 38). The size of individual clathrin lattices were much larger and are quantified in Figure 2B (compare t= 0 and 15). The absence of α-synuclein resulted in significantly smaller clathrin lattices than WT (t= 15, WT= 37,204± 2648 nm 2 and TKO= 25,624± 2049 nm 2, p< 0.001). Supplementation of TKO cytosol with mouse α-synuclein (2 μM; at endogenous concentrations) partially rescued the size of the clathrin lattices formed on the monolayer after 15 min incubation with an ATP reconstitution system (Fig.2B, t= 15, WT= 37,204± 2648 nm 2 and t= 15, TKO+α-syn= 30,783± 2379 nm 2, NS). This in vitro observation that addition of α-synuclein can rescue the reduced size of clathrin lattices formed on synthetic monolayers with TKO cytosol confirms the α-synuclein dependence of these effects.
Next, we examined the curvature of clathrin lattices formed by analyzing the fractions of pentagons and heptagons to hexagons in the clathrin formations (Fig.2C). The presence of pentagons and/or heptagons indicates that a curved lattice is being generated (39, 40). We found that incubation with TKO cytosol leads to fewer nonhexagonal clathrin formations (Fig.2, D and E) as compared to WT, indicating flatter clathrin lattices. The addition of recombinant mouse α-synuclein (2 μM) to TKO cytosol resulted in a greater number of lattices with pentagons and heptagons being present, shapes which are associated with curved lattices (Fig.2, D and E).
Taken together, these two complementary in vitro experiments show an effect of α-synuclein on the size and curvature of clathrin/AP180 lattices formed on synthetic membranes suggesting that α-synuclein may play a direct role in the regulation of clathrin lattices.
α-Synuclein localizes to large clathrin lattices on cell membranes
To test whether α-synuclein can affect clathrin lattices in a physiological context, we imaged α-synuclein localization relative to clathrin structures on cell membranes. We prepared adherent membranes by unroofing PTK2 (Potorous tridactylus epithelial kidney) cells grown on coverslips transfected with a construct expressing membrane-tethered GFP. Membranes were washed with cytosolic buffer, immunostained for clathrin heavy chain and α-synuclein, and imaged by confocal microscopy. Clathrin and α-synuclein were present in puncta on PTK2 membranes (Fig.3, A–C), similar to observations in other cell types (24, 41, 42). A significant fraction of these puncta contained both proteins. All such puncta displayed a distinctive pattern of a clathrin puncta with α-synuclein colocalized at its inner core (Fig.3B). Puncta containing only α-synuclein overall were smaller and more homogeneous in size (Fig.3C).
Figure 3.
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α-Synuclein and clathrin colocalize in puncta on the plasma membrane.A, membrane sheets from PTK2 cells immunostained for α-synuclein (red) and clathrin heavy chain (green). α-Synuclein and clathrin are present in puncta on the membrane, with some puncta containing both proteins. Scale bar represents 1 μm. B, enlarged view of a clathrin+α-synuclein puncta. Note that α-synuclein localizes to the center of the clathrin patch. C, enlarged view of α-synuclein–only puncta. D, plot of fluorescence of clathrin versus α-synuclein, colored by size of the clathrin puncta. The dashed lines (x= 0.5× 10 4 and y= 1× 10 4) indicate the cut-offs used to demarcate puncta into clathrin-only (x< 0.5, y > 1), α-synuclein–only (x > 0.5, y< 1), and clathrin+α-synuclein (x > 0.5, y > 1) puncta. Note that clathrin and α-synuclein colocalize predominantly in larger size puncta. E, intensity of clathrin and α-synuclein in different puncta sizes. F, percent of the three populations of puncta observed on the membrane. Clathrin-only (green), α-synuclein–only (red), and clathrin+α-synuclein (yellow; abbreviated α-syn/Cla). G, average size of clathrin-only, α-synuclein–only, and clathrin+α-synuclein puncta. N= 3 independent experiments. ∗∗∗∗p< 0.0001. Scale bar represents 1 μm.
To quantify the puncta, we measured fluorescence intensity and size of a linear region of interest (ROI) (cross-section) drawn across clathrin and α-synuclein puncta for all puncta on the membranes (Fig.3D). Then, we plotted the relative clathrin fluorescence (Y-axis) versus the relative α-synuclein fluorescence (X-axis) for all puncta (n= 2071). We set cut-offs (x= 0.5× 10 4 and y= 1 x10 4) by visually identifying two populations of puncta on the plot where the relative fluorescence of clathrin and α-synuclein were independent of each other (dashed lines, Fig.3D). Using these cut-offs, we classified the total population of puncta into three categories: clathrin-only, α-synuclein-only, and clathrin+α-synuclein puncta. We classified 59% as clathrin-only, 20% of puncta as α-synuclein-only, and 21% as clathrin+α-synuclein under steady state conditions (Fig.3F). The average diameters for clathrin-only, α-synuclein-only, and clathrin+α-synuclein puncta were 0.66 μm, 0.53 μm, and 0.96 μm, respectively (Fig.3G; n= 1174, p= 0.001), indicating that puncta where clathrin+α-synuclein are localized together are consistently larger than clathrin-only and α-synuclein-only puncta.
To understand the relationship between α-synuclein and clathrin puncta size in steady state conditions, we binned the observed puncta into five size categories based on the diameter of clathrin puncta (orange: 0.12–0.60 μm, light green: 0.72–1.08 μm, dark green: 1.20–1.56 μm, blue: 1.68–2.04 μm, and purple: 2.16–2.52 μm) (Fig.3D). While the measured diameters are an approximation—as the size measurement is distorted by the use of primary and dye-conjugated secondary antibodies, in addition to the resolution limitations of light microscopy—they are a useful measure to compare populations within the same experimental conditions. We plotted α-synuclein and clathrin intensity as a function of clathrin diameter, color coding the relative size categories (Fig.3E). The average clathrin fluorescence exceeded that of α-synuclein, suggesting that α-synuclein is found within clathrin puncta when they colocalize. Average clathrin fluorescence increased linearly with puncta size, as expected. However, average α-synuclein fluorescence was largely constant and increased only in very large puncta (diameter > 1.6 μm) (Fig.3E, blue and purple). This imaging data suggested that α-synuclein is colocalized with clathrin, preferentially in the larger clathrin puncta on physiological membranes, congruent with our in vitro data (Figs.1F and 2B).
Next, we incubated the PTK2 membranes with brain cytosol, an ATP regeneration system, and GTPγS to facilitate CCP formation and to test how the size and relative ratio of the three puncta classes changed over time (Fig.4). Addition of GTPγS blocks dynamin function and leads to predominantly larger, flatter clathrin lattices. From the total population of puncta observed with incubation of brain cytosol, we observed that the fraction of clathrin-only puncta decreases over time (Fig.4A: 59% at t= 0–0% at t= 30 min) with a concomitant increase in clathrin+α-synuclein puncta (Fig.4A: 20% at t= 0–50% at t= 30 min), with the 5 min time point showing an intermediate level. This indicates that the localization of α-synuclein is dynamic and that it can be recruited to clathrin puncta. Comparing t= 15 and t= 30 time points suggests that α-synuclein can also come off clathrin puncta (Fig.4A). The average size of α-synuclein-only and clathrin-only puncta remained constant, while the average clathrin+α-synuclein puncta diameter changed over time (Fig.4B; 0.96 μm at t= 0 and 0.77 μm at t= 30). These data show that initially after unroofing cells, a fraction of α-synuclein is localized within the larger clathrin puncta on the membranes, which dynamically changes over time with the introduction of native brain cytosol proteins and ATP.
Figure 4.
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Temporal changes of size and colocalization of α-synuclein and clathrin puncta. PTK2 cells were unroofed, incubated with brain cytosol with an ATP regeneration system for the denoted time, and processed to detect α-synuclein and clathrin. A, fraction of the three types of puncta as a function of time. Clathrin-only (green), α-synuclein–only (red), and clathrin+α-synuclein (yellow; abbreviated α-syn/Cla). B, size comparisons of clathrin-only, α-synuclein–only, and clathrin+α-synuclein puncta at the denoted time points. N= 3 independent experiments with at least three cells analyzed per experiment. Total puncta analyzed per time: 0 min= 1982, 5 min= 636, 15 min= 4459, and 30 min=1018.
Colocalization of α-synuclein and clathrin on the membrane depends on α-synuclein lipid/PI(4,5)P2 binding
Recently. it has been shown that α-synuclein colocalizes with PI(4,5)P2 in the membrane of many cell lines—A2780, HeLa, SH-SY5Y and SK-MEL-2 (43). To determine whether α-synuclein binds the membrane of PTK2 cells through PI(4,5)P2 when colocalized with clathrin, PTK2 membrane sheets (t= 0) were immunostained for clathrin heavy chain, α-synuclein, and PI(4,5)P2 and imaged by confocal microscopy (Fig 5A). As performed above (Figs.3 and 4), we manually selected circular independent puncta. We set absolute thresholds for the three channels (threshold of pixel intensity: α-Syn= 5,000, PI(4,5)P2= 5,000, clathrin= 6000) and classified the puncta into those containing individual constituents (clathrin, α-synuclein, or PI(4,5)P2 only) and two or more constituents. We quantified the relative distribution of clathrin and α-synuclein with respect to PI(4,5)P2 (n= 3123 puncta) (Fig.5, A and B). We observed that the majority of the PI(4,5)P2 puncta quantified were positive for α-synuclein (53%), suggesting α-synuclein possibly stabilizes these puncta. Clathrin and α-synuclein also colocalized together with PI(4,5)P2 (21%). There was a small percentage of clathrin puncta that colocalized with PI(4,5)P2 (7%), which are probably bound via AP2. These results in conjunction with earlier results (24, 43) suggest that one mechanism through which α-synuclein affects clathrin lattice formation is through its interaction with PI(4,5)P2.
Figure 5.
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α-Synuclein and PI(4,5)P2 colocalize on the plasma membrane.A, membrane sheets from PTK2 cells immunostained for α-synuclein (red), clathrin heavy chain (green), and PI(4,5)P2 (blue), with different classes of PI(4,5)P2 puncta highlighted (t= 0). B, relative distribution of α-synuclein and clathrin in the PI(4,5)P2 positive puncta. N= 2 independent experiments with four cells analyzed per experiment (3123 total puncta). Scale bar represents 1 μm.
To test the effect of α-synuclein lipid binding on its colocalization with clathrin, we performed an experiment in which brain cytosol was supplemented with recombinant, monomeric human α-synuclein (4 μM) or membrane-binding deficient human A30P α-synuclein (4 μM). We incubated the membrane sheets with the supplemented cytosol for 15 min and stained for clathrin and α-synuclein (Fig.6A). To ensure that successful invagination and vesicle formation occurred at 15 min, we performed immunostaining for dynamin in addition to clathrin and α-synuclein. We then quantified α-synuclein, clathrin, and α-synuclein/clathrin positive puncta that also colocalized with these dynamin-positive sites. We observed a larger fraction of puncta with both α-synuclein and clathrin when WT human α-synuclein was added for 15 min to brain cytosol compared to the control (WT α-syn 73% versus control 57%, Fig.6B). However, when we added 4 μM human A30P α-synuclein to brain cytosol, we observed no change in the colocalization of α-synuclein and clathrin (A30P α-syn 56% versus control 57%, Fig.6B). As A30P α-synuclein does not bind lipid membranes, this result suggests that α-synuclein binding to the membrane is necessary for its colocalization with clathrin.
Figure 6.
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α-Synuclein colocalization with clathrin on membranes depends on α-synuclein lipid binding.A, membrane sheets from PTK2 cells were immunostained for pan-dynamin (blue), clathrin heavy chain (green), and α-synuclein (red), with different classes of puncta highlighted, after incubation with no added α-synuclein (control), 4 μM WT human α-synuclein, and 4 μM human A30P α-synuclein for 15 min. B, relative distribution of α-synuclein and clathrin in dynamin-positive puncta in different conditions. N= 3 cells analyzed per experiment (ctrl= 3804 total puncta, WT= 4784 total puncta, A30P= 4271 total puncta). Scale bar represents 1 μm.
AP180 localizes at the center of the α-synuclein/clathrin large lattices on cell membranes
We extended the adherent membrane studies to include AP180, as it is a brain-specific clathrin adapter (44) and can transiently form a complex with α-synuclein (14). We incubated unroofed membranes with mouse brain cytosol, ATP reconstitution system, and GTPγS for 15 min, the time point we previously observed the maximum colocalization of clathrin+α-synuclein (Fig.4A). We then immunostained for clathrin, α-synuclein, and AP180 and observed, as expected, heterogenous population of puncta (Fig.7A) of varying sizes and protein content. We manually drew an ROI around each puncta and selected circular independent puncta. Then, we set absolute thresholds for the three channels (threshold of pixel intensity: α-Syn= 7000, AP180= 6,000, clathrin= 6000) and classified the puncta into those containing individual proteins (clathrin, α-synuclein, or AP180-only) and two or more proteins (Fig.7, A and B). After quantification (N= 4 experiments, 2637 puncta), we found six different types of puncta: clathrin+AP180+α-synuclein puncta (76.3%), clathrin+AP180 puncta (12.4%), clathrin+α-synuclein puncta (6.1%) clathrin-only puncta (4.4%), AP180-only puncta (0.4%), and α-synuclein-only puncta (0.1%) (Fig.7B). We did not find AP180+α-synuclein puncta (0%) congruent with our previous biochemical data suggesting AP180 and α-synuclein do not directly interact (14).
Figure 7.
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AP180 colocalizes with α-synuclein and clathrin puncta on the plasma membrane.A, membrane sheets from PTK2 cells immunostained for α-synuclein (red), clathrin heavy chain (green), and AP180 (blue) different classes of puncta are highlighted. B, relative distribution of all observed puncta. α-synuclein+AP180+Clathrin puncta are the most prevalent. C, image showing α-synuclein+AP180+Clathrin puncta. AP180 localizes to the center of the clathrin puncta and is surrounded by α-synuclein, denoted by dashed circles. Insets show individual channels. D, scheme showing colocalization of clathrin, α-synuclein, and AP180. E, average diameter of AP180, α-synuclein, and clathrin in triple colocalized puncta. F, size of clathrin puncta in the denoted types of puncta containing clathrin. G, size of AP180 puncta in the three types of puncta containing AP180. H, size of α-synuclein puncta in the three types of puncta containing α-synuclein. N= 3 independent experiments, total number of puncta analyzed= 1681. ∗p< 0.05 and ∗∗p< 0.01; Scale bar represents 1 μm.
We observed that the puncta containing all three proteins have a specific organization with AP180 in the center of the puncta, surrounded by a perimeter of α-synuclein within the clathrin pit (Fig.7C). To quantify the concentric organization of these proteins, we measured the longest diameter of each circular puncta containing all three proteins (Fig.7D). AP180 puncta which were located in the center had the smallest average diameter of 0.48 μm, with α-synuclein surrounding the AP180 puncta at an average diameter of 0.69 μm and with an outer clathrin shell diameter of 0.99 μm (Fig.7E). Using the average diameters, we calculated the average area of each protein using the formula for the area of a circle (A= πr 2). We found that AP180 occupied on average 23% of the clathrin puncta, while α-synuclein occupied on average 49% of clathrin puncta. These data indicate that AP180 localizes to the center of the larger clathrin/α-synuclein puncta on physiological membranes and that there is a hierarchy of diameters with clathrin >α-synuclein >AP180.
To understand whether clathrin puncta size is different when AP180 or α-synuclein are present, we measured the clathrin puncta diameter depending on its colocalization with AP180 and α-synuclein. The diameter of the clathrin puncta in clathrin+α-synuclein+AP180 puncta were significantly larger (1.07 μm± 0.1, p<0.05) than either clathrin+α-synuclein (0.71 μm± 0.03) or clathrin+AP180 puncta (0.71 μm± 0.55) and clathrin by itself (0.54 μm± 0.02) (Fig.7F). Next, we measured AP180 and α-synuclein puncta diameters within the various puncta classes. The diameter of AP180 puncta was larger in clathrin+α-synuclein+AP180 (0.45 μm± 0.06) than clathrin+AP180 (0.28 μm± 0.03) (Fig.7G). The same pattern was observed for α-synuclein puncta diameter in the different classes (clathrin+α-synuclein+AP180: 0.69 μm± 0.07 versus clathrin+α-synuclein: 0.31± 0.02, p< 0.05) (Fig.7H). These puncta size comparisons suggest that each of the protein (AP180 and α-synuclein) puncta is larger when they are colocalized with one another and clathrin. These results strongly agree with our in vitro experiments suggesting α-synuclein and AP180 act synergistically to increase clathrin lattice size.
Our data thus far suggests that α-synuclein colocalizes with clathrin and influences the size of clathrin structures on both synthetic and cellular membranes through a molecular mechanism involving α-synuclein binding to membranes via PI(4,5)P2 and an indirect interaction with AP180. As α-synuclein is a presynaptic protein, it could affect the size of clathrin structures on synaptic membranes and so we investigated this possibility next.
α-Synuclein is mobilized from SVs to synaptic membranes upon stimulation
α-Synuclein is predominantly found on SVs at rest (22, 45, 46), consistent with its preferential binding to highly curved membranes (35, 36). Upon synaptic stimulation, fusion of SVs to the membrane increases the membrane surface area and decreases the number of SVs. To determine whether α-synuclein localization is altered upon stimulation and mobilized from SVs to synaptic membranes, we prepared synaptosomes from WT mouse brains and tested three conditions: rest, depolarization with high K+ stimulation, and repolarization. We then fixed the synaptosomes using a hypotonic fixative as described by previous studies (47). This technique fixes and breaks the synaptosomes simultaneously to increase the penetration of antibodies and capture the state of mobile proteins. The broken synaptosomes were immediately embedded in agarose to enable their manipulation for immunolabeling. We used monoclonal α-synuclein antibodies (clone42, BD Biosciences) that we previously showed by immunogold EM of α-synuclein KO synaptosomes to be highly specific (Fig.8) (22). After incubation with gold-labeled secondary antibodies, the samples were embedded and processed for EM. Gold particles within a 9 nm radius of a vesicle, or membrane, were assigned to these subcellular categories, while the remaining particles were assigned to the cytosol. At rest, the majority of α-synuclein is present on SVs (61.1% on SVs, 25.9% on synaptic membranes, 5.9% in cytosol; Fig.8, A and B). This is similar to the integral SV protein synaptobrevin-2 (78.2% on SV, 12.0% on membranes, 3.1% in cytosol; Fig.8, A and C), in agreement with previous publications (22, 48). However, upon stimulating synaptosomes with high K+, α-synuclein relocates from SVs to the synaptic plasma membrane and cytosol (30.1% on SVs, 51.1% on membranes, 11.2% in the cytosol) This change was significant: α-synuclein on SVs at rest versus K+ stimulation, 61.1% versus 30.1%; p< 0.001; α-synuclein on membranes at rest versus K+ stimulation, 25.9% versus 51.1%; p< 0.001. This confirms previous confocal imaging which shows that α-synuclein becomes diffuse upon neuronal stimulation (48, 49, 50). In contrast, synaptobrevin-2 remains mainly on SVs when stimulated (rest versus K+ stimulation: 78.2% versus 68.4% on SVs; p= 0.095). Repolarization results in α-synuclein relocalization to the surface of SVs (70.1% on SVs, 18.2% on membranes, 3.8% in cytosol). The relocalization of α-synuclein was significant (K+ stimulation versus recovery: 30.1% versus 70.1% on SVs; p< 0.001). Thus, the subsynaptic location of α-synuclein is dynamic and is likely to be linked to the SV cycle. α-Synuclein localization on the synaptic plasma membrane upon neuronal stimulation supports the hypothesis that α-synuclein could localize with clathrin and AP180 in larger clathrin puncta at synaptic plasma membranes. Unfortunately, our attempts to double label clathrin or AP180 were unsuccessful, precluding us from showing that α-synuclein is recruited to CCPs. Overall, this experiment details the dynamic nature of α-synuclein’s localization and indicates that α-synuclein is likely to be on the plasma membrane, possibly through PI(4,5)P2 binding, allowing it to function in SVE.
Figure 8.
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α-Synuclein localization is dynamically regulated by neuronal activity.A, electron micrographs of permeabilized synaptosomes at rest, stimulated with 90 mM KCl, and upon recovery. Immunogold-labeled α-synuclein is localized to SVs at rest (arrowhead). During stimulation, α-synuclein disassociates from SVs and predominantly localizes to the synaptic membrane. During recovery, α-synuclein returns to the SV membrane. B, electron micrographs of permeabilized synaptosomes during rest, stimulation, and recovery conditions immunolabeled for synaptobrevin-2. Gold-labeled synaptobrevin-2 remains associated with the vesicle in all three conditions. C and D, quantification of gold label localization for α-synuclein and synaptobrevin-2. Bars represent percent gold label from n= 3 independent experiments for control conditions and n= 2 for remaining conditions. Twenty five to forty micrographs per condition were analyzed. ANOVA was used to determine significance. ∗∗∗p< 0.001 and ∗∗∗∗p< 0.0001. Scale bar represents 100 nm. SV, synaptic vesicle.
α-Synuclein affects the size of CCVs purified from brain
To assess the role of α-synuclein in regulating the size of CCVs in the brain, we purified CCVs from both WT and αβγ-synuclein TKO mouse brains. Purified CCVs were dropped onto a grid and processed for EM (Fig 9A). CCVs were manually and independently measured by two blind raters. Their results were paired based on the distance between their centroids. In order to avoid quantifying arbitrary structures as true CCVs, ROIs without paired centroids within their radii were excluded from analysis. For each CCV, perimeter was averaged between the two raters and the results were plotted (Fig.9B). A Mann–Whitney U test was performed to compare mean perimeter measurements between WT and TKO CCVs. The results indicated that the perimeter of WT CCVs was significantly greater than the perimeter of CCVs purified from TKO brain, U= 283,855, p<0.0001 (Fig.9C). This suggests a regulatory role for synucleins in the formation of CCVs during their development on the membrane.
Figure 9.
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Deletion of synucleins decreases clathrin-coated vesicle size.A, representative images of CCV purified from WT and αβγ-synuclein KO (TKO) mouse brain, negative stained with uranyl acetate for EM. Scale bar represents 1000 nm. B and C, frequency distribution and violin plots of clathrin-coated vesicle perimeters quantified from electron micrographs in panel A. (WT: Mean= 307.225 nm, 25% Percentile= 279.916, 75% Percentile= 333.48; TKO: Mean= 290.433 nm, 25% Percentile= 263.894, 75% Percentile= 311.599). Mann–Whitney U test was used to determine significance of difference in mean perimeters, U= 283,855, ∗∗∗∗ p< 0.0001. TKO, triple knockout.
Discussion
α-Synuclein function and CCPs formation models
In this study, we show that α-synuclein increases the size of clathrin structures in vitro and colocalizes preferentially with larger clathrin structures on physiological membranes. Our endocytic reconstitution experiments (Figs.1 and 2) demonstrate that recombinant α-synuclein can directly impact clathrin lattice formation in a minimal system, providing the first proof that α-synuclein could affect early steps of SVE. All previously published data inferred a role for α-synuclein in SVE through knockout and/or overexpression studies. Our cell membrane–based assays and CCV experiments indicate that α-synuclein impacts clathrin lattice size (Figure 3, Figure 4, Figure 7 and 9).
The clathrin structures we observe in our cell membrane assays are likely enlarged CCPs, whose size are increased because of membrane tension, unroofing, and the inclusion of GTP-γ-S to block dynamin function. However, the clathrin puncta also resemble flat clathrin lattices (also known as plaques), observed in a variety of cell lines, based on their larger area (>500 nm) (42, 51). Flat clathrin lattices are long-lived (42, 51, 52), though they turnover by budding CCPs at the edges (52) or bending the entire lattice (42). Thus, our findings suggest that α-synuclein could be involved in regulating the size of both CCP and flat clathrin lattices.
Presently, there are two main models to describe the growth of clathrin structures and the formation of CCPs: (1) the constant curvature model and (2) the constant area model. The constant curvature model proposes that curvature of CCPs develops, while endocytic protein adapters recruit and assemble clathrin triskelia to directly bend the plasma membrane, that is, curvature and clathrin assembly occur simultaneously. Consequently, a nucleation spot serves as a hub to develop a dome-like structure resembling a “rising sun”, which eventually is mechanically scissioned from the membrane by dynamin. In contrast, the constant area model suggests a flat clathrin lattice grows until it reaches sufficient size to bend and form a vesicle. Bending of the lattice would be achieved by late arrival of membrane-bending proteins. On membranes, a combination of the two models have been observed (40, 51, 53, 54, 55, 56). In this study, we show that inclusion of α-synuclein in the lipid monolayer assay increased both the size and curvature of the clathrin lattices formed in vitro (Fig.2), in support of the constant curvature model. In fact, we find the number of nonhexagonal clathrin structures/area of lattice to be unaltered (Fig.2B versusFig.2D), indicating they are changing in tandem. Furthermore, we find that α-synuclein and AP180 are present at the center of the clathrin puncta on membranes (Figs.3 and 6). As α-synuclein can sense and generate membrane curvature (33, 35, 36), this supports the hypothesis that α-synuclein is regulating both the size and curvature of clathrin lattices simultaneously. Additional evidence comes from our observation that AP180, α-synuclein, and clathrin are organized in a concentric manner in clathrin puncta. This phenomenon has also been observed by Sochacki et al. (57), where endocytic proteins maintain characteristic nanodomains to ensure proper sorting and membrane bending. Additionally, we find that α-synuclein is recruited from brain cytosol to the center of clathrin puncta (Figure 3, Figure 4, Figure 5, Figure 6).
Functions of α-synuclein with AP180, VAMP2, and PI(4,5)P2
Our data shows that α-synuclein affects clathrin lattice formation, and it colocalizes with AP180, clathrin, and PI(4,5)P2 on the membrane. This is consistent with the documented function of α-synuclein in SVE (14, 15, 16) where AP180 and clathrin are the main participants. Published studies have established that α-synuclein functionally interacts with clathrin machinery including clathrin heavy chain (CHC), AP2, AP180 (58, 59, 60). We have previously shown that α-synuclein is part of a protein complex with AP180 and VAMP2 (14). Our current data shows AP180 and α-synuclein colocalization in clathrin puncta, confirming the documented functional interactions. AP180 and VAMP2 have also been observed together in the center of clathrin lattices by Sochacki et al. (54). In those studies, AP180 and VAMP2 have a constrained distribution at the center of clathrin lattices and this is thought to underlie retrieval of the v-SNARE from the membrane. The AP180/VAMP2 distribution resembles our observations where α-synuclein, AP180, and clathrin are concentrically organized in large clathrin puncta. By extension, α-synuclein is also likely to participate in the retrieval of v-SNAREs from the membrane.
How do α-synuclein and AP180 affect clathrin puncta size? In Figure 7F, the size of clathrin structures is significantly larger only when colocalized with α-synuclein and AP180, suggesting that α-synuclein and AP180 cooperate to alter clathrin lattice size. This is further supported by the observations that α-synuclein and AP180 are never present individually in large clathrin structures. We hypothesize that when α-synuclein interacts with PI(4,5)P2 and AP180, the complex stabilizes clathrin triskelia on the membrane. This is supported by the data in Figure 1, where we showed that α-synuclein increases the size of clathrin lattices formed with AP180 and PI(4,5)P2 under minimal in vitro conditions.
While we see a significant fraction of clathrin puncta are colocalized with α-synuclein, a recent paper by Kaur et al. (41) showed that α-synuclein colocalizes with VAMP2 and syntaxin but not with CHC. This discrepancy is most likely due to the use of recombinant α-synuclein labeled with a dye and experimental conditions that did not include an ATP regeneration system. In our case, we investigated the localization of native α-synuclein on the membrane sheets which could explain the differences in our results. The labeled recombinant α-synuclein may not have been able to bind CHC in patches as they are already occupied by native α-synuclein. That aside, the observations by Kaur regarding VAMP2 colocalization with α-synuclein on membranes is consistent with reported literature, including our own observations on the αβγ-synuclein TKO model (22).
Recent publications have shown α-synuclein regulation of PI(4,5)P2 levels and clathrin-mediated endocytosis in cell lines and neurons (24). Increased α-synuclein levels correlate with increased PI(4,5)P2 levels and increased internalization of transferrin receptors and vice versa. In agreement, Jacob et al. (43) demonstrated α-synuclein is localized in subdomains of the membrane enriched in PI(4,5)P2. These findings align with our observations in Figure 5, wherein the majority of PI(4,5)P2-positive puncta colocalize with α-synuclein. Furthermore, addition of human WT α-synuclein increases colocalization of α-synuclein and clathrin, but the A30P mutant, which cannot bind lipids, fails to have this effect (Fig.6). In combination, these data allow us to hypothesize a mechanism by which PI(4,5)P2 levels might help to recruit α-synuclein and AP180 to the center of clathrin patches. This would be expected to promote an increase in the size and curvature of clathrin lattices that we have observed. However, the precise mechanism for α-synuclein’s role in endocytosis through PI(4,5)P2/AP180/clathrin requires further exploration in intact neurons and synapses.
α-Synuclein, clathrin-mediated endocytosis, and other endocytosis pathways in neurons
In previous work, we showed that α-synuclein regulates SVE (14). We tested different stimulation paradigms on αβγ-synuclein TKO that trigger different endocytic pathways. We observed endocytic deficits with electrical stimulation (100AP, 20 Hz) that triggers clathrin-mediated endocytosis and K+ stimulation that triggers bulk endocytosis. We went on to show by cholera toxin labeling and EM that synucleins regulate the kinetics of clathrin-mediated endocytosis. In this study, we discovered that α-synuclein dynamically moves from the SVs to the synaptic membranes in conditions that support bulk endocytosis. The large cisternae generated in bulk endocytosis can be part of the synaptic membranes we classified in Figure 8. Our results show evidence of regulation of large clathrin lattice formation in vitro and colocalization of α-synuclein, AP180, and clathrin on larger clathrin puncta aligning very well with the recent observation of large clathrin lattice formation on bulk endosomes (61, 62). Another interesting internalization pathway for SVs is ultra-fast endocytosis which has been recently described in the literature (63), yet there is no evidence of α-synuclein participation. We think investigating this avenue will be of great interest for the field.
Finally, we hypothesize that in pathological conditions, large clathrin lattices can be promoted by excess α-synuclein in the cell, comparable to our experiment in Figure 6, where an acute increase in α-synuclein levels increased the colocalization with clathrin. Stability and longevity of these large clathrin structures could lead to the deficit in internalization of SVs, an effect observed by Busch 2014 (15), where an acute overexpression of human α-synuclein blocked SVE. Together, these data open up new avenues for exploration, where α-synuclein, PI(4,5)P2, AP180, and clathrin function together in SVE and neurodegeneration.
Experimental procedures
Mice
WT and TKO mice were used in these studies. All experimental protocols involving animals were approved by the Institutional Animal Care & Use Committee at Yale University.
Lipid monolayer assay with purified proteins
An 8-well Teflon block was arranged in a humid chamber and wells were filled with 40 to 60 μl of HKM buffer (25 mM Hepes pH 7.4, 125 mM potassium acetate, 5 mM magnesium acetate, 1 mM DTT). Lipid mixture (10% cholesterol, 40% PE, 40% PC, and 10% PI(4,5)P2 to final concentration of 0.1 mg/ml in a 19:1 mixture of chloroform to methanol) was carefully pipetted onto the buffer in each well. The blocks were incubated in a humid chamber at room temperature for 1 h to evaporate methanol/chloroform. Carbon-coated copper grids were placed carbon-side down onto each well. Proteins were introduced via side-injection ports beside each well. Final protein concentrations per well were as follows: 2 μM AP180 and α-synuclein, 500 nM purified bovine/porcine clathrin. Grids were incubated for 60 min in a humid chamber at room temperature and then removed from the block and immediately negative stained with 1% uranyl acetate. After drying, grids were imaged on a Phillip CM10 transmission electron microscope at 46k, 80 kV. Clathrin lattices were outlined manually, and the areas were quantified in Image J. Nonhexagonal clathrin lattices could not be quantified due to the high contrast needed to visualize these structures in these experiments.
Purification of proteins
α-Synuclein and AP180 were recombinantly expressed and purified from BL21 (DE3) E.Coli as previously described (32, 33, 64). Clathrin was purified from porcine or bovine brains by purifying CCVs and disassembling cages as described (37). For the monolayer experiments, we used porcine and bovine clathrin interchangeably with no significant difference.
Lipid monolayer assay with mouse brain cytosol
Five mg/ml WT and αβγ-synuclein KO mouse brain cytosol was added to wells of the Teflon block. Lipid mixture (see above) was pipetted on top of the cytosol. Blocks were incubated in a humid chamber for 1 h to evaporate methanol/chloroform. Carbon-coated copper grids were placed on top of the wells, and an ATP regenerating system (see above) and 150 μM GTPγS were introduced through a side-injection port. Grids were incubated in a humid chamber at 37 °C for 0 and 15 min and then removed from the block and immediately negative stained as described above. Grids were imaged on a FEI Tecnai Biotwin at scope 42k, 80 kV. Clathrin lattices were outlined manually; the area and number of pentagons, heptagons and hexagons were quantified in Image J.
Reconstitution of endocytosis in PTK2 cells
PTK2 cells were transfected to express palmitoylated GFP and cultured for 48 h in a Mattek dish to confluence. Membrane sheets were prepared by sonication, using 20% output power, 1 brief pulse on the center of the well. The sheets were washed gently with cytosolic buffer and used within 20 min. The membranes were incubated for 0, 5, 15, 30 min with a mix of 2 mg/ml cytosol, 1.5 mM ATP, ATP reconstitution system (16.7 mM phosphocreatine, 16.7 U/ml creatine phosphokinase), and 150 μM GTPγS. The reaction was stopped by gently washing two times with cytosolic buffer and immediately fixed with 4% paraformaldehyde for 15 min in PBS. Membrane-bound proteins were detected using appropriate primary (clathrin: abcam ab2731, α-synuclein: Everest Biotech EB11713/Cell Signaling D37A6, and AP180: SCBT LP2D11) and secondary antibodies (ThermoFisher AlexaFluor 488,546,633). The membrane sheets were covered with cytosolic buffer. Z stack images were captured using spinning disc confocal microscopy, and analyses were performed in ImageJ using Fiji. Z slices were summed (Sum Slice) to create a single comprised of all channels. For each circular structure observed on membranes of interest, linear ROIs were drawn manually to each diameter of the puncta to analyze for size and fluorescence intensity.
All puncta were identified using analyze particle function Fiji with 1.0% as threshold in dark background. Puncta that overlapped with each other or with unusual shapes exceeding Fiji's threshold function of circularity 1.0 were excluded from puncta analysis (65). To determine puncta identity, we quantified the mean fluorescence for each protein for all ROIs. To identify whether a ROI was positive for a protein, we compared the mean fluorescence value of that protein against a cutoff value. The thresholds we choose for each channel were determined visually by the investigator and cross-checked with control puncta positive for individual proteins.
Some modifications were made for the membrane recruitment experiments with protein supplementation (α-synuclein overexpression) and the triple immunostaining for PI(4,5)P2 experiments. PTK2 cells were cultured for 24 h before sonication, as GFP transfection was not performed. For α-synuclein overexpression experiments, 4 μM recombinant human WT or A30P a-synuclein was added with the reconstitution mixture. For triple immunostaining for PI(4,5)P2, primary antibodies consisted of clathrin Abcam ab2731, α-synuclein Cell Signaling D37A6, PI(4,5)P2 Echelon Biosciences Z-P045; secondary antibodies consisted of ThermoFisher AlexaFluor 568, 633 and Abcam AlexaFluor 488. For α-synuclein overexpression experiments, primary antibodies consisted of clathrin Cell Signaling D3C6, α-synuclein Everest Biotech EB11713, and dynamin BD Transduction Clone 41. In the absence of GFP transfection, cells were incubated with Biotium CellBrite Steady 405 membrane dye to visualize membranes following immunostaining.
Cytosol purification
Mouse brains were removed and washed in washing buffer (23 mM Tris–HCl, pH 7.4, 320 mM sucrose) for 30 s. Two brains were homogenized at 2500 rpm in a 5 ml potter homogenizer with 2 ml of homogenization buffer (25 mM Tris–HCl, pH 8.0, 500 mM KCl, 250 mM sucrose, 2 mM EGTA, and 1 mM DTT), using ten strokes at 5000 rpm. The homogenate was transferred to a 3.5 ml ultracentrifuge tube and centrifuged at 160,000 g for 2 h at 4 °C. A PD-10 column was equilibrated with 25 ml cytosolic buffer, the supernatant was added to the PD-10 column, and then eluted with 3.5 ml of cytosolic buffer (25 mM Hepes–NaOH, pH 7.4, 120 mM potassium glutamate, 2.5 mM magnesium acetate, 20 mM KCl, and 5 mM EGTA–NaOH, pH 8.0, filtered and stored at 4 °C, with 1 mM DTT added immediately before use). Protein concentration was quantified by bicinchoninic acid assay. Mini cOmplete protease inhibitor cocktail was added, and 100 μl aliquots were flash frozen in liquid nitrogen and stored at−80 °C.
Immuno-EM of mouse brain synaptosomes
Synaptosome isolation
Brains from WT mice (n= 3) were harvested and washed in homogenizing buffer (0.32 M sucrose, 10 mM Tris–HCl pH 7.4, mini cOmplete protease inhibitor tablet). Cerebella were removed, hemispheres separated, and white matter removed before adding the remaining brain to 10 ml of homogenizing buffer. Brains were homogenized in a 55 ml Potter homogenizer at 80% power for eight strokes. Homogenate was centrifuged at 2330 g for 4 min at 4 °C in a JA-20 rotor. Pellet (P1) was discarded, and supernatant (S1) was transferred to a 50 ml centrifuge tube. S1 was recentrifuged at 18,850 g for 12 min at 4 °C. Supernatant (S2) was discarded and pellet (P2) was resuspended in 6 ml of homogenizing buffer. Gradients of Percoll solutions (from bottom to top: 2.5 ml 23%, 3 ml 10%, and 2.5 ml 3%) were prepared in 15 ml glass tubes. Two microliters of resuspended P2 was added to the top of each gradient, and tubes were centrifuged at 18,850 g for 12 min at 4 °C. Synaptosome fractions were collected by Pasteur pipette.
Stimulation
Washing buffer Krebs 1 (140 mM NaCl, 10 mM Tris–HCl pH 7.4, 5 mM KCl, 5 mM NaHCO 3, 1.3 mM MgSO 4, 1 mM Na-phosphate buffer) was added to the collected synaptosomes, and sample was centrifuged at 18,850 g for 12 min. Pellet was resuspended in 1.5 ml Krebs 1 and divided equally into three 1.5 ml microcentrifuge tubes labeled for rest, stimulation, and recovery. The tubes were centrifuged at 16,000 g for 3 min. Each pellet was resuspended in 500 μl of washing buffer Krebs 2 (10 mM glucose, 1.2 mM CaCl2 in Krebs 1) and incubated in a 37 °C water bath for 15 min. The rest condition was left in the bath, while the stimulation and recovery conditions were each removed, supplemented to 90 mM KCl final concentration, rotated by hand, and returned to the bath for 2 min. Stimulation and recovery tubes were then spun at 16,000 g for 2 min. The stimulation pellet was resuspended in 200 μl Krebs 1 and left on ice. The recovery pellet was resuspended in 500 μl of washing buffer Krebs 3 (10 mM Glucose, 1 mM EGTA, 1 phosSTOP tablet in Krebs 1) and returned to the bath for 10 min. The recovery and rest tubes were removed from the bath and spun at 16,000 g for 2 min. Pellets were resuspended in 200 μl of Krebs 1. All conditions were transferred into separate 50 ml centrifuge tubes filled with 30 ml of hypotonic fixative solution (3% paraformaldehyde, 0.25% glutaraldehyde in 5 mM Na-phosphate buffer) and incubated on ice for 30 min.
Agarose embedding
The samples were centrifuged in their 50 ml tubes at 18,850 g at 4 °C for 12 min. Each pellet was resuspended in 400 μl of 120 mM Na-phosphate buffer. One hundred eighty microliter aliquots of each sample were pipetted into glass test tubes and held on ice until mixed with agarose. Pasteur pipettes and glass slide frames were warmed in a 60 °C oven. 3% agarose solution was prepared in 5 mM Na-phosphate buffer, warmed to 95 °C until agarose dissolved, and kept in a 54 °C water bath. One hundred eighty microliters of agarose solution was pipetted over each broken synaptosome sample, while the tube was held in the 54 °C water bath and mixed by gentle pipetting with a Pasteur pipette. The agarose-embedded samples were pipetted into the warmed glass slide frames and solidified on ice. Once solid, the frames were disassembled, leaving a solid agarose gel sheet on one glass slide. The gel was cut into 3 mm 3 cubes with a razor blade and washed from the slide with 120 mM Na-phosphate buffer into petri dishes.
Immunolabeling and EM preparation
Each condition’s cubes were divided into five wells of glazed porcelain well plates. Cubes were incubated for 30 min with 5% BSA in solution A (0.5 M NaCl, 0.02 M Na-phosphate buffer) at room temperature. BSA solution was removed, and each well was incubated overnight with 200 μl of primary antibodies in solution A at 4 °C. Cubes were washed with 5 changes of solution A over the course of 1.5 h. Solution A was removed from the wells and cubes were incubated with secondary antibodies in solution A and 5% BSA for 6 h at room temperature. Cubes were washed with 5 changes of solution A and incubated at 4 °C overnight.
Agarose samples were prepared for resin embedding, sectioned, and stained with 2% uranyl acetate and 1% lead citrate before imaging on a FEI Tecnai Biotwin scope at 48k, 80 kV.
CCV purifications
We purified CCVs from adult WT and TKO brains following previously published protocols (14). The CCVs were spread on an EM grid and imaged as described in the above section.
Measurement and analysis of CCV dimensions
CCVs were independently and blindly quantified by two raters in ImageJ. Each CCV was approximated as a circle, and area and perimeter were measured. After ROIs were created, ROIs were paired between raters based on the distance between their centroids. In order to avoid quantifying arbitrary structures as true CCVs, ROIs without paired centroids within their radii were excluded from analysis. For each CCV, perimeter was averaged between the two raters and the results were plotted.
Statistics
All experimental analysis was done blinded to condition. Statistical analysis was performed by GraphPad Prism and R studio. Data is presented as average± SEM.
Data availability
All data are presented in the manuscript.
Conflict of interest
The authors declare that they have no conflicts of interest with the contents of this article.
Acknowledgments
We thank Aurelie Nardin, Jeffery Yong, Dr Pietro De Camilli, and his lab members for setting up and optimizing the PTK2 assay. We thank John. E. Lee and Sofia Massaro Tieze for editing the manuscript.
Author contributions
K. J. V., P. L. C., and S. S. C. methodology; K. J. V., P. L. C., E. G., J.-M. P., L. E. H., and S. S. C. investigation; K. J. V., P. L. C., E. G., J.-M. P., L. E. H., and S. S. C. formal analysis; K. J. V., P. L. C., E. G., J.-M. P., L. E. H., and S. S. C. writing–original draft; K. J. V., P. L. C., E. G., J.-M. P., L. E. H., and S. S. C. writing–review and editing.
Funding and additional information
This work was supported by NIH (R01 NS064963, R01 NS110354, R01 NS083846), Nina Compagnon Hirshfeld Parkinson’s Disease Research Fund, and DOD (W81XWH-17-1-0564) to S. S. C. This work was also supported by 1R03NS116646 to K. J. V. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Reviewed by members of the JBC Editorial Board. Edited by Phyllis Hanson
Footnotes
Present addresses for: Karina J. Vargas, Cell Biology Department, University of Pittsburgh, PA; P. L. Colosi, University of Pennsylvania, PA; Eric Girardi, University of Stony Brook, NY; Jae-Min Park, Van Andel Institute, MI.
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6505 | https://themathdistrict.com/algebra-1/1.4-Sets-Intersection-and-Union.pdf | TheMathDistrict.com Sets; Intersection and Union of Sets Definition of a Set A set is a collection of objects. The objects that belong to a set are called elements, or members of the given set. We use uppercase letters for the names of sets. One common method of describing a set, is the roster method. With this method, we include the elements of a set inside the braces { }. Examples of Sets 𝐴= {1, 5, 7, 8} 𝐵= {𝑎, 𝑐, 𝑓, ℎ, 𝑘} 𝑁= {1, 2, 3, 4, … } A set that has a finite number of elements is called a Finite Set (we can count the elements) A set that has an infinite number of elements is called an Infinite Set (we cannot count the elements) Examples 𝐵= {4, 6, 11} 𝐷= {1, 2, 3, 4, 5, 6} Examples 𝑁= {1, 2, 3, 4, … } 𝑂= {1, 3, 5, 7, … } The Symbol ∈ The symbol ∈ is used to indicate whether a certain object is an element of a set. We read ∈ - “is an element of” We read ∉ - “is not an element of” Example Let A = {2, 4, 5, 7, 8} be a set. 2 ∈A (we read it: “Two is an element of set A”). And it is true that 2 belongs to set A. 3 ∉A (we read it: “Three is not an element of set A”) And it is true that 3 does not belong to set A. TheMathDistrict.com A set that contains no elements is an empty set and is represented by ∅ or { }. Three most Common Ways of Describing a Set (using the elements 1, 2, 3, 4, 5) Roster Method (List all the elements inside the braces) Set-Builder Notation (Use a variable 𝑥) Word Description (use words to describe a set) 𝐴= {1, 2, 3, 4, 5} 𝐴= {𝑥|𝑥∈𝑁 𝑎𝑛𝑑 𝑥< 6} 𝐴 is the set of the natural numbers less than 6. How to read it: A is the set of all elements 𝑥, such that 𝑥 is an element of the natural numbers, and 𝑥 is less than 6. The Intersection ∩ of Two Sets The intersection of sets 𝐴 and 𝐵, written 𝐴∩𝐵, is the set of elements common to both set 𝐴 and set 𝐵. The Union ∪ of Two Sets The union of sets 𝐴 and 𝐵, written 𝐴∪𝐵, is the set of elements that are members of set 𝐴 or of set 𝐵 or of both sets. Example Two sets are given: 𝐴= {2, 3, 6, 8) 𝐵= {1, 2, 3, 4, 5, } The intersection of these two sets is: 𝐴∩𝐵= {2, 3} because the numbers 2 and 3 are common to both sets A and B. Example Two sets are given: 𝐴= {2, 3, 6, 8) 𝐵= {1, 2, 3, 4, 5, } The union of these two sets is: 𝐴∪𝐵= {1, 2, 3, 4, 5, 6, 8} because these numbers belong to either A or B or both. In other words, we list all the numbers together, starting with the smallest to the largest. And if some of them repeat, list them only once. TheMathDistrict.com Exercises Given the set 𝐴= {1, 4, 7, 9}, determine whether the number 4 is an element of this set. Solution Yes, 4 is an element of set A. 4 ∈𝐴 Given the set 𝐴= {1, 4, 7, 9}, determine whether the number 5 is an element of this set. Solution No, 5 is not an element of set A. 5 ∉𝐴 Two sets are given: 𝐴= {2, 4, 5, 9, 10} and 𝐵= {4, 5, 6}. Determine the intersection ∩ of these two sets. Solution The numbers that belong to both A and B, (or the common numbers) are 4 and 5. So, 𝐴∩𝐵= {4, 5} Two sets are given: 𝐴= {2, 4, 5, 9, 10} and 𝐵= {4, 5, 6}. Determine the union ∪ of these two sets. Solution The numbers that belong to either A or B, or both sets, are all the given numbers written from the smallest to the largest. (If some of them repeat, write them only once). So, 𝐴∪𝐵= {2, 4, 5, 6, 9, 10} Two sets are given: 𝐴= {1, 3, 5} and 𝐵= {2, 4, 6}. Determine the intersection ∩ of these two sets. Solution There are no common numbers to both A and B, so the intersection is the empty set. 𝐴∩𝐵= ∅ Two sets are given: 𝐴= {1, 3, 5} and 𝐵= {2, 4, 6}. Determine the union ∪ of these two sets. Solution The numbers that belong to either A or B, or both sets, are all the given numbers written from the smallest to the largest. So, 𝐴∪𝐵= {1, 2, 3, 4, 5, 6} Two sets are given: 𝐴= {7, 8, 11} and 𝐵= { }. Determine the intersection ∩ of these two sets. Solution Notice that set B is an empty set. There are no common numbers to both A and B, so the intersection is the empty set. 𝐴∩𝐵= { } or ∅ Two sets are given: 𝐴= {7, 8, 11} and 𝐵= { }. Determine the union ∪ of these two sets. Solution The numbers that belong to either A or B, or both sets, are all the given numbers written from the smallest to the largest. So, 𝐴∪𝐵= {7, 8, 11} |
6506 | https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Supplemental_Modules_(Organic_Chemistry)/Aldehydes_and_Ketones/Reactivity_of_Aldehydes_and_Ketones/Carbonyl_Group-Mechanisms_of_Addition | Skip to main content
Carbonyl Group-Mechanisms of Addition
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: Jan 23, 2023
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Carbonyl Group Reactions
Clemmensen Reduction
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: 751
( \newcommand{\kernel}{\mathrm{null}\,})
The Carbonyl Group is a polar functional group that is made up a carbon and oxygen double bonded together. There are two simple classes of the carbonyl group: Aldehydes and Ketones. Aldehydes have the carbon atom of the carbonyl group is bound to a hydrogen and ketones have the carbon atom of the carbonyl group is bound to two other carbons. Since the carbonyl group is extremely polar across the carbon-oxygen double bond, this makes it susceptible to addition reactions like the ones that occur in the pi bond of Alkenes, especially by nucleophilic and electrophilic attack.
Regions of Reactivity
There are three regions of reactivity for both aldehydes and ketones: the electron donating oxygen, the electron withdrawing carbon, and the carbon adjacent to the carbonyl group (labeled "alpha"). This module will only address the oxygen and electron withdrawing carbon areas of reactivity.
Figure 1: Regions of Reactivity
Resonance structure is defined as any of two or more possible structures of the same compound that have identical geometry but different arrangements of their paired electrons; none of the structures have physical reality or adequately account for the properties of the compound. Examination of the resonance structures of the carbonyl group clearly shows its polar nature, and highlights the areas for either electrophilic or nucleophilic attack in the addition reaction.
Figure 2: Resonance structures of the carbonyl group
Ionic Addition to Carbonyl Group
As a result of the dipole shown in the resonance structures, polar reagents such as LiAlH4 and NaBH4 (hydride reagents) or R'MgX (Grignard reagent) will reduce the carbonyl groups, and ultimately convert unsaturated aldehydes and ketones into unsaturated alcohols. Since these reagents are extremely basic, their addition reactions are irreversible.
There are, however, addition reactions with less basic nucleophiles such as water, thiols, and amines that are capable of establishing equilibria or reversible reactions. These less basic reagents can react with the carbonyl group via two pathways: nucleophilic addition-protonation and electrophilic addition-protonation.
Nucleophilic Addition-Protonation
Under neutral or basic conditions, nucleophilic attack of the electrophilic carbon takes place. As the nucleophile approaches the electrophilic carbon, two valence electrons from the nucleophile form a covalent bond to the carbon. As this occurs, the electron pair from the pie bond transfers completely over to the oxygen which produces the intermediate alkoxide ion. This alkoxide ion, with a negative charge on oxygen is susceptible to protonation from a protic solvent like water or alcohol, giving the final addition reaction.
Figure 3: Alkoxide Ion Intermediate
Electrophilic Addition-Protonation
Under acidic conditions, electrophilic attack of the carbonyl oxygen takes place. Initially, protonation of the carbonyl group at the oxygen takes place because of excess all around. Once protonation has occurred, nucleophilic attack by the nucleophile finishes the addition reaction. It should be noted that electrophilic attack is extremely unlikely, however, a few carbonyl groups do become protonated initially to initiate addition through electrophilic attack. This type of reaction works best when the reagent being used is a very mildly basic nucleophile.
Figure 4: Electrophilic Addition
Outside links
References
Vollhardt, K. P.C. & Shore, N. (2007). Organic Chemistry (5thEd.). New York: W. H. Freeman. p. 775-777
Otera, Junzo, ed. Modern Carbonyl Chemistry. Weinheim; Chichester: Wiley-VCH, 2000.
Problems
What is the best reagent to do the following conversion?
H2/Ni
Hg(OAc)2, H3O+
BH3, THF & 2.H2O2, NaOH
NaBH4, THF
What is the Product of this reaction?
What is the Product of this reaction?
What is the Product of this reaction?
What is the product of this reaction?
Solutions
D
Carbonyl Group Reactions
Clemmensen Reduction |
6507 | https://pubmed.ncbi.nlm.nih.gov/6964676/ | New attachment following surgical treatment of human periodontal disease - PubMed
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J Clin Periodontol
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. 1982 Jul;9(4):290-6.
doi: 10.1111/j.1600-051x.1982.tb02095.x.
New attachment following surgical treatment of human periodontal disease
S Nyman,J Lindhe,T Karring,H Rylander
PMID: 6964676
DOI: 10.1111/j.1600-051x.1982.tb02095.x
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New attachment following surgical treatment of human periodontal disease
S Nyman et al. J Clin Periodontol.1982 Jul.
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. 1982 Jul;9(4):290-6.
doi: 10.1111/j.1600-051x.1982.tb02095.x.
Authors
S Nyman,J Lindhe,T Karring,H Rylander
PMID: 6964676
DOI: 10.1111/j.1600-051x.1982.tb02095.x
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Abstract
The present experiment was undertaken to test the hypothesis that new connective tissue attachment may form on a previously periodontitis involved root surface provided cells originating from the periodontal ligament are enabled to repopulate the root surface during healing. A mandibular incisor with advanced periodontal disease of long standing (the distance between the cemento-enamel junction and the alveolar bone crest was 9 mm) was subjected to periodontal surgery using a technique which during healing prevented the dentogingival epithelium and the gingival connective tissue from reaching contact with the curetted root surface. Preference was hereby given to the periodontal ligament cells to repopulate the previously diseased root surface. After 3 months of healing a block biopsy containing the incisor and surrounding tissue was sampled. The histological analysis revealed that new cementum with inserting principal fibers had formed on the previously diseased root surface. This new attachment extended in coronal direction to a level 5 mm coronal to the alveolar bone crest. This finding suggests that new attachment can be achieved by cells originating from the periodontal ligament and demonstrates that the concept that the periodontitis affected root surface is a major preventive factor for new attachment is invalid.
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6508 | https://www.fda.gov/regulatory-information/search-fda-guidance-documents/recommendations-reduce-risk-transmission-mycobacterium-tuberculosis-mtb-human-cells-tissues-and | Published Time: Mon, 05/05/2025 - 07:33
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Docket Number:FDA-2024-D-3863Issued by:
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We, FDA, are issuing this guidance to assist you, establishments making donor eligibility (DE) determinations, in understanding the requirements in 21 CFR part 1271, subpart C (21 CFR part 1271, subpart C). The regulations under 21 CFR part 1271, subpart C, set out requirements for determining donor eligibility, including donor screening and testing, for donors of human cells, tissues, and cellular and tissue-based products (HCT/Ps). This guidance provides recommendations for screening donors for evidence of, and risk factors for, infection with Mycobacterium tuberculosis (Mtb), the organism that causes tuberculosis. The guidance also recommends additional steps that HCT/P establishments should take to reduce risk of transmission of Mtb until such time as appropriate FDA-licensed, approved, or cleared donor screening tests are available for use to test donors for Mtb infection.
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6509 | https://helpingwithmath.com/powers-bases-and-exponents/ | Skip to content
Home » Math Theory » Algebra » Powers (Bases and Exponents)
Powers (Bases and Exponents)
What is an exponent?
An exponent indicates how many times the number must be multiplied by itself. Exponents are also referred to as powers or indices. Usually, the exponent is written as a superscript to the right of the base.
Consider the number 53.
In this example, 5 is the base and 3 is the exponent. A base is a number that we will repeatedly multiply to itself, and the exponent will tell us how many times the base will be multiplied to itself. Thus, 53 means the same thing as 5 x 5 x 5.
For us to read 53, we can read as:
five raised to the power of three,
five raised to three, and
five to the power of three.
Exponentiation is a mathematical operation written in the form of xn, where x denotes the base and n represents the exponent.
What is the origin of exponents?
The word power came from the Latin word potentia, potestas, dignitas which is a mistranslation of the ancient Greek term δύναμις (dúnamis), which was employed for the square of a line by Euclid, following Hippocrates of Chios.
Meanwhile, Archimedes discovered and proved the laws of exponents in his work The Sand Reckoner. In his work, he was able to prove 10a x 10b = 10a + b. During the 19th century, Muhammad ibn Mūsā al-Khwārizmī introduced the terms مَال (māl, “possessions”, “property”) for a square.
Jost Bürgi utilized exponents for Roman numerals in the late 16th century. In the 15th century, Nicolas Chuquet created a type of exponential notation that was later utilized by Henricus Grammateus and Michael Stifel in the 16th century. Michael Stifel created the term “exponent” in 1544. In 1696, Samuel Jeake introduced the term indices.
René Descartes established the first form of our contemporary exponential notation in his treatise titled La Géométrie in the early 17th century; the notation is introduced in Book I of the text.
Leonhard Euler proposed variable exponents and, implicitly, non-integer exponents in 1748 when he wrote:
“consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant.”
What are the laws of exponent?
To answer and solve problems involving exponents, the laws of exponents are fundamental. These rules are needed to be followed to be able to simplify formulas with exponents. Let’s take a look at the following rules we have for exponents!
Product Rule
The product rule of exponents is used to multiply expressions that have the same bases. This rule states that for any non-zero term a where m and n are real numbers,
amxan=am+n
This means that, to get the product of an exponent that have the same base, we are going to simply copy the base and add their exponents.
Example #1
What is the product of 52 and 53?
Solution
| | |
--- |
| Process | Explanation |
| 52 x 53 | Set-up the multiplication process. |
| 52 x 53 = 52 + 3 | Using product rule, we will simply copy the base and add the exponents. |
| 52 + 3 = 55 | Add the exponents. Hence, 2 + 3 = 5 |
| 55 = 3125 | Evaluate 55. Hence, 5 x 5 x 5 x 5 x 5 = 3125 |
| | Therefore, the product of 52 and 53 is 55 or 3125. |
Example #2
Find the product of a and a6.
Solution
| | |
--- |
| Process | Explanation |
| a1 x a6 | Set-up the multiplication process. |
| a1 x a6 = a1 + 6 | Using product rule, we will simply copy the base and add the exponents. The exponent of a is 1. Hence, we will have a1 + 6 |
| a1 + 6 = a7 | Add the exponents. Hence, 1 + 6 = 7 |
| | Therefore, the product of a and a6 is a7. |
Example #3
Find the product of x3y and x3y2.
Solution
| | |
--- |
| Process | Explanation |
| (x3y)(x3y2) | Set-up the multiplication process. |
| (x3y)(x3y2) = x3 + 3y1 + 2 | Using product rule, we will simply copy the base and add the exponents. Since we have two variables, we will still copy both bases, but take note of the exponents of each variable. Hence, we will have x3 + 3y1 + 2 |
| x3 + 3y1 + 2 = x6y3 | Add the exponents. Hence, the exponent of x will be x3 + 3 = x6 , while the exponent of will be y1 + 2 = y3 |
| | Therefore, the product of x3y and x3y2 is x6y3. |
Quotient Rule
The quotient rule of exponents is used to simplify algebraic terms or expressions that have the same bases. This rule states that for any non-zero term a where m and n are real numbers,
$\frac{a^m}{a^n}= a^{m – n}$
This means that, to get the quotient of an exponent that has the same base, we are going to simply copy the base and subtract the exponent of the numerator by the exponent of the denominator.
Example #1
What is the quotient if 97 is divided by 93?
Solution
| | |
--- |
| Process | Explanation |
| $\frac{9^7}{9^3}$ | Set-up the division process. |
| $\frac{9^7}{9^3} = 9^{7 – 3}$ | Using quotient rule, simply copy the base and subtract the exponents. |
| 97 – 3 = 94 | Subtract the exponents. Hence,7 – 3 = 4 |
| 94 = 6561 | Evaluate 55. Hence,9 x 9 x 9 x 9 = 6561 |
| | Therefore, the quotient of 97 and 93 is 94 or 6561. |
Example #2
Determine the quotient if a10 is divided by a7.
Solution
| | |
--- |
| Process | Explanation |
| $\frac{a^10}{a^7}$ | Set-up the division process. |
| $\frac{a^10}{a^7} = a^{10 – 7}$ | Using quotient rule, simply copy the base and subtract the exponents. |
| a10 – 7 = a3 | Subtract the exponents. Hence,10 – 7 = 3 |
| | Therefore, the quotient of a10 and a7 is a3. |
Example #3
Find the quotient if x4y3 is divided by xy2.
Solution
| | |
--- |
| Process | Explanation |
| $\frac{x^{4}y^{3}}{xy^2}$ | Set-up the division process. |
| $\frac{x^{4}y^{3}}{xy^2} = x^{4 – 1}y^{3-2}$ | Using quotient rule, copy the base and subtract the exponents. Since we have two variables, we will copy both bases, but take note of the exponents of each variable. Hence, we will have x4 – 1y3 – 2 |
| $x^{4 – 1}y^{3-2}= x^3y$ | Subtract the exponents. Hence, the exponent of x will be x4 – 1 = x3, while the exponent of will be y3 – 2 = y1. Since the exponent of y is 1, we can simply write it as y instead of y1. |
| | Therefore, the quotient of x4y3 and xy2 is x3y. |
Zero Exponent Rule
The zero exponent rule is used to simplify algebraic terms or expressions that have an exponent of zero. This rule simply states that if the exponent of any non-zero base a is zero, then it is always equal to 1.
a0= 1
Example #1
What is the equivalent of 30?
Solution
| | |
--- |
| Process | Explanation |
| 30 = 1 | By zero exponent rule, 30 is simply equal to 1. |
| | Therefore, 30 is equal to 1. |
Example #2
Determine the value of a0.
Solution
| | |
--- |
| Process | Explanation |
| a0 = 1 | By zero exponent rule, a0 is simply equal to 1. |
| | Therefore, a0 is equal to 1. |
Example #3
Find the equivalent of x0y0.
Solution
| | |
--- |
| Process | Explanation |
| x0y0 = (1)(1) | By zero exponent rule, x0 = 1, y0 = 1. |
| (1)(1) = 1 | Get the product of 1 and 1. Thus, 1 x 1 = 1 |
| | Therefore, x0y0 is equal to 1. |
Negative Exponent Rule
The negative exponent rule is used to simplify algebraic terms or expressions that have a negative exponent. This rule states that if the exponent of any non-zero base a is negative, we need to convert it to a positive exponent by getting its reciprocal.
$a^{-m}= \frac{1}{a^m}$
Similarly,
$\frac{1}{a^{-m}} = a^{m}$
Example #1
What is the equivalent of 2-4?
Solution
| | |
--- |
| Process | Explanation |
| 2-4 = $\frac{1}{2^4}$ | By negative exponent rule, we will get the reciprocal of 2-4. Hence, we will have $\frac{1}{2^4}$. |
| $\frac{1}{2^4} = \frac{1}{16}$ | Evaluate the exponent. Hence, $\frac{1}{2 × 2 × 2 × 2}$ = $\frac{1}{16}$ |
| | Therefore, 2-4 is equal to $\frac{1}{2^4}$ or $\frac{1}{16}$. |
Example #2
Simplify a-2.
Solution
| | |
--- |
| Process | Explanation |
| a-2 = $\frac{1}{a^2}$ | By negative exponent rule, we will get the reciprocal of 2-4. Hence, we will have $\frac{1}{2^4}$. |
| | Therefore, 2-4 is equal to a-2 or $\frac{1}{a^2}$. |
Example #3
What is the equivalent of $\frac{x^{-3}}{y^{-4}}$?
Solution
| | |
--- |
| Process | Explanation |
| $\frac{x^{-3}}{y^{-4}} = (\frac{1}{y^{-4}})(\frac{1}{x^3})$ | By negative exponent rule, we will get the reciprocal of x-3. Hence, we will have $\frac{1}{x^3}$. Thus, $\frac{x^{-3}}{y^{-4}} = (\frac{1}{y^{-4}})(\frac{1}{x^3})$ |
| $(\frac{1}{y^{-4}})(\frac{1}{x^3}) = (\frac{1}{x^3})(y^4) = \frac{y^4}{x^3}$ | Since we still have $\frac{1}{y^{-4}}$, we also need to get its reciprocal. Hence, $\frac{1}{y^{-4}}= y4$. Thus, |
| | Therefore, $\frac{x^{-3}}{y^{-4}}$ is equal to $\frac{y^4}{x^3}$. |
Power of a Power Rule
The power of a power rule is used to simplify algebraic terms where the exponent of the base is raised to another exponent, we will get the product of the two exponents. This rule states that if a non-zero term a and m and n are integers,
(am)n= amn
Example #1
Simplify (43)2.
Solution
| | |
--- |
| Process | Explanation |
| (43)2 = 43 x 2 | To simplify (43)2, we will use the power of the power rule. Hence, (43)2 = 43 x 2 |
| 43 x 2 = 46 | Multiply the exponents. Thus, 3 x 2 = 6 |
| 46 = 4096 | Evaluate 46. Thus, 4 x 4 x 4 x 4 x 4 x 4 = 4096 |
| | Therefore, (43)2 is equal to 46 or 4096. |
Example #2
Simplify (a3)7.
Solution
| | |
--- |
| Process | Explanation |
| (a3)7= a3 x 7 | To simplify (a3)7, we will use the power of the power rule. Hence, (a3)7 = a3 x 7 |
| a3 x 7 = a21 | Multiply the exponents. Thus, 3 x 7 = 21 |
| | Therefore, (a3)7 is equal to a21. |
Power of a Product Rule
The power of a product rule is used to simplify algebraic terms with more than one variable that is being multiplied together and raised to an exponent. This rule states that if a non-zero term a and b and m is an integer,
(ab)m= ambm
Example #1
Simplify (ab)8.
Solution
| | |
--- |
| Process | Explanation |
| (ab)8 = a8y8 | To simplify (ab)8, we will use the power of a product rule. Hence, (ab)8 = a8y8 |
| | Therefore, (x4y5)3 is equal to x12y15. |
Example #2
Simplify (x4y5)3.
Solution
| | |
--- |
| Process | Explanation |
| (x4y5)3 = x4 x 3y5 x 3 | To simplify (a3)7, we will use the power of a product rule. Hence, (x4y5)3 = x4 x 3y5 x 3 |
| x4 x 3y5 x 3 = x12y15 | Multiply the exponents. Thus, for the exponent of x: 4 x 3 = 12 for the exponent of y: 5 x 3 = 15 |
| | Therefore, (x4y5)3 is equal to x12y15. |
Power of a Quotient Rule
The power of a quotient rule is used to simplify the quotient of an algebraic term with more than one variable by simply distributing the exponent to the numerator and the denominator. This rule states that if a non-zero term a and b where m is an integer,
$(\frac{a}{b})^{m}= \frac{a^m}{b^m}$
Example #1
Simplify $(\frac{4}{5})^2$ .
Solution
| | |
--- |
| Process | Explanation |
| $(\frac{4}{5})^2 = \frac{4^2}{5^2}$ | To simplify $(\frac{4}{5})^2$, we will use the power of a quotient rule. Hence, we will distribute the exponent of 2 to the numerator and denominator. Thus, $(\frac{4}{5})^2 = \frac{4^2}{5^2}$ |
| $\frac{4^2}{5^2} = \frac{16}{25}$ | Evaluate the exponents in the denominator and denominator. Thus, numerator: 42 = 4 x 4 = 16 denominator: 52 = 5 x 5 = 25 |
| | Therefore, $\frac{4}{5}^2$is equal to $\frac{4^2}{5^2} or \frac{16}{25}$. |
Example #2
Simplify $(\frac{a^7}{b^6})^4$
Solution
| | |
--- |
| Process | Explanation |
| $(\frac{a^7}{b^6})^4 = \frac{a^{7×4}}{b^{6×4}}$ | To simplify $(\frac{a^7}{b^6})^4$, we will use the power of a quotient rule. Hence, we will distribute the exponent of 4 to the numerator and denominator. Thus, $(\frac{a^7}{b^6})^4 = \frac{a^{7×4}}{b^{6×4}}$ |
| $\frac{a^{7×4}}{b^{6×4}} = (\frac{a^28}{b^24})$ | Get the product of the exponents in the numerator and denominator, respectively. Hence, numerator: 7 x 4 = 28 denominator: 6 x 4 = 24 |
| | Therefore, $(\frac{a^7}{b^6})^4$ is equal to $(\frac{a^28}{b^24})$. |
Example #3
Simplify $(\frac{x^3y^2}{wz^5})^6$
Solution
| | |
--- |
| Process | Explanation |
| $(\frac{x^3y^2}{wz^5})^6 = \frac{x^{3×6}y^{2×6}}{w^{1×6}z^{5×6}}$ | To simplify $(\frac{x^3y^2}{wz^5})^6$, we will use the power of a quotient rule. Hence, we will distribute the exponent of 6 to the numerator and denominator. Thus, $(\frac{x^3y^2}{wz^5})^6 = \frac{x^{3×6}y^{2×6}}{w^{1×6}z^{5×6}}$ |
| $\frac{x^{3×6}y^{2×6}}{w^{1×6}z^{5×6}}= \frac{x^{18}y^{12}}{w^{6}z^{30}}$ | Get the product of the exponents in the numerator and denominator, respectively. Hence, numerator (x): 3 x 6 = 18 numerator (y): 2 x 6 = 12 denominator (w): 1 x 6 = 6 denominator (z): 5 x 6 = 30 |
| | Therefore, $(\frac{x^3y^2}{wz^5})^6$ is equal to $\frac{x^{18}y^{12}}{w^{6}z^{30}}$. |
What are the common misconceptions in exponents?
When it comes to simplifying exponents, students commit errors that will result to a wrong answer. Some of these errors are listed below:
Multiplying the base and exponent. Usually, students simply multiply the base to the number written on the exponent.
Incorrect: 34 ≠3 x 4
Correct: 33 = 3 x 3 x 3 x 3
Applying the power of the product rule wrong. Some students distribute the exponent even though it is not a product of two or more variables.
Incorrect: (4 + 5)2 ≠ 42 + 52
Correct: (4 + 5)2 = (9)2 = 81
What is the significance of exponents?
Exponents are usually used as a shortcut to represent a number that is repeatedly multiplied by itself. Most of the time, we don’t really see or use exponents in our everyday lives, but they are very significant, especially in the sciences.
Exponents are used in showing square or cubic measurements such as square feet (ft2), square meters (m2), square inches (in2), square miles (mi2), cubic feet (ft3), cubic meters (m3). With the help of exponents, it is easier for us to distinguish square and cubic measurements. Taking measurements and calculating multi-dimensional quantities is one of the most prevalent real-world applications of exponents. Because area is a two-dimensional measure of space (length x breadth), it is always measured in square units such as square feet or square meters.
Scientists use exponents to show extraordinarily large or small numbers. With the help of exponents, writing numbers in a scientific notation became possible. In scientific notation, 150 000 can be written as 1.5 x 105 – which translates to 1.5 x 10 x 10 x 10 x 10 x 10.
Exponents are of great importance in any scientific measurement that uses a scale such as pH scale and Richter scale. The pH scale and the Richter scale are both logarithmic scales, with each whole number denoting a ten-fold rise over the previous value.
Say, for example, a scientist knows that a material with a pH of 4 indicates 104, but a substance with a pH of 8 represents 105. This suggests that a substance with a pH of 5 is ten times more basic than one with a pH of 4.
Exponents are also used in computer language. We frequently hear the terms megabytes, gigabytes, and terabytes. “Mega” refers to 106 or one million, “giga” refers to 109, and “tera” refers to 1012. More so, the prefixes mega- and giga- are also used in other domains such as megahertz indicates 106 or one million hertz.
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Additional Algebra Theory:
Linear Inequalities in two variables
Division of Radicals
Graphing Ordered Pairs
Algebraic Identities
The Coordinate Graphing System
Union of Sets
Commutative and Associative Property for Addition
One Variable Inequality
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6510 | https://www.maths.tcd.ie/~dwilkins/Courses/MA2C02/MA2C02_Section2_0910.pdf | Course MA2C02, Hilary Term 2010 Section 2: Trigonometric Identities, Complex Exponentials and Periodic Sequences David R. Wilkins Copyright c ⃝David R. Wilkins 2000–2010 Contents 2 Trigonometric Identities, Complex Exponentials and Peri-odic Sequences 20 2.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . .
20 2.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . .
22 2.3 Basic Properties of Complex Numbers . . . . . . . . . . . . .
23 2.4 Complex Numbers and Trigonometrical Identities . . . . . . .
25 2.5 The Exponential of a Complex Number . . . . . . . . . . . . .
25 2.6 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . .
26 2.7 Multiplication of Complex Exponentials . . . . . . . . . . . .
27 2.8 Complex Roots of Unity . . . . . . . . . . . . . . . . . . . . .
28 2.9 Representation of Periodic Sequences . . . . . . . . . . . . . .
29 2.10 Periodic Sequences of Real Numbers . . . . . . . . . . . . . .
31 i 2 Trigonometric Identities, Complex Expo-nentials and Periodic Sequences 2.1 Basic Trigonometric Identities An anticlockwise rotation about the origin through an angle of θ radians sends a point (x, y) of the plane to the point (x′, y′), where x′ = x cos θ −y sin θ y′ = x sin θ + y cos θ (9) (This follows easily from the fact that such a rotation takes the point (1, 0) to the point (cos θ, sin θ) and takes the point (0, 1) to the point (−sin θ, cos θ).) An anticlockwise rotation about the origin through an angle of φ radians then sends the point (x′, y′) of the plane to the point (x′′, y′′), where x′′ = x′ cos φ −y′ sin φ y′′ = x′ sin φ + y′ cos φ (10) Now an anticlockwise rotation about the origin through an angle of θ + φ radians sends the point (x, y), of the plane to the point (x′′, y′′), and thus x′′ = x cos(θ + φ) −y sin(θ + φ) y′′ = x sin(θ + φ) + y cos(θ + φ) (11) But if we substitute the expressions for x′ and y′ in terms of x, y and θ provided by equation (9) into equation (10), we find that x′′ = x(cos θ cos φ −sin θ sin φ) −y(sin θ cos φ + cos θ sin φ) y′′ = x(sin θ cos φ + cos θ sin φ) + y(cos θ cos φ −sin θ sin φ) (12) On comparing equations (11) and (12) we see that cos(θ + φ) = cos θ cos φ −sin θ sin φ, (13) and sin(θ + φ) = sin θ cos φ + cos θ sin φ.
(14) On replacing φ by −φ, and noting that cos(−φ) = cos φ and sin(−φ) = −sin φ, we find that cos(θ −φ) = cos θ cos φ + sin θ sin φ, (15) and sin(θ −φ) = sin θ cos φ −cos θ sin φ.
(16) 20 If we add equations (13) and (15) we find that cos θ cos φ = 1 2(cos(θ + φ) + cos(θ −φ)).
(17) If we subtract equation (13) from equation (15) we find that sin θ sin φ = 1 2(cos(θ −φ) −cos(θ + φ)).
(18) And if we add equations (14) and (16) we find that sin θ cos φ = 1 2(sin(θ + φ) + sin(θ −φ)).
(19) If we substitute φ = θ in equations (13) and (14), and use the identity cos2 θ + sin2 θ = 1, we find that sin 2θ = 2 sin θ cos θ (20) and cos 2θ = cos2 θ −sin2 θ = 2 cos2 θ −1 = 1 −2 sin2 θ.
(21) It then follows from equation (21) that sin2 θ = 1 2(1 −cos 2θ) (22) cos2 θ = 1 2(1 + cos 2θ).
(23) Remark Equations (9) and (10) may be written in matrix form as follows: x′ y′ = cos θ −sin θ sin θ cos θ x y , x′′ y′′ = cos φ −sin φ sin φ cos φ x′ y′ .
Also equation (11) may be written x′′ y′′ = cos(θ + φ) −sin(θ + φ) sin(θ + φ) cos(θ + φ) x y .
It follows from basic properties of matrix multiplication that cos(θ + φ) −sin(θ + φ) sin(θ + φ) cos(θ + φ) = cos φ −sin φ sin φ cos φ cos θ −sin θ sin θ cos θ , and therefore cos(θ + φ) = cos θ cos φ −sin θ sin φ sin(θ + φ) = sin θ cos φ + cos θ sin φ.
This provides an alternative derivation of equations (13) and (14).
21 2.2 Basic Trigonometric Integrals On differentiating the sine and cosine function, we find that d dx sin kx = k cos kx (24) d dx cos kx = −k sin kx.
(25) for all real numbers k.
It follows that Z sin kx = −1 k cos kx + C (26) Z cos kx = 1 k sin kx + C, (27) for all non-zero real numbers k, where C is a constant of integration.
Theorem 2.1 Let m and n be positive integers. Then Z π −π cos nx dx = 0, (28) Z π −π sin nx dx = 0, (29) Z π −π cos mx cos nx dx = π if m = n, 0 if m ̸= n, (30) Z π −π sin mx sin nx dx = π if m = n, 0 if m ̸= n, (31) Z π −π sin mx cos nx dx = 0.
(32) Proof First we note that Z π −π cos nx dx = 1 n sin nx π −π = 1 n (sin nπ −sin(−nπ)) = 0 and Z π −π sin nx dx = −1 n cos nx π −π = −1 n (cos nπ −cos(−nπ)) = 0 for all non-zero integers n, since cos nπ = cos(−nπ) = (−1)n and sin nπ = sin(−nπ) = 0 for all integers n.
22 Let m and n be positive integers. It follows from equations (17) and (18) that Z π −π cos mx cos nx dx = 1 2 Z π −π (cos((m −n)x) + cos((m + n)x)) dx.
and Z π −π sin mx sin nx dx = 1 2 Z π −π (cos((m −n)x) −cos((m + n)x)) dx But Z π −π cos((m + n)x) dx = 0 (since m + n is a positive integer, and is thus non-zero). Also Z π −π cos((m −n)x) dx = 0 if m ̸= n, and Z π −π cos((m −n)x) dx = 2π if m = n (since cos((m −n)x) = 1 when m = n). It follows that Z π −π cos mx cos nx dx = Z π −π sin mx sin nx dx = 1 2 Z π −π cos((m −n)x) dx = π if m = n; 0 if m ̸= n.
Using equation (19), we see also that Z π −π sin mx cos nx dx = 1 2 Z π −π (sin((m + n)x) + sin((m −n)x)) dx = 0 for all positive integers m and n. (Note that sin((m −n)x) = 0 in the case when m = n).
2.3 Basic Properties of Complex Numbers We shall extend the definition of the exponential function so as to define a value of ez for any complex number z. First we note some basic properties of complex numbers.
A complex number is a number that may be represented in the form x+iy, where x and y are real numbers, and where i2 = −1. The real numbers x 23 and y are referred to as the real and imaginary parts of the complex number x + iy, and the symbol i is often denoted by √−1. One adds or subtracts complex numbers by adding or subtracting their real parts, and adding or subtracting their imaginary parts. Thus (x+iy)+(u+iv) = (x+u)+i(y+v).
(x+iy)−(u+iv) = (x−u)+i(y−v).
Multiplication of complex numbers is defined such that (x + iy) × (u + iv) = (xu −yv) + i(xv + uy).
The reciprocal (x + yi)−1 of a non-zero complex number x + iy is given by the formula (x + iy)−1 = x x2 + y2 −i y x2 + y2.
Complex numbers may be represented by points of the plane (through the Argand diagram). A complex number x + iy represents, and is represented by, the point of the plane whose Cartesian coordinates are (x, y). One often therefore refers to the set of all complex numbers as the complex plane. This complex plane is pictured as a flat plane, containing lines, circles etc., and distances and angles are defined in accordance with the usual principles of plane geometry and trigonometry.
The modulus of a complex number x + iy is defined to be the quantity p x2 + y2: it represents the distance of the corresponding point (x, y) of the complex plane from the origin (0, 0). The modulus of a complex number z is denoted by |z|.
Let z and w be complex numbers. Then z lies on a circle of radius |z| centred at 0, and the point z + w lies on a circle of radius |w| centred at z. But this circle of radius |w| centred at z is contained within the disk bounded by a circle of radius |z| + |w| centred at the origin, and therefore |z + w| ≤|z| + |w|. This basic inequality is essentially a restatement of the basic geometric result that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides. Indeed the complex numbers 0, z and z + w represent the vertices of a triangle in the complex plane whose sides are of length |z|, |w| and |z+w|. The inequality is therefore often referred to as the Triangle Inequality.
Let z and w be complex numbers, and let z = x + iy and w = u + iv.
Then zw = (xu −yv) + i(xv + yu) and therefore |zw|2 = (xu −yv)2 + (xv + yu)2 = (x2u2 + y2v2 −2xyuv) + (x2v2 + y2u2 + 2xyuv) = (x2 + y2)(u2 + v2) = |z|2|w|2.
24 It follows that |zw| = |z| |w| for all complex numbers z and w. A straight-forward proof by induction on n then shows that |zn| = |z|n for all complex numbers z and non-negative integers n.
2.4 Complex Numbers and Trigonometrical Identities Let θ and ϕ be real numbers, and let z = cos θ + i sin θ, w = cos ϕ + i sin ϕ, where i = √−1. Then zw = (cos θ cos ϕ −sin θ sin ϕ) + i(sin θ cos ϕ + cos θ sin ϕ) = cos(θ + ϕ) + i sin(θ + ϕ).
2.5 The Exponential of a Complex Number Let z be a complex number, and, for each non-negative integer m, let pm(z) = m X n=0 zn n! .
Then p0(z), p1(z), p2(z), . . . is an infinite sequence of complex numbers. More-over one can show that, as the integer m increases without limit, the value of the complex number pm(z) approaches a limiting value p∞(m), so that, given any strictly positive real number ε (no matter how small), there exists some positive integer M such that |pm(z) −p∞(z)| < ε whenever m ≥M. (The quantity |pm(z) −p∞(z)| measures the distance in the complex plane from pm(z) to p∞(z), and thus quantifies the error that results on approximating the quantity p∞(z) by pm(z). The size of this error can be made as small as we please, provided that we choose a value of m that is sufficiently large.) This limiting value exp p∞(z) is said to be the limit lim m→+∞pm(z) of pm(z) as m tends to +∞. The exponential ez of the complex number z is defined to be the value of this limit. Thus ez = +∞ X n=0 zn n! = p∞(z) = lim m→+∞pm(z) = lim m→+∞ m X n=0 zn n!
!
.
We may also write ez = +∞ X n=0 zn n! = 1 + z + z2 2! + z3 3! + z4 4! + · · · , 25 The exponential ez of the complex number z is also denoted by exp z. The exponential function exp: C →C, mapping the set of complex numbers to itself, which sends each complex number z to its exponents ez.
2.6 Euler’s Formula Theorem 2.2 (Euler’s Formula) eiθ = cos θ + i sin θ for all real numbers θ.
Proof Let us take the real and imaginary parts of the infinite series that defines eiθ. Now i2 = −1, i3 = −i and i4 = 1, and therefore eiθ = ∞ X n=0 inθn n!
= C(θ) + iS(θ), where C(θ) = 1 −θ2 2! + θ4 4! −θ6 6! + θ8 8! −θ10 10! + θ12 12! −· · · S(θ) = θ −θ3 3! + θ5 5! −θ7 7! + θ9 9! −θ11 11! + θ13 13! −· · · .
However the infinite series that define these functions C(θ) and S(θ) are the Taylor series for the trigonometric functions cos θ and sin θ. Thus C(θ) = cos θ and S(θ) = sin θ for all real numbers θ, and therefore eiθ = cos θ+i sin θ, as required.
Note that if we set θ = π in Euler’s formula we obtain the identity eiπ + 1 = 0.
The following identities follow directly from Euler’s formula.
Corollary 2.3 cos θ = 1 2 eiθ + e−iθ , sin θ = 1 2i eiθ −e−iθ for all real numbers θ.
It is customary to define the values cos z and sin z of the cosine and sine functions at any complex number z by the formulae cos z = 1 2 eiz + e−iz , sin z = 1 2i eiz −e−iz .
Corollary 2.3 ensures that the cosine and sine functions defined for complex values of the argument in this fashion agree with the standard functions for real values of the argument defined through trigonometry.
26 2.7 Multiplication of Complex Exponentials Let z and w be complex numbers. Then ez ew = ∞ X j=0 zj j!
! ∞ X k=0 wk k!
!
= ∞ X j=0 ∞ X k=0 zjwk j!k! .
Thus the value of the product ez ew is equal to the value of the infinite double sum that is obtained on adding together the quantities zjwk/(j!k!) for all ordered pairs (j, k) of non-negative integers. A fundamental result in the theory of infinite series ensures that, in this case, the value of this infinite double sum is independent of the order of summation, and that, in particular, we can evaluate this double sum by first adding together, for each non-negative integer n, the values of the quantities zjwk/(j!k!) for all ordered pairs (j, k) of negative numbers with j + k = n, and then adding together the resultant quantities for all non-negative values of the integer n. Thus ezew = ∞ X n=0 X (j,k) j+k=n zjwk j!k!
= ∞ X n=0 1 n! n X j=0 n!
j!(n −j)!zjwn−j !
.
(Here we have used the fact that if j + k = n then k = n −j.) Now the quantity n!
j!(n −j)! is the binomial coefficient n j . It follows from the Binomial Theorem that n X j=0 n!
j!(n −j)!zjwn−j = (z + w)n.
If we substitute this identity in the formula for the product ezew, we find that ezew = ∞ X n=0 (z + w)n n!
= ez+w.
We have thus obtained the following result.
Theorem 2.4 ezew = ez+w for all complex numbers z and w.
On combining the results of Theorem 2.4 and Euler’s Formula (Theo-rem 2.2), we obtain the following identity for the value of the exponential of a complex number.
27 Corollary 2.5 ex+iy = ex(cos y + i sin y) for all complex numbers x + iy.
2.8 Complex Roots of Unity Lemma 2.6 Let ω be a complex number satisfying the equation ωn = 1 for some positive integer n. Then ω = e 2πmi n = cos 2πm n + i sin 2πm n for some integer m.
Proof The modulus |ω| of ω is a positive real number satisfying the equation |ω|n = |ωn| = 1.
It follows that ω = eiθ = cos θ + i sin θ for some real number θ. Now (eiθ)2 = eiθeiθ = e2iθ, (eiθ)3 = e2iθeiθ = e3iθ, etc., and a straightforward proof by induction on r shows that (eiθ)r = eriθ = cos rθ + i sin rθ for all positive integers r. Now ωn = 1. It follows that 1 = (eiθ)n = eniθ = cos nθ + i sin nθ, and thus cos nθ = 1 and sin nθ = 0. But these conditions are satisfied if and only if nθ = 2πm for some integer m, in which case ω = e2πmi/n, as required.
We see that, for any positive integer n, there exist exactly n complex num-bers ω satisfying ωn = 1. These are of the form e2πmi/n for m = 0, 1, . . . , n−1.
They lie on the unit circle in the complex plane (i.e., the circle of radius 1 centred on 0 in the complex plane) and are the vertices of a regular n-sided polygon in that plane.
28 2.9 Representation of Periodic Sequences Definition A doubly-infinite sequence (zn : n ∈Z) of complex numbers associates to every integer n a corresponding complex number zn.
Definition We say that doubly-infinite sequence (zn : n ∈Z) of complex numbers is m-periodic if zn+m = zn for all integers n.
Lemma 2.7 Let m be a positive integer, and let ωm = e2πi/m. Then the value of m−1 P k=0 ωkn m is determined, for any integer n, as follows: m−1 X k=0 ωkn m = m if n is divisible by m; 0 if n is not divisible by m.
Proof The complex number ωm has the property that ωm m = 1. Also (1 −z)(1 + z + z2 + · · · + zm−1) = 1 −zm for any complex number z. It follows that (1 −ωn m) m−1 X k=0 ωkn m = 1 −ωmn m = 0 for all integers n, and therefore m−1 X k=0 ωkn m = 0 provided that ωn m ̸= 1.
Now ωn m = 1 if and only if the integer n is divisible by m. We can therefore conclude that m−1 X k=0 ωkn m = m if n is divisible by m, 0 if n is not divisible by m, as required.
Theorem 2.8 Let (zn : n ∈Z) be a doubly-infinite sequence of complex numbers which is m-periodic . Then zn = m−1 X k=0 ckωkn m , for all integers n, where ωm = e2πi/m and ck = 1 m m−1 X j=0 zjω−kj m .
29 Proof It follows from the definition of the numbers ck that m−1 X k=0 ckωkn m = 1 m m−1 X k=0 m−1 X j=0 zjω−kj m ωkn m = 1 m m−1 X j=0 zj m−1 X k=0 ω(n−j)k m !
, for all integers n. Now it follows from Lemma 2.7 that m−1 X k=0 ω(n−j)k m = 0 unless n −j is divisible by m, in which case m−1 X k=0 ω(n−j)k m = m.
Moreover, given any integer n, there is a unique integer r between 0 and m −1 for which n −r is divisible by m. It follows that m−1 X k=0 ckωkn m = zr where 0 ≤r < m and r ≡n (mod m).
Moreover zr = zn, because the sequence (zn : n ∈Z) is m-periodic. Thus m−1 X k=0 ckωkn m = zn for all integers n, as required.
Example Let (zn : n ∈Z) be an 3-periodic sequence with z0 = 2, z1 = 4, z2 = 5. Let ω = ω3 = e2πi/3. It follows from Theorem 2.8 that zn = c0 + c1ωn + c2ω2n for all integers n, where ωm = e2πi/m and ck = 1 3 z0 + z1ω−k + z2ω−2k .
for k = 0, 1, 2. Now ω−1 = ω2 and ω−2 = ω, because ω3 = 1. Therefore ck = 1 3 z0 + z1ω2k + z2ωk , 30 and thus c0 = 1 3(2 + 4 + 5) = 11 3 , c1 = 1 3(2 + 4ω2 + 5ω), c2 = 1 3(2 + 4ω + 5ω2).
Now ω = cos 2π 3 + i sin 2π 3 = 1 2(−1 + √ 3 i), ω2 = cos 4π 3 + i sin 4π 3 = 1 2(−1 − √ 3 i).
It follows that c1 = 1 6(−5 + √ 3 i), c2 = 1 6(−5 − √ 3 i).
Example Let (zn : n ∈Z) be an 4-periodic sequence with z0 = 2, z1 = 4, z2 = 5, z3 = 1. Now if ω4 is defined as in the statement of Theorem 2.8 then ω4 = e2πi/4 = i. It follows from Theorem 2.8 that zn = c0 + c1in + c2(−1)n + c3(−i)n for all integers n, where ωm = e2πi/m and ck = 1 4 z0 + z1i−k + z2i−2k + z3i−3k = 1 4 2 + 4 × (−i)k + 5 × (−1)k + ik .
Thus c0 = 3, c1 = −3 4 −3 4i, c2 = 1 2, c3 = −3 4 + 3 4i.
2.10 Periodic Sequences of Real Numbers Theorem 2.9 Let (xn : n ∈Z) be a doubly-infinite sequence of real numbers which is m-periodic . Then xn = m−1 X k=0 pk cos 2πkn m + qk sin 2πkn m , for all integers n, where ωm = e2πi/m and pk = 1 m m−1 X j=0 xj cos 2πkj m , qk = 1 m m−1 X j=0 xj sin 2πkj m .
31 Proof It follows from Theorem 2.8 that xn = m−1 X k=0 ckωkn m , for all integers n, where ωm = e2πi/m and ck = 1 m m−1 X j=0 xjω−kj m .
Now ωn m = cos 2nπ m + i sin 2nπ m ω−n m = cos 2nπ m −i sin 2nπ m for all integers n. Now ck = pk −qki for k = 0, 1, . . . , m −1, where pk = 1 m m−1 X j=0 xj cos 2πkj m , qk = 1 m m−1 X j=0 xj sin 2πkj m .
(Note that pk and qk are real numbers for all k. It follows that xn = Re m−1 X k=0 ckωkn m !
= m−1 X k=0 pk cos 2πkn m + qk sin 2πkn m , where Re m−1 X k=0 ckωkn m !
denotes the real part of m−1 X k=0 ckωkn m .
Problems 1. Let (zn : n ∈Z) be the doubly-infinite 3-periodic sequence with z0 = 1, z1 = 2 and z2 = 6. Find values of a0, a1 and a2 such that zn = a0 + a1ωn + a2ω2n for all integers n, where ω = e2πik/3. (Note that ω = 1 2(−1 + √ 3 i), ω2 = e−2πik/3 = 1 2(−1 − √ 3 i) and thus ω3 = 1 and ω + ω2 = −1.) 32 |
6511 | https://flexbooks.ck12.org/cbook/ck-12-math-analysis-concepts/section/6.6/primary/lesson/hyperbola-equations-and-the-focal-property-mat-aly/ | Skip to content
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6.6
Hyperbola Equations and the Focal Property
Written by:Raja Almukkahal | Larame Spence |
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
Adrian and Evan were discussing the math class they just completed. The class focused on hyperbolas (pun intended!), and reviewed the properties of hyperbolas.
Adrian thinks that hyperbolas are very similar to the parabolas that they studied last week, and believes that the shapes are really the same. Evan thinks that it does not matter what the parts of the hyperbola look like, since it is the complete shape that they are studying.
Who is correct?
Hyperbola Equations and the Focal Property
Compared to parabolas and ellipses, hyperbolas may seem... messy! Not only is it an infinite shape, but there are two pieces that aren’t even connected! Hyperbolas are conic sections though: when the plane slices through two parts of the cone, the two infinite “U”-shaped parts are together called a hyperbola.
In this section we will see that this sprawling shape actually has some beautiful properties that make it as noble as its cousins.
The Focal Property
Even though this shape seems much harder to conceive of than an ellipse, the hyperbola has a defining focal property that is as simple as the ellipse’s. Remember, an ellipse has two foci and the shape can be defined as the set of points in a plane whose distances to these two foci have a fixed sum.
Hyperbolas also have two foci, and they can be defined as the set of points in a plane whose distances to these two points have the same difference. So in the picture below, for every point @$\begin{align}P\end{align}@$ on the hyperbola, @$\begin{align}|d_2 - d_1| = C\end{align}@$ for some constant @$\begin{align}C\end{align}@$.
The general form for a hyperbola that opens upwards and downwards and whose foci lie on the @$\begin{align}y-\end{align}@$axis is:
@$$\begin{align}\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\end{align}@$$
Switching @$\begin{align}x\end{align}@$ and @$\begin{align}y\end{align}@$ we have hyperbolas that open rightwards and leftwards and whose foci lie on the @$\begin{align}x-\end{align}@$axis.
@$$\begin{align}\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\end{align}@$$
For a hyperbola that is centered around the point @$\begin{align}(h,k)\end{align}@$ we have the shifted equations:
@$$\begin{align}\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\end{align}@$$
for a hyperbola opening up and down, and
@$$\begin{align}\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\end{align}@$$
for a hyperbola opening left and right.
Examples
Example 1
Earlier, you were asked whether Evan or Adrian is correct.
Evan and Adrian are both correct in their own ways. Adrian is correct that a hyperbola is just two parabolas in opposite directions, which becomes clear when you consider that a parabola is created by slicing a single cone, and a hyperbola by slicing two identical cones at the same time. Evan is correct that although a hyperbola is constructed of two parabolas, it is the complete shape that they are studying, and many or most of the formulas and definitions that they will consider only apply to the complete shape.
Example 2
The hyperbola is infinite in size. In mathematics this is called unbounded, which means no circle, no matter how large, can enclose the shape. Explain why a focal property involving a difference results in an unbounded shape, while a focal property involving a sum results in a bounded shape.
In the case of an ellipse, we had two distances summing to a constant. Since the distances are both positive then there is a limit to the size of the numbers. In the case of hyperbolas, two very large positive numbers can have a much smaller difference, infinitely small in fact.
Example 3
Show that the following equation is a hyperbola. Graph it, and show its foci.
@$$\begin{align}144x^2 - 576x - 25y^2 - 150y - 3249 = 0\end{align}@$$
The positive leading coefficient for the @$\begin{align}x^2\end{align}@$ term and the negative leading coefficient for the @$\begin{align}y^2\end{align}@$ term indicate that this is a hyperbola that is horizontally oriented. Grouping and completing the square, we have:
@$$\begin{align}144(x^2-4x)-25(y^2+6y)& = 3249 \
144(x^2 - 4x+4) - 25(y^2+6y+9)& = 3249 + 576 -225 \
144(x-2)^2-25(y+3)^2 & = 3600 \
\frac{(x-2)^2}{5^2} - \frac{(y+3)^2}{12^2} & = 1\end{align}@$$
So our hyperbola is centered at (2,-3). Its vertices are 5 units to the right and left of (2,-3), or at the points (7,-3) and (-3,-3). It opens to the right and left from these vertices. It’s foci are @$\begin{align}c\end{align}@$ units to the left and right of (2,-3), where @$\begin{align}c = \sqrt{a^2 + b^2} = \sqrt{5^2 + 12^2} = 13\end{align}@$. So it’s foci are at (15,-3) and (-11,-3). Plotting a few points near (7,-3) and (-3,-3), the graph looks like:
Example 4
Graph the following hyperbola and mark its foci: @$\begin{align}16x^2 + 64x-9y^2+90y - 305=0\end{align}@$.
The positive leading coefficient for the term and the negative leading coefficient for the term indicate that this is a hyperbola that is horizontally oriented. Grouping and completing the square, we have:
@$\begin{align}16(x^2 + 4x +4) -9(y^2 -10y +25) -305 = 64 -225\end{align}@$
Factoring and combining like terms:
@$\begin{align}16(x + 2)^2 -9(y - 5)^2 = 144\end{align}@$
Divide both sides by 144 and rewrite 9 and 16 as 32 and 42:
@$\begin{align}\frac{(x + 2)^2}{3^2} - \frac{(y - 5)^2}{4^2} = 1\end{align}@$
Example 5
Graph the following hyperbola and mark its foci: @$\begin{align}x^2 - 8x - y^2 - 4y = -28\end{align}@$.
To graph @$\begin{align}x^2 - 8x - y^2 + 4y = -28\end{align}@$:
@$\begin{align}(x - 4)^2 - (y - 2)^2 = -16\end{align}@$ ..... complete the square to factor
@$\begin{align}\frac{(y - 2)^2}{16} - \frac{(x - 4)^2}{16} = 1\end{align}@$ ..... rewrite in standard form
@$\begin{align}x - 4 = 0 \to x = 4\end{align}@$ and @$\begin{align}y - 2 = 0 \to y = 2\end{align}@$ ..... therefore the center is (4, 2)
Mark 4 units left and right of (4, 2) and Mark 2 units above and below (4, 2), use those four points to define the sides of a box.
Connect the corners of the box to illustrate the asymptotes.
Since the "y" term is the positive one, the hyperbola opens up and down.
Graph the hyperbola, it should look like this:
Example 6
Graph the following hyperbola: @$\begin{align}x^2-8x-y^2-2y=-14\end{align}@$.
To graph @$\begin{align}x^2 - 8x - y^2 - 2y = -14\end{align}@$:
@$\begin{align}(x - 4)^2 - (y + 1)^2 = 1\end{align}@$ ..... complete the square to factor@$\begin{align}\frac{(x - 4)^2}{1} - \frac{(y + 1)^2}{1} = 1\end{align}@$ ..... rewrite in standard form
@$\begin{align}x - 4 = 0 \to x = 4\end{align}@$ and @$\begin{align}y + 1 = 0 \to y = -1\end{align}@$ ..... therefore the center is (4, -1)
Mark 4 units left and right of (4, -1) and Mark 1 unit above and below (4, -1), use those four points to define the sides of a box.
Connect the corners of the box to illustrate the asymptotes.
Since the "x" term is the positive one, the hyperbola opens left and right.
Graph the hyperbola, it should look like this:
Example 7
Find the equation for the following hyperbola:
@$\begin{align}\frac{(x - 4)^2}{4} - \frac{(y + 2)^2}{45} = 1\end{align}@$
Review
Sketch the hyperbolas.
@$\begin{align}\frac{(x + 4)^2}{4} - \frac{(y - 1)^2}{9} = 1\end{align}@$
@$\begin{align}\frac{(y + 3)^2}{4} - \frac{(x - 4)^2}{9} = 1\end{align}@$
@$\begin{align}\frac{(y + 4)^2}{16} - \frac{(x - 1)^2}{4} = 1\end{align}@$
@$\begin{align}(x - 2)^2 - 4y^2 = 16\end{align}@$
@$\begin{align}\frac{y^2}{4} - \frac{(x - 1)^2}{4} = 1\end{align}@$
@$\begin{align}\frac{(x - 2)^2}{16} - \frac{(y + 4)^2}{1} = 1\end{align}@$
@$\begin{align}\frac{(x + 2)^2}{9} - \frac{(y + 2)^2}{16} = 1\end{align}@$
@$\begin{align}\frac{(x + 4)^2}{9} - \frac{(y - 2)^2}{4} = 1\end{align}@$
Graph the hyperbola and mark its foci: @$\begin{align}9y^2+18y-x^2+4x-4=0\end{align}@$
Graph the hyperbola and mark its foci: @$\begin{align}25x^2+150x-4y^2+24y+89=0\end{align}@$
Identify the equation of the hyperbola using the image.
Review (Answers)
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6513 | https://chemistry.stackexchange.com/questions/81602/co-co%E2%82%82-ratio-when-given-amounts-of-carbon-and-oxygen-react-with-each-other | physical chemistry - CO : CO₂ ratio when given amounts of carbon and oxygen react with each other - Chemistry Stack Exchange
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CO : CO₂ ratio when given amounts of carbon and oxygen react with each other
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By the reaction of carbon and oxygen a mixture of C O C O and C O X 2 C O X 2 is obtained. What is the composition of the mixture by mass obtained when 20 grams of O X 2 O X 2 reacts with 12 grams of carbon?
I wrote the equation as
4 C+3 O X 2⟶2 C O+2 C O X 2 4 C+3 O X 2⟶2 C O+2 C O X 2
and identified O X 2 O X 2 as the limiting reagent then the masses of C O C O formed will be 11.66 grams and mass of C O X 2 C O X 2 formed will be 18.33 grams, which gives the ratio as 7:11, but the answer is given as 21:11.
Can somebody please tell me where am I going wrong?
physical-chemistry
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edited Aug 25, 2017 at 7:26
Pritt says Reinstate Monica
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asked Aug 22, 2017 at 15:06
GemGem
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2 What was your motivation for writing the equation that way? That automatically assumes an (incorrect) answer.Chet Miller –Chet Miller 2017-08-22 15:31:47 +00:00 Commented Aug 22, 2017 at 15:31
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First and foremost refer to the tip given in the answer of @ChesterMiller: use separate reactions for each given process, and don't rely on limiting reagent concept when more than one reaction is possible:
C x+0.5 O X 2 0.5 x C y+O X 2 y⟶C O x⟶C O X 2 y(1)(2)(1)C x+0.5 O X 2 0.5 x⟶C O x(2)C y+O X 2 y⟶C O X 2 y
let x x mol be the amount of carbon participating in forming carbon monoxide C O C O, and y y mol -- the amount of carbon participating in forming carbon dioxide C O X 2 C O X 2.
We have the following amounts of reactants in the system:
n(C)=m(C)M(C)=12 g 12 g m o l−1=1 m o l n(C)=m(C)M(C)=12 g 12 g m o l−1=1 m o l
n(O X 2)=m(O X 2)M(O X 2)=20 g 32 g m o l−1=0.625 m o l n(O X 2)=m(O X 2)M(O X 2)=20 g 32 g m o l−1=0.625 m o l
where m m -- mass; M M -- molecular weight.
Taking into account reactants ratios, the following system of equations is relevant:
{x+y 0.5 x+y=1=0.625 total amount of carbon total amount of oxygen{x+y=1 total amount of carbon 0.5 x+y=0.625 total amount of oxygen
Solving this system of equations, one gets x=0.75 x=0.75 (mol) and y=0.25 y=0.25 (mol). Hence the masses of the products are:
m(C O)m(C O X 2)=x⋅M(C O)=y⋅M(C O X 2)=0.75 m o l⋅28 g m o l−1=0.25 m o l⋅44 g m o l−1=21 g=11 g m(C O)=x⋅M(C O)=0.75 m o l⋅28 g m o l−1=21 g m(C O X 2)=y⋅M(C O X 2)=0.25 m o l⋅44 g m o l−1=11 g
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answered Aug 22, 2017 at 16:45
andselisk♦andselisk
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Is this actually a plausible reaction? Could we also get more C O 2 C O 2 but some unburnt C C?badjohn –badjohn 2017-08-25 08:28:13 +00:00 Commented Aug 25, 2017 at 8:28
@badjohn The way the problem is stated (conditions were such that only C O C O + C O X 2 C O X 2 are presented in the system) suggests only this solution.andselisk –andselisk♦ 2017-08-25 10:04:06 +00:00 Commented Aug 25, 2017 at 10:04
@andselisk Yes but it raised this related question in my mind. Maybe, I should start a new question.badjohn –badjohn 2017-08-25 10:55:43 +00:00 Commented Aug 25, 2017 at 10:55
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You should have written:
C+O X 2⟶C O X 2 C+O X 2⟶C O X 2
and
C+1 2 O X 2⟶C O C+1 2 O X 2⟶C O
Then you should have determined what fraction of the carbon reacts by by the first reaction and what fraction of the carbon reacts by the second reaction.
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edited Aug 25, 2017 at 7:19
Nilay Ghosh
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answered Aug 22, 2017 at 15:35
Chet MillerChet Miller
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How can I find that?Gem –Gem 2017-08-22 15:51:13 +00:00 Commented Aug 22, 2017 at 15:51
1 Let f be fraction of C that forms CO2 and let (1-f) be the fraction of C that forms CO. Solve for f such that 20 grams of O2 are consumed.Chet Miller –Chet Miller 2017-08-22 15:58:19 +00:00 Commented Aug 22, 2017 at 15:58
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6514 | https://www.fq.math.ca/Papers1/48-3/Knopfmacher_Robbins.pdf | SOME PROPERTIES OF CYCLIC COMPOSITIONS ARNOLD KNOPFMACHER AND NEVILLE ROBBINS Abstract. Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let n k denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We prove some theorems concerning n k .
1. Introduction If k, n are integers such that 1 ≤k ≤n, let C(n, k) denote the set of all compositions of n into k parts. (It is well-known that the cardinality of C(n, k) is n −1 k −1 ). Now consider the relation R defined on C(n, k) as follows: two elements of C(n, k) are related if and only if they differ only by a cyclic shift. For example, we have [1 + 2 + 3]R = [2 + 3 + 1]R = [3 + 1 + 2]R ̸= [1 + 3 + 2]R.
Let n k denote the number of distinct equivalence classes of C(n, k) with respect to R. We refer to n k as the number of cyclic compositions of n with k parts. These combinatorial objects were first investigated more than a century ago. (See and the references contained therein.) Suppose that for each n ≥2 and for each k such that 1 ≤k ≤n −1, we list all the n k in a horizontal row. We then obtain a triangular array similar to Pascal’s triangle, namely AUGUST 2010 249 THE FIBONACCI QUARTERLY 1 1 1 1 2 1 1 2 2 1 1 3 4 3 1 1 3 5 5 3 1 1 4 7 10 7 4 1 1 4 10 14 14 10 4 1 1 5 12 22 26 22 12 5 1 1 5 15 30 42 42 30 15 5 1 1 6 19 43 66 80 66 43 19 6 1 1 6 22 55 99 132 132 99 55 22 6 1 1 7 26 73 143 217 246 217 143 73 26 7 1 · · · Remark 1. This table appears in as A037306. Note that for aesthetic reasons we omit the additional column n n = 1 from each row above.
In this note, we derive an explicit formula for n k . We also give two proofs that the triangular array shown above is symmetric, that is, n k = n n −k for all k such that 1 ≤k ≤n−1. (One proof is based on generating functions while the second is combinatorial.) We prove some additional parity-related results, including the following: 2| 2n n for all n ≥2. (The latter result is analogous to a similar property of binomial coefficients stated in (3) below.) 2. Preliminaries Theorem 1. If p is prime and 1 ≤k ≤p −1, then p k = 1 p p k . This theorem, proven by Sommerville in , will be generalized in Corollary 1 below.
Definition 1. Let D(n) denote the total number of cyclic equivalence classes of compositions of n, that is D(n) = n X k=1 n k .
It is known that D(n) = −1 + 1 n X d|n φ(d)2 n d .
(This follows from [2, p. 48].) 250 VOLUME 48, NUMBER 3 SOME PROPERTIES OF CYCLIC COMPOSITIONS Definition 2. Let D∗(n) denote the total number of necklaces of n beads in two colors, say black and white. It is known that D∗(n) = 1 + D(n) .
Definition 3. Let D∗∗(n) denote the total number of necklaces of n beads in two colors, with the proviso that at least one bead must be black.
Clearly, D∗∗(n) = D(n), but we will prove more, namely, we will demonstrate a bijection between cyclic compositions of n into k parts and necklaces with beads of which k are black, the rest white, where 1 ≤k ≤n −1.
Definition 4. Let t2(n) denote the sum of the binary digits of the natural number n.
Definition 5. Let o2(n) = m if 2m|n but 2m+1 ̸ |n.
Definition 6. Let [x] denote the integer part of the real number x.
Identities. For all n ≥1, we have: X k odd n k = 2n−1 (1) o2 n k = t2(k) + t2(n −k) −t2(n) (2) 2 2n n (3) 2n −1 k ≡1 (mod 2) for all k (4) If t is odd and a > b then o2 2a 2bt = o2 2a−b t .
(5) Remark 2. Identity (2) follows from the formula op(n!) = n −tp(n) p −1 which may be found on p. 26 of ; (3), (4), and (5) follow from (2).
3. The Main Results We begin with our main result, which is an explicit formula for n k .
Theorem 2. If 1 ≤k ≤n, then n k = 1 n X j| gcd(n,k) φ(j) n/j k/j .
Proof. We use the general expression for the bivariate generating function of cycles of unlabeled combinatorial structures, as given in [2, p. 729], specialized to the case of cyclic compositions (the corresponding univariate generating function appears on [2, p. 48]). This gives C(z, u) := X n≥0 X k≥0 n k ukzn = X j≥1 φ(j) j log 1 −ujzj 1 −zj −1 .
AUGUST 2010 251 THE FIBONACCI QUARTERLY Now C(z, u) = X j≥1 φ(j) j log(1 −zj) −log(1 −(1 + uj)zj) = X j≥1 φ(j) j − X m≥1 zjm m + X m≥1 (1 + uj)zjm m = X j≥1 X m≥1 φ(j) jm zjm (1 + uj)m −1 .
Setting n = jm we get n k = 1 n X j|n φ(j)[uk] (1 + uj)n/j −1 = 1 n X j| gcd(n,k) φ(j) n/j k/j .
□ As a corollary of Theorem 2 we obtain the following result, which generalizes Theorem 1.
We also give a direct combinatorial proof thereof.
Corollary 1. If 1 ≤k ≤n and (k, n) = 1, then n k = 1 n n k .
Proof. (First.) This follows directly from the hypothesis and Theorem 2.
□ Proof. (Second.) Suppose that a cyclic equivalence class of compositions of n into k parts has order d. Then d must divide the order of the cyclic group of order k, that is, d|k. Then the composition, viewed as a sequence, has a subsequence of order k d, and furthermore, k d|n. Since (k, n) = 1 by hypothesis, this implies k d = 1, that is, d = k. Since each of the n k cyclic equivalence classes has order k, and since the total number of compositions of n into k parts is n −1 k −1 , we have k n k = n −1 k −1 → n k = 1 k n −1 k −1 = 1 n n k .
□ Another consequence of Theorem 2 is Corollary 2 below, which is analogous to the corre-sponding property of binomial coefficients given in (1).
Corollary 2.
X k odd 2n k = 22n−n−1.
Proof. By the hypothesis and Theorem 2, we have 2n k = 1 2n 2n k 252 VOLUME 48, NUMBER 3 SOME PROPERTIES OF CYCLIC COMPOSITIONS so that, invoking (1), we obtain: X k odd 2n k = X k odd 1 2n 2n k = 1 2n X k odd 2n k = 1 2n (22n−1) = 22n−n−1.
□ As a further consequence of Theorem 2, we obtain the symmetry property of cyclic compo-sition coefficients, namely: Theorem 3. If 1 ≤k ≤n −1, then n n −k = n k .
Proof. (First.) This follows immediately from Theorem 2, to wit: n n −k = 1 n X j|(n−k,n) φ(j) n/j (n −k)/j = 1 n X j|(k,n) φ(j) n/j k/j = n k .
□ We now offer an alternate proof of Theorem 3, based on combinatorics.
Proof. (Second.) There is a bijection between necklaces of n beads of which k are black, the rest white, and cyclic compositions of n into k parts as follows: Given such a necklace, start at any black bead. Let n1 denote the number of beads traversed before coming to the second black bead. Let n2 denote the number of beads traversed before coming to the third black bead, etc. Since there are a total of k black beads by hypothesis, we get a composition of n into k parts, namely n = n1 +n2 +· · · +nk. Conversely, given a composition of n into k parts, start with a black bead, add n1 −1 white beads, then a second black bead, then n2 −1 white beads, etc., finally nk −1 white beads to separate the kth and first black beads.
Note that each necklace with k black beads makes a contribution of 1 to n k and each necklace with n −k black beads makes a contribution of 1 to n n −k . But a necklace with n −k black nodes also has k white nodes. Clearly, there is a bijection between necklaces with k black nodes and those with k white nodes. The conclusion now follows.
□ Remark 3. An alternate proof of Theorem 3 using lattice path representations of cyclic com-positions was given by D. Wasserman .
Let Cn = 1 2n+1 2n n denote the nth Catalan number.
Our triangular array of cyclic composition coefficients contains the Catalan numbers as the following theorem asserts.
Theorem 4. 2n + 1 n = Cn for all n ≥1.
Proof. Since (n, 2n + 1) = 1, Lemma 1 implies that 2n + 1 n = 1 2n + 1 2n + 1 n = 1 2n + 1 (2n + 1)!
n!(n + 1)!
= 1 n + 1 (2n)!
(n!)2 = 1 n + 1 2n n = Cn.
□ AUGUST 2010 253 THE FIBONACCI QUARTERLY There is also a connection between our triangular array of cyclic composition coefficients and the pentagonal numbers which play an important role in the theory of partitions. Recall that the pentagonal numbers are defined for k ∈Z by ω(k) = k(3k −1)/2.
Theorem 5. 3n + 1 3 = ω(n); 3n + 2 3 = ω(−n).
Proof. Theorem 2 implies 3n + 1 3 = 1 3n + 1 3n + 1 3 = n(3n −1) 2 = ω(n).
Likewise 3n + 2 3 = 1 3n + 2 3n + 2 3 = n(3n + 1) 2 = ω(−n).
□ 4. Parity Considerations The next theorem, which concerns the parity of the central cyclic composition coefficient, is analogous to a corresponding property of binomial coefficients given in (3) above: Theorem 6. 2 2n n for all n ≥2.
Proof. The second author has shown that D(n) −1 is even for all n > 2 . Thus, invoking the identity that follows Definition 1, and noting that n n = 1, we have D(n) −1 = n−1 X k=1 n k is even for all n ≥2.
Now D(2n) −1 = 2n−1 X k=1 2n k = n−1 X k=1 2n k + 2n n + 2n−1 X k=n+1 2n k .
By Theorem 3 the two sums that appear in the right member of the last identity are equal.
The conclusion now follows.
□ The last theorem states a parity property of cyclic composition coefficients that again recalls a similar parity property of binomial coefficients.
Theorem 7.
(a) 2n −1 k ≡1 (mod 2) for 1 ≤k ≤2n −1.
(b) 2n + 1 k ≡0 (mod 2) for 2 ≤k ≤2n and n ≥2.
Proof. Theorem 2 implies (2n ± 1) 2n ± 1 k = X j|(k,2n±1) φ(j) (2n ± 1)/j k/j 254 VOLUME 48, NUMBER 3 SOME PROPERTIES OF CYCLIC COMPOSITIONS → 2n ± 1 k ≡ X j|(k,2n±1) φ(j) (2n ± 1)/j k/j (mod 2).
Now j|(2n ± 1) ⇒j is odd ⇒φ(j) ≡0 (mod 2) unless j = 1, so we have 2n ± 1 k ≡ 2n ± 1 k (mod 2).
Now (a) follows from (4).
To prove (b), we must show that 2n + 1 k ≡0 (mod 2) for 2 ≤k ≤2n, or equivalently, that o2 2n + 1 k ≥1. Now (2) implies o2 2n + 1 k = t2(k) + t2(2n + 1 −k) −t2(2n + 1) = t2(k) + t2(2n + 1 −k) −2.
If t2(k) ≥2, then o2 2n + 1 k ≥t2(2n + 1 −k) ≥1. If t2(k) = 1, then k = 2m and o2 2n + 1 2m = t2(2n + 1 −2m) −1.
If t2(2n + 1 −2m) = 1, then 2n + 1 −2m = 2r, so m = 0 or m = n, and hence, k = 1 or k = 2n, contrary to the hypothesis. Therefore t2(2n + 1 −2m) ≥2, so o2 2n + 1 2m ≥1.
□ References R. D. Carmichael, The Theory of Numbers, John Wiley & Sons, 1914.
P. Flajolet and R. Sedgwick, Analytic Combinatorics, Cambridge University Press, 2009.
N. Robbins, Some congruence properties of the number of cyclic compositions, (to appear in Japan J. of Algebra, Number Theory, and Applications).
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,
D. Y. M. Sommerville, On certain perodic properties of cyclic compositions of integers, Proc. London Math.
Soc., (1909), 263–313.
D. Wasserman, link from A037306 in Online Encyclopedia of Integer Sequences,
MSC2010: 11B65, 11A07 School of Computational and Applied Mathematics, University of the Witwatersrand, Jo-hannesburg, Wits 2050, South Africa E-mail address: arnoldk@cam.wits.ac.za Department of Mathematics, San Francisco State University, San Francisco, CA 94132 E-mail address: robbins@math.sfsu.edu AUGUST 2010 255 |
6515 | https://www.endocrine-abstracts.org/ea/0073/ea0073aep603 | Swyer Syndrome, 46, XY complete gonadal dysgenesis in a patient with spontaneous menarche | ECE2021 | European Congress of Endocrinology 2021 | Endocrine Abstracts
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Endocrine Abstracts (2021) 73 AEP603 | DOI: 10.1530/endoabs.73.AEP603
ECE2021Audio Eposter PresentationsReproductive and Developmental Endocrinology (55 abstracts)
Swyer Syndrome, 46, XY complete gonadal dysgenesis in a patient with spontaneous menarche
Dana Larisa Geru1, Alina Maria Dumitru2, Georgiana Cristina Taujan3 &Adina Simona Dragomir1
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1 C.I. Parhon National Institute of Endocrinology, Infertility and Gonadal Pathology, Bucharest, Romania; 2 C.I. Parhon National Institute of Endocrinology, Adrenal and Bone Metabolism, Bucharest, Romania; 3 Centre Hospitalier Universitaire Brugmann, Brussels, Belgium
46, XY complete gonadal dysgenesis, also known as Swyer Syndrome, is characterized by the presence of normal female external genitalia at birth, late puberty and primary amenorrhea. Spontaneous menses (due to hormone-secreting tumor) and breasts development occur in rare cases. With proper hormonal substitution, patients could carry pregnancies achieved through IVF with donor oocytes.
Case presentation
Female patient, aged 22, addresses endocrinology specialist for a routine examination. She mentions having first menses when she was 14, with spontaneous menses for the next 5 years followed by secondary amenorrhea, for which she started estroprogestative substitution after medical recommendation, resulting in regular menstrual cycles and normal breast development. She came to our clinic for amenorrhea after stopping the hormone substitution therapy. Clinical investigations contributed to the diagnosis of hypergonadotropic hypogonadism, with testosterone in normal range and low estradiol. A karyotype test was performed with the result of 46, XY genotype without structural or numerical abnormalities. FISH testing revealed the presence of SRY, DXZ1 and DYZ1, therefore, the patient was advised to seek further genetic analysis. MRI showed Mullerian structures, infantile uterus with Fallopian tubes but instead of gonads, fibrotic bands could be seen. Considering the risk of developing a gonadoblastoma or even a dysgerminoma in a patient with Disorders of Sexual Development, the patient was advised to present to the Surgical Department to undergo exploratory laparotomy and resection of dysgenetic gonads, followed by estroprogestative substitution. Histopathological examination and immunohistochemistry analysis revealed that one of the dysgenetic gonad was a dysgerminoma with Ki67 positive in 17% of tumor cells, and the other one was a testes having Wolffian and Mullerian structures with no tumor characteristics.
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6516 | https://pmc.ncbi.nlm.nih.gov/articles/PMC7891488/ | Physiological dead space and alveolar ventilation in ventilated infants - PMC
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Pediatr Res
. 2021 Feb 18;91(1):218–222. doi: 10.1038/s41390-021-01388-8
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Physiological dead space and alveolar ventilation in ventilated infants
Emma Williams
Emma Williams
1 Department of Women and Children’s Health, School of Life Course Sciences, Faculty of Life Sciences and Medicine, King’s College London, London, UK
2 Asthma UK Centre for Allergic Mechanisms in Asthma, King’s College London, London, UK
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1,2, Theodore Dassios
Theodore Dassios
1 Department of Women and Children’s Health, School of Life Course Sciences, Faculty of Life Sciences and Medicine, King’s College London, London, UK
3 Neonatal Intensive Care Unit, King’s College Hospital NHS Foundation Trust, London, UK
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1,3, Paul Dixon
Paul Dixon
4 Individual Consultant, London, UK
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4, Anne Greenough
Anne Greenough
1 Department of Women and Children’s Health, School of Life Course Sciences, Faculty of Life Sciences and Medicine, King’s College London, London, UK
2 Asthma UK Centre for Allergic Mechanisms in Asthma, King’s College London, London, UK
5 NIHR Biomedical Centre at Guy’s and St Thomas NHS Foundation Trust, King’s College London, London, UK
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1 Department of Women and Children’s Health, School of Life Course Sciences, Faculty of Life Sciences and Medicine, King’s College London, London, UK
2 Asthma UK Centre for Allergic Mechanisms in Asthma, King’s College London, London, UK
3 Neonatal Intensive Care Unit, King’s College Hospital NHS Foundation Trust, London, UK
4 Individual Consultant, London, UK
5 NIHR Biomedical Centre at Guy’s and St Thomas NHS Foundation Trust, King’s College London, London, UK
✉
Corresponding author.
Received 2020 Nov 17; Accepted 2021 Jan 18; Issue date 2022.
© The Author(s) 2021, corrected publication 2021
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 license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license 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 license, visit
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This article has been corrected. See Pediatr Res. 2021 Aug 17;91(1):265.
Abstract
Background
Dead space is the volume not taking part in gas exchange and, if increased, could affect alveolar ventilation if there is too low a delivered volume. We determined if there were differences in dead space and alveolar ventilation in ventilated infants with pulmonary disease or no respiratory morbidity.
Methods
A prospective study of mechanically ventilated infants was undertaken. Expiratory tidal volume and carbon dioxide levels were measured. Volumetric capnograms were constructed to calculate the dead space using the modified Bohr–Enghoff equation. Alveolar ventilation (V A) was also calculated.
Results
Eighty-one infants with a median (range) gestational age of 28.7 (22.4–41.9) weeks were recruited. The dead space [median (IQR)] was higher in 35 infants with respiratory distress syndrome (RDS) [5.7 (5.1–7.0) ml/kg] and in 26 infants with bronchopulmonary dysplasia (BPD) [6.4 (5.1–7.5) ml/kg] than in 20 term controls with no respiratory disease [3.5 (2.8–4.2) ml/kg, p< 0.001]. Minute ventilation was higher in both infants with RDS or BPD compared to the controls. V A in infants with RDS or BPD was similar to that of the controls [p = 0.54].
Conclusion
Prematurely born infants with pulmonary disease have a higher dead space than term controls, which may influence the optimum level during volume-targeted ventilation.
Impact
Measurement of the dead space was feasible in ventilated newborn infants.
The physiological dead space was a significant proportion of the delivered volume in ventilated infants.
The dead space (per kilogram) was higher in ventilated infants with respiratory distress syndrome or evolving bronchopulmonary dysplasia compared to term controls without respiratory disease.
The dead space volume should be considered when calculating the most appropriate volume during volume-targeted ventilation.
Introduction
Newborn infants often require respiratory support with invasive mechanical ventilation, unfortunately such infants can develop chronic respiratory morbidity.1 Use of volume-targeted ventilation is a lung protective strategy as it potentially avoids too high or too low delivered volumes.2 Delivery of inappropriately large tidal volumes can lead to alveolar over-distension and development of chronic respiratory morbidity.3 Inappropriately small tidal volumes can be associated with a prolonged duration of mechanical ventilation, a pro-inflammatory state and an increased work of breathing.4,5
An important influence on the appropriate size of the delivered volume during mechanical ventilation is the size of the dead space, the volume of inhaled gas that does not take part in gas exchange. The physiological dead space is the anatomical dead space plus alveolar dead space. The anatomical dead space is the total volume of the conducting airways from the nose or mouth to the terminal bronchioles, and in ventilated infants includes the apparatus dead space (endotracheal tube and flow sensor). Alveolar dead space comprises alveoli which are ventilated, but not supplied by the pulmonary arterial circulation, or alveoli which are atelectatic.6 In pulmonary diseases in infants, there are a variety of pathologies and hence the dead space is not predictable and has rarely been reported.7–9
Minute ventilation fails to adequately describe ventilation efficiency as it does not differentiate between alveolar (effective) and dead space ventilation. In contrast, knowledge of the alveolar ventilation (the volume of air that reaches the alveoli per minute) provides information on the volume of gas taking part in gas exchange at the alveolar–capillary interface.10 A previous study has shown that a reduction in alveolar ventilation resulted in hypercapnia in ventilated infants with respiratory distress syndrome (RDS).11
The aim of this study was to determine if the physiological dead space in prematurely born infants with RDS or evolving/established bronchopulmonary dysplasia (BPD) was greater than in term controls with no respiratory disease. We further aimed to determine whether the size of the dead space influenced the alveolar ventilation in mechanically ventilated infants.
Methods
Study design and participants
A prospective study of ventilated infants at King’s College Hospital NHS Foundation Trust neonatal intensive care unit was undertaken. Infants were recruited from 1 January 2019–1 September 2020 after written informed parental consent was obtained. Approval was given by the London Camden and King’s Cross Research Ethics Committee (REC reference: 18/LO/1602).
Term and preterm infants were eligible for recruitment into the study if they were receiving invasive mechanical ventilation, but did not have major congenital or chromosomal abnormalities. Infants were supported by volume-targeted or pressure-controlled time-cycled ventilation using the SLE6000 neonatal ventilator (SLE, Croydon UK). Infants were intubated, as per unit policy, with shouldered Cole’s endotracheal tubes (ETT).12 All preterm infants who required intubation and ventilation received surfactant.
Term born infants with no underlying respiratory disease who required invasive mechanical ventilatory support for poor perinatal adaptation (term controls), preterm infants with acute RDS, and preterm infants supported by invasive ventilation for longer than 1 week who were classified as having evolving9 or established BPD were recruited.
Protocol
Baseline demographic data were collected including gestational age, birth weight and sex. On the day of study postmenstrual age, postnatal age and current weight were recorded. The average respiratory rate was recorded from the bedside monitor prior to measuring each infant. The partial arterial or capillary pressure of carbon dioxide (PaCO 2) was documented from the latest blood gas analysis which was routinely performed by the clinical team and where possible corresponded with the timing of the research recordings. To collect the ventilatory and respiratory measurement data a 10-min recording was made using the Capnostat-5 sensor, which was inserted between the ETT and ventilator circuit.
Respiratory measurements
An NM3 respiratory profile monitor (Philips Respironics, CT), connected to a combined pressure and flow sensor with mainstream capnograph (Capnostat-5), was used [Fig.1]. The flow and pressure measurements were from a fixed orifice pneumotachometer and differential pressure sensor, respectively. The expired CO 2 was measured using infrared absorption spectroscopy. The respiratory monitor was automatically calibrated for pressure, flow and CO 2 according to the factory-stored calibrations within the monitor. The NM3 monitor was connected to a Laptop (Dell Latitude, UK) with customised Spectra software (3.0.1.4; Grove Medical, UK). The data were exported from Spectra to Microsoft Excel (for Office 365, 2007 version) with a sampling output of 100 Hz for analysis and construction of volumetric capnograms.
Fig. 1. Position of monitoring device within the circuit.
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a combined pneumotachograph and mainstream capnograph b Endotracheal tube c Ventilator tubing.
Volumetric capnography
The start of expiration was defined as the start of negative flow with the end of expiration defined as the end of negative flow. There was no delay between the CO 2 and flow signals with the maximal end-tidal CO 2 aligning with the end of expiration as determined by the flow. Flow data were used to calculate the expired volume during each expiratory phase where flow rate was the change in volume per unit of time. Integration of the volume (derived from the flow signal) and the corresponding CO 2 signal was used to calculate the mean CO 2 of the mixed expired air:
To manually construct the volumetric capnograms for each individual breath, the calculated expired volume was plotted on the x-axis against the corresponding partial pressure of exhaled CO 2 on the y-axis.
The waveforms of flow and CO 2 were deemed to be of sufficient quality for analysis if, after capnogram construction, there were five consecutive breaths of ventilator inflations that were free from artefact interference and allowed for adequate definition between the three phases of the volumetric capnogram. The three phases were related to firstly the emptying of CO 2 free gas from the conducting airways, secondly the mixing of gas from the conducting airways with CO 2 rich alveolar gas, and finally during the last phase of expiration which represents pure alveolar CO 2.13 The calculated values for each of the five breaths were averaged for each infant and subsequently used for the dead space calculations.
Dead space and alveolar ventilation calculations
The dead space was calculated from the Enghoff modification of the Bohr equation by substituting the partial pressure of alveolar carbon dioxide (P A CO 2) with arterial partial pressure (P a CO 2).14 The physiological dead space (V Dphys) (ml) was calculated from the following equation:
The anatomical dead space (V Dana) (ml) was calculated from the following equation:
The alveolar dead space (V Dalv) (ml) was calculated from the following:
where V te is the expired tidal volume in ml, P a CO 2 is the partial pressure of CO 2 in the blood in mmHg, P Et CO 2 is the maximal end-tidal CO 2 in mmHg, and P emean CO 2 is the mean CO 2 of the mixed expired air (mmHg).15
Minute and alveolar ventilation were also calculated:
where V te is the expired tidal volume in ml/kg, V Dphys is the physiological dead space in ml/kg and frequency is the respiratory rate in breaths per minute.
Apparatus dead space
The apparatus dead space consisted of the ETT and mainstream capnograph with the combined flow and CO 2 sensor (Capnostat-5). The Capnostat-5 had a dead space volume of 0.8 ml as measured by water displacement and confirmed by the manufacturer. The ETT “dead space” was also measured using the water displacement technique and was 1.2 ml for a size 2.0 mm ETT, 1.8 ml for 2.5 mm ETT, 2 ml for 3.0 mm ETT, 3 ml for 3.5 mm ETT and 3.2 ml for 4.0 mm ETT.
Sample size calculation
Previous studies have reported a difference in the physiological dead space of 0.8 ml/kg between preterm and term infants16 with a standard deviation of the physiological dead space in preterm infants of 0.78 ml.8 Therefore, to detect a difference in the physiological dead space of 0.8 ml/kg with 90% power at the 5% significance level a minimum of 20 infants within each group was required. Infants were recruited consecutively into the study until the minimum size in each group had been achieved.
Statistical analysis
The data were tested for normality using the Kolmogorov–Smirnov test and found to be non-normally distributed. Data are therefore presented as median values with ranges or interquartile ranges (range or IQRs). The Kruskall–Wallis test was used to assess if there were significant differences in the dead space and alveolar ventilation between the three groups, and if significant differences were identified Mann Whitney U tests were performed to determine between which groups the differences occurred. Statistical analysis was performed using the SPSS software version 25.0 (SPSS Inc., Chicago IL).
Results
Eighty-one infants were recruited into the study with a median (range) gestational age of 28.7 (22.4–41.9) weeks and a birth weight of 1.0 (0.43–4.6) kg. Twenty term infants had no underlying lung pathology (controls), 35 preterm infants had RDS and 26 preterm infants had evolving/established BPD. Two infants subsequently needed a tracheostomy and 21 infants developed severe BPD. The control infants were studied at a median (interquartile range) postnatal age of 4 (2–7) days, infants with RDS at 3 (2–4) days and infants with BPD at 16 (10–25) days [Table1]. At the time of study, the median (IQR) FiO 2 in the RDS group was 0.24 (0.21–0.30) and in the BPD group was 0.34 (0.26–0.49) (p = 0.01).
Table 1.
Baseline demographics of infants in each group.
| | Control (n = 20) | RDS (n = 35) | BPD (n = 26) | p value |
--- ---
| Gestational age at birth (weeks) | 38.4 (36.7–40.2) | 29.1 (25.9–31.7) | 25.8 (25.0–27.3) | <0.001 |
| Birth weight (kg) | 3.30 (2.75–3.66) | 1.00 (0.74–1.41) | 0.70 (0.58–0.82) | <0.001 |
| BW Z-score | −0.07 (−0.72 to 0.63) | −0.41 (−1.56 to 0.00) | −0.76 (−1.66 to −0.19) | 0.066 |
| Postnatal day of life at study (days) | 4 (2–7) | 3 (2–4) | 16 (10–25) | <0.001 |
| Postmenstrual age at study (weeks) | 39.8 (37.6–40.9) | 29.6 (26.1–31.9) | 28.3 (26.9–30.2) | <0.001 |
| Weight at study (kg) | 3.36 (2.85–3.78) | 1.03 (0.74–1.44) | 0.91 (0.77–1.18) | <0.001 |
| End-tidal carbon dioxide (mmHg) | 31.3 (28.3–39.9) | 29.3 (24.2–33.4) | 33.7 (28.2–38.4) | 0.030 |
| Mean expired carbon dioxide (mmHg) | 18.2 (15.4–21.2) | 10.6 (7.4–12.7) | 11.8 (9.9–15.1) | <0.001 |
| Partial arterial pressure of carbon dioxide (mmHg) | 40.4 (34.0–43.1) | 40.0 (34.5–43.7) | 51.4 (47.8–59.0) | <0.001 |
| PaCO 2–EtCO 2 gradient (mmHg) | 5.2 (2.8–12.2) | 10.0 (5.5–16.3) | 17.9 (14.8–27.0) | <0.001 |
| Peak inspiratory pressure (cmH 2 O) | 18.7 (16.5–21.5) | 16.9 (13.1–20.3) | 23.8 (18.8–25.6) | 0.006 |
| Positive end expiratory pressure (cmH 2 O) | 5.9 (5.4–6.0) | 5.4 (5.1–5.9) | 6.1 (5.4–6.7) | 0.005 |
| Respiratory rate (breaths per minute) | 35 (31–39) | 45 (40–58) | 49 (42–55) | <0.001 |
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Data presented as median with interquartile range (IQR) in brackets.
The control infants had expired tidal volumes of 6.4 (5.4–7.6) ml/kg; which were lower than the expired tidal volumes of infants with RDS [8.5 (6.6–9.6) ml/kg] or BPD [8.2 (7.5–10.3) ml/kg] (p = 0.009). Respiratory rate was 35 (31–39) bpm in the term infants and was higher in preterm infants with RDS [45 (40–58) bpm] or BPD [49 (42–55) bpm; p< 0.001]. Term infants had similar PaCO 2 levels [40.4 (34.0–43.1) mmHg] to infants with RDS [40.0 (34.5–43.7) mmHg; p = 0.997], but infants with BPD had higher P aCO 2 levels [51.4 (47.8–59.0) mmHg; p< 0.001] [Table1].
The physiological dead space was higher in infants with RDS [5.7 (5.1–7.0) ml/kg] or in those with evolving/established BPD [6.4 (5.1–7.5) ml/kg] than in term controls [3.5 (2.8–4.3) ml/kg] (<0.001) [Fig.2]. There was no significant difference (p = 0.37) in the physiological dead space between infants with RDS or BPD. Within the BPD group, there was no significant difference in the physiological dead space in those who survived to discharge versus those who died before discharge (p = 0.915). There were also no significant differences in the physiological dead space between infants with mild, moderate or severe BPD (p = 0.187). In the controls, the alveolar dead space was 0.49 (0.30–1.16) ml/kg which was lower than in infants with BPD [0.99 (0.75–1.46) ml/kg; p = 0.006], but similar to infants with RDS [0.68 (0.24–1.46) ml/kg; p = 0.506]. Alveolar dead space was higher in infants with BPD compared to infants with RDS (p = 0.03) [Table2].
Fig. 2. Box and whisker plot of dead space volumes within the three groups.
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The physiological, anatomical and apparatus dead space contributions are shown for term controls and preterm infants with respiratory distress syndrome and evolving bronchopulmonary dysplasia. The ends of each box are the upper and lower quartiles, with the median marked by a horizontal line inside the box. The whiskers mark the minimum and maximum dead space values.
Table 2.
Dead space and alveolar ventilation parameter of infants in each group.
| | Control (n = 20) | RDS (n = 35) | BPD (n = 26) | p value | Control–RDS | Control–BPD | RDS–BPD |
--- --- --- --- |
| Physiological dead space (ml/kg) | 3.5 (2.8–4.3) | 5.7 (5.1–7.0) | 6.4 (5.1–7.5) | <0.001 | <0.001 | <0.001 | 0.372 |
| Total anatomical dead space (ml/kg) | 2.7 (2.4–3.2) | 5.1 (4.7–5.9) | 5.4 (4.4–6.1) | <0.001 | <0.001 | <0.001 | 0.811 |
| Apparatus dead space (ml/kg) | 1.2 (1.1–1.4) | 2.8 (2.0–3.8) | 3.1 (2.8–3.7) | <0.001 | <0.001 | <0.001 | 0.353 |
| Anatomical minus apparatus dead space (ml/kg) | 1.6 (1.1–2.2) | 2.4 (1.5–3.2) | 2.0 (1.4–2.9) | 0.171 | | | |
| Alveolar dead space (ml/kg) | 0.5 (0.3–1.2) | 0.7 (0.2–1.5) | 1.0 (0.8–1.4) | 0.018 | 0.506 | 0.006 | 0.030 |
| Tidal volume (ml/kg) | 6.4 (5.4–7.6) | 8.5 (6.6–9.6) | 8.2 (7.5–10.3) | 0.009 | 0.008 | 0.004 | 0.745 |
| Dead space to tidal volume ratio | 0.52 (0.43–0.62) | 0.75 (0.64–0.81) | 0.77 (0.71–0.82) | <0.001 | <0.001 | <0.001 | 0.220 |
| Minute ventilation (ml/kg/min) | 239 (193–378) | 369 (311–450) | 403 (321–505) | 0.003 | 0.002 | 0.001 | 0.580 |
| Alveolar ventilation (ml/kg/min) | 111 (81–159) | 83 (66–137) | 82 (62–133) | 0.537 | | | |
Open in a new tab
Data presented as median with interquartile range (IQR) in brackets.
The total anatomical dead space in term infants was 2.7 (2.8–4.3) ml/kg, which was lower than in infants with RDS [5.1 (4.7–5.9) ml/kg; p< 0.001] and infants with BPD [5.4 (4.4–6.1) ml/kg; p< 0.001]. The apparatus dead space was lower in term infants [1.2 (1.1–1.4) ml/kg] than preterm infants with either RDS [2.8 (2.0–3.8) ml/kg; p< 0.001] or BPD [3.1 (2.8–3.7); p< 0.001]. After subtracting the apparatus contribution from the total anatomical dead space there was no significant difference in the remaining anatomical dead space between the three groups (p = 0.17) [Table2].
Minute ventilation was higher in both infants with RDS [369 (311–450) ml/kg/min] or BPD [403 (321–505) ml/kg/min] compared to the controls [239 (193–378) ml/kg/min; p = 0.003]. Alveolar ventilation in preterm infants with RDS [83 (66–137) ml/kg/min] or BPD [83 (62–133) ml/kg/min] was not significantly different to that of the term controls [111 (81–159) ml/kg/min; p = 0.54] [Table2].
Discussion
We have demonstrated that in ventilated newborns, preterm infants with RDS or evolving/established BPD had a higher physiological dead space than term infants without respiratory disease. Furthermore, there was an increase in minute ventilation in preterm infants with underlying lung pathology and as a consequence the alveolar ventilation was similar in the three groups.
The increase in alveolar dead space in infants with evolving/established BPD arises from the underlying disease process. BPD is a heterogeneous lung disease characterised by simplified alveoli with areas of cystic change.17 Computerised tomography studies have correlated the severity of BPD with pulmonary hyper-expansion, emphysematous changes and interstitial abnormalities seen on imaging.18 Cystic areas developing within the lung parenchyma in infants with evolving/established BPD may explain therefore our findings of an increase in alveolar dead space in comparison to healthy term controls. The BPD and RDS patients had similar V/A, yet the BPD patients had higher pCO 2 levels. We suggest the mechanism underlying this finding is that infants with BPD develop non-functional cystic areas which do not contribute to gas exchange and clearance of carbon dioxide.
The preterm infants with either RDS or BPD had a greater apparatus dead space volume per kilogram. This finding emphasises the concerns regarding the contribution of the apparatus dead space (ETT and flow sensor) in extremely low birth weight infants who have small tidal volumes.9,19
The values of physiological dead space in our study are higher than previously reported.8,20 One explanation is that infants in our study were less mature and of lower birth weight than those previously reported and were requiring invasive ventilatory support for underlying lung disease. Furthermore, used the Enghoff modification of the Bohr equation to calculate the dead space—this modification substitutes alveolar CO 2 with arterial CO 2 due to the technical and methodological difficulties in obtaining accurate alveolar CO 2 values. The Bohr equation gives the true dead space, whereas the Enghoff modification may overestimate the dead space as it reflects not only the dead space, but also the degree of intrapulmonary shunt, diffusion impairment and ventilation-perfusion inhomogeneity.21
Our study has strengths and some limitations. The European Respiratory Society/American Thoracic Society (ERS/ATS) task force standards for infant respiratory function testing state that the additional dead space of the measuring apparatus should be not more than 2 ml/kg.22 We used a combined pressure and flow sensor with mainstream capnograph to calculate the physiological measurements which had a dead space below that recommended for all infants recruited.22 A further strength is that we have used the Bohr–Enghoff equation to calculate dead space parameters as opposed to Fletcher’s method, which uses Fowler’s equal triangles to calculate the anatomical dead space and may prove not-applicable in low birth weight infants with high respiratory rates and short expiratory times.15 The former method has previously been shown to be superior in ventilated preterm infants and in one study could be used for all infant dead space calculations irrespective of the capnogram shape.20 In that study20 the single breath CO 2 test was used which has been superseded by low dead space, fast response CO 2 analysers which were employed in the present study. Our method of volumetric capnography, however, does require infants to be invasively ventilated and thus is not generalisable to the whole population of preterm infants. Knowledge of the physiological dead space is most relevant in ventilated infants to determine the optimum delivered volume. A further limitation is that we use shouldered endotracheal tubes which are not routinely employed in other units, it would, therefore, be interesting to repeat our study in a unit which uses straight endotracheal tubes. We do not have the exact timings of blood gas analysis, but blood gases were done on a clinical basis and the research measurements were performed as close in time as possible, usually within two hours.
Our study has important clinical implications suggesting that the dead space should be considered alongside other clinical parameters when deciding the magnitude of targeted tidal volumes required to maintain effective alveolar ventilation. Our results suggest that premature infants with RDS or evolving BPD might need higher tidal volume targeting levels compared to term infants with no respiratory disease. It would be interesting to assess whether these early measures of physiological dead space could predict longer term outcomes, this was not possible in this cohort as all but one infant with evolving BPD went on to develop established BPD.
In conclusion we have reported that the physiological dead space is higher in ventilated preterm infants with RDS or evolving/established BPD compared to term controls without respiratory disease. These findings should be taken into consideration when determining the magnitude of the volumes that should be delivered in volume-targeted modes of ventilation.
Acknowledgments
This research was supported by the National Institute for Health Research (NIHR) Biomedical Research Centre at Guy’s and St Thomas’ NHS Foundation Trust and King’s College London. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health.
Author contributions
T.D., A.G. and E.W. conceptualised and designed the study. E.W. acquired the data. E.W., P.D. and T.D. analysed the data. E.W., T.D. and A.G. interpreted the data. E.W. wrote the first draft of the manuscript. All authors were involved revising the manuscript and approved the final version.
Competing interests
The authors declare no competing interests.
Consent statement
Written informed parental consent was obtained for all infants recruited into the study (REC reference: 18/LO/1602).
Footnotes
The original online version of this article was revised due to a retrospective Open Access order.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Change history
8/17/2021
A Correction to this paper has been published: 10.1038/s41390-021-01676-3
References
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6517 | https://www.mathway.com/popular-problems/Calculus/520793 | Calculus Examples
limx→∞ln(x)x
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
limx→∞ln(x)limx→∞x
Step 1.1.2
As log approaches infinity, the value goes to ∞.
∞limx→∞x
Step 1.1.3
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
∞∞
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Infinity divided by infinity is undefined.
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Step 1.2
Since ∞∞ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
limx→∞ln(x)x=limx→∞ddx[ln(x)]ddx[x]
Step 1.3
Find the derivative of the numerator and denominator.
Tap for more steps...
Step 1.3.1
Differentiate the numerator and denominator.
limx→∞ddx[ln(x)]ddx[x]
Step 1.3.2
The derivative of ln(x) with respect to x is 1x.
limx→∞1xddx[x]
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Differentiate using the Power Rule which states that ddx[xn] is nxn−1 where n=1.
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Multiply 1x by 1.
limx→∞1x
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Since its numerator approaches a real number while its denominator is unbounded, the fraction 1x approaches 0.
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6518 | https://texasgateway.org/resource/43-newtons-second-law-motion | Skip to main content
Learning Objectives
Describing Newton’s Second Law of Motion
Applying Newton’s Second Law
Practice Problems
Check Your Understanding
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
Describe Newton’s second law, both verbally and mathematically
Use Newton’s second law to solve problems
Section Key Terms
| freefall | Newton’s second law of motion | weight |
Describing Newton’s Second Law of Motion
Describing Newton’s Second Law of Motion
Newton’s first law considered bodies at rest or bodies in motion at a constant velocity. The other state of motion to consider is when an object is moving with a changing velocity, which means a change in the speed and/or the direction of motion. This type of motion is addressed by Newton’s second law of motion, which states how force causes changes in motion. Newton’s second law of motion is used to calculate what happens in situations involving forces and motion, and it shows the mathematical relationship between force, mass, and acceleration. Mathematically, the second law is most often written as
4.2
Fnet=maorΣF=ma,
where Fnet (or ∑F) is the net external force, m is the mass of the system, and a is the acceleration. Note that Fnet and ∑F are the same because the net external force is the sum of all the external forces acting on the system.
First, what do we mean by a change in motion? A change in motion is simply a change in velocity: the speed of an object can become slower or faster, the direction in which the object is moving can change, or both of these variables may change. A change in velocity means, by definition, that an acceleration has occurred. Newton’s first law says that only a nonzero net external force can cause a change in motion, so a net external force must cause an acceleration. Note that acceleration can refer to slowing down or to speeding up. Acceleration can also refer to a change in the direction of motion with no change in speed, because acceleration is the change in velocity divided by the time it takes for that change to occur, and velocity is defined by speed and direction.
From the equation Fnet=ma,we see that force is directly proportional to both mass and acceleration, which makes sense. To accelerate two objects from rest to the same velocity, you would expect more force to be required to accelerate the more massive object. Likewise, for two objects of the same mass, applying a greater force to one would accelerate it to a greater velocity.
Now, let’s rearrange Newton’s second law to solve for acceleration. We get
4.3
a=Fnetmora=ΣFm.
In this form, we can see that acceleration is directly proportional to force, which we write as
4.4
a∝Fnet,
where the symbol ∝ means proportional to.
This proportionality mathematically states what we just said in words: acceleration is directly proportional to the net external force. When two variables are directly proportional to each other, then if one variable doubles, the other variable must double. Likewise, if one variable is reduced by half, the other variable must also be reduced by half. In general, when one variable is multiplied by a number, the other variable is also multiplied by the same number. It seems reasonable that the acceleration of a system should be directly proportional to and in the same direction as the net external force acting on the system. An object experiences greater acceleration when acted on by a greater force.
It is also clear from the equation a=Fnet/m that acceleration is inversely proportional to mass, which we write as
4.5
a ∝ 1m.
Inversely proportional means that if one variable is multiplied by a number, the other variable must be divided by the same number. Now, it also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. This relationship is illustrated in Figure 4.6, which shows that a given net external force applied to a basketball produces a much greater acceleration than when applied to a car.
Figure 4.6 The same force exerted on systems of different masses produces different accelerations. (a) A boy pushes a basketball to make a pass. The effect of gravity on the ball is ignored. (b) The same boy pushing with identical force on a stalled car produces a far smaller acceleration (friction is negligible). Note that the free-body diagrams for the ball and for the car are identical, which allows us to compare the two situations.
Applying Newton’s Second Law
Applying Newton’s Second Law
Before putting Newton’s second law into action, it is important to consider units.
The equation Fnet=ma is used to define the units of force in terms of the three basic units of mass, length, and time (recall that acceleration has units of length divided by time squared). The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of 1 m/s2. That is, because Fnet=ma, we have
4.6
1 N=1 kg×1 m/s2=1 kg⋅ms2.
One of the most important applications of Newton’s second law is to calculate weight (also known as the gravitational force), which is usually represented mathematically as W. When people talk about gravity, they don’t always realize that it is an acceleration. When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that the net external force acting on an object is responsible for the acceleration of the object. If air resistance is negligible, the net external force on a falling object is only the gravitational force (i.e., the weight of the object).
Weight can be represented by a vector because it has a direction. Down is defined as the direction in which gravity pulls, so weight is normally considered a downward force. By using Newton’s second law, we can figure out the equation for weight.
Consider an object with mass m falling toward Earth. It experiences only the force of gravity (i.e., the gravitational force or weight), which is represented by W. Newton’s second law states that Fnet=ma. Because the only force acting on the object is the gravitational force, we have Fnet=W. We know that the acceleration of an object due to gravity is g, so we have a=g. Substituting these two expressions into Newton’s second law gives
4.7
W=mg.
This is the equation for weight—the gravitational force on a mass m. On Earth, g=9.80 m/s2, so the weight (disregarding for now the direction of the weight) of a 1.0-kg object on Earth is
4.8
W=mg=(1.0 kg)(9.80 m/s2)=9.8 N.
Although most of the world uses newtons as the unit of force, in the United States the most familiar unit of force is the pound (lb), where 1 N = 0.225 lb.
Recall that although gravity acts downward, it can be assigned a positive or negative value, depending on what the positive direction is in your chosen coordinate system. Be sure to take this into consideration when solving problems with weight. When the downward direction is taken to be negative, as is often the case, acceleration due to gravity becomes
g = −9.8 m/s2.
When the net external force on an object is its weight, we say that it is in freefall. In this case, the only force acting on the object is the force of gravity. On the surface of Earth, when objects fall downward toward Earth, they are never truly in freefall because there is always some upward force due to air resistance that acts on the object (and there is also the buoyancy force of air, which is similar to the buoyancy force in water that keeps boats afloat).
Gravity varies slightly over the surface of Earth, so the weight of an object depends very slightly on its location on Earth. Weight varies dramatically away from Earth’s surface. On the moon, for example, the acceleration due to gravity is only 1.67 m/s2. Because weight depends on the force of gravity, a 1.0-kg mass weighs 9.8 N on Earth and only about 1.7 N on the moon.
It is important to remember that weight and mass are very different, although they are closely related. Mass is the quantity of matter (how much stuff) in an object and does not vary, but weight is the gravitational force on an object and is proportional to the force of gravity. It is easy to confuse the two, because our experience is confined to Earth, and the weight of an object is essentially the same no matter where you are on Earth. Adding to the confusion, the terms mass and weight are often used interchangeably in everyday language; for example, our medical records often show our weight in kilograms, but never in the correct unit of newtons.
Snap Lab
Mass and Weight
In this activity, you will use a scale to investigate mass and weight.
Materials:
1 bathroom scale
1 table
What do bathroom scales measure?
When you stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that compress in proportion to your weight—similar to rubber bands expanding when pulled.
The springs provide a measure of your weight (provided you are not accelerating). This is a force in newtons (or pounds). In most countries, the measurement is now divided by 9.80 to give a reading in kilograms, which is a of mass. The scale detects weight but is calibrated to display mass.
If you went to the moon and stood on your scale, would it detect the same mass as it did on Earth?
Grasp Check
While standing on a bathroom scale, push down on a table next to you. What happens to the reading? Why?
The reading increases because part of your weight is applied to the table and the table exerts a matching force on you that acts in the direction of your weight.
The reading increases because part of your weight is applied to the table and the table exerts a matching force on you that acts in the direction opposite to your weight.
The reading decreases because part of your weight is applied to the table and the table exerts a matching force on you that acts in the direction of your weight.
The reading decreases because part of your weight is applied to the table and the table exerts a matching force on you that acts in the direction opposite to your weight.
Tips For Success
Only net external force impacts the acceleration of an object. If more than one force acts on an object and you calculate the acceleration by using only one of these forces, you will not get the correct acceleration for that object.
Watch Physics
Newton’s Second Law of Motion
This video reviews Newton’s second law of motion and how net external force and acceleration relate to one another and to mass. It also covers units of force, mass, and acceleration, and reviews a worked-out example.
Click to view content
Grasp Check
True or False—If you want to reduce the acceleration of an object to half its original value, then you would need to reduce the net external force by half.
True
False
Worked Example
What Acceleration Can a Person Produce when Pushing a Lawn Mower?
Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N parallel to the ground. The mass of the mower is 240 kg. What is its acceleration?
Figure 4.7
Strategy
Because Fnet and m are given, the acceleration can be calculated directly from Newton’s second law: Fnet = ma.
Solution
Solving Newton’s second law for the acceleration, we find that the magnitude of the acceleration, a, is a=Fnetm. Entering the given values for net external force and mass gives
4.9
a=51 N240 kg
Inserting the units kg⋅m/s2 for N yields
4.10
a=51 kg⋅m/s2240 kg=0.21 m/s2.
Discussion
The acceleration is in the same direction as the net external force, which is parallel to the ground and to the right. There is no information given in this example about the individual external forces acting on the system, but we can say something about their relative magnitudes. For example, the force exerted by the person pushing the mower must be greater than the friction opposing the motion, because we are given that the net external force is in the direction in which the person pushes. Also, the vertical forces must cancel if there is no acceleration in the vertical direction (the mower is moving only horizontally). The acceleration found is reasonable for a person pushing a mower; the mower’s speed must increase by 0.21 m/s every second, which is possible. The time during which the mower accelerates would not be very long because the person’s top speed would soon be reached. At this point, the person could push a little less hard, because he only has to overcome friction.
Worked Example
What Rocket Thrust Accelerates This Sled?
Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on humans at high accelerations. Rocket sleds consisted of a platform mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust, T, for the four-rocket propulsion system shown below. The sled’s initial acceleration is 49 m/s2, the mass of the system is 2,100 kg, and the force of friction opposing the motion is 650 N.
Figure 4.8
Strategy
The system of interest is the rocket sled. Although forces act vertically on the system, they must cancel because the system does not accelerate vertically. This leaves us with only horizontal forces to consider. We’ll assign the direction to the right as the positive direction. See the free-body diagram in Figure 4.8.
Solution
We start with Newton’s second law and look for ways to find the thrust T of the engines. Because all forces and acceleration are along a line, we need only consider the magnitudes of these quantities in the calculations. We begin with
4.11
Fnet=ma,
where Fnet is the net external force in the horizontal direction. We can see from Figure 4.8 that the engine thrusts are in the same direction (which we call the positive direction), whereas friction opposes the thrust. In equation form, the net external force is
4.12
Fnet=4T−f .
Newton’s second law tells us that Fnet= ma, so we get
4.13
ma=4T−f .
After a little algebra, we solve for the total thrust 4T:
4.14
4T=ma+f ,
which means that the individual thrust is
4.15
T=ma+f4 .
Inserting the known values yields
4.16
T=(2100 kg)(49 m/s2)+650 N4=2.6×104 N .
Discussion
The numbers are quite large, so the result might surprise you. Experiments such as this were performed in the early 1960s to test the limits of human endurance and to test the apparatus designed to protect fighter pilots in emergency ejections. Speeds of 1000 km/h were obtained, with accelerations of 45 g. (Recall that g, the acceleration due to gravity, is 9.80 m/s2. An acceleration of 45 g is 45×9.80 m/s2, which is approximately 440 m/s2.) Living subjects are no longer used, and land speeds of 10,000 km/h have now been obtained with rocket sleds. In this example, as in the preceding example, the system of interest is clear. We will see in later examples that choosing the system of interest is crucial—and that the choice is not always obvious.
Practice Problems
Practice Problems
If 1 N is equal to 0.225 lb, how many pounds is 5 N of force?
0.045 lb
1.125 lb
2.025 lb
5.000 lb
How much force needs to be applied to a 5-kg object for it to accelerate at 20 m/s2?
1 N
10 N
100 N
1,000 N
Check Your Understanding
Check Your Understanding
Exercise 9
What is the mathematical statement for Newton’s second law of motion?
F = ma
F = 2ma
F=ma
F = ma2
Exercise 10
Newton’s second law describes the relationship between which quantities?
Force, mass, and time
Force, mass, and displacement
Force, mass, and velocity
Force, mass, and acceleration
Exercise 11
What is acceleration?
Acceleration is the rate at which displacement changes.
Acceleration is the rate at which force changes.
Acceleration is the rate at which velocity changes.
Acceleration is the rate at which mass changes. |
6519 | https://math.stackexchange.com/questions/3958878/derive-rodrigues-formula-for-laguerre-polynomials | special functions - Derive Rodrigues’ formula for Laguerre polynomials - Mathematics Stack Exchange
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Derive Rodrigues’ formula for Laguerre polynomials
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Derive Rodrigues’ formula for Laguerre polynomials
L n(x)=e x n!.d n d x n(x n e−x)L n(x)=e x n!.d n d x n(x n e−x)
The Rodrigues’ formula for Hermite polynomials can be obtained by taking n t h n t h order partial derivatives of its generatig function
g(x,t)=∑n=0∞H n(x)t n n!=1+t H 1(x)+t 2 2!H 2(x)+⋯⋯⋯+t n n!H n(x)+⋯⋯⋯∂n∂t n(e 2 x t−t 2)=H n(x)+(n+1)n(n−1)⋯2(n+1)!t H n+1(x)+⋯H n(x)=[∂n∂t n(e 2 x t−t 2)]t=0=e x 2[∂n∂t n(e−(x−t)2)]t=0=(−1)n e x 2[∂n∂x n(e−(x−t)2)]t=0=(−1)n e x 2[∂n∂x n(e−x 2)]g(x,t)=∑n=0∞H n(x)t n n!=1+t H 1(x)+t 2 2!H 2(x)+⋯⋯⋯+t n n!H n(x)+⋯⋯⋯∂n∂t n(e 2 x t−t 2)=H n(x)+(n+1)n(n−1)⋯2(n+1)!t H n+1(x)+⋯H n(x)=[∂n∂t n(e 2 x t−t 2)]t=0=e x 2[∂n∂t n(e−(x−t)2)]t=0=(−1)n e x 2[∂n∂x n(e−(x−t)2)]t=0=(−1)n e x 2[∂n∂x n(e−x 2)]
Generating function for laguerre polynomials is g(x,t)=e−x t 1−t 1−t=∑∞n=0 L n(x)t n g(x,t)=e−x t 1−t 1−t=∑n=0∞L n(x)t n
I do not think the same technique applies in the case of Laguerre polynomials. So how do I derive that for Laguerre polynomials ?
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edited Dec 24, 2020 at 10:33
GNUSupporter 8964民主女神 地下教會
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asked Dec 22, 2020 at 22:05
SOORAJ SOMANSOORAJ SOMAN
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Just done it another way.metamorphy –metamorphy 2021-01-20 18:03:54 +00:00 Commented Jan 20, 2021 at 18:03
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Note that the Taylor-Maclaurin formula cannot be used directly (as in the case of Hermite polynomials) as there is some n n dependency inside the n t h n t h derivative. How can we get around this ? ... Let use a coefficient extrator
x n=[u 0]:u−n 1−x u.x n=[u 0]:u−n 1−x u.
We have
∑n=0∞L n(x)t n===[u 0]:e x∑n=0∞(t u)n 1 n!d n d x n e−x 1−x u[u 0]:e x e−(x+t/u)1−u(x+t/u)1 1−t[u 0]:e−t/u 1−u x 1−t.∑n=0∞L n(x)t n=[u 0]:e x∑n=0∞(t u)n 1 n!d n d x n e−x 1−x u=[u 0]:e x e−(x+t/u)1−u(x+t/u)=1 1−t[u 0]:e−t/u 1−u x 1−t.
Expand these two functions and observe that the central term is ... what we want
(∑i=0∞(−t/u)i i!)(∑j=0∞(u x 1−t)j)=⋯+∑k=0∞1 k!(−x t 1−t)k+⋯(∑i=0∞(−t/u)i i!)(∑j=0∞(u x 1−t)j)=⋯+∑k=0∞1 k!(−x t 1−t)k+⋯
Now reverse engineer all this ... and your result follows.
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edited Dec 24, 2020 at 12:57
answered Dec 22, 2020 at 23:07
Donald SplutterwitDonald Splutterwit
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Could you please elaborate a little more as I have difficulty in following your approach. The coefficient corresponds to u 0 u 0 in the expansion of u−n 1−x u u−n 1−x u is 1 n!x n 1 n!x n, right ?SOORAJ SOMAN –SOORAJ SOMAN 2020-12-23 06:46:39 +00:00 Commented Dec 23, 2020 at 6:46
The coefficient of u 0 u 0 in u−n(1+x u+x 2 u 2+⋯x n u n+⋯)u−n(1+x u+x 2 u 2+⋯x n u n+⋯) would be x n x n. (not sure where the 1/n!1/n! came from ?) The coefficient of u 0 u 0 in u−n e x u u−n e x u would be x n/n!x n/n!Donald Splutterwit –Donald Splutterwit 2020-12-23 12:37:08 +00:00 Commented Dec 23, 2020 at 12:37
ohh thats my mistake, its an infinite GP right, I got confused by thinking about binomial expansion of the expression.SOORAJ SOMAN –SOORAJ SOMAN 2020-12-23 14:23:51 +00:00 Commented Dec 23, 2020 at 14:23
I also have trouble understanding how the expression [u 0]:e x∑∞n=0(t u)n e x n!d n d x n e−x 1−x u[u 0]:e x∑n=0∞(t u)n e x n!d n d x n e−x 1−x u becomes [u 0]:e x e−(x+t/u)1−u(x+t/u)[u 0]:e x e−(x+t/u)1−u(x+t/u), rest seems fine, could you please clarify on that ?SOORAJ SOMAN –SOORAJ SOMAN 2020-12-23 14:39:01 +00:00 Commented Dec 23, 2020 at 14:39
This is the Taylor-Maclaurin Series f(x+a)=f(x)+a f′(x)+a 2 2!f′′(x)+a 3 3!f(3)(x)+⋯f(x+a)=f(x)+a f′(x)+a 2 2!f″(x)+a 3 3!f(3)(x)+⋯ see en.wikipedia.org/wiki/Taylor_seriesDonald Splutterwit –Donald Splutterwit 2020-12-23 14:43:18 +00:00 Commented Dec 23, 2020 at 14:43
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If you are allowed to use the explicit representation for Laguerre polynomials,
L n(x)=∑k=0 n(n k)(−x)k k!,L n(x)=∑k=0 n(n k)(−x)k k!,
then the result follows easily from the Leibniz differentiation rule for products
(f(x)g(x))(n)=∑k=0 n(n k)f(k)(x)g(n−k)(x).(f(x)g(x))(n)=∑k=0 n(n k)f(k)(x)g(n−k)(x).
Refer to the Rodrigues formula you want to prove. Let f(x)=x n f(x)=x n and g(x)=e−x.g(x)=e−x. Then f(k)(x)=x n−k n!/(n−k)!f(k)(x)=x n−k n!/(n−k)! and g(n−k)(x)=(−1)n−k e−x.g(n−k)(x)=(−1)n−k e−x. Now clean it up and redo the sum in reverse order, i.e., k→n−k,k→n−k, and you'll get the top formula.
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answered Dec 23, 2020 at 19:12
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6520 | https://www.splashlearn.com/math-vocabulary/geometry/isosceles-triangle | Published Time: 2022-04-21T04:55:53+00:00
Isosceles Triangle - Definition, Angles, Properties, Examples
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Isosceles Triangles – Definition, Parts, Types, Properties, Examples
Home » Math Vocabulary » Isosceles Triangles – Definition, Parts, Types, Properties, Examples
What Is an Isosceles Triangle?
Parts of an Isosceles Triangle
Isosceles Triangle Formulas
Solved Examples on Isosceles Triangle
Practice Problems on Isosceles Triangles
Frequently Asked Questions about Isosceles Triangle
What Is an Isosceles Triangle?
A triangle with two sides of equal length is an isosceles triangle.
Examples of Isosceles Triangle:
Non-example of Isosceles Triangle:
Examples of Isosceles Triangles in Real Life:
Many things in the world have the shape of an isosceles triangle. Some popular examples of these triangles in real life are:
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Parts of an Isosceles Triangle
Source: Parts of an isosceles triangle
1. Legs: The two equal sides of an isosceles triangle are called “legs.” In triangle ABC, sides AB and BC are the two legs of the isosceles triangle.
2. Base: The ‘base’ of an isosceles triangle is the third and unequal side. Here, side BC is the base of the isosceles triangle ABC.
3. Vertex angle: The vertex angle is formed by two equal sides of an isosceles triangle. ∠BAC is the vertex angle of the isosceles triangle.
4. Base angles: The ‘base angles’ are the angles that involve the base of an isosceles triangle. ∠ABC and ∠ACB are the two base angles of the isosceles triangle.
Properties of an Isosceles Triangle
Here is a list of some properties of isosceles triangles:
In an isosceles triangle, if two sides are equal, then the angles opposite to the two sides correspond to each other and are also always equal.
In the given isosceles triangle ABC, the two angles ∠B and ∠C, opposite to the equal sides AB and BC, are equal to each other.
The sum of three angles of an isosceles triangle is always 180°.
Types of Isosceles Triangles
Generally, isosceles triangles are classified into three different types:
Isosceles acute triangle: An isosceles acute triangle is a triangle in which all three angles are less than 90 degrees, and at least two of its angles are equal in measurement. One example of isosceles acute triangle angles is 50°, 50°, and 80°.
Isosceles right triangle: This is a right triangle with two legs (and their corresponding angles) of equal measure.
Isosceles obtuse triangle: An isosceles obtuse triangle is a triangle in which one of the three angles is obtuse (lies between 90 degrees and 180 degrees), and the other two acute angles are equal in measurement. One example of isosceles obtuse triangle angles is 30°, 30°, and 120°.
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Isosceles Triangle Formulas
Area of an Isosceles Triangle: The area of an isosceles triangle is given by the following formula:
Area = 12 × base × height
A = 12 × b × H
Perimeter of an Isosceles Triangle: The perimeter of the isosceles triangle is given by
Perimeter = Sum of equal sides + base
Perimeter = 2a + b
Here, ‘a’ refers to the length of the equal sides of the isosceles triangle, and ‘b’ refers to the length of the third unequal side.
Solved Examples on Isosceles Triangle
Example 1: What is the height of an isosceles triangle with an area of 12 sq. cm and a base of 6 cm?
Solution:
Area of isosceles triangle = ½ × base × height
12 = ½ × 6 × height
12 = 3 × height
Height = 4 cm
Example 2: How do you calculate the perimeter of an isosceles triangle with two sides of length ‘a’ cm each and a third side of length ‘b’ cm?
Solution:
Perimeter of an isosceles triangle = sum of its sides
Perimeter of an isosceles triangle = (a + a + b) cm
Perimeter = (2a + b) cm
Example 3: Find the perimeter of an isosceles triangle if the base is 16 cm and the equal sides are 24 cm each.
Solution:
The formula of the perimeter of an isosceles triangle is
P = 2a + b
Here, a (length of equal sides) = 24 cm and b (base) = 16 cm
Therefore, the perimeter of an isosceles triangle = P= 2(24) + 16 = 64 cm
Hence, the perimeter is 64 cm.
Practice Problems on Isosceles Triangles
Isosceles Triangles
Attend this Quiz & Test your knowledge.
1
What is the height of an isosceles triangle with an area of 10 sq. cm and a base of 5 cm?
10 cm
5 cm
2 cm
4 cm
CorrectIncorrect
Correct answer is: 4 cm
Area of an isosceles triangle = 12 × base × height
10 cm2 = 12 × 5 cm × height
Height = 4 cm
2
In ΔABC, if ∠A = ∠B, then
AC ≠ BC
AC = BC
AB = AC
AB = BC
CorrectIncorrect
Correct answer is: AC = BC
Given: ∠A = ∠B
Sides opposite to equal angles are also equal.
In ΔABC, side BC is opposite to ∠A, and side AC is opposite to angle B.
Therefore, AC = BC.
3
What is the area of the given isosceles triangle?
21 cm2
45 cm2
90 cm2
180 cm2
CorrectIncorrect
Correct answer is: 45 cm2
Area of an isosceles triangle = 12 × base × height = 12 × 15 cm × 6 cm = 45 cm2
Frequently Asked Questions about Isosceles Triangle
How do we know if a triangle is isosceles?
A triangle is said to be an isosceles triangle if any of its two sides are equal. Let’s take a triangle that has AB, BC, and CA as its three sides. If any of these are true—AB = BC, BC = CA or CA = AB—then the triangle is isosceles.
Can a right triangle also be an isosceles triangle?
Yes, a right triangle or right-angle triangle can be an isosceles triangle. An isosceles right triangle will have 1 right angle and 2 other angles as equal angles.
Can you find all the angles of an isosceles triangle if you know one of the equal angles?
Yes! We know that the sum of all angles of a triangle is equal to 180 degrees. If we know the two equal angles, we can easily add them and subtract the sum from 180 degrees to find the third angle.
What are some of the properties of an isosceles triangle?
It has two sides of equal length.
The angles opposite to the equal sides are also equal in measure.
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6521 | https://www.dummies.com/article/academics-the-arts/math/algebra/remove-a-radical-from-an-equation-by-squaring-both-sides-138671/ | Book & Article Categories
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Remove a Radical from an Equation by Squaring Both Sides
Radicals can make equations very difficult to work with. Fortunately, it's possible to get rid of a radical by squaring both sides of the equation; for this method to work, the equation must be in the form
The only problem arises when you end up with an extraneous root. Consider the non-equation –3 = 3. You know that the equation isn't correct, but what happens when you square both sides of this statement? You get (–3)2 = (3)2, or 9 = 9. Now you have an equation. Squaring both sides can mask or hide an incorrect statement.
Much like the process of getting rid of fractions in equations, the method of squaring both sides is the easiest way to deal with radicals in equations. You just accept that you always have to watch for extraneous roots when solving equations by squaring.
For example, to solve the equation
follow these steps:
Change the equation so that the radical term is by itself on the left.
In this case, you add 6 to each side of the equation.
Square both sides of the equation.
Note that a very common error when squaring problems is to square the binomial on the right incorrectly. Don't forget the middle term — you can't just square the two terms alone [(a + b)2 = a2 + 2ab + b2].
Simplify the result and solve the resulting equation.
In this example, you have a quadratic equation. Set it equal to zero and solve it:
When x + 3 = 0, x = –3. When x + 5 = 0, x = –5.
Check to see whether your solutions fit with the original equation.
Plug your answers in the original equation,
When x = –3,
It works. Checking x = –5, you get
This solution works, too.
Both solutions working out is more the exception rather than the rule when you're dealing with radicals. Most of the time, one solution or the other works, but not both. And, unfortunately, sometimes you go through all the calculations and find that neither solution works in the original equation. You get an answer, of course (that there is no answer), but it isn't very fulfilling.
About This Article
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About the book author:
Mary Jane Sterling (Peoria, Illinois) is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and many other For Dummies books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics.
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6522 | https://books.google.com/books/about/Pocket_Companion_to_Robbins_Cotran_Patho.html?id=YBm-EAAAQBAJ | Pocket Companion to Robbins & Cotran Pathologic Basis of Disease: First ... - Richard N. Mitchell, Vinay Kumar, Abul K. Abbas, Jon C. Aster - Google Books
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Pocket Companion to Robbins & Cotran Pathologic Basis of Disease: First South Asia Edition - E-book =================================================================================================== Richard N. Mitchell, Vinay Kumar, Abul K. Abbas, Jon C. Aster Elsevier Health Sciences, Sep 13, 2016 - Medical - 892 pages This pocket companion offers rapid, portable access to the most important pathology facts and concepts from Robbins and Cotran Pathologic Basis of Disease, First South Asia Edition. It distills the key concepts and principles of pathology into a condensed, at-a-glance format, making it the perfect pocket-sized reference for quick review anytime! - Access key concepts and principles of pathology in a condensed, at-a-glance format that fits in your pocket. - Locate additional information with abundant page references to the parent text. More » Preview this book »
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abnormalacidactivationacuteadenomasand/oranemiaantibodiesantigensapoptosisarteryassociatedautoimmuneautosomalbacterialbenignbindingbloodbonecancercarcinomacardiaccausecellscellularcharacterizedchromosomechronicClinical Featurescollagencommoncomplexcongenitalcysticcystscytokinesdamagedefectsdeficiencydevelopdiabetesdiseasedisordersedemaendothelialendotheliumenzymesepithelialepitheliumfibrosisfunctiongenegeneticgrowth factorhemorrhagehepatitishistologichormonehyperplasiahypertensionimmuneincreasedinduceinfarctioninfectionsinfiltratesinflammationinflammatoryinjuryintracellularinvolvedischemiakinaseleadslesionsleukocytesliverlosslunglymphocyteslymphomalysosomalmacrophagesmalignantmarrowmediatorsmembranemetabolicmetastasesMicroscopicmoleculesMorphologymucosalmusclemutationsnecrosisneoplasmsneuronsneutrophilsnormalobstructionoccurosteoclastpancreaticPathogenesispathwayspatientsperipheralplasmaplateletprimaryproductionprogressiveproliferationproteinpulmonaryreceptorrenalresponseresultrisksecondaryskinsquamoussyndromesynthesissystemicT-celltherapythyroidtiontissuetranscriptiontranscription factortumorstypicallyvascularvesselsviralvirusvitamin
About the author(2016)
Richard N Mitchell, MD, PhD, Lawrence J. Henderson Professor of Pathology and Health Sciences and Technology, Department of Pathology, Harvard Medical School, Staff Pathologist, Brigham and Women's Hospital, Boston, MassachusettsVinay Kumar, MBBS, MD, FRCPath, Alice Hogge and Arthur A. Baer Distinguished Service Professor of Pathology, Biologic Sciences Division and the Pritzker Medical School, The University of Chicago, Chicago, Illinois, USAAbul K. Abbas, MBBS, Distinguished Professor and Chair, Department of Pathology, University of California San Francisco, San Francisco, CaliforniaJon C. Aster, MD, PhD, Professor of Pathology, Harvard Medical School; Brigham and Women's Hospital, Boston, Massachusetts.
Bibliographic information
Title Pocket Companion to Robbins & Cotran Pathologic Basis of Disease: First South Asia Edition - E-book
AuthorsRichard N. Mitchell, Vinay Kumar, Abul K. Abbas, Jon C. Aster
Publisher Elsevier Health Sciences, 2016
ISBN 8131247252, 9788131247259
Length 892 pages
SubjectsMedical
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Pathology
Medical / Pathology
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6523 | https://www.investopedia.com/terms/z/zscore.asp | Skip to content
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Table of Contents
Table of Contents
What Is Z-Score?
Understanding Z-Score
Z-Score Formula
How to Calculate Z-Score
How the Z-Score Is Used
Z-Scores vs. Standard Deviation
How Is Z-Score Calculated?
Z-Score FAQs
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What Is Z-Score?
Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean. In investing and trading, Z-scores are measures of an instrument's variability and can be used by traders to help determine volatility.
Key Takeaways
A Z-Score is a statistical measurement of a score's relationship to the mean in a group of scores.
A Z-score can reveal to a trader if a value is typical for a specified data set or if it is atypical.
In general, a Z-score of -3.0 to 3.0 suggests that a stock is trading within three standard deviations of its mean.
Traders have developed many methods that use z-score to identify correlations between trades, trading positions, and evaluate trading strategies.
Understanding Z-Score
Z-score is a statistical measure that quantifies the distance between a data point and the mean of a dataset. It's expressed in terms of standard deviations. It indicates how many standard deviations a data point is from the mean of the distribution.
If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.
The Z-score is sometimes confused with the Altman Z-score, which is calculated using factors taken from a company's financial reports. The Altman Z-score is used to calculate the likelihood that a business will go bankrupt within the range of one to 10 years, while the Z-score can be used to determine how far a stock's return differs from it's average return—and much more.
Fast Fact
Z-score is also known as the standard score.
Z-Score Formula
The statistical formula for a value's z-score is calculated using the following formula:
z = ( x - μ ) / σ
Where:
z = Z-score
x = the value being evaluated
μ = the mean
σ = the standard deviation
How to Calculate Z-Score
Z-Score
Calculating a z-score requires that you first determine the mean and standard deviation of your data. Once you have these figures, you can calculate your z-score. So, assume you have the following variables:
x = 57
μ = 52
σ = 4
You would use the variables in the formula:
z = ( 57 - 52 ) / 4
z = 1.25
So, your selected value has a z-score that indicates it is 1.25 standard deviations from the mean.
Spreadsheets
To determine z-score using a spreadsheet, you'll need to input your values and determine the average for the range and the standard deviation. Using the formulas:
=AVERAGE(A2:A7)
=STDEV(A2:A7)
You'll find that the following values have a mean of 12.17 and a standard deviation of 6.4.
| A | B | C |
---
| 1 | Factor (x) | Mean (μ) | St. Dev. (σ) |
| 2 | 3 | 12.17 | 6.4 |
| 3 | 13 | 12.17 | 6.4 |
| 4 | 8 | 12.17 | 6.4 |
| 5 | 21 | 12.17 | 6.4 |
| 6 | 17 | 12.17 | 6.4 |
| 7 | 11 | 12.17 | 6.4 |
Using the z-score formula, you can figure out each factor's z-score. Use the following formula in D2, then D3, and so on:
Cell D2 = ( A2 - B2 ) / C2
Cell D3 = ( A3 - B3 ) / C3
| A | B | C | D |
--- --- |
| 1 | Factor (x) | Mean (μ) | St. Dev. (σ) | Z-Score |
| 2 | 3 | 12.17 | 6.4 | -1.43 |
| 3 | 13 | 12.17 | 6.4 | 0.13 |
| 4 | 8 | 12.17 | 6.4 | -0.65 |
| 5 | 21 | 12.17 | 6.4 | 1.38 |
| 6 | 17 | 12.17 | 6.4 | 0.75 |
| 7 | 11 | 12.17 | 6.4 | -0.18 |
How the Z-Score Is Used
In it's most basic form, the z-score allows you determine how far (measured in standard deviations) the returns for the stock you're evaluating are from the mean of a sample of stocks. The average score you have could be the mean of a stock's annual return, the average return of the index it is listed on, or the average return of a selection of stocks you've picked.
Some traders use the z-scores in more advanced evaluation methods, such as weighting each stock's return to use factor investing, where stocks are evaluated based on specific attributes using z-scores and standard deviation. In the forex markets, traders use z-scores and confidence limits to test the capability of a trading system to generate winning and losing streaks.
Z-Scores vs. Standard Deviation
In most large data sets (assuming a normal distribution of data), 99.7% of values lie between -3 and 3 standard deviations, 95% between -2 and 2 standard deviations, and 68% between -1 and 1 standard deviations.
Standard deviation indicates the amount of variability (or dispersion) within a given data set. For instance, if a sample of normally distributed data had a standard deviation of 3.1, and another had one of 6.3, the model with a standard deviation (SD) of 6.3 is more dispersed and would graph with a lower peak than the sample with an SD of 3.1.
A distribution curve has negative and positive sides, so there are positive and negative standard deviations and z-scores. However, this has no relevance to the value itself other than indicating which side of the mean it is on. A negative value means it is on the left of the mean, and a positive value indicates it is on the right.
The z-score shows the number of standard deviations a given data point lies from the mean. So, standard deviation must be calculated first because the z-score uses it to communicate a data point's variability.
How Is Z-Score Calculated?
The Z-score is calculated by finding the difference between a data point and the average of the dataset, then dividing that difference by the standard deviation to see how many standard deviations the data point is from the mean.
How Is Z-Score Used in Real Life?
A z-score is used in many real-life applications, such as medical evaluations, test scoring, business decision-making, and investing and trading opportunity measurements. Traders that use statistical measures like z-scores to evaluate trading opportunities are called quant traders (quantitative traders).
What Is a Good Z-Score?
The higher (or lower) a z-score is, the further away from the mean the point is. This isn't necessarily good or bad; it merely shows where the data lies in a normally distributed sample. This means it comes down to preference when evaluating an investment or opportunity. For example, some investors use a z-score range of -3.0 to 3.0 because 99.7% of normally distributed data falls in this range, while others might use -1.5 to 1.5 because they prefer scores closer to the mean.
Why Is Z-Score So Important?
A z-score is important because it tells where your data lies in the data distribution. For example, if a z-score is 1.5, it is 1.5 standard deviations away from the mean. Because 68% of your data lies within one standard deviation (if it is normally distributed), 1.5 might be considered too far from average for your comfort.
The Bottom Line
A z-score is a statistical measurement that tells you how far away from the mean (or average) your datum lies in a normally distributed sample. At its most basic level, investors and traders use quantitative analysis methods such as a z-score to determine how a stock performs compared to other stocks or its own historical performance. In more advanced z-score uses, traders weigh investments based on desirable criteria, develop other indicators, or even try to predict the outcome of a trading strategy.
Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
PennState, Eberly College of Science. "STAT 500: 3.3.3 - Probabilities for Normal Random Variables (Z-scores)."
Zhao, Tianyang. "The Relationship Between Z-Score and Stock Prices." April 2015, pp. 1-36. Download PDF.
Business Compass, LLC. "Corporate Distress & Bankruptcy: Altman Z-Score+."
PennState, Eberly College of Science. "STAT 500: 3.3.4 - The Empirical Rule."
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6524 | https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:multiplication-of-vector-by-a-scalar/v/collinearity-in-3d | Use of cookies
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6525 | https://www.openmiddle.com/multiplying-fractions-to-make-a-whole-number/ | Open Middle®
Take The Online Workshop
Multiplying Fractions to Make a Whole Number
Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a whole number product.
Hint
How can you use equivalent fractions to find the answer?
Answer
6/2 1/3 = 1, 8/4 1/2 = 1, 2/4 6/3 = 1
Source: Owen Kaplinsky
Tags 5.NF.4 DOK 2: Skill / Concept Owen Kaplinsky
Previous Multiplying 3 Fractions to Get 1
Next Open Number Line
Check Also
Add and Subtract Mixed Numbers
Directions: Using the digits 1-9 at most one time each, create an equation using addition …
13 comments
Heather
February 4, 2020 at 2:02 pm
My students came up with more solutions:
2/1 X 9/6 = 3
4/1 X 6/3 = 8
6/4 X 2/3 = 1
3/2 X 6/9 = 1
4/2 X 3/1 = 6
3/6 X 4/2 = 1
2/3 X 9/6 = 1
4/8 X 6/3 = 1
9/3 X 4/2 = 6
Reply
2. Heather
March 24, 2020 at 8:47 am
I did 2/1 times 9/6=3
Reply
3. Heather
March 24, 2020 at 8:48 am
ok
Reply
4. Reed Kahn
April 3, 2020 at 8:17 am
I had 6/2 x 3/1 = 9. I got this by finding a number that times by something would be under ten. I first got 6/2 and that got me 3. Then i got 3 from 3/1. Lastly i multiplied 3 by 3 and got 9. I agree it was a challenging puzzle.
Reply
5. Max D. Wong
April 13, 2020 at 10:05 am
I got 8/1 x 6/2 = 21
Reply
6. Ariana Grande
May 28, 2020 at 10:05 am
This is not what I was looking for please be more accurate with the answers I really HATE it when I get the answers wrong. 🙁
Reply
7. Mithun
June 8, 2020 at 6:07 am
2/3×1/3=3/3 or 1
Reply
Sarah
October 21, 2020 at 12:10 pm
2/3 x 1/3 is 2/9, not 1.
Reply
8. Julie Barrett
March 24, 2022 at 1:07 pm
I don’t agree with the first two provided answers. If you use a unit fraction as a factor, you cannot reuse the “1” tile in the answer. In fact, I don’t think there is an answer with a unit fraction as a factor. The third supplied answer works, though.
Reply
9. JK
February 22, 2023 at 11:05 am
I got 8/6 x 3/4=1. Im 10! This is not really 5th grade level if you know how to multiply fractions.
Reply
10. Vanessa Churchill
May 10, 2023 at 6:13 am
Today we found:
9/6 x 2/3 = 1
6/9 x 3/2 = 1
6/3 x 2/4 = 1
6/8 x 4/3 = 1
4/6 x 3/2 = 1
1/8 x 4/3 = 6
6/4 x 8/3 = 2
Reply
Rudolf Österreicher
July 10, 2023 at 3:30 pm
The 24 unique solutions are
2/3 6/4 = 1
2/3 9/6 = 1
3/4 8/6 = 1
3/6 4/2 = 1
3/9 6/2 = 1
4/8 6/3 = 1
3/6 4/1 = 2
3/9 6/1 = 2
4/3 9/6 = 2
2/4 6/1 = 3
2/6 9/1 = 3
4/6 9/2 = 3
4/8 6/1 = 3
8/6 9/4 = 3
2/1 6/3 = 4
3/6 8/1 = 4
8/6 9/3 = 4
2/1 9/3 = 6
3/2 4/1 = 6
3/4 8/1 = 6
4/2 9/3 = 6
8/4 9/3 = 6
4/3 6/1 = 8
3/1 6/2 = 9
You can always swap the fractions, the numerators or the denominators to get another solution.
Reply
12. mytayvion
September 17, 2025 at 6:13 am
I’m having trouble like with the box when u color it and getting yo answer
Reply
Leave a Reply Cancel reply |
6526 | https://www.youtube.com/watch?v=pUQvhCmfAgg | Coordinate Geometry | Point of intersection of circle and y-axis
Tambuwal Maths Class
313000 subscribers
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Posted: 3 Apr 2023
3 comments
Transcript:
hello good day viewers still on coordinate geometry here's our problem for today the cycle C has an equation x squared plus three X Plus y squared plus 6y equal to 3x minus 2y minus seven we're asked to find the center and radius of that cycle we're asked to find the point of intersection of the cycle and y-axis that is y-intercept and also we have to show that the cycle does not intersect x-axis right let's get started I think for the first question I am going to show you two different ways to do that we are going to use the general form and the standard form the equation we are given is x squared plus 3x Plus y squared plus 6y this is equal to 3x minus 2y minus seven let us bring every time to the left hand side we have x squared here we have plus 3x if this cross is over it becomes negative 3x then we have y squared we have 6y if this crosses over it becomes positive we have plus 2y if this crosses over it becomes positive as well and the whole of this equal to zero now let us simplify this we have x squared plus 0x for a reason Plus y squared plus eight y then plus seven the whole of this equal to zero this equation is in general form which we can write as what x squared plus 2 f x Plus y squared plus 2 g y then plus C the whole of this equal to zero where the center of that cycle is giving us let me write it here Center the center has the coordinates negative f comma negative G and the radius can be obtained by taking the square root of F squared plus G squared minus C this is our value of C which corresponds to 7 here all right let us find these values if we start with f you can see the coefficient of x here is zero while the coefficient of x here is 2f so it means that 2f is equal to 0 which means that F also must be equal to zero and the coefficient of Y is 8 here while the coefficient of Y here is 2G so we have 2G equal to 8 and if you divide both sides by 2 you have G to be equal to 4. so what will be the center the center of that cycle will be negative f f is 0 negative 0 is still zero then comma negative g g is 4 and negative of that will be negative four and hence this is the center of that cycle so what about radius to find the radius R this is equal to the square root of f squared that is 0 squared plus G squared which is 4 squared the minus c c is seven you can see that here minus 7. R is not equal to the square root of what this is 0 this is 16. minus 7 is equal to 9 so this is square root of 9 and square root of 9 is equal to three remember R is radius which is a distance so we should expect a positive result so this is all about a part and I promise to show you two different ways to do that right so I would like to copy this equation this is the equation I would like to transform into standard form so this is x squared Plus 0x to make it a complex squared you have to take the square of half of the coefficient of x the coefficient of x is zero half of that is still zero squared is also zero so it's like we are adding 0 here which we don't have to then Plus y squared plus 8y to make this a complete Square you take half of this which is 4 you square that you get 16 right remember we are taking seven to the other side it becomes negative seven and since we have added 16 to the left hand side we have to add it to the right hand side to make the equation balanced right so you know 0x is still zero so we have x squared here plus if you want to factorize this you take y then half of this is four we have plus four whole squared and this is equal to negative 7 plus 16 will give us 9 so this is nine so this is our equation in standard form which if you can remember we can write as x minus a whole squared Plus y minus B whole squared equal to r squared where we say that the center of that cycle will be what a comma B so now let's start with the center a here you can see we have no a here so we are going to consider it as zero so the center x coordinate is zero then B is this value here that is negative 4. we have negative four you can see they're exactly the same with the previous one zero comma negative four and the right hand side you can see that r squared will be equal to what 9. so r squared equal to 9 which implies that R must be equal to three you see the radius is equal to 3. so this is all about the first question then going to the second question we asked to find the points of intersection of the cycle and y axis so we are looking for y-intercept that's exactly where the cycle crosses y axis and it can be obtained by setting the value of x equal to zero we can take either of these equations either in standard form who in general form it doesn't matter so I think let me take the equation in standard form so B part we are looking for y intercepts that's exactly where X is equal to zero so if we set X to be equal to zero the whole of this will be zero right we only have y plus 4 whole squared equal to 9. let's take square root of both sides we have y plus 4 equal to plus o minus the square root of nine y plus 4 is equal to plus o minus three therefore y will be equal to minus four if you take this to the other side plus or minus 3. meaning we have two values of Y y could be negative 4 plus 3 which is going to give us what negative one so we have negative one or negative 4 negative three which is negative seven so this is negative seven so these are exactly the points where the cycle crosses y-axis all at the negative side then the last question here we asked to show that the cycle does not intersect x-axis meaning there is no x-intercept you know x-intercept is the value of x when Y is equal to zero so now let us pretend as if we have x-intercept by setting y to be equal to zero so this is the C part we are looking for X intercept that is at y equal to zero which means what x squared plus 0 plus 4 because Y is zero this is equal to 9 0 plus 4 is 4 squared is 16. so we have x squared plus 16 equal to 9 if you take 16 to the other side you have x squared equal to 9 minus 16 and 9 minus 16 is negative seven x squared is equal to negative seven and hence you can see that this statement is not true because there is no any real value of x you can square to get a negative result and if you so which you can take the square root of both sides you have X to be equal to plus or minus the square root of negative seven a square root of negative 70 remember is imaginary so x equal to plus o minus the square root of 7 I meaning the cycle does not crosses the x axis so this is all about this question thank you for watching do share to your learning colleagues and don't forget to subscribe to my YouTube channel for more exciting videos the only thing you can make a conclusion here since x value is imaginary it means what the cycle does not crosses the x-axis thank you for watching bye bye |
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6528 | https://www.gigacalculator.com/articles/when-to-use-mean-median-or-mode/ | When to Use Mean, Median, or Mode
The mean, mode and median are the most well-known and widely used arithmetic averages. But what exactly are each used for? While all of them are measures of central tendency in a distribution of values – obtained from different kinds of measurements or observations – the question of when to use mean, median or mode is not properly understood even by seasoned practitioners and statisticians.
If you’ve ever wondered when to use mean, median, or mode, it’s time to answer these questions.
What is Mean, Median, Mode?
The mean, median, and mode are all summary statistics describing a property of a set of values. Each of them may be called an “average” depending on the context, but typically – unless otherwise specified – “average” refers to the arithmetic mean. In this article “average” is used to refer to all three as a group.
All of the three averages are lossy, meaning that information is inevitably lost in the process of compressing a whole distribution into a single number.
Mean
The arithmetic mean is defined mathematically as the sum of all values divided by their count. It has the useful property of minimizing the mean squared error, meaning that the mean value is such that it is the point of balance of the distribution. It exactly balances the sums of distances of all numbers from itself. In this sense it is the single best predictor of the set, so if you were tasked with guessing what a randomly drawn value from the distribution is, choosing the mean would minimize your error in either under or overestimating the true value.
Median
The median, on the other hand, is the value which splits the distribution in two equal parts. The number of members of the set to the left of the median equals the number of members of the set to the right of it. It has the property of minimizing the average distance between itself and each number of that set.
Mode
The mode is the value which occurs the most times out of all the values of a set. If you want to know which is the most frequent value, the mode is the answer. It should be noted that a distribution might have multiple modes if many values occur with the same frequency.
For a very in-depth explanation of the three concepts, including simulations, consider this article.
When to Use Mean, Median, or Mode? – The Textbook Approach
Generally, most articles and textbooks on mathematics and statistics recommend using the median for most purposes. Some even go as far as saying it is “the best average” due to it being least affected by adding new values to a data set. For example, adding the number 100 to the set [1,2,3,4,5,6,7] only shifts the median from 4 to 4.5 while the mean jumps from 4 to 16.
The mode is less often recommended even though it is even less affected by adding new values, on average, since a distribution can have multiple modes.
The arithmetic mean is only recommended as a good average to use if a distribution is normal or close to normally distributed. Its use is discouraged in the presence of high skewness. This appears to be mainly on the grounds of how often the mean lies in the range where the majority of values in a distribution are, as well as how it responds to new values further from where the majority of values lie. Fancier wording is that the mean is very sensitive to outliers hence one should use an average which is “more robust”, like the median.
You can use our mean, median, and mode calculator for quick estimation of these summary statistics of a data set.
Examined closely, these general recommendations amount to only using the mean when it is identical or nearly identical to the median. Therefore, most textbooks could have just as well said that one should use the median in every case.
This attitude seems to stem from a mistaken implicit assumption that when looking at an average, most people are interested in where most values lie, instead of what each of these averages can convey. Some interpret measures of central tendency as representing values “typical” for the distribution, but no such quality is present in the description of the mean or the median, and it applies to the mode only in the narrowest sense. Such qualities only arise when the distribution is normal-like, or at least symmetrical around the mean, and this may or may not apply to any given case.
The Proper Way to Decide When to Use the Mean, Median, or Mode
The textbook recommended approach often “forgets” that what is of primary interest is whether the chosen statistic corresponds to the question posed to the data or not. Sensitivity to outliers and other characteristics are of secondary concern, if at all.
Hence, a proper choice of descriptive statistic should be informed only by the question asked of the data it summarizes. It does not matter how robust to “outliers” or how close to the majority of the values it is.
The best way to understand when to use mean, median, or mode is by looking at a real-life scenario. For instance, let’s assume an analyst is examining a country’s car market and works with a distribution of prices paid by new car owners in a given period, say, a year. For the purpose of illustration, assume the following dataset of forty sales in total, with the following prices:
Plotting the distribution of prices results in the following histogram:
Here, the mean is $47,368, the median is $35,500 and the mode is $30,000. They all fall within the first bin of the histogram in the graph above.
When the mean answers the question
If an analyst wants to determine the price which is the closest to all the prices paid, then the only statistic which answers that question is the mean. It is often what one is interested in since the mean reflects many different changes to buying behavior, be it more purchases of cheaper cars, purchases of fewer but more expensive cars, etc. Comparisons of means across periods, segments, markets, etc. are therefore quite informative.
Neither the median nor the mode can provide the answer to the above question, nor would they reflect changes in the underlying distribution in that same responsive way.
When to use the median
In another inquiry using the same data, one might be interested in what is the price point that half of the consumers are not able or willing to pay above. This would be answered by the median and no other average will do.
When to choose the mode
If, instead, one is interested in the price most commonly paid for a new car, then neither the arithmetic mean nor the median can answer that question, only the mode does.
Note that in the above example we have the same data, but different questions and the choice of the average to use was guided entirely by the question posed, and not by the distribution of the data which is of course one and the same.
The above example should make it obvious that there is no such thing as the “best average” or “best measure of central tendency”. There are measures fitting the question, a.k.a. appropriate averages, and unfitting or inappropriate ones.
When Neither Average Does The Job
Examining just the mean, median, or mode may be sufficient in many situations. However, since these summary statistics are lossy, in some inquiries neither of them would contain sufficient information to answer the question of interest.
When that is the case, one can resort to using quantiles or frequency cut-offs such as percentiles. For example, one might be interested in changes of the mean price in each quintile or in the price that 95% of buyers are paying below ($250,000 in this data).
Sometimes the price range of 90% of new car sales might be what’s of interest. In this case that is between $27,000 and $200,000. The upper and lower values of the quantiles can also be useful:
In a year over year analysis a comparison of the whole distribution of current year sales versus last year’s sales might be the most informative, but the outcome of such comparisons is difficult to communicate.
Takeaways
After all this, how does one answer the question “When to use the mean, median, and mode“?
The proper average to choose is informed entirely by the question being asked of the data under analysis. The interpretation of the chosen average should match the question at hand. The mean minimizes the average distance to all points, the median splits the set in two halves, and the mode (or modes) is the most frequently occurring value. If the question corresponds to one of these qualities, the appropriate average is self-evident.
Switching from an appropriate average to an inappropriate one based on considerations such as how influenced by “outliers” it is or how close it is to the majority of the values in a distribution makes no sense. The temptation to introduce such considerations hints at an inadequate transformation of the business question.
Sometimes an inquiry cannot be satisfactorily answered by any average, be it the mean, median, or mode, due to their inherently lossy nature. Quantile ranges and their means, as well as percentiles, while also lossy, provide a different balance between interpretability and information retained that may often be appropriate. In yet other situations frequency ranges or even whole distribution plots might be required, depending on the task at hand.
The article will use mean as a shorthand for “arithmetic mean”. Other means such as the geometric mean and the harmonic mean are not subject of the current piece.
Georgi Georgiev
An applied statistician, data analyst, and optimizer by calling, Georgi has expertise in web analytics, statistics, design of experiments, and business risk management. He covers a variety of topics where mathematical models and statistics are useful. Georgi is also the author of “Statistical Methods in Online A/B Testing”.
This entry was posted in Mathematics and tagged arithmetic average, arithmetic mean, central tendency, median, mode, statistics. By Georgi Georgiev
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Published: December 13, 2022
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6529 | https://www.cuemath.com/geometry/bisect/ | Bisect
Bisect
Imagine a lovely cake with delicious frosting that needs to be divided at a birthday. The person cutting the cake will not divide the cake into multiple pieces, as it will create quite the mess.
Instead, the person will first divide the cake into two equal halves. This is called bisection and it is an important part of geometry and how we study angles.
In this lesson, we will learn how to bisect a segment, how to bisect lines, and the rules that are applied while bisecting angles.
Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
| | |
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| 1. | What is Bisect? |
| 2. | Thinking out of Box! |
| 3. | Important Notes on Bisect |
| 4. | Solved Examples on Bisect |
| 5. | Interactive Questions on Bisect |
What is Bisect?
Bisect: Definition
Bisect means to cut or divide into two equal parts.
Bisecting a Line Segment
Let's do an activity to understand the meaning of bisecting a line segment.
Fold a sheet of paper and let ¯¯¯¯¯¯¯¯AB be the line of fold.
Put an ink-dot on any one side of ¯¯¯¯¯¯¯¯AB and name it as X
Can you find the mirror image of X?
Name the mirror image of X as X′.
Let ¯¯¯¯¯¯¯¯AB and ¯¯¯¯¯¯¯¯¯¯¯XX′ intersect at O.
Is ¯¯¯¯¯¯¯¯¯OX and ¯¯¯¯¯¯¯¯¯¯OX′ equal?
This means ¯¯¯¯¯¯¯¯AB divides ¯¯¯¯¯¯¯¯¯¯¯XX′ into two equal parts.
So, here ¯¯¯¯¯¯¯¯AB bisects ¯¯¯¯¯¯¯¯¯¯¯XX′ and ¯¯¯¯¯¯¯¯AB is a bisector of ¯¯¯¯¯¯¯¯¯¯¯XX′.
Use the simulation below to learn how to bisect a line segment using a ruler and a compass.
Bisecting an Angle
Let's do another activity to understand how to bisect an angle.
Mark a point O on a paper.
Taking O as the initial point, draw two rays ¯¯¯¯¯¯¯¯OA and ¯¯¯¯¯¯¯¯OB
You will get ∠AOB
Now, fold the sheet through O such that the rays ¯¯¯¯¯¯¯¯OA and ¯¯¯¯¯¯¯¯OB overlap each other.
Name the line of the crease on the paper as ¯¯¯¯¯¯¯¯OC
Can you measure the angles ∠AOC and ∠BOC? Are the measures equal?
Yes, both the angles are equal and OC is the angle bisector of ∠AOB.
Use the simulation below to learn the way to bisect an angle using a ruler and a compass.
Think Tank
A ray BD bisects an angle ∠ABC. If ∠ABC=120∘, what would be the measure of the angles ∠ABD and ∠DBC?
Find the coordinates of the midpoint of the line segment ¯¯¯¯¯¯¯¯AB joining the points (−5,3) and (3,2).
Bisecting a Shape
Some shapes can also be bisected.
Look at the shapes shown below.
A line segment bisects each shape into two equal parts.
Important Notes
Bisect means to cut or divide something into two equal parts.
You can use a compass and a ruler to bisect a line segment or an angle.
The bisector of a line segment is called a perpendicular bisector.
Solved Examples
| |
| Example 1 |
Ryan is flying a kite.
The kite has two angles bisected as shown below.
Can you find the measure of the angles ∠EKI and ∠ITE?
Solution
The angles ∠EKI and ∠ITE are bisected by the line KT
KT divides the angles ∠EKI and ∠ITE in two equal angles respectively.
Thus,
∠EKI=2×45∘=90∘ and
∠ITE=2×27∘=54∘
| |
| ∴ ∠EKI=90∘ and ∠ITE=54∘. |
| |
| Example 2 |
In the diagram shown below, RQ bisects the angle ∠PRS.
Can you find the value of x?
Solution:
Since RQ bisects the angle ∠PRS,
∠PRQ=∠QRS
Using this information, we can solve for x.
∠PRQ=∠QRSx+40=3x−2040+20=3x−x2x=60x=30
| |
| ∴ The value of x is 90∘ |
Interactive Questions
Here are few activities for you to practice. Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about Bisect with the simulations and practice questions. Now you will be able to easily solve problems and understand bisect definition, bisect symbol, bisect geometry definition, bisect a segment, bisecting lines, and bisecting angles.
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Frequently Asked Questions (FAQs)
1. How to bisect a line?
Do the following steps to draw a bisector of a line.
Draw a line segment ¯¯¯¯¯¯¯¯AB on a paper.
Taking A as a center, draw a circle. Just make sure that the radius of the circle should be more than half of the line segment ¯¯¯¯¯¯¯¯AB.
Now, taking B as a center and with the same radius, draw another circle. Let this circle intersect the previous circle at the points C and D.
Join the points C and D.
The line (\overline{CD}) is the bisector of the line ¯¯¯¯¯¯¯¯AB.
2. What happens when you bisect an angle?
When we bisect an angle, the angle is divided into two equal smaller angles.
3. How do you bisect an angle of 60 degrees?
Let's consider an angle of measure 60∘
Do the following steps to draw a bisector of an angle.
Keeping the sharp end of your compass at B, draw an arc on BC and mark it as S.
Similarly, keeping the sharp end of your compass at B, draw an arc on AB and mark it as T.
Now, keeping the sharp end of your compass at S, draw an arc within AB and BC
Repeat the third step at T
Join the point B and the intersection of the two arcs.
The line is the angle bisector of ∠ABC.
The line is the angle bisector of ∠ABC=60∘.
4. How do you bisect an angle of 45 degrees?
Assume you have constructed an angle of measure 45∘
Do the following steps to draw a bisector of this angle.
Keeping the sharp end of your compass at B, draw an arc on BC and mark it as S.
Similarly, keeping the sharp end of your compass at B, draw an arc on AB and mark it as T.
Now, keeping the sharp end of your compass at S, draw an arc within AB and BC
Repeat the third step at T
Join the point B and the intersection of the two arcs.
The line is the angle bisector of ∠ABC.
The line is the angle bisector of ∠ABC=45∘.
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6530 | https://math.libretexts.org/Courses/Kansas_State_University/Your_Guide_to_Intermediate_Algebra/01%3A_Foundations/1.05%3A_Solve_Linear_Inequalities | Skip to main content
1.5: Solve Linear Inequalities
Last updated
: May 27, 2023
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1.4E: Exercises
1.5E: Exercises
Page ID
: 109045
( \newcommand{\kernel}{\mathrm{null}\,})
Learning Objectives
By the end of this section, you will be able to:
Graph inequalities on the number line
Solve linear inequalities
Translate words to an inequality and solve
Solve applications with linear inequalities
Warm-up
Solve
Solution
2
-2
3
Graph Inequalities on the Number Line
What number would make the inequality true? Are you thinking, " could be four"? That’s correct, but could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality . We show all the solutions to the inequality on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.
We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express as . The symbol is read as “infinity.” It is not an actual number. Figure shows both the number line and the interval notation.
We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.
The inequality means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at . We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions (Figure ). There is no lower end to those numbers. We write in interval notation as . The symbol is read as “negative infinity.”
Figure shows both the number line and interval notation.
INEQUALITIES, NUMBER LINES, AND INTERVAL NOTATION
The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.
Example
Graph each inequality on the number line and write in interval notation.
Answer
: ⓐ
Solution to a.
| | |
| Shade to the right of , and put a bracket at . | |
| Write in interval notation. | |
ⓑ
Solution to b.
| | |
| Shade to the left of 2.5 and put a parenthesis at 2.5. | |
| Write in interval notation. | |
ⓒ
Solution to c.
| | |
| Shade to the left of , and put a bracket at . | |
| Write in interval notation. | |
Graph each inequality on the number line and write in interval notation:
.
Answer
: ⓐ
ⓑ
ⓒ
Example
Graph each inequality on the number line and write in interval notation:
.
Answer
: ⓐ
ⓑ
ⓒ
What numbers are greater than two but less than five? Are you thinking say, ? We can represent all the numbers between two and five with the inequality . We can show on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See Figure.
Graph each inequality on the number line and write in interval notation.
ⓐ ⓑ ⓒ
Answer
: ⓐ
Solution to a.
| | | |
| Shade between and 4. Put a parentheses at and 4. | | |
| Write in interval notation. | | |
ⓑ
Solution to b.
| | | | |
| Shade between and −1. Put a bracket at , and a parenthesis at −1. | | | |
| Write in interval notation. | | | |
ⓒ
Solution to c.
| | | | |
| Shade between 0 and 2.5. Put a bracket at 0 and at 2.5. | | | |
| Write in interval notation. | | | |
Example
Graph each inequality on the number line and write in interval notation:
ⓐ ⓑ ⓒ
Answer
: ⓐ
ⓑ
ⓒ
Example
Graph each inequality on the number line and write in interval notation:
ⓐ ⓑ ⓒ
Answer
: ⓐ
ⓑ
ⓒ
Solve Linear Inequalities
A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms, , , , or .
Definition: LINEARE INEQUALITY
A linear inequality is an inequality in one variable that can be written in one of the following forms where and are real numbers and :
When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:
Notice that the inequality sign stayed the same.
This leads us to the Addition and Subtraction Properties of Inequality.
Definition: ADDITION AND SUBTRACTION PROPERTY OF INEQUALITY
For any numbers and if , then
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality
What happens to an inequality when we divide or multiply both sides by a constant?
Let’s first multiply and divide both sides by a positive number.
The inequality signs stayed the same.
Does the inequality stay the same when we divide or multiply by a negative number?
Notice that when we filled in the inequality signs, the inequality signs reversed their direction.
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
This gives us the Multiplication and Division Property of Inequality.
Definition: MULTIPLICATION AND DIVISION PROPERTY OF INEQUALITY
For any numbers and
When we divide or multiply an inequality by :
positive number, the inequality stays the same.
negative number, the inequality reverses.
Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”
Example
Solve each inequality. Graph the solution on the number line, and write the solution in interval notation.
ⓐ ⓑ ⓒ
Answer
: ⓐ
Solution to a.
| | |
| Add 3838 to both sides of the inequality. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
ⓑ
Solution to b.
| | |
| Divide both sides of the inequality by 9; since 9 is positive, the inequality stays the same. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
ⓒ
Solution to c.
| | |
| Multiply both sides of the inequality by . Since is positive, the inequality stays the same. | |
| Simplify. | |
| Rewrite with the variable on the left. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
Solve each inequality, graph the solution on the number line, and write the solution in interval notation:
ⓐ ⓑ ⓒ
Answer
: ⓐ
ⓑ
ⓒ
Example
Solve each inequality, graph the solution on the number line, and write the solution in interval notation:
ⓐ ⓑ ⓒ
Answer
: ⓐ
ⓑ
ⓒ
Be careful when you multiply or divide by a negative number—remember to reverse the inequality sign.
Example
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
ⓐ ⓑ
Answer
: ⓐ
Solution to a.
| | |
| Divide both sides of the inequality by . Since is a negative, the inequality reverses. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
ⓑ
Solution to b.
| | |
| Multiply both sides of the inequality by . Since is a negative, the inequality reverses. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
Example
Solve each inequality, graph the solution on the number line, and write the solution in interval notation:
ⓐ ⓑ .
Answer
: ⓐ
ⓑ
Example
Solve each inequality, graph the solution on the number line, and write the solution in interval notation:
ⓐ ⓑ .
Answer
: ⓐ
ⓑ
Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but make sure to pay close attention when we multiply or divide to isolate the variable.
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solution
Solution to the example.
| | |
| Subtract 11y from both sides to collect the variables on the left. | |
| Simplify. | |
| Divide both sides of the inequality by −5, and reverse the inequality. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
Exercise
Solve the inequality, graph the solution on the number line, and write the solution in interval notation: .
Answer
Example
Solve the inequality, graph the solution on the number line, and write the solution in interval notation: .
Answer
When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don’t have to multiply or divide by a negative—which means we don’t have to remember to reverse the inequality sign.
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
: Solution to the example.
| | |
| Simplify each side as much as possible. | |
| Distribute. | |
| Combine like terms. | |
| Subtract from both sides to collect the variables on the left, since . | |
| Simplify. | |
| Add to both sides to collect the constants on the right. | |
| Simplify. | |
| Divide both sides of the inequality by ; the inequality stays the same. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | |
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
: Solution to the example.
| Simplify each side as much as possible. | | |
| Distribute. | | |
| Combine like terms. | | |
| Subtract from both sides to collect the variables on the left. | | |
| Simplify. | | |
| The 's are gone, and we have a true statement. | | The inequality is an identity. The solution is all real numbers. |
| Graph the solution on the number line. | | |
| Write the solution in interval notation. | | |
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
: Solution to the example.
| | |
| Multiply both sides by the LCD, 24, to clear the fractions. | |
| Simplify. | |
| Combine like terms. | |
| Subtract from both sides to collect the variables on the left. | |
| Simplify. | |
| The statement is false. | The inequality is a contradiction. There is no solution. |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | There is no solution. |
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
Example
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Answer
Key Concepts
Inequalities, Number Lines, and Interval Notation
Linear Inequality
A linear inequality is an inequality in one variable that can be written in one of the following forms where a, b, and care real numbers and :
Addition and Subtraction Property of Inequality
For any numbers a, b, and c, if a<b,thena<b,then
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.
Multiplication and Division Property of Inequality
For any numbers a, b, and c,
1.4E: Exercises
1.5E: Exercises |
6531 | https://www.jstor.org/stable/3620186 | Review: [Untitled] on JSTOR
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Review: [Untitled]
Reviewed Work: The Joy of π David Blatner
Review by: Steve Abbott
The Mathematical Gazette
Vol. 82, No. 493 (Mar., 1998), pp. 145-147 (3 pages)
Published By: The Mathematical Association
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The Mathematical Gazette is the original journal of the Mathematical Association and it is now over a century old. Its readership is a mixture of school teachers, college and university lecturers, educationalists and others with an interest in mathematics. One of the world's leading journals in its field, it publishes articles about the teaching and learning of mathematics, with a focus on the 15-20 age range, and expositions of attractive areas of mathematics. Regular sections include letters, extensive book reviews and a problem corner.
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6532 | https://www.tutorchase.com/answers/igcse/chemistry/what-is-the-relationship-between-atomic-number-and-electronic-configuration | What is the relationship between atomic number and electronic configuration? | TutorChase
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Article
What is the relationship between atomic number and electronic configuration?
The atomic number of an element determines its electronic configuration.
The atomic number, also known as the proton number, is a fundamental property of an atom. It is equal to the number of protons in the nucleus of an atom. This number is unique for each element and it determines the identity of an element. For example, the atomic number of hydrogen is 1, helium is 2, and so on.
The electronic configuration of an atom refers to the arrangement of electrons in the atom's electron shells or energy levels. The electrons in an atom are arranged in a specific way, which is determined by the atomic number. The first energy level can hold up to 2 electrons, the second can hold up to 8, and the third can hold up to 8 as well. This pattern continues for the higher energy levels.
The atomic number of an element tells us the total number of electrons in a neutral atom of that element. For instance, an atom of hydrogen has an atomic number of 1, which means it has one electron. An atom of helium has an atomic number of 2, which means it has two electrons. These electrons are arranged in the energy levels according to the rules of electronic configuration.
Therefore, the atomic number directly influences the electronic configuration of an atom. By knowing the atomic number of an element, we can predict its electronic configuration. This is a fundamental concept in chemistry, as the electronic configuration of an atom determines its chemical properties and reactivity. For example, elements with similar electronic configurations often have similar chemical properties. This is the basis of the periodic table, where elements are grouped together based on their electronic configurations.
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6533 | https://www.mathcelebrity.com/search.php?q=tan%286%29 | Crop Image
tan(6)
Answer
↓Steps Explained:↓
↓Steps Explained:↓
Calculate tan(6)
tan is found using Opposite/Adjacent
Determine quadrant:
Determine quadrant:
Since 0 ≤ 6 ≤ 90 degrees
it is in Quadrant I
sin, cos and tan are positive.
Determine angle type:
Determine angle type:
6 < 90°, so it is acute
tan(6) = 0.10510423514469
Write tan(6) in terms of cot
Write tan(6) in terms of cot
Since 6° is less than 90...
We can express this as a cofunction
tan(6) = cot(90 - 6)
tan(6) = cot(84)
Special Angle Values
Special Angle Values
θ° | θrad | sin(θ) | cos(θ) | tan(θ) | csc(θ) | sec(θ) | cot(θ) || 0° | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | π/2 | 1 | 0 | N/A | 1 | 0 | N/A |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 | 2√3/3 | -2 | -√3/3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 | 2 | -2√3/3 | -√3 |
| 180° | π | 0 | -1 | 0 | 0 | -1 | N/A |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 | -2 | -2√3/3 | √3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 | -√2 | -√2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 | -2√3/3 | -2 | √3/3 |
| 270° | 3π/2 | -1 | 0 | N/A | -1 | 0 | N/A |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 | -2√3/3 | 2 | -√3/3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 | -√2 | √2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 | -2 | 2√3/3 | -√3 |
θ° | θrad | sin(θ) | cos(θ) | tan(θ) | csc(θ) | sec(θ) | cot(θ) || 0° | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | π/2 | 1 | 0 | N/A | 1 | 0 | N/A |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 | 2√3/3 | -2 | -√3/3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 | 2 | -2√3/3 | -√3 |
| 180° | π | 0 | -1 | 0 | 0 | -1 | N/A |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 | -2 | -2√3/3 | √3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 | -√2 | -√2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 | -2√3/3 | -2 | √3/3 |
| 270° | 3π/2 | -1 | 0 | N/A | -1 | 0 | N/A |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 | -2√3/3 | 2 | -√3/3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 | -√2 | √2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 | -2 | 2√3/3 | -√3 |
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Final Answer |
6534 | https://en.wikipedia.org/wiki/Constraint_(mathematics) | Jump to content
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Contents
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1 Example
2 Terminology
3 Hard and soft constraints
4 Global constraints
5 See also
6 References
7 Further reading
8 External links
Constraint (mathematics)
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From Wikipedia, the free encyclopedia
Condition of an optimization problem which the solution must satisfy
For constraints in Hamiltonian mechanics, see constraint (classical mechanics), first class constraint, primary constraint, and holonomic constraint.
| |
| This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (September 2016) (Learn how and when to remove this message) |
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.
Example
[edit]
The following is a simple optimization problem:
subject to
and
where denotes the vector (x1, x2).
In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions.
Without the constraints, the solution would be (0,0), where has the lowest value. But this solution does not satisfy the constraints. The solution of the constrained optimization problem stated above is , which is the point with the smallest value of that satisfies the two constraints.
Terminology
[edit]
If an inequality constraint holds with equality at the optimal point, the constraint is said to be binding, as the point cannot be varied in the direction of the constraint even though doing so would improve the value of the objective function.
If an inequality constraint holds as a strict inequality at the optimal point (that is, does not hold with equality), the constraint is said to be non-binding, as the point could be varied in the direction of the constraint, although it would not be optimal to do so. Under certain conditions, as for example in convex optimization, if a constraint is non-binding, the optimization problem would have the same solution even in the absence of that constraint.
If a constraint is not satisfied at a given point, the point is said to be infeasible.
Hard and soft constraints
[edit]
If the problem mandates that the constraints be satisfied, as in the above discussion, the constraints are sometimes referred to as hard constraints. However, in some problems, called flexible constraint satisfaction problems, it is preferred but not required that certain constraints be satisfied; such non-mandatory constraints are known as soft constraints. Soft constraints arise in, for example, preference-based planning. In a MAX-CSP problem, a number of constraints are allowed to be violated, and the quality of a solution is measured by the number of satisfied constraints.
Global constraints
[edit]
Global constraints are constraints representing a specific relation on a number of variables, taken altogether. Some of them, such as the alldifferent constraint, can be rewritten as a conjunction of atomic constraints in a simpler language: the alldifferent constraint holds on n variables , and is satisfied if the variables take values which are pairwise different. It is semantically equivalent to the conjunction of inequalities . Other global constraints extend the expressivity of the constraint framework. In this case, they usually capture a typical structure of combinatorial problems. For instance, the regular constraint expresses that a sequence of variables is accepted by a deterministic finite automaton.
Global constraints are used to simplify the modeling of constraint satisfaction problems, to extend the expressivity of constraint languages, and also to improve the constraint resolution: indeed, by considering the variables altogether, infeasible situations can be seen earlier in the solving process. Many of the global constraints are referenced into an online catalog.
See also
[edit]
Constraint algebra
Karush–Kuhn–Tucker conditions
Lagrange multipliers
Level set
Linear programming
Nonlinear programming
Restriction
Satisfiability modulo theories
References
[edit]
^ Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. p. 61. ISBN 0-521-31498-4.
^ Rossi, Francesca; Van Beek, Peter; Walsh, Toby (2006). "7". Handbook of constraint programming (1st ed.). Amsterdam: Elsevier. ISBN 9780080463643. OCLC 162587579.
^ Rossi, Francesca (2003). Principles and Practice of Constraint Programming CP 2003 00 : 9th International Conference, CP 2003, Kinsale, Ireland, September 29 October 3, 2003. Proceedings. Berlin: Springer-Verlag Berlin Heidelberg. ISBN 9783540451938. OCLC 771185146.
Further reading
[edit]
Beveridge, Gordon S. G.; Schechter, Robert S. (1970). "Essential Features in Optimization". Optimization: Theory and Practice. New York: McGraw-Hill. pp. 5–8. ISBN 0-07-005128-3.
External links
[edit]
Nonlinear programming FAQ Archived 2019-10-30 at the Wayback Machine
Mathematical Programming Glossary Archived 2010-03-28 at the Wayback Machine
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6535 | https://artofproblemsolving.com/wiki/index.php/Category:Functional_Equation?srsltid=AfmBOorrMkXp9itLsiqytUOoHMxSxnSnvAC2H2dZupro8W4Z9EJqGtvH | Art of Problem Solving
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6536 | https://www.gauthmath.com/solution/1803967938599942/3-Let-R-be-the-region-bounded-by-the-graph-y-square-root-of-x-and-y-x3-Find-the- | Question
Solution
Larry will have $1011.24 in total after 2 years.
Step 1: Calculate the interest earned in the first year by multiplying $900 by 6%: $900 6% = $54.
Step 2: Add the interest earned in the first year to the initial amount: $900 + $54 = $954.
Step 3: Calculate the interest earned in the second year on $954 at a 6% rate: $954 6% = $57.24.
Step 4: Add the interest earned in the second year to the total amount after the first year: $954 + $57.24 = $1011.24.
contact@gauthmath.com |
6537 | https://www.quora.com/How-do-I-convert-m-2-to-cm-2-And-cm-2-to-m-2 | Something went wrong. Wait a moment and try again.
Area (mathematics)
Metric Systems
Square Meter
Measurement Conversion
Units of Area
Unit Conversions
Area Computation
Conversion (math)
5
How do I convert m^2 to cm^2? And cm^2 to m^2?
Sort
Assistant
Bot
·
1y
To convert between square meters (m²) and square centimeters (cm²), you can use the following relationships:
From m² to cm²:
- Since 1 meter = 100 centimeters, then:
1
m
2
=
(
100
cm
)
2
=
10
,
000
cm
2
- To convert m² to cm², multiply by 10,000:
Area in cm
2
=
Area in m
2
×
10
,
000
2. From cm² to m²:
- To convert cm² to m², divide by 10,000:
Area in m
2
=
Area in cm
2
10
,
000
Example:
Convert 2 m² to cm²:
2
m
2
×
10
,
000
=
20
,
000
cm
2
Convert 50,000 cm² to m²:
\frac{50,000 \, \text{cm}
To convert between square meters (m²) and square centimeters (cm²), you can use the following relationships:
From m² to cm²:
- Since 1 meter = 100 centimeters, then:
1
m
2
=
(
100
cm
)
2
=
10
,
000
cm
2
- To convert m² to cm², multiply by 10,000:
Area in cm
2
=
Area in m
2
×
10
,
000
2. From cm² to m²:
- To convert cm² to m², divide by 10,000:
Area in m
2
=
Area in cm
2
10
,
000
Example:
Convert 2 m² to cm²:
2
m
2
×
10
,
000
=
20
,
000
cm
2
Convert 50,000 cm² to m²:
50
,
000
cm
2
10
,
000
=
5
m
2
Related questions
How do you convert cm^2 to m^2?
How do you convert cm (centimeter) to m^2 (meter square)?
How can one convert cm^-2 to meters?
When we convert 1m^2 to centimeters, why is it 10,000 cm^2 when it should be 100cm^2? How does the length change to 10,000cm?
How do you convert 10 m^2 to cm?
Lecta Orinosto
Why? Why? WHY ask homework questions on Quora?!
·
Author has 793 answers and 2.1M answer views
·
9y
Step 1: 100 cm = 1 m.
What do you want? The exchange rate of cm^2 to m^2. So square both sides.
Step 2: (100 cm)^2 = (1 m )^2
Step 3 (simplify): 10^4 cm^2 = 1 m^2
So if you want to convert cm^2 to m^2, divide by 10000. If you want to convert the other way around, multiply by 10000.
Promoted by Coverage.com
Johnny M
Master's Degree from Harvard University (Graduated 2011)
·
Updated Sep 9
Does switching car insurance really save you money, or is that just marketing hype?
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I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take.
But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious.
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If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying.
Rob Hooft
Manager, Scientist, Programmer.
·
Author has 5K answers and 9.2M answer views
·
9y
Draw it out: draw a square and put 1m at the sides. That is your square meter. Now inside, draw a small square that measures 1 cm. that will be your square centimeter. Next, realize that horizontally a hundred small squares fit to fill the large square. And similarly one hundred of those rows fit inside vertically. The total scale between the two units is therefore 100
2
.
Ilkka Vuorio
PhD in Physics, Massachusetts Institute of Technology (Graduated 1987)
·
Author has 17.7K answers and 2.2M answer views
·
11mo
The only things you need to know is that c = centi = 1/100. And that it’s attached to the “m” following it by a multiplication sign (which isn’t written down). Once you know these elementary things, all the unit conversion in the SI system become trivial. WAY easier than Imperial.
Related questions
How do you convert cm^2 to m^2?
How do you convert cm (centimeter) to m^2 (meter square)?
How can one convert cm^-2 to meters?
When we convert 1m^2 to centimeters, why is it 10,000 cm^2 when it should be 100cm^2? How does the length change to 10,000cm?
How do you convert 10 m^2 to cm?
How do you convert m to cm and mm?
How many cm are in 1 2 inches?
How many 1/4 cups does it take to make a 2/3 cup?
How many cm are there in 1 m^2?
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How can you convert from m/s to m/s^2?
What is 1 cm for every 5 km?
An inch is 2.5 cm. That means 180 cm is 72 inches. If 72 inches is 6 feet, why is 180 cm 5 feet 11?
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6538 | https://www.youtube.com/watch?v=q9YupUH37NQ | Inclusive vs. Exclusive Interquartile Range
PsychExamReview
81100 subscribers
99 likes
Description
5300 views
Posted: 15 Feb 2023
In this video I explain the difference between calculating an exclusive interquartile range and an inclusive interquartile range. While we generally use an exclusive interquartile range with an even set of scores, when we have an odd-numbered set of scores we can choose which method we'd like to use. The inclusive method will tend to give us a slightly smaller interquartile range by increasing the 25th percentile and decreasing the 75th percentile.
13 comments
Transcript:
Hi, I'm Michael Corayer and this is Psych Exam
Review. In the video explaining the interquartile range I didn't go into detail on the fact that
there's actually two different calculations for finding the interquartile range. These are
an inclusive method and an exclusive method, and this refers to whether we're including the
median as being part of our upper and lower halves of the data or if we're excluding the median. Now
if we're excluding the median what we're really saying is that the lower half of the data is less
than the median and the upper half is greater than the median. But if we're using an inclusive method
what we're saying is the lower half of our data is less than or equal to the median and the upper
half of the data is greater than or equal to the median. This will slightly change the boundaries
of where our 25th and 75th percentiles lie and this means that it will slightly change the
interquartile range. If we have an even number of scores then we naturally tend to exclude
the median because it falls between two of our values. It's the mean of the two middle scores;
it's not actually one of the values of our data, so it makes sense to exclude it from being part of
the upper and lower halves of our data. So let's take a look at a set of 10 scores and see what I
mean. Since we have 10 scores our median is going to be the mean of the 5th and 6th positions: 9
and 11. So this gives us a median of 10. And we can see that this line for 10 neatly divides our
data into two halves and now we take the median of each of those two halves to find the 25th and
75th percentiles. So we have a 25th percentile of 6 and a 75th percentile of 15. This gives us
an interquartile range of 15 minus 6 equals 9. Now in this case we're excluding the median
because it's not actually one of our values. So it makes sense to say that well 10 is not really
part of the lower half of the data and it's not part of the upper half of the data. But in the
case of an odd number of scores we can see that our median is going to be one of our values and
that means we might want to include it as being part of the lower and upper halves of the data.
So let's take a look at a group of 11 scores. In this case our median is going to be the
6th position, which gives us a median of 11. Now if we want to find an exclusive interquartile
range we're going to ignore this 11. We're going to define the lower half as being below this and
the upper half as being above it. So now we find the medians of those two halves and we get a 25th
percentile of 4 and a 75th percentile of 18. So to find our exclusive interquartile range we take
18 minus 4 equals 14. But we could decide to do an inclusive interquartile range here. This
means we'd say that the lower half of our data extends up to and including 11 and the upper
half of our data starts at 11 and moves up. This means that the 25th percentile will
be the mean of 4 and 5, which would be 4.5 and the 75th percentile would be the
mean of 16 and 18 which would be 17. So now we'd have an inclusive interquartile
range of 17 minus 4.5 equals 12.5. As we can now see, if we use this inclusive method
we'll tend to get a slightly smaller interquartile range. This is because by including the median
as part of the lower and upper halves we're moving the 25th percentile up and we're pulling
the 75th percentile down, so we're looking at a narrower range of our data now. Depending on
our data, this distinction between an inclusive and exclusive interquartile range might be
trivial or we might find if we have repeated scores around those points that it actually
doesn't change the interquartile range at all. This distinction isn't always fully explained
in introductory statistics textbooks but I think it's important to understand that there's two
calculations and to understand the difference between them. It can also help you understand why
the same data set in different software programs might lead to slightly different interquartile
ranges if one is using an inclusive method and the other is using an exclusive method.
I hope you found this helpful. If so, let me know in the comments, like and
subscribe, and make sure to check out the hundreds of other psychology tutorials that
I have on the channel. Thanks for watching! |
6539 | https://www.xconvert.com/unit-converter/feet-per-second-to-metres-per-second | Feet per second (ft/s) to Metres per second (m/s) conversion
Feet per second to Metres per second conversion table
| Feet per second (ft/s) | Metres per second (m/s) |
--- |
| 0 | 0 |
| 1 | 0.30479991872 |
| 2 | 0.60959983744 |
| 3 | 0.91439975616 |
| 4 | 1.21919967488 |
| 5 | 1.5239995936 |
| 6 | 1.82879951232 |
| 7 | 2.13359943104 |
| 8 | 2.43839934976 |
| 9 | 2.74319926848 |
| 10 | 3.0479991872 |
| 20 | 6.0959983744 |
| 30 | 9.1439975616 |
| 40 | 12.1919967488 |
| 50 | 15.239995936 |
| 60 | 18.2879951232 |
| 70 | 21.3359943104 |
| 80 | 24.3839934976 |
| 90 | 27.4319926848 |
| 100 | 30.479991872 |
| 1000 | 304.79991872 |
How to convert feet per second to metres per second?
Converting between feet per second (fps) and meters per second (m/s) is a common task in physics and engineering. This section explains how to perform these conversions, providing formulas and examples for practical understanding.
Understanding the Conversion Factor
The core of the conversion lies in the relationship between feet and meters. One foot is defined as exactly 0.3048 meters. This constant is the key to converting between the two units of speed.
Therefore, to convert feet per second to meters per second, you multiply by this conversion factor.
Converting Feet per Second to Meters per Second
To convert from feet per second to meters per second, use the following formula:
For example, to convert 1 foot per second to meters per second:
So, 1 foot per second is equal to 0.3048 meters per second.
Converting Meters per Second to Feet per Second
To convert from meters per second to feet per second, you divide by the same conversion factor (0.3048):
Or, equivalently, multiply by the reciprocal of 0.3048, which is approximately 3.28084:
For example, to convert 1 meter per second to feet per second:
So, 1 meter per second is approximately equal to 3.28084 feet per second.
Notable Figures and Historical Context
While the conversion itself is straightforward, the use of these units is rooted in different measurement systems. The foot is a customary unit primarily used in the United States and a few other countries, while the meter is a base unit in the International System of Units (SI), used globally for scientific and most engineering applications. The standardization of these measurements has been influenced by scientific figures like the members of the French Academy of Sciences, who helped establish the metric system in the late 18th century.
Real-World Examples
Running Speed: An average jogging speed might be around 6 mph, which is approximately 8.8 feet per second. In meters per second:
Wind Speed: A moderate breeze might be 15 mph, which is approximately 22 feet per second. In meters per second:
Car Speed: A car traveling at 60 mph is roughly 88 feet per second. In meters per second:
See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the Metres per second to other unit conversions.
What is Feet per Second?
Feet per second (fps) is a unit of speed expressing the distance in feet traveled per second. It's commonly used in various fields, including sports, engineering, and meteorology, to quantify how fast an object is moving.
Formation of Feet per Second
Feet per second is derived from two fundamental units:
Feet (ft): A unit of length in the imperial and US customary systems.
Second (s): The base unit of time in the International System of Units (SI).
The unit is formed by dividing a distance measured in feet by a time measured in seconds:
Real-World Examples and Applications
Sports: A baseball pitch can reach speeds of 90-100 mph, which is approximately 132-147 fps.
Meteorology: Wind speed is often measured in feet per second, especially in aviation and weather forecasting. For example, a moderate breeze might be around 15-20 fps.
Engineering: Calculating the speed of a vehicle or fluid flow in a pipe. For instance, water flowing through a pipe at 10 fps.
Ballistics: Measuring the muzzle velocity of a bullet, which can range from 800 fps to over 3000 fps depending on the firearm and ammunition.
Relationship to Other Speed Units
Feet per second can be converted to other common units of speed using the following relationships:
Miles per hour (mph): 1 fps ≈ 0.681818 mph
Meters per second (m/s): 1 fps ≈ 0.3048 m/s
Kilometers per hour (km/h): 1 fps ≈ 1.09728 km/h
Notable Associations and Facts
While there isn't a specific law or person directly associated with the invention or formalization of feet per second, its usage is deeply ingrained in fields that involve motion and speed measurements.
Galileo Galilei: His work on motion and free fall laid the groundwork for understanding speed and acceleration, though he didn't use the modern unit of feet per second.
Speed of Sound: At sea level and 20°C, the speed of sound is approximately 1125 fps. This is a critical value in acoustics and aerodynamics. Read more about speed of sound from NASA Glenn Research Center.
Quick Conversions
Here are some common speeds converted to feet per second:
30 mph ≈ 44 fps
60 mph ≈ 88 fps
100 mph ≈ 146.67 fps
What is metres per second?
What is Metres per second?
Metres per second (m/s) is the standard unit of speed (scalar) or velocity (vector) in the International System of Units (SI). It represents the distance traveled in metres during each second. Understanding this unit is crucial in physics and everyday applications for quantifying motion.
Understanding Metres per Second
Definition
Metres per second (m/s) is a derived unit, meaning it's defined in terms of base SI units: the metre (m) for length and the second (s) for time. It expresses how many metres an object travels in one second.
Formula
Speed or velocity is calculated as:
Where:
= speed or velocity (m/s)
= distance traveled (m)
= time taken (s)
Formation of Metres per Second
The unit is formed by dividing a distance measured in metres by a time measured in seconds. This gives you the rate at which an object is moving.
For example, if a car travels 100 metres in 10 seconds, its average speed is:
Notable Associations
Galileo Galilei
Galileo Galilei, a prominent figure in physics and astronomy, significantly contributed to our understanding of motion. While he didn't use the modern unit of m/s, his experiments with falling objects and motion on inclined planes laid the groundwork for understanding speed and acceleration, concepts directly related to metres per second.
Isaac Newton
Isaac Newton used the concepts of distance and time in his laws of motion. The first law states that an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force. This constant speed is very related to meters per second.
Real-World Examples
Walking Speed
A typical walking speed is around 1.4 m/s.
Sprinting Speed
An Olympic sprinter can reach speeds of around 10-12 m/s.
Car Speed
A car traveling at 60 km/h is moving at approximately 16.67 m/s. ()
Speed of Sound
The speed of sound in dry air at 20°C is approximately 343 m/s.
Orbital Speed
The International Space Station orbits Earth at approximately 7,660 m/s.
Conversion to Other Units
Metres per second can be converted to other common units of speed:
Kilometres per hour (km/h): Multiply m/s by 3.6. ()
Miles per hour (mph): Multiply m/s by 2.237. ()
Knots (kn): Multiply m/s by 1.944. ()
Complete Feet per second conversion table
| Convert 1 ft/s to other units | Result |
--- |
| Feet per second to Metres per second (ft/s to m/s) | 0.30479991872 |
| Feet per second to Kilometres per hour (ft/s to km/h) | 1.097279707392 |
| Feet per second to Millimeters per hour (ft/s to mm/h) | 1097279.707392 |
| Feet per second to Miles per hour (ft/s to mph) | 0.681818 |
| Feet per second to Knots (ft/s to knot) | 0.5924838739671 |
| Feet per second to Feet per minute (ft/s to ft/min) | 60.000176000563 |
| Feet per second to Inches per hour (ft/s to in/h) | 43207.73130545 |
Speed conversions
Feet per second to Metres per second (ft/s to m/s)
Feet per second to Kilometres per hour (ft/s to km/h)
Feet per second to Millimeters per hour (ft/s to mm/h)
Feet per second to Miles per hour (ft/s to mph)
Feet per second to Knots (ft/s to knot)
Feet per second to Feet per minute (ft/s to ft/min)
Feet per second to Inches per hour (ft/s to in/h) |
6540 | https://quicktakes.io/learn/chemistry/questions/what-distinguishes-monoprotic-acids-from-diprotic-and-triprotic-acids | Student Question : What distinguishes monoprotic acids from diprotic and triprotic acids? | Chemistry | QuickTakes
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What distinguishes monoprotic acids from diprotic and triprotic acids?
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Answer
Monoprotic, diprotic, and triprotic acids are distinguished by the number of ionizable hydrogen ions (protons) they can donate per molecule during acid-base reactions.
Monoprotic Acids: These acids can donate only one proton (H+) per molecule. Examples include hydrochloric acid (HCl), hydrobromic acid (HBr), and nitric acid (HNO3). During an acid-base titration, they transfer a single H+ ion.
Diprotic Acids: These acids can donate two protons per molecule. They undergo ionization in two steps, with the first ionization typically occurring to a greater extent than the second. A common example is sulfuric acid (H2SO4), which can release two protons sequentially.
Triprotic Acids: These acids can donate three protons per molecule. They also ionize in a stepwise manner, with each subsequent loss of a proton becoming less favorable. An example of a triprotic acid is phosphoric acid (H3PO4).
In summary, the key distinction lies in the number of protons each type of acid can donate: monoprotic acids donate one, diprotic acids donate two, and triprotic acids donate three.
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Central Limit Theorem | Formula, Definition & Examples
Central Limit Theorem | Formula, Definition & Examples
Published on July 6, 2022 by Shaun Turney. Revised on June 22, 2023.
The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the population isn’t normally distributed.
Table of contents
What is the central limit theorem?
Central limit theorem formula
Sample size and the central limit theorem
Conditions of the central limit theorem
Importance of the central limit theorem
Central limit theorem examples
Practice questions
Other interesting articles
Frequently asked questions about the central limit theorem
What is the central limit theorem?
The central limit theorem relies on the concept of a sampling distribution, which is the probability distribution of a statistic for a large number of samples taken from a population.
Imagining an experiment may help you to understand sampling distributions:
Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean.
Now you draw another random sample of the same size, and again calculate the mean.
You repeat this process many times, and end up with a large number of means, one for each sample.
The distribution of the sample means is an example of a sampling distribution.
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.
A normal distribution is a symmetrical, bell-shaped distribution, with increasingly fewer observations the further from the center of the distribution.
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Central limit theorem formula
Fortunately, you don’t need to actually repeatedly sample a population to know the shape of the sampling distribution. The parameters of the sampling distribution of the mean are determined by the parameters of the population:
The mean of the sampling distribution is the mean of the population.
The standard deviation of the sampling distribution is the standard deviation of the population divided by the square root of the sample size.
We can describe the sampling distribution of the mean using this notation:
Where:
X̄ is the sampling distribution of the sample means
~ means “follows the distribution”
N is the normal distribution
µ is the mean of the population
σ is the standard deviation of the population
n is the sample size
Sample size and the central limit theorem
The sample size (n) is the number of observations drawn from the population for each sample. The sample size is the same for all samples.
The sample size affects the sampling distribution of the mean in two ways.
1. Sample size and normality
The larger the sample size, the more closely the sampling distribution will follow a normal distribution.
When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”
By convention, we consider a sample size of 30 to be “sufficiently large.”
When n < 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.
When n ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.
2. Sample size and standard deviations
The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is).
When n is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.
When n is high, the standard deviation is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.
Conditions of the central limit theorem
The central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions:
The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30.
The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the sampling is random.
The population’s distribution has finite variance. Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.
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Importance of the central limit theorem
The central limit theorem is one of the most fundamental statistical theorems. In fact, the “central” in “central limit theorem” refers to the importance of the theorem.
Central limit theorem examples
Applying the central limit theorem to real distributions may help you to better understand how it works.
Continuous distribution
Suppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this:
Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.
Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired.
Example: Central limit theorem; sample of n = 5
| | | | | |
--- ---
| 68 | 73 | 70 | 62 | 63 |
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.
Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean.
Example: Central limit theorem; means of 10 small samples
| | | | | | | | | | |
--- --- --- --- --- |
| 60.8 | 57.8 | 62.2 | 68.6 | 67.4 | 67.8 | 68.3 | 65.6 | 66.5 | 62.1 |
If you repeat the procedure many more times, a histogram of the sample means will look something like this:
Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew.
Notice also that the spread of the sampling distribution is less than the spread of the population.
The central limit theorem says that the sampling distribution of the mean will always follow a normal distribution when the sample size is sufficiently large. This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.
Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.
Example: Central limit theorem; sample of n = 50
| | | | | | | | | | |
--- --- --- --- --- |
| 73 | 49 | 62 | 68 | 72 | 71 | 65 | 60 | 69 | 61 |
| 62 | 75 | 66 | 63 | 66 | 68 | 76 | 68 | 54 | 74 |
| 68 | 60 | 72 | 63 | 57 | 64 | 65 | 59 | 72 | 52 |
| 52 | 72 | 69 | 62 | 68 | 64 | 60 | 65 | 53 | 69 |
| 59 | 68 | 67 | 71 | 69 | 70 | 52 | 62 | 64 | 68 |
The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large.
Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:
In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.
The standard deviation of this sampling distribution is 0.85 years, which is less than the spread of the small sample sampling distribution, and much less than the spread of the population. If you were to increase the sample size further, the spread would decrease even more.
We can use the central limit theorem formula to describe the sampling distribution:
µ = 65
σ = 6
n = 50
Discrete distribution
Approximately 10% of people are left-handed. If we assign a value of 1 to left-handedness and a value of 0 to right-handedness, the probability distribution of left-handedness for the population of all humans looks like this:
The population mean is the proportion of people who are left-handed (0.1). The population standard deviation is 0.3.
Imagine that you take a random sample of five people and ask them whether they’re left-handed.
Example: Central limit theorem; sample of n = 5
| | | | | |
--- ---
| 0 | 0 | 0 | 1 | 0 |
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.
Imagine you repeat this process 10 times, randomly sampling five people and calculating the mean of the sample. This is a sampling distribution of the mean.
Example: Central limit theorem; means of 10 small samples
| | | | | | | | |
--- --- --- --- |
| 0 | 0 | 0.4 | 0.2 | 0.2 | 0 | 0.4 | 0 |
If you repeat this process many more times, the distribution will look something like this:
The sampling distribution isn’t normally distributed because the sample size isn’t sufficiently large for the central limit theorem to apply.
As the sample size increases, the sampling distribution looks increasingly similar to a normal distribution, and the spread decreases:
n = 10
n = 20
n = 30
n = 100
The sampling distribution of the mean for samples with n = 30 approaches normality. When the sample size is increased further to n = 100, the sampling distribution follows a normal distribution.
We can use the central limit theorem formula to describe the sampling distribution for n = 100.
µ = 0.1
σ = 0.3
n = 100
Practice questions
Other interesting articles
If you want to know more about statistics, methodology, or research bias, make sure to check out some of our other articles with explanations and examples.
Statistics
Confidence interval
Kurtosis
Descriptive statistics
Measures of central tendency
Correlation coefficient
p value
Methodology
Cluster sampling
Stratified sampling
Types of interviews
Case study
Cohort study
Thematic analysis
Research bias
Implicit bias
Cognitive bias
Survivorship bias
Availability heuristic
Nonresponse bias
Regression to the mean
Frequently asked questions about the central limit theorem
What is a normal distribution?
: In a normal distribution, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.
The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.
What are the three types of skewness?
: The three types of skewness are:
Right skew (also called positive skew). A right-skewed distribution is longer on the right side of its peak than on its left.
Left skew (also called negative skew). A left-skewed distribution is longer on the left side of its peak than on its right.
Zero skew. It is symmetrical and its left and right sides are mirror images.
Why are samples used in research?
: Samples are used to make inferences about populations. Samples are easier to collect data from because they are practical, cost-effective, convenient, and manageable.
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Shaun Turney
During his MSc and PhD, Shaun learned how to apply scientific and statistical methods to his research in ecology. Now he loves to teach students how to collect and analyze data for their own theses and research projects.
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6542 | https://orinanobworld.blogspot.com/2012/08/step-functions-and-mixed-functions.html | OR in an OB World: Step Functions and Mixed Functions
OR in an OB World
A mix of operations research items and software tricks that I'll probably forget if I don't write them down somewhere.
Monday, August 27, 2012
Step Functions and Mixed Functions
Someone asked me (elsewhere) about modeling, in a mathematical program, a piece-wise linear function that was a mixture of steps and linear segments. Since it contains step functions, it is neither continuous nor convex, but nonetheless it can be modeled within a mixed-integer linear program (MILP). Here is a sketch of a sample function.
I drew this with one diagonal linear segment, at the right end, but what follows will work for any mixture of steps and linear segments. If you are wondering why the heights are listed at y 2,y 4,y 6 y 2,y 4,y 6, it will become clear in a minute. The labeling is a bit fuzzy (sorry about that). The annotation of the diagonal segment is y=m x+b y=m x+b.
The key to modeling this is recognizing that every point on the graph is a linear combination of two adjacent endpoints. (The endpoints are the ones represented by dots on the graph.) There are nine endpoints on the graph, which we will designate as (x 1,y 1),…,(x 9,y 9)(x 1,y 1),…,(x 9,y 9), as shown in the second sketch. For the modeling trick to work, it is necessary that the domain of the function be bounded (in this case 0≤x≤x 9 0≤x≤x 9).
Hopefully the reason for the subscripts of the ordinates of the horizontal segments is now apparent
Let y y be the value of the function. We introduce continuous variables μ 1,…,μ 9∈[0,1]μ 1,…,μ 9∈[0,1], which will be the weights of a convex combination of the graph points. Both x x and y y are expressed using these weights:x=μ 1 x 1+⋯+μ 9 x 9 y=μ 1 y 1+⋯+μ 9 y 9 μ 1+⋯+μ 9=1.x=μ 1 x 1+⋯+μ 9 x 9 y=μ 1 y 1+⋯+μ 9 y 9 μ 1+⋯+μ 9=1.
What makes this work is that we also make the weights {μ 1,…,μ 9}{μ 1,…,μ 9} a type 2 special ordered set (SOS2), specified in index order. This tells the browser that at most two of the weights can be nonzero and, if there are two nonzero weights, they must be consecutive in the stated order (e.g., μ 3=0.4,μ 4=0.6 μ 3=0.4,μ 4=0.6 is fine but μ 2=0.4,μ 7=0.6 μ 2=0.4,μ 7=0.6 is invalid). The SOS2 restriction essentially forces the solver to select one segment of the graph, whether horizontal, diagonal or even vertical; the weights then select one point on the segment. If y y is part of the objective, and the chosen segment happens to be vertical, it's a safe bet that one of the weights will be 1 and the rest 0 (picking whichever endpoint of the segment better suits the objective direction).
Although the weight variables are continuous, introducing the SOS2 constraint will effectively turn the problem into a MILP even if it otherwise would have been a linear program (LP).
There are other ways to model this type of function. In particular, anything done with SOS1 or SOS2 constraints can be done with binary variables and linear constraints involving those binaries.
Related posts:
Piecewise-linear Functions in Math Programs
Modeling the All Units Discount
Piecewise-linear Functions Redux
Modeling Absolute Values
Piecewise Linear Functions in CPLEX
Posted by Paul A. Rubin at 6:31 PM
Labels: math programming
7 comments:
UnknownAugust 28, 2012 at 3:23 PM
Professor Rubin,
Nice article indeed. Thanks a lot.
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1. Image 6 Paul A. RubinAugust 28, 2012 at 3:31 PM
You're very welcome.
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DewanSeptember 11, 2012 at 11:21 AM
Dear Paul,
If a convex continues curve instead of step function is in two adjacent endpoints, then is it possible to apply your proposed techniques to model it?
Regards,
Dewan
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Replies
1. Image 8 Paul A. RubinSeptember 11, 2012 at 11:36 AM
That depends what you mean by "curve". The technique above works for any piecewise-linear function (any mix of steps and linear segments). If your "curve" is nonlinear, you can do the same things, but you are now dealing with an MINLP (mixed integer nonlinear program), and you need a solver that can solve MINLPs. I don't work with MINLPs, so I don't know which solvers are commonly used and how (or if) they handle SOS constraints.
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2. Image 9 DewanSeptember 11, 2012 at 11:57 AM
Thanks for your quick reply.
In my case, "curve" is a quadratic equation, which is nonlinear for sure. But, the quadratic equation can be piecewise linearized. In details, a function which consists of some finite number of segments (which are known) and all the segments are quadratic equation. I think in this case the problem can be with a MILP solver. My idea is: solver should select the optimal segment at first, then piecewise linear technique is applied to the selected segment to find the optimal solution. Here I want to mention that the whole function at once is a non-convex non differential function. But each segment individually is convex. I am afraid that the could be full of SOS variables. Can I apply your proposed techniques to model it?
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3. Image 10 Paul A. RubinSeptember 12, 2012 at 2:58 PM
When you say "the quadratic equation can be piecewise linearized", you mean the function (or each quadratic segment of the function) can be approximated by a piecewise-linear function, right? If so, then yes, you can use SOS2 to convert the approximate problem into a MILP. When you say "solver should select the optimal segment at first, then piecewise linear technique is applied to the selected segment to find the optimal solution" you lose me. If you mean select the optimal quadratic segment, no; that would require a MINLP solver. If you are talking about first doing a pw-linear approximation, then solving, then refining the approximation near the "optimal" solution and solving again, you can try that. It likely will produce a good answer, but I'm not at all sure you can prove that it converges to an optimal solution to the original (nonlinear, nonconvex) problem.
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4. Image 11 DewanSeptember 13, 2012 at 8:21 AM
Thank you again. Yes, I will give a try to first model the piecewise-linear approximation, then will send them to find a optimal solution of the approximated function.
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About Me
Paul A. RubinI am Professor Emeritus of Management Science, retired from the Eli Broad College of Business at Michigan State University. My primary research interest is in the application of integer programming models and algorithms. In addition to research interests, I am active in the INFORMS Pro Bono Analytics project.View my complete profile
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6543 | https://www.mathsisfun.com/algebra/trig-solving-ssa-triangles.html | Solving SSA Triangles
Triangle Solving Practice
The Law of Sines
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Solving SSA Triangles
"SSA" means "Side, Side, Angle"
"SSA" is when we know two sides and an angle that is not the angle between the sides.
To solve an SSA triangle
use The Law of Sines first to calculate one of the other two angles
then use the three angles add to 180° to find the other angle
finally use The Law of Sines again to find the unknown side
Example 1
In this triangle we know
angle B = 31°
b = 8
and c = 13
In this case, we can use The Law of Sines first to find angle C:
sin(C)/c = sin(B)/b
sin(C)/13 = sin(31°)/8
sin(C) = (13×sin(31°))/8
sin(C) = 0.8369...
C = sin−1(0.8369...)
C = 56.818...° ( but see below)
C =56.8° to one decimal place
Next, we can use the three angles add to 180° to find angle A:
A = 180° − 31° − 56.818...°
A = 92.181...° =92.2° to one decimal place
Now we can use The Law of Sines again to find a:
a/sin(A) = b/sin(B)
a/sin(92.181...°) = 8/sin(31°)
Notice that we didn't use A = 92.2°, that angle is rounded to 1 decimal place. It's much better to use the unrounded number 92.181...° which should still be on our calculator from the last calculation.
a = (sin(92.181...°) × 8)/sin(31°)
a =15.52 to 2 decimal places
So, we have completely solved the triangle ...
... or have we?
Back when we calculated:
C = sin−1(0.8369...)
C = 56.818...°
We didn't include that sin−1(0.8369...) might have two answers (see Law of Sines):
The other answer for C is 180° − 56.818...°
Here you can see why we have two possible answers:
By swinging side "8" left and right we can
join up with side "a" in two possible locations.
So let's go back and continue our example:
The other possible angle is:
C = 180° − 56.818...°
C =123.2° to one decimal place
With a new value for C we will have new values for angle A and side a
Use "the three angles add to 180°" to find angle A:
A = 180° − 31° − 123.181...°
A = 25.818...°
A =25.8° to one decimal place
Now we can use The Law of Sines again to find a:
a/sin(A) = b/sin(B)
a/sin(25.818...°) = 8/sin(31°)
a = (sin(25.818...°)×8)/sin(31°)
a =6.76 to 2 decimal places
So the two sets of answers are:
C = 56.8°, A = 92.2°, a = 15.52
C = 123.2°, A = 25.8°, a = 6.76
Example 2
This is also an SSA triangle.
In this triangle we know angle M = 125°, m = 12.4 and l = 7.6
We will use The Law of Sines to find angle L first:
sin(L)/l = sin(M)/m
sin(L)/7.6 = sin(125°)/12.4
sin(L) = (7.6×sin(125°))/12.4
sin(L) = 0.5020...
L = 30.136...°
L =30.1° to one decimal place
Next, we will use "the three angles add to 180°" to find angle N:
N = 180° − 125° − 30.136...°
N = 24.863...°
N =24.9° to one decimal place
Now we will use The Law of Sines again to find n:
n/sin(N) = m/sin(M)
n/sin(24.863...°) = 12.4/sin(125°)
n = (sin(24.863...°)×12.4)/sin(125°)
n =6.36 to 2 decimal places
Note there is only one answer in this case. The "12.4" line only joins up one place.
The other possible answer for L is 149.9°. But that is impossible because we already have M = 125° and a triangle can't have two angles greater than 90°.
Conclusion:
When solving a "Side, Side, Angle" triangle we start with the Law of Sines, but need to check if there is another possible answer!
Mathopolis:Q1)Q2)Q3)Q4)Q5)Q6)Q7)Q8)Q9)Q10)
Triangle Solving PracticeThe Law of SinesThe Law of CosinesSolving TrianglesTrigonometry IndexAlgebra Index
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6544 | https://www.eecis.udel.edu/~weile/ELEG648Spring06/Resources/Excitation.pdf | Waveguides Free Space Modal Excitation Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648—Modal Excitation in Cartesian Coordinates D. S. Weile Modal Excitation Waveguides Free Space Outline 1 Waveguides Aperture Excitation Current Excitation D. S. Weile Modal Excitation Waveguides Free Space Outline 1 Waveguides Aperture Excitation Current Excitation 2 Free Space Apertures in a Ground Plane Plane Current Sheets D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Outline 1 Waveguides Aperture Excitation Current Excitation 2 Free Space Apertures in a Ground Plane Plane Current Sheets D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Excitation of Waveguides Our discussion so far has ignored how energy might be coupled into guides.
After all a mode is by definition a solution that can exist without a source.
How are modes excited?
Let us consider our standard guide. Over the z = 0 plane, let us assume Ex = 0 Ey = f(x, y) We further assume the waveguide is matched somewhere for z > 0 and seek the solution for the modes launched.
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Modal Expansion Since Ex = 0 we look for TEx solutions. We can thus write a Modal Expansion for ψ ψ = ∞ X m=1 ∞ X n=0 Amn sin mπx a cos nπy b e−γmnz To find the Amn we must impose f(x, y) = Ey|z=0 = ∞ X m=1 ∞ X n=0 γmnAmn sin mπx a cos nπy b How do we find the coefficients?
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Aside: Useful Notation Two “functions” that often arise in mathematical physics are the Kronecker Delta δmn = 1 if m = n 0 otherwise and the ever present and strangely named Neumann’s Number ϵn = 1 if n = 0 2 if n ̸= 0 D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Aside: Orthogonal Functions Note the Orthogonality Relations a Z 0 sin mπx a sin nπx a dx = a 2δmn for m, n = 1, 2, . . .
a Z 0 cos mπx a cos nπx a dx = a ϵm δmn for m, n = 0, 1, 2, . . .
a Z 0 sin mπx a cos nπx a dx = 0 for m, n = 0, 1, 2, . . .
Know these!
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Finding the Coefficients To find the coefficients, we multiply and integrate: a Z 0 b Z 0 f(x, y) sin m′πx a cos n′πy b dxdy = ∞ X m=1 ∞ X n=0 γmnAmn a Z 0 b Z 0 sin mπx a cos nπy b sin m′πx a cos n′πy b dxdy = ∞ X m=1 ∞ X n=0 γmnAmn ab 2ϵn′ δmm′δnn′ = γm′n′Am′n′ ab 2ϵn′ Thus Amn = 2ϵn abγmn a Z 0 b Z 0 f(x, y) sin mπx a cos nπy b dxdy D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Observations A similar procedure can be followed if the magnetic field is given.
Excitation by a field with both x- and y-components can be solved by superposition.
Note that A finite number of modes propagates.
An infinite number of modes evanesces.
How can we excite a given mode? Can we avoid exciting some modes entirely? How?
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Power Flow Because our modes are TEx, we can compute the power flow in the guide using P = − a Z 0 b Z 0 [EyH∗ x]z=0 dxdy.
Plugging in P = a Z 0 dx b Z 0 dy " ∞ X m=1 ∞ X n=0 Emn sin mπx a cos nπy b # × ∞ X p=1 ∞ X q=0 k2 − pπ a 2 jωµγ∗ pq E∗ pq sin pπx a cos qπy b Why do we use different variables in the two summations?
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Power Flow Thus P = ∞ X m=1 ∞ X n=0 ∞ X p=1 ∞ X q=0 k2 − pπ a 2 jωµγ∗ pq E∗ pqEmn a Z 0 dx sin mπx a sin pπx a × b Z 0 dy cos nπy b cos qπy b = ∞ X m=1 ∞ X n=0 ab 2ϵn k2 − mπ a 2 jωµγ∗ mn |Emn|2 = ∞ X m=1 ∞ X n=0 ab 2ϵn (Y0)∗ mn |Emn|2 where (Y0)mn is the modal admittance.
Thus, each mode carries its own power.
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Waveguide Junction Consider the junction between the waveguides below. Their width is a.
y z c b We assume the field in the junction is Ey|z=0 ≈ sin πx a y < c 0 y > c D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Waveguide Junction Now, we have seen that the field in the junction is given by f(x, y) = Ey|z=0 = ∞ X m=1 ∞ X n=0 γmnAmn sin mπx a cos nπy b and that the coefficients can be computed from γmnAmn = Emn = 2ϵn ab a Z 0 b Z 0 f(x, y) sin mπx a cos nπy b dxdy Immediately, we see the only excited modes have m = 1 by orthogonality.
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Computing the Coefficients Plugging in we find E10 = 2 ab a Z 0 c Z 0 sin2 πx a dxdy = c b, and E1n = 4 ab a Z 0 c Z 0 sin2 πx a cos nπy b dxdy = 2 nπ sin nπc b .
Most of these modes are of course evanescent.
D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Power Flow and Admittance Applying our power formula P = ac2 2b (Y0)∗ 10 + 2 ∞ X n=1 (Y0)∗ 1n " sin nπ b nπc b #2 We can also define an aperture admittance by defining the aperture “voltage” as the line integral of E across the center of the aperture. Then Ya = 2P∗ |V|2 D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Outline 1 Waveguides Aperture Excitation Current Excitation 2 Free Space Apertures in a Ground Plane Plane Current Sheets D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Current Excitation of Waveguides In aperture excitation, we match a mode series to a given field in the junction.
In current excitation, we are given a current.
The electric field is continuous at the current.
The magnetic field is discontinuous at the current.
Note that in this way, aperture excitation can be thought of as magnetic current excitation.
Suppose for simplicity we assume a current sheet in the z = 0 plane with Js = uxf(x, y) and that the guide is matched in both directions. We can assume the waves generated are TMx. (Why?) D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Current Excitation of Waveguides The most general solution for TM waves can be written as ψ+ = ∞ X m=0 ∞ X n=1 B+ mn cos mπx a sin nπy b e−γmnz z > 0 ψ− = ∞ X m=0 ∞ X n=1 B− mn cos mπx a sin nπy b e+γmnz z < 0 Continuity of the tangential E-field (which is proportional to ψ and its transverse derivatives) requires Bmn = B+ mn = B− mn D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Current Excitation of Waveguides The magnetic field has to be discontinuous by the current, i.e.
uz × H+ −H− = Js Thus, f(x, y) = H− y −H+ y z=0 = ∞ X m=0 ∞ X n=1 2γmnBmn cos mπx a sin nπy b and we have the Field Coefficients 2γmnBmn = Jmn = 2ϵm ab a Z 0 dx b Z 0 dy f(x, y) cos mπx a sin nπy b D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation Power Flow The power supplied can be computed from what we know of the Poynting Theorem: P = − a Z 0 dx b Z 0 dy ExJ∗ x z=0 After substituting the Fourier series for Ex and Jx and appealing to orthogonality, we find an expression for Power Flow P = ∞ X m=0 ∞ X n=1 ab 4ϵm (Z0)mn |Jmn|2 D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation An Example Consider a coaxially fed waveguide with the probe affixed to the opposite wall: a b c x y The probe acts as an antenna; by image theory, we expect the current to be approximately Probe Current Jx = I0 cos[k(a −x)]δ(y −c) D. S. Weile Modal Excitation Waveguides Free Space Aperture Excitation Current Excitation An Example Jmn = 2ϵm ab a Z 0 dx b Z 0 dy f(x, y) cos mπx a sin nπy b = 2I0ϵm ab sin nπc b a Z 0 dx cos[k(a −x)] cos mπx a = 2I0ϵmka sin ka sin nπc b b[(ka)2 −(mπ)2] The impedance seen by the probe can be computed in the usual way: Zi = 2P |Ii|2 where Ii is the feed current Ii = I0 cos ka D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Outline 1 Waveguides Aperture Excitation Current Excitation 2 Free Space Apertures in a Ground Plane Plane Current Sheets D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Have We Seen This Before?
We already know one way to compute the field radiated from an aperture, but consider: All methods must give the same result because of uniqueness.
This method is a spectral method. Spectral methods converge fastest when spatial methods converge slowest.
Why?
D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Have We Seen This Before?
We already know one way to compute the field radiated from an aperture, but consider: All methods must give the same result because of uniqueness.
This method is a spectral method. Spectral methods converge fastest when spatial methods converge slowest.
Why?
This method elucidates different physics.
Thus, suppose we are given the aperture field Ex(x, 0, z) and we want the field for y > 0. How should we proceed?
D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Modal Expansion As before, we can expand the field in modes. Now, however, we have a continuum of modes, rather than a discrete set.
Therefore, we choose the Wave Function ψ(x, y, z) = 1 4π2 ∞ Z −∞ dkx ∞ Z −∞ dkz f(kx, kz)ejkxxejkyyejkzz Note that this is subject as usual to the usual separation or dispersion relation k2 x + k2 y + k2 z = k2 Our unknown function is the Fourier Transform of ψ: ˜ ψ(kx, kz) = f(kx, kz)e−jkyy D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Modal Expansion Given Ex in a constant y plane, we choose TEz modes. From before, these modes are related to the wave function with Ex = −∂ψ ∂y Hx = 1 jωµ ∂2ψ ∂x∂z Ey = ∂ψ ∂x Hy = 1 jωµ ∂2ψ ∂y∂z Ez = 0 Hz = 1 jωµ ∂2 ∂z2 + k2 ψ We can easily compute the Spectral Domain TEz Equations ˜ Ex = −jky ˜ ψ ˜ Hx = −kxkz jωµ ˜ ψ ˜ Ey = jkx ˜ ψ ˜ Hy = −kykz jωµ ˜ ψ ˜ Ez = 0 ˜ Hz = k2−k2 z jωµ ˜ ψ D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets The Solution Since we know the electric field in the aperture, we can write ˜ Ex(kx, kz) y=0 = −jkyf(kx, kz) or f(kx, kz) = −1 jky ˜ Ex(kx, kz) y=0 Of course, care must be taken in the computation of ky: ky = ( j p k2 x + k2 z −k2 for k < p k2 x + k2 z p k2 −k2 x −k2 z for k > p k2 x + k2 z How are the signs chosen?
What does this say about determining the current far away from it?
D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets An Example: The Infinite Slot To simplify the example, we use a 2D problem independent of z. Consider the infinite slot shown below: x y a If we consider the TEM mode, H is z directed everywhere. We therefore use Hz as our wave function: Hz = 1 2π ∞ Z −∞ f(kx)ejkxxejkyydkx D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets The Infinite Slot From this expression, we see immediately ˜ Hz = f(kx)ejkyy To get the electric field, we use Ampere’s Law: ∇× H = jωϵE Now, in the spectral domain, this becomes k × ˜ H = jωϵ˜ E From symmetry, this equation becomes −kxHz = jωϵEy kyHz = jωϵEx D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets The Infinite Slot Now, assume we know the electric field in the slot. We have just shown ˜ Ex y=0 = ky ωϵf(kx) Now, given the incident wave and the symmetry in the z-direction, we can approximate Ex|y=0 ≈ E0 |x| < a 2 0 |x| > a 2 Therefore ˜ Ex y=0 = E0 a 2 Z −a 2 e−jkxxdx = 2E0 kx sin kxa 2 D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets The Infinite Slot From this we find that f(kx) = ωϵ ky 2E0 kx sin kxa 2 The remaining field can be found from this via the formulas on the previous pages.
As usual, the value of ky is dictated by ensure that the wave Evanesces (and does not explode) in the direction of propagation, or, Travels away from (and not towards) the aperture.
This implies ky = ( j p k2 x −k2 for k < |kx| p k2 −k2 x for k > |kx| D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Outline 1 Waveguides Aperture Excitation Current Excitation 2 Free Space Apertures in a Ground Plane Plane Current Sheets D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Plane Current Sheets Plane current sheets excitation of free space can be computed in the same way. Assuming the sheet is in the y = 0 plane, and A = µψuz, we can write the Fourier transformed (with respect to x and z) equations for TMz propagation: Hx = jky ˜ ψ Ex = −kxkz jωϵ ˜ ψ Hy = −jkx ˜ ψ Ey = −kykz jωϵ ˜ ψ Hz = 0 Ez = k2−k2 z jωϵ ˜ ψ D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Plane Current Sheets We need to expand ˜ ψ on either side of the sheet: ˜ ψ+ = f +(kx, kz)ejk+ y y for y > 0 ˜ ψ− = f −(kx, kz)ejk− y y for y < 0 Of course, k+ y = −k− y = ( j p k2 x + k2 z −k2 for k < p k2 x + k2 z p k2 −k2 x −k2 z for k > p k2 x + k2 z D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets Boundary Condition At the sheet Ex and Ez have to be continuous, so f +(kx, kz) = f −(kx, kz) = f(kx, kz) We also need uy × H+ −H− = J jk+ y ˜ ψ −jk− y ˜ ψ y=0 = ˜ Jx(kx, kz) 2jk+ y f(kx, kz) = ˜ Jx(kx, kz) f(kx, kz) = −1 2jk+ y ˜ Jx(kx, kz) where ˜ Jx(kx, kz) = Z ∞ −∞ Z ∞ −∞ Jx(kx, kz)e−jkxxe−jkzzdxdz D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets An Important Example Consider a current sheet with Jz = Iℓδ(x)δ(z).
We already know this creates a magnetic vector potential with Az = µψ and ψ = Iℓ e−jk√ x2+y2+z2 4π p x2 + y2 + z2 On the other hand, we can treat it as a current sheet. The transform is ˜ Jz = 1 4π2 Z ∞ −∞ Z ∞ −∞ Jx(kx, kz)e−jkxxe−jkzzdxdz = Iℓ 4π2 D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets An Important Example Given this f(kx, kz) = 1 2jk+ y Iℓ 4π2 so that, for y > 0, ˜ ψ(kx, y, kz) = jIℓ 8π2ky ejkyy where ky = k+ y = ( j p k2 x + k2 z −k2 for k < p k2 x + k2 z p k2 −k2 x −k2 z for k > p k2 x + k2 z D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets An Important Example Therefore, we can find ψ from an inverse transform: For y > 0 ψ(x, y, z) = jIℓ 8π2 Z ∞ −∞ Z ∞ −∞ 1 ky ejkxxejkyyejkzzdxdz For y < 0, both y and ky change sign, so we can write ψ(x, y, z) = jIℓ 8π2 Z ∞ −∞ Z ∞ −∞ 1 ky ejkxxejky|y|ejkzzdxdz without caveat, with the understanding that ky changes sign at the y = 0 plane.
D. S. Weile Modal Excitation Waveguides Free Space Apertures in a Ground Plane Plane Current Sheets The Weyl Identity Invoking equivalence (how?) we can equate the two different expressions for ψ. This gives rise to the Weyl Identity e−jkr r = 1 2πj Z ∞ −∞ Z ∞ −∞ 1 ky ejkxxejky|y|ejkzzdxdz What does this formula mean?
D. S. Weile Modal Excitation |
6545 | https://library.fiveable.me/key-terms/college-algebra/exponent-rule | Exponent Rule - (College Algebra) - Vocab, Definition, Explanations | Fiveable | Fiveable
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College Algebra
Exponent Rule
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key term - Exponent Rule
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Definition
An exponent rule is a mathematical principle that describes how to perform operations involving exponents. Exponents are used to represent repeated multiplication of a number, and exponent rules provide a systematic way to manipulate and simplify expressions containing exponents.
5 Must Know Facts For Your Next Test
Exponent rules allow for the simplification of complex expressions involving multiplication, division, and powers of the same base.
The product rule states that when multiplying powers with the same base, the exponents are added: $a^m \cdot a^n = a^{m+n}$.
The quotient rule states that when dividing powers with the same base, the exponents are subtracted: $\frac{a^m}{a^n} = a^{m-n}$.
The power rule states that when raising a power to a power, the exponents are multiplied: $(a^m)^n = a^{m\cdot n}$.
Rational exponents, such as $a^{\frac{1}{2}}$, represent the $\sqrt{a}$ or square root of $a$, and follow the same exponent rules.
Review Questions
Explain how the product rule for exponents can be used to simplify an expression.
The product rule for exponents states that when multiplying powers with the same base, the exponents are added. For example, if we have the expression $2^3 \cdot 2^4$, we can use the product rule to simplify it to $2^{3+4} = 2^7$. This rule allows us to condense repeated multiplication of the same base into a single term with the sum of the exponents.
Describe how the quotient rule for exponents can be used to simplify an expression involving division.
The quotient rule for exponents states that when dividing powers with the same base, the exponents are subtracted. For instance, if we have the expression $\frac{5^8}{5^3}$, we can use the quotient rule to simplify it to $5^{8-3} = 5^5$. This rule enables us to cancel out common factors in the numerator and denominator of a fraction containing exponents.
Analyze how the power rule for exponents can be applied to expressions involving raising a power to a power.
The power rule for exponents states that when raising a power to a power, the exponents are multiplied. For example, if we have the expression $(2^3)^4$, we can use the power rule to simplify it to $2^{3\cdot 4} = 2^{12}$. This rule allows us to condense nested exponents into a single term by multiplying the exponents together. Understanding the power rule is crucial for working with rational exponents and simplifying complex expressions involving exponents.
Related terms
Power:The exponent, or the number that represents how many times the base is multiplied by itself.
Base:
The number that is being raised to a power.
Radical:
The symbol used to represent a root, which is the inverse operation of raising a number to a power.
"Exponent Rule" also found in:
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how to make a truth table from an boolean expression
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I am trying to make a truth table from an SOP boolean algebra expression. I understand AND, OR, NOT truth tables. I just don't understand these types of tables and their outputs.
This is the expression:
A′B D′+B C D+A B C′+A B′D=A′B D′+B C D+A B C′+A B′D+B C′D′+A′B C+A B D.A′B D′+B C D+A B C′+A B′D=A′B D′+B C D+A B C′+A B′D+B C′D′+A′B C+A B D.
I can use either side whichever is easier. Just let me know which side.
Would A′A′ be a 1 1 and the others be a zero? I am also not sure how they get the output?
I understand the outputs of a AND, OR truth tables.
But I can't figure out these outputs. Would this be considered an OR table since the expression is ++?
Would I just construct A A, B B, and D D with nots = 1 or zero? Then, how do I determine the output?
```
A | B | D | output
| 1 | 0 | 1 | 1? | A'BD'
| 0 | 0 | 0 | 0? | BCD
| 0 | 0 | 1 | 1? | ABC'
```
something like that.
What I am trying to achieve is how the below expression is true using theorems.
A′B D′+B C D+A B C′+A B′D=A′B D′+B C D+A B C′+A B′D+B C′D′+A′B C+A B D A′B D′+B C D+A B C′+A B′D=A′B D′+B C D+A B C′+A B′D+B C′D′+A′B C+A B D
boolean-algebra
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edited Mar 5, 2018 at 20:10
amWhy
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asked Oct 22, 2010 at 3:48
Corey J MattisCorey J Mattis
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A, B, C, and D are boolean variables, meaning that each takes the value "true" or "false". More complex expressions have value "true" or "false" depending on the values of these variables, so for example A'BD' is true if A is false, B is true, and D is false, and C is either true or false.
To say that F = G, where F and G are complex expressions, means that, no matter what values the boolean variables have, the value of F is the same as the value of G (that is, F and G are either both true or both false). So, for example, we have
AB + AC = A(B+C)
because if A is true and either B or C is true, then both sides are true, and in any other case both sides are false---there's no way to assign values to A, B, C such that the two side come out differently.
As previous posters have mentioned, you can always prove (or disprove!) an equality by going through all possible assignments of "true" and "false" to the variables. The goal seems to be to prove "by theorems", that is, using operations previously proven true. As you say, we can either manipulate one side of the equation until it has the form of the other side, or we could manipulate both sides and get them into a common form.
In this case, the first thing to notice is that the right-hand side (RHS) is a copy of the LHS with some extra stuff tacked onto the end. For this reason it's simplest to manipulate just the RHS to get rid of the surplussage! We can start with this basic theorem:
if Q is true whenever P is true, then Q = Q + P
We can prove this theorem by systematically considering all possibilities for P and Q. Or look at it this way: If Q is true, then both sides of the equation are true. And if Q is false, then P must be false (since, by assumption, if P were true Q would be true) hence both sides are false.
Given this theorem we prove:
XY + X'Z = XY + X'Z + YZ
Proof: Let P be YZ and let Q be XY+X'Z. Suppose P is true, that is, Y and Z are both true. But if Y and Z are both true, then XY+X'Z must be true. (Reason: If X is true then, since Y is true, XY is true. And if X is false then, since Z is true, X'Z is true. So in either case XY+X'Z is true.) So we've shown that Q is true whenever P is true, hence by the previous theorem Q = Q + P, which in this case is what we wanted to prove.
Now we can prove
A'D' + AC' = A'D' + AC' + C'D'
This is just the same as the previous theorem, putting A for X, D' for Y, and C' for Z.
This last theorem implies
B(A'D' + AC') = B(A'D' + AC' + C'D')
which is the same as
A'BD' + ABC' = A'BD' + ABC' + BC'D'
Given this, we can take the RHS of the original and substitute A'BD' + ABC' for A'BD' + ABC' + BC'D', which is to say, we can drop the term BC'D'.
With steps similar to the above we can prove these two theorems:
A'BD' + BCD = A'BD' + BCD + A'BC
BCD + ABC' = BCD + ABC' + ABD
which permit us to drop the final two terms of the RHS of the original, completing the proof.
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answered Oct 22, 2010 at 13:12
Larry DenenbergLarry Denenberg
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Thank you so much, things make more sense. Looking at my original uquation everything is the same except the last 3 peices. Should those cancel out? without those they equal exactly!Corey J Mattis –Corey J Mattis 2010-10-22 19:22:58 +00:00 Commented Oct 22, 2010 at 19:22
So, what your telling me is since the left hand side equals the right hand side except the last 3 i can just drop the last 3?A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + //BC'D' + A'BC + ABD // all variables are the same except these Corey J Mattis –Corey J Mattis 2010-10-22 19:28:17 +00:00 Commented Oct 22, 2010 at 19:28
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Your truth table will have 2^4=16 lines. Start with four columns labeled A,B,C,D. On each line put a different combination of truth values for A,B,C, and D. Then add more columns for the pieces of your expression. So to build up the left hand side, make a column A'B and put in the truth values of that combination. The prime on A is not and the multiply is AND. So this column would show (NOT A) AND B for the values of A and B on that line. Continue adding columns for other terms until you have built up the entire left hand side. Then start on the terms that make up right hand side. When you have built up the right hand side, compare the left hand side column and the right hand side column. If they agree on all 16 lines, your equation is correct. For a simple example, say we want to prove (A+B)'=A'B' The table would look like this
As the (A+B)' column and the A'B' column agree everywhere, the equation is correct.
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answered Oct 22, 2010 at 4:02
Ross MillikanRoss Millikan
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How are we to do an output? like a function Corey J Mattis –Corey J Mattis 2010-10-22 10:28:40 +00:00 Commented Oct 22, 2010 at 10:28
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This is more of an extended remark out of academic curiosity. I thought I would try to use Prover9 to prove the given identity. [here v denotes "or" and ^ denotes "and"]
To test this, I inputted the assumptions into Prover9 (this is the OP's list of axioms from here):
```
x v x' = y v y'. % false
x ^ x' = y ^ y'. % true
y v (x ^ x') = y.
y v (x v x') = y.
x v x = x.
x v y = y v x.
(x v y) v z = x v (y v z).
x ^ (x v y) = x.
(x ^ y) v (x ^ y') = x.
```
from which it proved the OP's identity (and if I understand correctly, it was by contradiction):
((((w' ^ x) ^ z') v ((x ^ y) ^ z)) v ((w ^ x) ^ y')) v ((w ^ x') ^ z) = (((((((w' ^ x) ^ z') v ((x ^ y) ^ z)) v ((w ^ x) ^ y')) v ((w ^ x') ^ z)) v ((x ^ y') ^ z')) v ((w' ^ x) ^ y)) v ((w ^ x) ^ z).
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edited Apr 13, 2017 at 12:20
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answered Oct 23, 2010 at 3:24
Douglas S. StonesDouglas S. Stones
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That is an interesting piece of software... just for that link, +1.J. M. ain't a mathematician –J. M. ain't a mathematician 2010-10-23 03:28:26 +00:00 Commented Oct 23, 2010 at 3:28
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I think the "using theorems" part of the question is intended to prevent you from using the exhaustive "try every possibility" technique (so don't use this technique if it's a homework question).
However, this technique will work fine, you can just run through the 2 4=16 2 4=16 possible values of (A,B,C,D) and check that both sides of the equation match. This will be a rigorous proof of the identity -- just not a particularly intelligent one.
Assuming I understand the notation correctly:
concatenation (i.e. writing the letters next to each other) is short for "and", for example ABC is "A and B and C",
addition (i.e. +) is short for "or", for example AB+CD is "(A and B) or (C and D)"
adding a dash ' is the notation for negation, i.e. "not", for example AB' is "A and (not B)".
So A'BD' is the same as (not A) and B and (not D). In your table, the first row is wrong since (not A) is false (because A is true=1). The second row is correct since B is false=0 (regardless of what value C has). The third row is wrong since A is false=0 (again, regardless of what values C has).
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answered Oct 22, 2010 at 4:02
Douglas S. StonesDouglas S. Stones
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This is a homework question. We are to use theorems to Show the above expression is true using theorems. Not sure how to tackle this problem?Corey J Mattis –Corey J Mattis 2010-10-22 04:13:49 +00:00 Commented Oct 22, 2010 at 4:13
I admit it is tricky, my first instinct would be to prove a smaller identity, such as ABC'+A'BD'=ABC'+A'BD'+BC'D', then use it to eliminate the BC'D' from the right-hand-side of the equation. Then repeat with similar identities.Douglas S. Stones –Douglas S. Stones 2010-10-22 04:28:18 +00:00 Commented Oct 22, 2010 at 4:28
im going to give up because i dont even know what the = means? does the left side equal the right side? or are they two different equations? does the lfet side when worked out equal the right side? im lost..Corey J Mattis –Corey J Mattis 2010-10-22 05:37:03 +00:00 Commented Oct 22, 2010 at 5:37
The = implies that the left-hand side is true whenever the right-hand side is true (and the left-hand side is false whenever the right-hand side is false) [so yes, when worked out]. It is just one equation, your goal is to prove that it's always correct.Douglas S. Stones –Douglas S. Stones 2010-10-22 05:44:37 +00:00 Commented Oct 22, 2010 at 5:44
what does true mean? do i have to work both sides? Why is this so hard? The book gives us some theorems with two variables and he gives us this monster to work on. So, in other words i have to work both sides of the = ? "your goal is to prove that it's always correct" what does correct mean? i dont mean to sound stupid but i am brand new to this stuff. I am a Software engineering student!Corey J Mattis –Corey J Mattis 2010-10-22 10:33:32 +00:00 Commented Oct 22, 2010 at 10:33
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I colored in gray the general table.This would be the point were you start from.
You take each term and calculate the value for each combination.
You do the SUM or PRODUCT depending your equation Y.
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answered Oct 22, 2019 at 8:38
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6547 | https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/ContinuedFractions.pdf | Chapter 17 Continued fractions 17.1 Finite continued fractions Definition 17.1. A finite continued fraction is an expression of the form a0 + 1 a1 + 1 a2 + 1 · · · + 1 an , where ai ∈Z with a1, . . . , an ≥1. We denote this fraction by [a0, a1, . . . , an].
Example: The continued fraction [2, 1, 3, 2] = 2 + 1 1 + 1 3 + 1 2 represents the rational number 2 + 1 1 + 2 7 = 2 + 7 9 = 25 9 .
83 Conversely, suppose we start with a rational number, say 57 33.
To convert this to a continued fraction: 57 33 = 1 + 14 33.
Now invert the remainder: 33 14 = 2 + 5 14.
Again: 14 5 = 2 + 4 5, and again: 5 4 = 1 + 1 4, and finally: 4 1 = 4.
Thus 57 33 = [1, 2, 2, 1, 4].
Note that the numbers 1, 2, 2, 1, 4 in the continued fraction are precisely the quotients that would arise if we used the Euclidean Algorithm to compute gcd(57, 33).
We can consider continued fractions — particularly when we come to in-finite continued fractions — as a generalisation or extension of the Euclidean Algorithm.
17.2 The p’s and q’s We can consider [a0, a1, a2, . . . , an] as a function of the variables a0, a1, . . . , an. Evidently [a0, a1, a2, . . . , an] = P Q, 84 where P, Q are polynomials in a0, a1, . . . , an with integer coefficients. This does not define P, Q precisely; but we shall give a precise recursive definition below, using induction on the length n of the continued fraction.
We start with the continued fraction [a0] = a0 = a0 1 , setting p = a0, q = 1, Now suppose that we have defined p, q for continued fractions of length < n; and suppose that under this definition α1 = [a1, a2, . . . , an] = p′ q′ .
Then α = a0 + 1 α1 = a0 + q′ p′ = a0p′ + q′ p′ .
So we set p = a0p′ + q′, q = p′ as the definition of p, q for a continued fraction of length n. We set this out formally in Definition 17.2. The ‘canonical representation’ of a continued fraction [a0, a1, a2, . . . , an] = p q is defined by induction on n, setting p = a0p′ + q′, q = p′, where [a1, a2, . . . , an] = p′ q′ is the canonical representation for a continued fraction of length n −1. The induction is started by setting [a0] = a0 1 .
Henceforth if we write [a0, a1, a2, . . . , an] = p q, then p, q will refer to the canonical representation defined above.
85 17.3 Successive approximants Definition 17.3. If α = [a0, a1, . . . , an] then we call [a0, a1, . . . , ai] = pi qi the ith convergent or approximant to α (for 0 ≤i ≤n).
Example: Continuing the previous example, the successive approximants to 57 33 = [1, 2, 2, 1, 4] are p0 q0 = = 1 1, p1 q1 = [1, 2] = 1 + 1 2 = 3 2, p2 q2 = [1, 2, 2] = [1, 5/2] = 1 + 2 5 = 5 7, p3 q3 = [1, 2, 2, 1] = [1, 2, 3] = [1, 7/3] = 1 + 3 7 = 10 7 , p4 q4 = [1, 2, 2, 1, 4] = [1, 2, 2, 5/4] = [1, 2, 14/5] = [1, 33/14] = 57 33.
Note that while we normally assume that the entries an in continued fractions are integers (with an ≥1 for n ≥1), it makes sense to use fractional (or even variable) entries, using our recursive formulae for pn, qn as functions of a0, a1. . . . . Usually this will only involve the last entry, where [a0, . . . , an−1, an, x] = [a0, . . . , an−1, an + 1/x].
Note too that p0 q0 < p2 q2 < p4 q4 < p3 q3 < p1 q1 ; first we get the even convergents, increasing, and then the odd convergents, in reverse order, with the actual number sandwiched in between.
As we shall see, this is the general situation; moreover, the successive convergents are very good approximants to the given number.
Theorem 17.1. If α = [a0, a1, . . . , an] 86 then pi = aipi−1 + pi−2, qi = aiqi−1 + qi−2, for i = 2, 3, . . . , n.
Proof. We argue by induction on n.
The result follows by induction for i ̸= n, since the convergents involved are — or can be regarded as — convergents to [a0, a1, . . . , an−1], covered by our inductive hypothesis.
It remains to prove the result for i = n. In this case, by the inductive definition of p, q, pn = a0p′ n−1 + q′ n−1, pn−1 = a0p′ n−2 + q′ n−2, pn−2 = a0p′ n−3 + q′ n−3.
But now by our inductive hypothesis, p′ n−1 = anp′ n−2 + p′ n−3, q′ n−1 = anq′ n−2 + q′ n−3, since a′ n−1 = an, ie the (n −1)th entry in α′ is the nth entry in α.
Hence pn = a0p′ n−1 + q′ n−1, = a0(anp′ n−2 + p′ n−3) + (anq′ n−2 + q′ n−3), = an(a0p′ n−2 + q′ n−2) + (a0p′ n−3 + q′ n−3), = anpn−1 + pn−2; with the second result qn = anqn−1 + qn−2 following in exactly the same way.
We can regard this as a recursive definition of pi qi , starting with p0 q0 = a0 1 , p1 q1 = a0a1 + 1 a1 , and defining p2 q2 , p3 q3 , p4 q4 , . . .
successively.
Actually, we can go back two futher steps.
87 Proposition 17.1. If we set p−2 = 1, q−2 = 0, p−1 = 0, q−1 = 1, then pi = aipi−1 + pi−2, qi = aiqi−1 + qi−2, for all i ≥0.
One more or less obvious result.
Proposition 17.2. Both the p’s and q’s are strictly increasing: 0 < q0 < q1 · · · < qn, p0 < p1 · · · < pn.
Proof. This follows at once by repeated application of the recursive identities pi = aipi−1 + pi−2, qi = aiqi−1 + qi−2, since a1, a2, . . . , an > 0 and q0 = 1, q1 = a1,.
17.4 Uniqueness Consider the continued fraction for a rational number x. If n > 0 and an > 1 then [a0, a1, . . . , an] = [a0, a1, . . . , an −1, 1].
And if n = 0, ie x ∈Z, then [x0] = [x0 −1, 1].
Thus with our example above, 57 33 = [1, 2, 2, 1, 4] = [1, 2, 2, 1, 3, 1].
So there are at least 2 ways of expressing x as a continued fraction.
Proposition 17.3. A rational number x ∈Q has just two representations as a continued fraction: one with n = 0 or n > 1, an > 1, and the other with n > 0 and an = 1.
88 Proof. It is sufficient to show that x has just one representation of the first kind. Suppose x = [a0, a1, . . . , am] = [b0, b1, . . . , bn], We may assume that m ≤n.
We argue by induction on n. The result is trivial if m = n = 0.
Lemma 17.1. If n > 0 and an > 1 then a0 < [a0, a1, a2, . . . , an] < a0 + 1.
Proof. We argue, as usual, by induction on n. This tells us that [a1, a2, . . . , an] > 1, from which the result follows, since [a0, a1, a2, . . . , an] = a0 + 1 [a1, a2, . . . , an].
It follows that [x] = a0 = b0.
Thus x−a0 = 1 [a1, a2, . . . , am] = 1 [b1, b2, . . . , bn] = ⇒[a1, a2, . . . , am] = [b1, b2, . . . , bn], from which the result follows by induction.
We will take the first form for the continued fraction of a rational number as standard, ie we shall assume that the last entry an > 1 unless the contrary is stated.
17.5 A fundamental identity Theorem 17.2. Successive convergents pi/qi, pi+1/qi+1 to the continued frac-tion [a0, a1, . . . , an] satisfy the identity piqi+1 −qipi+1 = (−1)i+1.
Proof. We argue by induction on i, using the relations pi = aipi−1 + pi−2, qi = aiqi−1 + qi−2.
89 Thus piqi+1 −qipi+1 = pi(ai+1qi + qi−1) = qi(ai+1pi + pi−1) = piqi−1 −qipi−1 = −(pi−1qi −qi−1pi) = −(−1)i = (−1)i+1.
The result holds for i = −2 since p−2q−1 −q−2p−1 = 0 · 0 −1 · 1 = (−1)−1.
We conclude that the result holds for all i ≥0.
Proposition 17.4. The even convergents are monotonically increasing, while the odd convergents are monotonically decreasing: p0 q0 < p2 q2 < · · · ≤x ≤· · · < p3 q3 < p1 q1 .
Proof. By the last Proposition, if i is even then pi+1 qi+1 −pi qi = pi+1qi −piqi+1 qiqi+1 = 1 qiqi+1 .
Thus pi qi < pi+1 qi+1 .
Moreover, pi+1 qi+1 −pi+2 qi+2 = pi+1qi+2 −pi+2qi+1 qi+2qi+1 = 1 qi+2qi+1 < 1 qi+2qi+1 .
It follows that pi+2/qi+2 is closer than pi/qi to pi+1/qi+1. Hence pi qi < pi+2 qi+2 < pi+1 qi+1 .
So the even convergents are increasing; and similarly the odd convergents are decreasing.
Also, any even convergent is less than any odd convergent; for if i is even and j is odd then pi qi < pi+j−1 qi+j−1 < pi+j qi+j < pj qj .
And since x is equal to the last convergent, it must be sandwiched between the even and odd convergents.
90 17.6 Infinite continued fractions So far we have been considering continued fraction expansions of rational numbers. But the concept extends to any real number α ∈R.
Suppose α is irrational. We set a0 = [α], and let α1 = 1 α −a0 .
Then we define a1, a2, . . . , successively, setting a1 = [α1], α2 = 1 α1 −a1 , a2 = [α2], α3 = 1 α2 −a2 , and so on.
Proposition 17.5. Suppose a0, a1, a2, · · · ∈Z with a1, a2, · · · > 0.
Let [a0, a1, . . . , ai] = pi qi .
Then the sequence of convergents converges: pi qi →x as i →∞.
Proof. It follows from the finite case that the even convergents are increasing, and the odd convergents are decreasing, with the former bounded by the latter, and conversely: p0 q0 < p2 q2 < p4 q4 < · · · < p5 q5 < p3 q3 < p1 q1 .
It follows that the even convergents must converge, to α say, and the odd convergents must also converge, to β say.
But if i is even, pi qi −pi+1 qi+1 = 1 qiqi+1 .
91 Since pi qi < α ≤β < pi+1 qi+1 , it follows that 0 ≤β −α < 1 qiqi+1 < 1 q2 i .
Hence α = β, ie the convergents tend to a limit α ∈R.
Proposition 17.6. Each irrational number α ∈R has a unique expression as an infinite continued fraction α = [a0, a1, a2, . . . ].
Proof. One could argue that this follows from the algorithm above for con-structing the continued fractions of α.
Express each rational numbers x < α as a continued fraction. For sim-plicity, let us choose the version with final entry an > 1.
Lemma 17.2. Suppose α = [a0, a1, . . . , ], β = [b0, b1, . . . , ]; and suppose a0 = b0, . . . , an−1 = bn−1, an < bn.
Then α < β if n is even, α > β if n is odd.
Proof. This follows easily from the fact that even convergents are increasing, odd convergents decreasing.
Now let a0 be the largest first entry among rational x < α; let a1 be the least second entry among those rationals with a0 as first entry; let a2 be the largest third entry among those rationals with a0, a1 as first two entries; and so on. Then it is a simple exercise to show that α = [a0, a1, a2, dots].
(Note that if the an (with given a0, . . . , an−1) at the (n + 1)th stage were unbounded then it would follow that α is rational, since [a0, . . . , an−1, x] →[a0, . . . , an−1] if x →∞.) 92 17.7 Diophantine approximation Theorem 17.3. If pn/qn is a convergent to α = [a0, a1, a2, . . . ] then |α −pn qn | ≤1 q2 n .
Proof. Recall that α lies between successive convergents pn/qn, pn+1/qn+1.
Hence |α −pn qn | ≤|pn+1 qn+1 −pn qn | = 1 qnqn+1 ≤1 q2 n .
Remarks: 1. There is in fact inequality in the theorem except in the very special case where α is rational, pn/qn is the last but one convergent, and an+1 = 1; for except in this case qn < qn+1.
2. Since 1 qnqn+1 = 1 qn(anqn + qn−1) ≤ 1 anq2 n , if an > 1 then |α −pn qn | ≤ 1 2q2 n .
In particular, if α is irrational then there are an infinity of convergents satisfying |α −p q| ≤ 1 2q2 unless an = 1 for all n ≥N.
In this case α = [a0, a1, . . . , an, φ] = pnφ + pn−1 qnφ + qn−1 ∈Q( √ 5).
93 We have seen that the convergents are good approximations to α. The next result shows that, conversely, good approximations are necessarily con-vergents.
Theorem 17.4. If |α −p q| ≤ 1 2q2 then p/q is a convergent to α.
Proof. Let us express p/q as a continued fraction: p q = pn qn = [a0, a1, . . . , an].
We want to express α in the form α = [a0, . . . , an, x] = [a0, . . . , an + 1 x].
In this case α = pn + pn−1x qn + qn−1x .
Solving for x, x = − qnα −pn qn−1α −pn−1 = − α −pn/qn α −pn−1/qn−1 We want to ensure that x > 0. This will be the case if (α −pn qn ) and (α −pn−1 qn−1 ) are of opposite sign, ie α lies between the two convergents.
At first this seems a matter of good or bad luck. But recall that there are two ways of representing p/q as a continued fraction, one of even length and one odd. (One has last entry an > 1, and the other has last entry 1.) We can at least ensure in this way that α lies on the same side of pn/qn as pn−1/qn−1, since even convergents are < odd convergents; so if α > p/q then we choose n to be even, while if α < p/q we choose n to be odd.
This ensures that x > 0. Now we must show that x ≥1; for then if x = [b0, b1, b2, . . . ] we have α = [a0, . . . , an, b0, b1, b2, . . . ], 94 and p q = [a0, . . . , an] is a convergent to α, as required.
But now |α −pn qn | ≤ 1 2q2 n ; and since |pn qn −pn−1 qn−1 | = 1 qnqn−1 it follows that |α −pn−1 qn−1 | ≥|pn qn −pn−1 qn−1 | −|α −pn qn | ≥ 1 qnqn−1 −1 2q2 n ≥1 q2 n −1 2q2 n = 1 2q2 n , and so |x| = |α −pn/qn| |α −pn−1/qn−1| ≥1.
17.8 Quadratic surds and periodic continued fractions Recall that a quadratic surd is an irrational number of the form α = x + y √ d, where x, y ∈Q, and d > 1 is square-free. In other words, α ∈Q( √ d) \ Q for some quadratic field Q( √ d).
Theorem 17.5. The continued fraction of α ∈R is periodic if and only if x is a quadratic surd.
95 Proof. Suppose first that α has periodic continued fraction: α = [a0, . . . , am, b0, . . . , bn, b0, . . . , bn, . . . ].
Let β = [b0, . . . , bn, β] = βp′ n + p′ n−1 βq′ n + q′ n−1 be the purely periodic part. Then β satisfies the quadratic equation p′ n−1β2 + (q′ n−1 −p′ n)β −q′ n = 0, and so is a quadratic surd. And since α = [a0, . . . , am, β] = βpm + pm−1 βqm + qm−1 , it too is a quadratic surd.
The converse is more difficult. Suppose α = [a0, a1, . . . ] satisfies the quadratic equation F(x) ≡Ax2 + 2Bx + C = 0 (A, B, C ∈Z).
Let αn = [an, an+1, . . . ].
We have to show that αm+n = αn for some m, n ∈N, m > 0.
We shall do this by showing that αn satisfies a quadratic equation with bounded coefficients.
Writing θ for an+1, for simplicity, α = [a0, . . . , an, θ] = θpn + pn−1 θqn + qn−1 .
Thus A(θpn + pn−1)2 + 2B(θpn + pn−1)(θqn + qn−1) + C(θqn + qn−1)2 = 0, 96 ie A′θ2 + 2B′θ + C′, where A′ = Ap2 n + 2Bpnqn + Cq2 n, B′ = Apnpn−1 + 2B(pnqn−1 + pn−1qn) + Cqnqn−1, C′ = Ap2 n−1 + 2Bpn−1qn−1 + Cq2 n−1.
Now A′ = q2 nF(pn/qn).
Since F(α) = 0 and pn/qn is close to α, F(pn/qn) is small.
More precisely, since |α −pn qn | ≤1 q2 n , it follows by the Mean Value Theorem that F(pn/qn) = −(F(α) −F(pn/qn)) = −F ′(t)(α −pn/qn), where t ∈[α, α + pn/qn].
Thus if we set M = max t∈[α−1,α+1] |F ′(t)| then |F(pn/qn)| ≤M q2 n and so |A′| ≤M.
Similarly |C′| ≤M.
Much the same argument applies to B′ = qnqn−1F +(pn/qn, pn−1/qn−1, where F +(x, y) = Axy + B(x + y) + C is the ‘polarised’ form of the quadratic form F(x).
97 Note that F(x2) −F +(x, y) = (x −y)(Ax + B) = 1 2(x −y)F ′(x).
Hence F(pn/qn) −F +(pn/qn, pn−1/qn−1) = 1 2(pn/qn −pn−1/qn−1)F ′(pn/qn), and so |F(pn/qn) −F +(pn/qn, pn−1/qn−1)| ≤ M 2qnqn−1 .
Since |F(pn/qn)| ≤M q2 n < M qnqn−1 , we conclude that |B′| = qnqn−1|F +(pn/qn, pn−1/qn−1| ≤3 2M.
Thus A′, B′, C′ are bounded for all n. We conclude that one (at least) of these equations occurs infinitely often; and so one of the αn occurs infinitely often, ie α is periodic.
Example: Let us determine the continued fraction for √ 3. We have √ 3 = 1 + ( √ 3 −1), 1 √ 3 −1 = √ 3 + 1 2 = 1 + √ 3 −1 2 , 2 √ 3 −1 = √ 3 + 1 = 2 + ( √ 3 −1), 1 √ 3 −1 = 1 + √ 3 −1 2 , . . .
Thus √ 3 = [1, 1, 2], where we have overlined the periodic part.
98 |
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Solving Linear Equations
Solve Absolute Value Inequalities
Learning Objectives
By the end of this section, you will be able to:
Solve absolute value equations
Solve absolute value inequalities with “less than”
Solve absolute value inequalities with “greater than”
Solve applications with absolute value
Before you get started, take this readiness quiz.
Evaluate:
If you missed this problem, review (Figure).
2. Fill in or for each of the following pairs of numbers.
ⓐ ⓑ ⓒ ⓓ
If you missed this problem, review (Figure).
3. Simplify:
If you missed this problem, review (Figure).
Solve Absolute Value Equations
As we prepare to solve absolute value equations, we review our definition of absolute value.
Absolute Value
The absolute value of a number is its distance from zero on the number line.
The absolute value of a number n is written as and for all numbers.
Absolute values are always greater than or equal to zero.
We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:
is 5 units away from 0, so
is 5 units away from 0, so
(Figure) illustrates this idea.
The numbers 5 and are both five units away from zero.
For the equation we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and are five units from zero on the number line. They are the solutions to the equation.
The solution can be simplified to a single statement by writing This is read, “x is equal to positive or negative 5”.
We can generalize this to the following property for absolute value equations.
Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
Solve: ⓐ ⓑ ⓒ
ⓐ
ⓑ
Since an absolute value is always positive, there are no solutions to this equation.
ⓒ
Both equations tell us that and so there is only one solution.
Solve: ⓐ ⓑ ⓒ
ⓐⓑ no solution ⓒ 0
Solve: ⓐ ⓑ ⓒ
ⓐⓑ no solution ⓒ 0
To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
How to Solve Absolute Value Equations
Solve
Solve:
Solve:
The steps for solving an absolute value equation are summarized here.
Solve absolute value equations.
Isolate the absolute value expression.
Write the equivalent equations.
Solve each equation.
Check each solution.
Solve
| | | |
---
| | | |
| Isolate the absolute value expression. | | |
| | | |
| Write the equivalent equations. | or | |
| Solve each equation. | or | |
| Check: | |
Solve:
Solve:
Remember, an absolute value is always positive!
Solve:
Solve:
No solution
Solve:
No solution
Some of our absolute value equations could be of the form where u and v are algebraic expressions. For example,
How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if then or
This tell us that
This leads us to the following property for equations with two absolute values.
Equations with Two Absolute Values
For any algebraic expressions, u and v,
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.
Solve:
Solve:
Solve:
Solve Absolute Value Inequalities with “Less Than”
Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.
Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation we saw that both 5 and are five units from zero on the number line. They are the solutions to the equation.
What about the inequality Where are the numbers whose distance is less than or equal to 5? We know and 5 are both five units from zero. All the numbers between and 5 are less than five units from zero. See (Figure).
In a more general way, we can see that if then See (Figure).
This result is summarized here.
Absolute Value Inequalities with or
For any algebraic expression, u, and any positive real number, a,
After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.
Solve Graph the solution and write the solution in interval notation.
| | |
--- |
| | |
| Write the equivalent inequality. | |
| Graph the solution. | |
| Write the solution using interval notation. | |
Check:
To verify, check a value in each section of the number line showing the solution. Choose numbers such as 1, and 9.
Graph the solution and write the solution in interval notation:
Graph the solution and write the solution in interval notation:
Solve Graph the solution and write the solution in interval notation.
| | |
--- |
| Step 1. Isolate the absolute value expression. It is isolated. | |
| Step 2. Write the equivalent compound inequality. | |
| Step 3. Solve the compound inequality. | |
| Step 4. Graph the solution. | |
| Step 5. Write the solution using interval notation. | |
| Check: The check is left to you. | |
Solve Graph the solution and write the solution in interval notation:
Solve Graph the solution and write the solution in interval notation:
Solve absolute value inequalities with < or ≤.
Isolate the absolute value expression.
Write the equivalent compound inequality.
Solve the compound inequality.
Graph the solution
Write the solution using interval notation.
Solve Absolute Value Inequalities with “Greater Than”
What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.
We started with the inequality We saw that the numbers whose distance is less than or equal to five from zero on the number line were and 5 and all the numbers between and 5. See (Figure).
Now we want to look at the inequality Where are the numbers whose distance from zero is greater than or equal to five?
Again both and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than and greater than 5 on the number line. See (Figure).
In a more general way, we can see that if then or See (Figure).
This result is summarized here.
Absolute Value Inequalities with > or ≥
For any algebraic expression, u, and any positive real number, a,
Solve Graph the solution and write the solution in interval notation.
| | |
--- |
| | |
| Write the equivalent inequality. | |
| Graph the solution. | |
| Write the solution using interval notation. | |
| Check: | |
To verify, check a value in each section of the number line showing the solution. Choose numbers such as 0, and 7.
Solve Graph the solution and write the solution in interval notation.
Solve Graph the solution and write the solution in interval notation.
Solve Graph the solution and write the solution in interval notation.
| | |
--- |
| | |
| Step 1. Isolate the absolute value expression. It is isolated. | |
| Step 2. Write the equivalent compound inequality. | |
| Step 3. Solve the compound inequality. | |
| Step 4. Graph the solution. | |
| Step 5. Write the solution using interval notation. | |
| Check: The check is left to you. | |
Solve Graph the solution and write the solution in interval notation.
Solve Graph the solution and write the solution in interval notation.
Solve absolute value inequalities with > or ≥.
Isolate the absolute value expression.
Write the equivalent compound inequality.
Solve the compound inequality.
Graph the solution
Write the solution using interval notation.
Solve Applications with Absolute Value
Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.
The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?
The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?
The diameter of the rod can be between 79.991 and 80.009 mm.
The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?
The diameter of the rod can be between 74.95 and 75.05 mm.
Access this online resource for additional instruction and practice with solving linear absolute value equations and inequalities.
Solving Linear Absolute Value Equations and Inequalities
Key Concepts
Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as and for all numbers.
Absolute values are always greater than or equal to zero.
Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
How to Solve Absolute Value Equations
1. Isolate the absolute value expression.
2. Write the equivalent equations.
3. Solve each equation.
4. Check each solution.
Equations with Two Absolute Values
For any algebraic expressions, u and v,
Absolute Value Inequalities with or
For any algebraic expression, u, and any positive real number, a,
How To Solve Absolute Value Inequalities with or
1. Isolate the absolute value expression.
2. Write the equivalent compound inequality.
3. Solve the compound inequality.
4. Graph the solution
5. Write the solution using interval notation
Absolute Value Inequalities with or
For any algebraic expression, u, and any positive real number, a,
How To Solve Absolute Value Inequalities with or
1. Isolate the absolute value expression.
2. Write the equivalent compound inequality.
3. Solve the compound inequality.
4. Graph the solution
5. Write the solution using interval notation
Section Exercises
Practice Makes Perfect
Solve Absolute Value Equations
In the following exercises, solve.
ⓐⓑ
ⓒ
ⓐⓑ
ⓒ
ⓐⓑ no solution ⓒ
ⓐⓑ
ⓒ
ⓐⓑ
ⓒ
ⓐⓑ no solution ⓒ
no solution
no solution
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.
A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?
An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?
The minimum to maximum expected production is 207,500 to 2,225,000 bottles
In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as ?100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?
At Lilly’s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?
The acceptable weight is 22.5 to 25.5 ounces.
Writing Exercises
Write a graphical description of the absolute value of a number.
In your own words, explain how to solve the absolute value inequality,
Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Chapter Review Exercises
Use a General Strategy to Solve Linear Equations
Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, determine whether each number is a solution to the equation.
no
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
contradiction; no solution
Solve Equations with Fraction or Decimal Coefficients
In the following exercises, solve each equation.
Use a Problem-Solving Strategy
Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems.
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
There are 116 people.
There are nine saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players.
Solve Number Word Problems
In the following exercises, solve each number word problem.
The sum of a number and three is forty-one. Find the number.
38
One number is nine less than another. Their sum is negative twenty-seven. Find the numbers.
One number is two more than four times another. Their sum is negative thirteen. Find the numbers.
The sum of two consecutive integers is Find the numbers.
Find three consecutive even integers whose sum is 234.
76, 78, 80
Find three consecutive odd integers whose sum is 51.
Koji has ?5,502 in his savings account. This is ?30 less than six times the amount in his checking account. How much money does Koji have in his checking account?
?922
Solve Percent Applications
In the following exercises, translate and solve.
What number is 67% of 250?
12.5% of what number is 20?
160
What percent of 125 is 150?
In the following exercises, solve.
The bill for Dino’s lunch was ?19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?
Dolores bought a crib on sale for ?350. The sale price was 40% of the original price. What was the original price of the crib?
Jaden earns ?2,680 per month. He pays ?938 a month for rent. What percent of his monthly pay goes to rent?
35%
Angel received a raise in his annual salary from ?55,400 to ?56,785. Find the percent change.
Rowena’s monthly gasoline bill dropped from ?83.75 last month to ?56.95 this month. Find the percent change.
32%
Emmett bought a pair of shoes on sale at 40% off from an original price of ?138. Find ⓐ the amount of discount and ⓑ the sale price.
Lacey bought a pair of boots on sale for ?95. The original price of the boots was ?200. Find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)
ⓐ ?105 ⓑ
Nga and Lauren bought a chest at a flea market for ?50. They re-finished it and then added a 350% mark-up. Find ⓐ the amount of the mark-up and ⓑ the list price.
Solve Simple Interest Applications
In the following exercises, solve.
Winston deposited ?3,294 in a bank account with interest rate 2.6% How much interest was earned in five years?
?428.22
Moira borrowed ?4,500 from her grandfather to pay for her first year of college. Three years later, she repaid the ?4,500 plus ?243 interest. What was the rate of interest?
Jaime’s refrigerator loan statement said he would pay ?1,026 in interest for a four-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?
?1,900
Solve a formula for a Specific Variable
Solve a Formula for a Specific Variable
In the following exercises, solve the formula for the specified variable.
Solve the formula
for L.
Solve the formula
for
Solve the formula
for t.
Solve the formula
for y.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
What is the height of a triangle with area square meters and base 9 meters?
The measure of the smallest angle in a right triangle is less than the measure of the next larger angle. Find the measures of all three angles.
The perimeter of a triangle is 97 feet. One side of the triangle is eleven feet more than the smallest side. The third side is six feet more than twice the smallest side. Find the lengths of all sides.
Find the length of the hypotenuse.
26
Find the length of the missing side. Round to the nearest tenth, if necessary.
Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is eight feet tall and Sergio has 10 feet of wire. How far from the base of the antenna can he attach the wire? Approximate to the nearest tenth, if necessary.
6 feet
Seong is building shelving in his garage. The shelves are 36 inches wide and 15 inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?
The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.
cm, cm
The width of a rectangle is three more than twice the length. The perimeter is 96 inches. Find the length and the width.
The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.
9 ft, 14 ft, 12 ft
Solve Mixture and Uniform Motion Applications
Solve Coin Word Problems
In the following exercises, solve.
Paulette has ?140 in ?5 and ?10 bills. The number of ?10 bills is one less than twice the number of ?5 bills. How many of each does she have?
Lenny has ?3.69 in pennies, dimes, and quarters. The number of pennies is three more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?
nine pennies, six dimes, 12 quarters
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Tickets for a basketball game cost ?2 for students and ?5 for adults. The number of students was three less than 10 times the number of adults. The total amount of money from ticket sales was ?619. How many of each ticket were sold?
125 tickets were sold for the jazz band concert for a total of ?1,022. Student tickets cost ?6 each and general admission tickets cost ?10 each. How many of each kind of ticket were sold?
57 students, 68 adults
Yumi spent ?34.15 buying stamps. The number of ?0.56 stamps she bought was 10 less than four times the number of ?0.41 stamps. How many of each did she buy?
Solve Mixture Word Problems
In the following exercises, solve.
Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost ?3.45 per pound and nuts cost ?7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him ?6.96 per pound?
lbs of raisins, lbs of nuts
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tile. She will use basic tiles that cost ?8 per square foot and decorator tiles that cost ?20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be ?10 per square foot?
Enrique borrowed ?23,500 to buy a car. He pays his uncle 2% interest on the ?4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total ?23,500? (Round your answer to the nearest tenth of a percent.)
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Solve Uniform Motion Applications
In the following exercises, solve.
When Gabe drives from Sacramento to Redding it takes him 2.2 hours. It takes Elsa two hours to drive the same distance. Elsa’s speed is seven miles per hour faster than Gabe’s speed. Find Gabe’s speed and Elsa’s speed.
Louellen and Tracy met at a restaurant on the road between Chicago and Nashville. Louellen had left Chicago and drove 3.2 hours towards Nashville. Tracy had left Nashville and drove 4 hours towards Chicago, at a speed one mile per hour faster than Louellen’s speed. The distance between Chicago and Nashville is 472 miles. Find Louellen’s speed and Tracy’s speed.
Louellen 65 mph, Tracy 66 mph
Two busses leave Amarillo at the same time. The Albuquerque bus heads west on the I-40 at a speed of 72 miles per hour, and the Oklahoma City bus heads east on the I-40 at a speed of 78 miles per hour. How many hours will it take them to be 375 miles apart?
Kyle rowed his boat upstream for 50 minutes. It took him 30 minutes to row back downstream. His speed going upstream is two miles per hour slower than his speed going downstream. Find Kyle’s upstream and downstream speeds.
upstream 3 mph, downstream 5 mph
At 6:30, Devon left her house and rode her bike on the flat road until 7:30. Then she started riding uphill and rode until 8:00. She rode a total of 15 miles. Her speed on the flat road was three miles per hour faster than her speed going uphill. Find Devon’s speed on the flat road and riding uphill.
Anthony drove from New York City to Baltimore, which is a distance of 192 miles. He left at 3:45 and had heavy traffic until 5:30. Traffic was light for the rest of the drive, and he arrived at 7:30. His speed in light traffic was four miles per hour more than twice his speed in heavy traffic. Find Anthony’s driving speed in heavy traffic and light traffic.
heavy traffic 32 mph, light traffic 66 mph
Solve Linear Inequalities
Graph Inequalities on the Number Line
In the following exercises, graph the inequality on the number line and write in interval notation.
Solve Linear Inequalities
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate Words to an Inequality and Solve
In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.
Five more than z is at most 19.
Three less than c is at least 360.
Nine times n exceeds 42.
Negative two times a is no more than eight.
Solve Applications with Linear Inequalities
In the following exercises, solve.
Julianne has a weekly food budget of ?231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?
?33 per day
Rogelio paints watercolors. He got a ?100 gift card to the art supply store and wants to use it to buy 12″ × 16″ canvases. Each canvas costs ?10.99. What is the maximum number of canvases he can buy with his gift card?
Briana has been offered a sales job in another city. The offer was for ?42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least ?66,500. What would her total sales need to be for her to move?
at least ?300,000
Renee’s car costs her ?195 per month plus ?0.09 per mile. How many miles can Renee drive so that her monthly car expenses are no more than ?250?
Costa is an accountant. During tax season, he charges ?125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are ?6,000. How many tax returns must he do if he wants to make a profit of at least ?8,000?
at least 112 jobs
Jenna is planning a five-day resort vacation with three of her friends. It will cost her ?279 for airfare, ?300 for food and entertainment, and ?65 per day for her share of the hotel. She has ?550 saved towards her vacation and can earn ?25 per hour as an assistant in her uncle’s photography studio. How many hours must she work in order to have enough money for her vacation?
Solve Compound Inequalities
Solve Compound Inequalities with “and”
In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
and
and
and
and
Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
or
or
or
or
Solve Applications with Compound Inequalities
In the following exercises, solve.
Liam is playing a number game with his sister Audry. Liam is thinking of a number and wants Audry to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Liam might be thinking of.
Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.
Solve Absolute Value Inequalities
Solve Absolute Value Equations
In the following exercises, solve.
no solution
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve. Graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.
A craft beer brewer needs 215,000 bottle per day. But this total can vary by as much as 5,000 bottles. What is the maximum and minimum expected usage at the bottling company?
The minimum to maximum expected usage is 210,000 to 220,000 bottles
At Fancy Grocery, the ideal weight of a loaf of bread is 16 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?
Practice Test
In the following exercises, solve each equation.
contradiction; no solution
Solve the formula
for y.
In the following exercises, graph the inequality on the number line and write in interval notation.
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
and
or
or
In the following exercises, translate to an equation or inequality and solve.
Four less than twice x is 16.
Find the length of the missing side.
One number is four more than twice another. Their sum is Find the numbers.
The sum of two consecutive odd integers is Find the numbers.
Marcus bought a television on sale for ?626.50 The original price of the television was ?895. Find ⓐ the amount of discount and ⓑ the discount rate.
Bonita has ?2.95 in dimes and quarters in her pocket. If she has five more dimes than quarters, how many of each coin does she have?
12 dimes, seven quarters
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs ?6.04 per gallon and the soda costs ?4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs ?5.71 per gallon?
The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.
The length of a rectangle is five feet more than four times the width. The perimeter is 60 feet. Find the dimensions of the rectangle.
Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?
Leon drove from his house in Cincinnati to his sister’s house in Cleveland, a distance of 252 miles. It took him hours. For the first half hour, he had heavy traffic, and the rest of the time his speed was five miles per hour less than twice his speed in heavy traffic. What was his speed in heavy traffic?
Sara has a budget of ?1,000 for costumes for the 18 members of her musical theater group. What is the maximum she can spend for each costume?
At most ?55.56 per costume. |
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Based on the information entered, the body mass index (BMI) is 0. This child's BMI-for-age is at above less than the 0 th percentile, which means that this child's BMI is less greater than the BMIs of 0% of 0-year-old s in the reference population used to create CDC's growth charts. This child's BMI falls into the category, defined as a BMI that is less than the 5 th percentile at or above the 5 th to less than the 85 th percentile at or above the 85 th to less than the 95 th percentile at or above the 95 th percentile at or above 120% of the 95 th percentile for sex and age or a BMI that is at or above 35 kg/m 2.
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BMI Category Percentile Ranges Table| BMI Category | BMI Range |
--- |
| Underweight | Less than the 5 th percentile |
| Healthy Weight | 5 th percentile to less than the 85 th percentile |
| Overweight | 85 th percentile to less than the 95 th percentile |
| Obesity | 95 th percentile or greater |
| Severe Obesity | 120% of the 95 th percentile or greater OR a BMI of 35 kg/m 2 or greater |
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Discuss the BMI results with your healthcare provider as BMI can relate to overall health and well-being. Learn more about Defining Child BMI Categories. Encourage children and teens to build and practice healthy routines.
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6550 | https://www.intmath.com/functions-and-graphs/angle-bisector.php | Interactive Mathematics
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What is Angle Bisector in Geometry?
Geometry is the branch of mathematics that deals with the properties of shapes and relationships between them. It's a fascinating subject that provides students with a wealth of knowledge and skills to help them understand the world around them and make sense of it. One of the important concepts in geometry is understanding angles and their properties. An angle bisector is an important concept in geometry that is used to divide angles into equal parts.
What is an Angle Bisector?
An angle bisector is a line segment that divides an angle into two equal parts. It's also known as the bisector of an angle. The line segment is perpendicular to the angle's two sides, and its length is always equal to half of the angle's total length. The angle bisector theorem states that the angle bisector divides the angle into two equal angles, each with a measure of half the original angle.
For example, consider an angle with a measure of 90 degrees. The angle bisector would divide this angle into two angles of 45 degrees each. Similarly, an angle with a measure of 60 degrees would be divided into two angles of 30 degrees each.
How to Bisect an Angle?
Bisecting an angle is the process of dividing it into two equal parts using an angle bisector. This can be done by drawing a straight line through the center of the angle, making sure that it is perpendicular to both sides of the angle. The line should be drawn so that it divides the angle into two equal parts.
To bisect an angle, first draw a line that is perpendicular to both sides of the angle. Then, draw another line that is parallel to one of the sides of the angle and intersects the first line. This will divide the angle into two equal parts.
The Angle Bisection Theorem
The angle bisection theorem states that the angle bisector divides the angle into two equal angles. This means that if an angle is bisected, the two resulting angles will be equal in measure. This theorem is especially useful when constructing shapes with specific angles, such as a triangle or a square.
To prove the angle bisection theorem, we can use the properties of similar triangles. If two angles of a triangle are equal, then the sides opposite those angles are also equal. This means that if two angles of a triangle are equal, then the triangle must be an isosceles triangle. Therefore, if an angle is bisected, the two resulting angles must be equal.
Practice Problems
To practice bisecting angles, we can use the following practice problems.
Problem 1
Bisect an angle with a measure of 30 degrees.
Solution: The angle can be bisected by drawing a line perpendicular to both sides of the angle. This will divide the angle into two angles of 15 degrees each.
Problem 2
Bisect an angle with a measure of 45 degrees.
Solution: The angle can be bisected by drawing a line perpendicular to both sides of the angle. This will divide the angle into two angles of 22.5 degrees each.
Problem 3
Bisect an angle with a measure of 60 degrees.
Solution: The angle can be bisected by drawing a line perpendicular to both sides of the angle. This will divide the angle into two angles of 30 degrees each.
Problem 4
Bisect an angle with a measure of 90 degrees.
Solution: The angle can be bisected by drawing a line perpendicular to both sides of the angle. This will divide the angle into two angles of 45 degrees each.
Problem 5
Bisect an angle with a measure of 120 degrees.
Solution: The angle can be bisected by drawing a line perpendicular to both sides of the angle. This will divide the angle into two angles of 60 degrees each.
Summary
In this lesson, we discussed angle bisector in geometry. We defined it as a line segment that divides an angle into two equal parts, and we discussed the angle bisection theorem. We also looked at how to bisect an angle, and we practiced bisecting angles with a few practice problems.
Angle bisector is an important concept in geometry, and understanding it will help students understand angles and their properties. With this knowledge, students will be able to construct shapes with specific angles, such as a triangle or a square.
FAQ
What is an angle bisector in geometry?
An angle bisector in geometry is a line segment or a ray that divides an angle into two equal parts.
Which is the best definition for angle bisector?
The best definition for angle bisector is a line segment or a ray that divides an angle into two equal parts.
How do you find the angle bisector in geometry?
To find the angle bisector in geometry, draw two lines that form the desired angle, then draw a line segment or a ray that divides the angle into two equal parts.
How does an angle bisector work?
An angle bisector works by dividing an angle into two equal parts. It is a line that bisects the angle and divides it into two equal parts.
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6551 | https://en.wikipedia.org/wiki/Tantalum_pentoxide | Jump to content
Tantalum pentoxide
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From Wikipedia, the free encyclopedia
Tantalum pentoxide
| | |
| | |
| Names | |
| IUPAC name Tantalum(V) oxide | |
| Systematic IUPAC name Ditantalum pentaoxide | |
| Identifiers | |
| CAS Number | 1314-61-0Y |
| 3D model (JSmol) | Interactive image |
| ChemSpider | 452513N |
| ECHA InfoCard | 100.013.854 |
| PubChem CID | 518712 |
| UNII | OEZ64Z53M4Y |
| CompTox Dashboard (EPA) | DTXSID10893849 |
| InChI InChI=1S/5O.2Ta | |
| SMILES O=TaOTa=O | |
| Properties | |
| Chemical formula | Ta2O5 |
| Molar mass | 441.893 g/mol |
| Appearance | white, odorless powder |
| Density | β-Ta2O5 = 8.18 g/cm3 α-Ta2O5 = 8.37 g/cm3 |
| Melting point | 1,872 °C (3,402 °F; 2,145 K) |
| Solubility in water | negligible |
| Solubility | insoluble in organic solvents and most mineral acids, reacts with HF |
| Band gap | 3.8–5.3 eV |
| Magnetic susceptibility (χ) | −32.0×10−6 cm3/mol |
| Refractive index (nD) | 2.275 |
| Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa). N verify (what is YN ?) Infobox references | |
Chemical compound
Tantalum pentoxide, also known as tantalum(V) oxide, is the inorganic compound with the formula Ta
2O
5. It is a white solid that is insoluble in all solvents but is attacked by strong bases and hydrofluoric acid. Ta
2O
5 is an inert material with a high refractive index and low absorption (i.e. colourless), which makes it useful for coatings. It is also extensively used in the production of capacitors, due to its high dielectric constant.
Preparation
[edit]
Occurrence
[edit]
Tantalum occurs in the minerals tantalite and columbite (columbium being an archaic name for niobium), which occur in pegmatites, an igneous rock formation. Mixtures of columbite and tantalite are called coltan. Tantalum was discovered in Tantalite in 1802 by Anders Gustaf Ekeberg at Ytterby, Sweden, and Kimoto, Finland. The minerals microlite and pyrochlore contain approximately 70% and 10% Ta, respectively.
Refining
[edit]
Tantalum ores often contain significant amounts of niobium, which is itself a valuable metal. As such, both metals are extracted so that they may be sold. The overall process is one of hydrometallurgy and begins with a leaching step; in which the ore is treated with hydrofluoric acid and sulfuric acid to produce water-soluble hydrogen fluorides, such as the heptafluorotantalate. This allows the metals to be separated from the various non-metallic impurities in the rock.
: (FeMn)(NbTa)2O6 + 16 HF → H2[TaF7] + H2[NbOF5] + FeF2 + MnF2 + 6 H2O
The tantalum and niobium hydrogenfluorides are then removed from the aqueous solution by liquid-liquid extraction using organic solvents, such as cyclohexanone or methyl isobutyl ketone. This step allows the simple removal of various metal impurities (e.g. iron and manganese) which remain in the aqueous phase in the form of fluorides. Separation of the tantalum and niobium is then achieved by pH adjustment. Niobium requires a higher level of acidity to remain soluble in the organic phase and can hence be selectively removed by extraction into less acidic water.
The pure tantalum hydrogen fluoride solution is then neutralised with aqueous ammonia to give hydrated tantalum oxide (Ta2O5(H2O)x), which is calcinated to tantalum pentoxide (Ta2O5) as described in these idealized equations:
: H2[TaF7] + 5 H2O + 7 NH3 → 1/2 Ta2O5(H2O)5 + 7 NH4F
: Ta2O5(H2O)5 → Ta2O5 + 5 H2O
Natural pure tantalum oxide is known as the mineral tantite, although it is exceedingly rare.
From alkoxides
[edit]
Tantalum oxide is frequently used in electronics, often in the form of thin films. For these applications it can be produced by MOCVD (or related techniques), which involves the hydrolysis of its volatile halides or alkoxides:
: Ta2(OEt)10 + 5 H2O → Ta2O5 + 10 EtOH
: 2 TaCl5 + 5 H2O → Ta2O5 + 10 HCl
Structure and properties
[edit]
The crystal structure of tantalum pentoxide has been the matter of some debate. The bulk material is disordered, being either amorphous or polycrystalline; with single crystals being difficult to grow. As such Xray crystallography has largely been limited to powder diffraction, which provides less structural information.
At least 2 polymorphs are known to exist. A low temperature form, known as L- or β-Ta2O5, and the high temperature form known as H- or α-Ta2O5. The transition between these two forms is slow and reversible; taking place between 1000 and 1360 °C, with a mixture of structures existing at intermediate temperatures. The structures of both polymorphs consist of chains built from octahedral TaO6 and pentagonal bipyramidal TaO7 polyhedra sharing opposite vertices; which are further joined by edge-sharing. The overall crystal system is orthorhombic in both cases, with the space group of β-Ta2O5 being identified as Pna2 by single crystal X-ray diffraction.
A high pressure form (Z-Ta2O5) has also been reported, in which the Ta atoms adopt a 7 coordinate geometry to give a monoclinic structure (space group C2).
Purely amorphous tantalum pentoxide has a similar local structure to the crystalline polymorphs, built from TaO6 and TaO7 polyhedra, while the molten liquid phase has a distinct structure based on lower coordination polyhedra, mainly TaO5 and TaO6.
The difficulty in forming material with a uniform structure has led to variations in its reported properties. Like many metal oxides Ta2O5 is an insulator and its band gap has variously been reported as being between 3.8 and 5.3 eV, depending on the method of manufacture. In general the more amorphous the material the greater its observed band gap.
These observed values are significantly higher than those predicted by computational chemistry (2.3 - 3.8 eV).
Its dielectric constant is typically about 25 although values of over 50 have been reported. In general tantalum pentoxide is considered to be a high-k dielectric material.
Reactions
[edit]
Ta2O5 does not react appreciably with either HCl or HBr, however it will dissolve in hydrofluoric acid, and reacts with potassium bifluoride and HF according to the following equation:
: Ta2O5 + 4 KHF2 + 6 HF → 2 K2[TaF7] + 5 H2O
Ta2O5 can be reduced to metallic Ta via the use of metallic reductants such as calcium and aluminium.
: Ta2O5 + 5 Ca → 2 Ta + 5 CaO
Uses
[edit]
In electronics
[edit]
Owing to its high band gap and dielectric constant, tantalum pentoxide has found a variety of uses in electronics, particularly in tantalum capacitors. These are used in automotive electronics, cell phones, and pagers, electronic circuitry; thin-film components; and high-speed tools. In the 1990s, interest grew in the use of tantalum oxide as a high-k dielectric for DRAM capacitor applications.
It is used in on-chip metal-insulator-metal capacitors for high frequency CMOS integrated circuits. Tantalum oxide may have applications as the charge trapping layer for non-volatile memories. There are applications of tantalum oxide in resistive switching memories.
In optics
[edit]
Due to its high refractive index, Ta2O5 has been utilized in the fabrication of the glass of photographic lenses.
It can also be deposited as an optical coating with typical applications being antireflection and multilayer filter coatings in near UV to near infrared.
Ta2O5 has also been found to have a high nonlinear refractive index, on the order of three times that of silicon nitride, which has led to interest in the utilization of Ta2O5 in photonic integrated circuits. Ta2O5 has been recently utilized as the material platform for the generation of supercontinuum and Kerr frequency combs in waveguides and optical ring resonators. Through the addition of rare-earth dopants in the deposition process, Ta2O5 waveguide lasers have been presented for a variety of applications, such as remote sensing and LiDAR.
References
[edit]
^ Reisman, Arnold; Holtzberg, Frederic; Berkenblit, Melvin; Berry, Margaret (20 September 1956). "Reactions of the Group VB Pentoxides with Alkali Oxides and Carbonates. III. Thermal and X-Ray Phase Diagrams of the System K2O or K2CO3 with Ta2O5". Journal of the American Chemical Society. 78 (18): 4514–4520. Bibcode:1956JAChS..78.4514R. doi:10.1021/ja01599a003.
^ Jump up to: a b Fairbrother, Frederick (1967). The Chemistry of Niobium and Tantalum. New York: Elsevier Publishing Company. pp. 1–28. ISBN 978-0-444-40205-9.
^ Anthony Agulyanski (2004). "Fluorine chemistry in the processing of tantalum and niobium". In Anatoly Agulyanski (ed.). Chemistry of Tantalum and Niobium Fluoride Compounds (1st ed.). Burlington: Elsevier. ISBN 9780080529028.
^ "Tantite: Tantite mineral information and data". Mindat.org. Retrieved 2016-03-03.
^ Jump up to: a b Askeljung, Charlotta; Marinder, Bengt-Olov; Sundberg, Margareta (1 November 2003). "Effect of heat treatment on the structure of L-Ta2O5". Journal of Solid State Chemistry. 176 (1): 250–258. Bibcode:2003JSSCh.176..250A. doi:10.1016/j.jssc.2003.07.003.
^ Stephenson, N. C.; Roth, R. S. (1971). "Structural systematics in the binary system Ta2O5–WO3. V. The structure of the low-temperature form of tantalum oxide L-Ta2O5". Acta Crystallographica Section B. 27 (5): 1037–1044. Bibcode:1971AcCrB..27.1037S. doi:10.1107/S056774087100342X.
^ Wells, A.F. (1947). Structural Inorganic Chemistry. Oxford: Clarendon Press.
^ Wolten, G. M.; Chase, A. B. (1 August 1969). "Single-crystal data for β Ta2O5 and A KPO3". Zeitschrift für Kristallographie. 129 (5–6): 365–368. Bibcode:1969ZK....129..365W. doi:10.1524/zkri.1969.129.5-6.365.
^ Hummel, Hans-U.; Fackler, Richard; Remmert, Peter (1992). "Tantaloxide durch Gasphasenhydrolyse, Druckhydrolyse und Transportreaktion aus 2H-TaS2: Synthesen von TT-Ta2O5 und T-Ta2O5 und Kristallstruktur von T-Ta2O5". Chemische Berichte. 125 (3): 551–556. doi:10.1002/cber.19921250304.
^ Zibrov, I. P.; Filonenko, V. P.; Sundberg, M.; Werner, P.-E. (1 August 2000). "Structures and phase transitions of B-Ta2O5 and Z-Ta2O5: two high-pressure forms of Ta2O5". Acta Crystallographica Section B. 56 (4): 659–665. doi:10.1107/S0108768100005462. PMID 10944257. S2CID 22330435.
^ Alderman, O. L. G.; Benmore, C.J.; Neuefeind, J.; Coillet, E.; Mermet, A.; Martinez, V.; Tamalonis, A.; Weber, R. (2018). "Amorphous tantala and its relationship with the molten state". Physical Review Materials. 2 (4): 043602. Bibcode:2018PhRvM...2d3602A. doi:10.1103/PhysRevMaterials.2.043602.{{cite journal}}: CS1 maint: article number as page number (link)
^ Kukli, Kaupo; Aarik, Jaan; Aidla, Aleks; Kohan, Oksana; Uustare, Teet; Sammelselg, Väino (1995). "Properties of tantalum oxide thin films grown by atomic layer deposition". Thin Solid Films. 260 (2): 135–142. Bibcode:1995TSF...260..135K. doi:10.1016/0040-6090(94)06388-5.
^ Fleming, R. M.; Lang, D. V.; Jones, C. D. W.; Steigerwald, M. L.; Murphy, D. W.; Alers, G. B.; Wong, Y.-H.; van Dover, R. B.; Kwo, J. R.; Sergent, A. M. (1 January 2000). "Defect dominated charge transport in amorphous Ta2O5 thin films". Journal of Applied Physics. 88 (2): 850. Bibcode:2000JAP....88..850F. doi:10.1063/1.373747.
^ Murawala, Prakash A.; Sawai, Mikio; Tatsuta, Toshiaki; Tsuji, Osamu; Fujita, Shizuo; Fujita, Shigeo (1993). "Structural and Electrical Properties of Ta2O5 Grown by the Plasma-Enhanced Liquid Source CVD Using Penta Ethoxy Tantalum Source". Japanese Journal of Applied Physics. 32 (Part 1, No. 1B): 368–375. Bibcode:1993JaJAP..32..368M. doi:10.1143/JJAP.32.368. S2CID 97813703.
^ Ramprasad, R. (1 January 2003). "First principles study of oxygen vacancy defects in tantalum pentoxide". Journal of Applied Physics. 94 (9): 5609–5612. Bibcode:2003JAP....94.5609R. doi:10.1063/1.1615700.
^ Sawada, H.; Kawakami, K. (1 January 1999). "Electronic structure of oxygen vacancy in Ta2O5". Journal of Applied Physics. 86 (2): 956. Bibcode:1999JAP....86..956S. doi:10.1063/1.370831.
^ Nashed, Ramy; Hassan, Walid M. I.; Ismail, Yehea; Allam, Nageh K. (2013). "Unravelling the interplay of crystal structure and electronic band structure of tantalum oxide (Ta2O5)". Physical Chemistry Chemical Physics. 15 (5): 1352–7. Bibcode:2013PCCP...15.1352N. doi:10.1039/C2CP43492J. PMID 23243661.
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^ Agulyansky, A (2003). "Potassium fluorotantalate in solid, dissolved and molten conditions". J. Fluorine Chem. 123 (2): 155–161. Bibcode:2003JFluC.123..155A. doi:10.1016/S0022-1139(03)00190-8.
^ Brauer, Georg (1965). Handbook of preparative inorganic chemistry. Academic Press. p. 256. ISBN 978-0-12-395591-3.
^
Ezhilvalavan, S.; Tseng, T. Y. (1999). "Preparation and properties of tantalum pentoxide (Ta2O5) thin films for ultra large scale integrated circuits (ULSIs) application - a review". Journal of Materials Science: Materials in Electronics. 10 (1): 9–31. doi:10.1023/A:1008970922635. S2CID 55644772.
^
Chaneliere, C; Autran, J L; Devine, R A B; Balland, B (1998). "Tantalum pentoxide (Ta2O5) thin films for advanced dielectric applications". Materials Science and Engineering: R. 22 (6): 269–322. doi:10.1016/S0927-796X(97)00023-5.
^
Wang, X; et al. (2004). "A Novel MONOS-Type Nonvolatile Memory Using High-κ Dielectrics for Improved Data Retention and Programming Speed". IEEE Transactions on Electron Devices. 51 (4): 597–602. Bibcode:2004ITED...51..597W. doi:10.1109/TED.2004.824684.
^
Zhu, H; et al. (2013). "Design and Fabrication of Ta2O5 Stacks for Discrete Multibit Memory Application". IEEE Transactions on Nanotechnology. 12 (6): 1151–1157. Bibcode:2013ITNan..12.1151Z. doi:10.1109/TNANO.2013.2281817. S2CID 44045227.
^
Lee, M-.J; et al. (2011). "A fast, high-endurance and scalable non-volatile memory device made from asymmetric Ta2O5−x/TaO2−x bilayer structures". Nature Materials. 10 (8): 625–630. Bibcode:2011NatMa..10..625L. doi:10.1038/NMAT3070. PMID 21743450.
^ Musikant, Solomon (1985). "Optical Glas Composition". Optical Materials: An Introduction to Selection and Application. CRC Press. p. 28. ISBN 978-0-8247-7309-0.
^ "Tantalum Oxide for Optical Coating Applications". Materion. Retrieved April 1, 2021.
^ Tai, Chao-Yi; Wilkinson, James S.; Perney, Nicolas M. B.; Netti, M. Caterina; Cattaneo, F.; Finlayson, Chris E.; Baumberg, Jeremy J. (2004-10-18). "Determination of nonlinear refractive index in a Ta2O5 rib waveguide using self-phase modulation". Optics Express. 12 (21): 5110–5116. Bibcode:2004OExpr..12.5110T. doi:10.1364/OPEX.12.005110. ISSN 1094-4087. PMID 19484065.
^ Jump up to: a b Jung, Hojoong; Yu, Su-Peng; Carlson, David R.; Drake, Tara E.; Briles, Travis C.; Papp, Scott B. (2021-06-20). "Tantala Kerr nonlinear integrated photonics". Optica. 8 (6): 811–817. arXiv:2007.12958. Bibcode:2021Optic...8..811J. doi:10.1364/OPTICA.411968. ISSN 2334-2536. S2CID 220793938.
^ Woods, Jonathan R. C.; Daykin, Jake; Tong, Amy S. K.; Lacava, Cosimo; Petropoulos, Periklis; Tropper, Anne C.; Horak, Peter; Wilkinson, James S.; Apostolopoulos, Vasilis (2020-10-12). "Supercontinuum generation in tantalum pentoxide waveguides for pump wavelengths in the 900 nm to 1500 nm spectral region". Optics Express. 28 (21): 32173–32184. Bibcode:2020OExpr..2832173W. doi:10.1364/OE.403089. ISSN 1094-4087. PMID 33115180.
^ Fan, Ranran; Lin, Yuan-Yao; Chang, Lin; Boes, Andreas; Bowers, John; Liu, Jia-Wei; Lin, Chao-Hong; Wang, Te-Keng; Qiao, Junpeng; Kuo, Hao-Chung; Lin, Gong-Ru; Shih, Min-Hsiung; Hung, Yung-Jr; Chiu, Yi-Jen; Lee, Chao-Kuei (2021-04-12). "Higher order mode supercontinuum generation in tantalum pentoxide (Ta2O5) channel waveguide". Scientific Reports. 11 (1): 7978. Bibcode:2021NatSR..11.7978F. doi:10.1038/s41598-021-86922-8. ISSN 2045-2322. PMC 8042067. PMID 33846403.
^ Tong, Amy S. K.; Mitchell, Colin J.; Aghajani, Armen; Sessions, Neil; Senthil Murugan, G.; Mackenzie, Jacob I.; Wilkinson, James S. (2020-09-01). "Spectroscopy of thulium-doped tantalum pentoxide waveguides on silicon". Optical Materials Express. 10 (9): 2201. Bibcode:2020OMExp..10.2201T. doi:10.1364/OME.397011. ISSN 2159-3930.
^ Aghajani, A; Murugan, G S; Sessions, N P; Apostolopoulos, V; Wilkinson, J S (2015-06-17). "Spectroscopy of high index contrast Yb:Ta 2 O 5 waveguides for lasing applications". Journal of Physics: Conference Series. 619 (1): 012031. Bibcode:2015JPhCS.619a2031A. doi:10.1088/1742-6596/619/1/012031. ISSN 1742-6596.{{cite journal}}: CS1 maint: article number as page number (link)
^ Subramani, Ananth Z.; Oton, Claudio J.; Shepherd, David P.; Wilkinson, James S. (November 2010). "Erbium-Doped Waveguide Laser in Tantalum Pentoxide". IEEE Photonics Technology Letters. 22 (21): 1571–1573. Bibcode:2010IPTL...22.1571S. doi:10.1109/LPT.2010.2072495. ISSN 1041-1135. S2CID 28849615.
| v t e Tantalum compounds | |
--- |
| Tantalum(II) | TaB2 |
| Tantalum(III) | TaAl3 TaAs TaN TaCl3 | | | --- | | Organotantalum(III) | - Ta(SiOtBu)3 | |
| Tantalum(IV) | TaS2 TaSe2 TaI4 TaC TaTe2 Ta4HfC5 |
| Tantalum(V) | TaF5 TaCl5 TaBr5 TaI5 Ta2O5 LiTaO3 NdTaO4 LuTaO4 K2TaF7 H2TaF7 K2Ta2O3F6 | | | --- | | Organotantalum(V) | - Ta2(OC2H5)10 - TaC5H15 - TaCl2(CH3)3 - Ta(C2H6N)5 - TaCp2H3 | |
| v t e | |
--- |
| Mixed oxidation states | Antimony tetroxide (Sb2O4) Boron suboxide (B12O2) Carbon suboxide (C3O2) Chlorine perchlorate (Cl2O4) Chloryl perchlorate (Cl2O6) Cobalt(II,III) oxide (Co3O4) Dichlorine pentoxide (Cl2O5) Iron(II,III) oxide (Fe3O4) Lead(II,IV) oxide (Pb3O4) Manganese(II,III) oxide (Mn3O4) Mellitic anhydride (C12O9) Praseodymium(III,IV) oxide (Pr6O11) Silver(I,III) oxide (Ag2O2) Terbium(III,IV) oxide (Tb4O7) Tribromine octoxide (Br3O8) Triuranium octoxide (U3O8) |
| +1 oxidation state | Aluminium(I) oxide (Al2O) Copper(I) oxide (Cu2O) Caesium monoxide (Cs2O) Dibromine monoxide (Br2O) Dicarbon monoxide (C2O) Dichlorine monoxide (Cl2O) Gallium(I) oxide (Ga2O) Iodine(I) oxide (I2O) Lithium oxide (Li2O) Mercury(I) oxide (Hg2O) Nitrous oxide (N2O) Potassium oxide (K2O) Rubidium oxide (Rb2O) Silver oxide (Ag2O) Thallium(I) oxide (Tl2O) Sodium oxide (Na2O) Water (hydrogen oxide) (H2O) |
| +2 oxidation state | Aluminium(II) oxide (AlO) Barium oxide (BaO) Berkelium monoxide (BkO) Beryllium oxide (BeO) Boron monoxide (BO) Bromine monoxide (BrO) Cadmium oxide (CdO) Calcium oxide (CaO) Carbon monoxide (CO) Chlorine monoxide (ClO) Chromium(II) oxide (CrO) Cobalt(II) oxide (CoO) Copper(II) oxide (CuO) Dinitrogen dioxide (N2O2) Disulfur dioxide (S2O2) Europium(II) oxide (EuO) Germanium monoxide (GeO) Iron(II) oxide (FeO) Iodine monoxide (IO) Lead(II) oxide (PbO) Magnesium oxide (MgO) Manganese(II) oxide (MnO) Mercury(II) oxide (HgO) Nickel(II) oxide (NiO) Nitric oxide (NO) Niobium monoxide (NbO) Palladium(II) oxide (PdO) Phosphorus monoxide (PO) Polonium monoxide (PoO) Protactinium monoxide (PaO) Radium oxide (RaO) Silicon monoxide (SiO) Strontium oxide (SrO) Sulfur monoxide (SO) Thorium monoxide (ThO) Tin(II) oxide (SnO) Titanium(II) oxide (TiO) Vanadium(II) oxide (VO) Yttrium(II) oxide (YO) Zirconium monoxide (ZrO) Zinc oxide (ZnO) |
| +3 oxidation state | Actinium(III) oxide (Ac2O3) Aluminium oxide (Al2O3) Americium(III) oxide (Am2O3) Antimony trioxide (Sb2O3) Arsenic trioxide (As2O3) Berkelium(III) oxide (Bk2O3) Bismuth(III) oxide (Bi2O3) Boron trioxide (B2O3) Californium(III) oxide (Cf2O3) Cerium(III) oxide (Ce2O3) Chromium(III) oxide (Cr2O3) Cobalt(III) oxide (Co2O3) Curium(III) oxide (Cm2O3) Dinitrogen trioxide (N2O3) Dysprosium(III) oxide (Dy2O3) Einsteinium(III) oxide (Es2O3) Erbium(III) oxide (Er2O3) Europium(III) oxide (Eu2O3) Gadolinium(III) oxide (Gd2O3) Gallium(III) oxide (Ga2O3) Gold(III) oxide (Au2O3) Holmium(III) oxide (Ho2O3) Indium(III) oxide (In2O3) Iron(III) oxide (Fe2O3) Lanthanum oxide (La2O3) Lutetium(III) oxide (Lu2O3) Manganese(III) oxide (Mn2O3) Neodymium(III) oxide (Nd2O3) Nickel(III) oxide (Ni2O3) Phosphorus trioxide (P4O6) Praseodymium(III) oxide (Pr2O3) Promethium(III) oxide (Pm2O3) Rhodium(III) oxide (Rh2O3) Samarium(III) oxide (Sm2O3) Scandium oxide (Sc2O3) Terbium(III) oxide (Tb2O3) Thallium(III) oxide (Tl2O3) Thulium(III) oxide (Tm2O3) Titanium(III) oxide (Ti2O3) Tungsten(III) oxide (W2O3) Vanadium(III) oxide (V2O3) Ytterbium(III) oxide (Yb2O3) Yttrium(III) oxide (Y2O3) |
| +4 oxidation state | Americium dioxide (AmO2) Berkelium(IV) oxide (BkO2) Bromine dioxide (BrO2) Californium dioxide (CfO2) Carbon dioxide (CO2) Carbon trioxide (CO3) Cerium(IV) oxide (CeO2) Chlorine dioxide (ClO2) Chromium(IV) oxide (CrO2) Curium(IV) oxide (CmO2) Dinitrogen tetroxide (N2O4) Germanium dioxide (GeO2) Iodine dioxide (IO2) Iridium dioxide (IrO2) Hafnium(IV) oxide (HfO2) Lead dioxide (PbO2) Manganese dioxide (MnO2) Molybdenum dioxide (MoO2) Neptunium(IV) oxide (NpO2) Nitrogen dioxide (NO2) Niobium dioxide (NbO2) Osmium dioxide (OsO2) Platinum dioxide (PtO2) Plutonium(IV) oxide (PuO2) Polonium dioxide (PoO2) Praseodymium(IV) oxide (PrO2) Protactinium(IV) oxide (PaO2) Rhenium(IV) oxide (ReO2) Rhodium(IV) oxide (RhO2) Ruthenium(IV) oxide (RuO2) Selenium dioxide (SeO2) Silicon dioxide (SiO2) Sulfur dioxide (SO2) Technetium(IV) oxide (TcO2) Tellurium dioxide (TeO2) Terbium(IV) oxide (TbO2) Thorium dioxide (ThO2) Tin dioxide (SnO2) Titanium dioxide (TiO2) Tungsten(IV) oxide (WO2) Uranium dioxide (UO2) Vanadium(IV) oxide (VO2) Xenon dioxide (XeO2) Zirconium dioxide (ZrO2) |
| +5 oxidation state | Antimony pentoxide (Sb2O5) Arsenic pentoxide (As2O5) Bismuth pentoxide (Bi2O5) Dinitrogen pentoxide (N2O5) Diuranium pentoxide (U2O5) Neptunium(V) oxide (Np2O5) Niobium pentoxide (Nb2O5) Phosphorus pentoxide (P2O5) Protactinium(V) oxide (Pa2O5) Tantalum pentoxide (Ta2O5) Tungsten pentoxide (W2O5) Vanadium(V) oxide (V2O5) |
| +6 oxidation state | Chromium trioxide (CrO3) Molybdenum trioxide (MoO3) Polonium trioxide (PoO3) Rhenium trioxide (ReO3) Selenium trioxide (SeO3) Sulfur trioxide (SO3) Tellurium trioxide (TeO3) Tungsten trioxide (WO3) Uranium trioxide (UO3) Xenon trioxide (XeO3) |
| +7 oxidation state | Dichlorine heptoxide (Cl2O7) Manganese heptoxide (Mn2O7) Rhenium(VII) oxide (Re2O7) Technetium(VII) oxide (Tc2O7) |
| +8 oxidation state | Iridium tetroxide (IrO4) Osmium tetroxide (OsO4) Ruthenium tetroxide (RuO4) Xenon tetroxide (XeO4) Hassium tetroxide (HsO4) |
| Related | Oxocarbon Suboxide Oxyanion Ozonide Peroxide Superoxide Oxypnictide |
| Oxides are sorted by oxidation state. Category:Oxides | |
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6553 | https://www.youtube.com/watch?v=t3DduGxXhlY | 42 - AC CIRCUITS PROBLEM 10 & 11 | Find the Impedance, current, and the phase angle #knust
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42 - AC CIRCUITS PROBLEM 10 & 11 | Find the Impedance current and the phase angle #knust
In this video, we are going to solve ac circuit problems, finding the impedance, current and phase angle between the current and the source voltage.
Finding the impedance of an rlc circuit.
Z = R + jXL - jXc, where
R is the resistance
XL is the inductive reactance
and
Xc is the capacitive reactance
00:00 - Problem 10
05:12 - Problem 11
4 comments
Transcript:
Problem 10 now let's try this set of questions question number 10 a circuit has two parallel parts and their impedances are z1 equals 10 plus j20 ohms z2 equals negative j 12.5 ohms calculate the combined impedance now let's solve this question together so according to the question we have two parallel parts z1 and z2 so let this be z1 which is equal to 10 plus j 20 ohms and then we have this to be z2 equals negative j 12.5 ohms and we have been asked to find the combined now two resistors connected in parallel is given by rt equals r1 times r2 divided by r1 plus r2 now the same applies to two impedances in parallel so the combined impedance that is zt is given by z1 times z2 divided by z1 plus z2 so let's input the values of z1 and then z2 so we have z1 to be 10 plus j20 and then z2 to be negative j 12.5 divided by z1 10 plus j20 plus z2 which is negative j 12.5 now let's try to simplify this we have 10 times negative j 12.5 and that gives negative j 125 plus 20 times 12.5 is 250 and then we are left with j and then negative j so j times negative j gives negative j squared we have this all divided by 10 plus 0 is 10 and then j 20 plus negative j 12.5 gives j 7.5 now let's simplify further we know that j square is equal to negative one therefore negative j square is equal to negative times negative one and that is equal to one so 1 times 250 is 250 so we have 250 and then minus j125 or divided by 10 plus j 7.5 now at this point we are going to convert these values from their rectangular forms to their polar forms now if you convert 250 minus j125 to the polar form then you have 279.508 polar negative 26.565 divided by 10 plus j 7.5 also gives 12.5 pula 36.870 so we divide 279.50 by 12.5 and then we have 22.361 and for the angle because we are dividing we are going to subtract so we have negative 26.565 minus 36.870 and that gives negative 63.43 now let's convert this from the polar form back to the rectangular form when you do that you have 10 minus j 20 ohms therefore we have the combined impedance z t to be equal to 10 minus j 20 ohms now let's move on to the next question Problem 11 so to question 11 a capacitor of 400 over pi microfarad is connected in parallel with a 40 ohm resistor across a 200 volt 50 has supply a find the total supply current and then b determine the power factor of the whole circuit now let's represent the information we have in a circuit so we have been told that we have a capacitor of 400 over pi microfarad and that is connected in parallel with a 40 ohm resistor across a 200 volt 50 has supply so we have this to be the ac source which is 200 volts 50 hairs supply and then we have a capacitor in parallel with a 40 ohm resistor so we have this to be the capacitor and that is in parallel with the resistor so we have r to be 40 ohms and then we have the capacitor to be 400 over pi microfarad and then we are going to find e the total supply current now to find the total supply current we first of all need to find the total impedance of this circuit and then we can divide the voltage that is 200 by the total impedance of the circuit now let the impedance due to the resistor be z1 and also the impedance due to the capacitor bz2 so let's set one equals r and that is equal to 40 ohms and also let z2 equals now because z2 is the impedance due to the capacitor that's going to be negative j x c so let's find x c that is the capacitive reactance so we have x c equals 1 over omega c and we can simplify this as 2 pi f c that is 1 over 2 pi fc now from the question we are giving the frequency to be 50 heads so we have 2 pi times 50 times the value for the capacitance which is 400 over pi microfarad so we have 400 over pi times 10 exponent negative 6. so when you simplify the denominator we are going to get 0.04 and then 1 divided by 0.04 is 25 so we have x c to be 25 ohms therefore representing the impedances in the rectangular form we have z1 equals 40 ohms and then z2 equals negative j 25 ohms so let's move on as we find the combined impedance so the combined impedance is giving us z t and that is equal to now because the two impedances are in parallel we are going to have z1 times z2 divided by z1 plus z2 and that is equal to we have z1 to be 40 z2 to be negative j 25 divided by 40 plus negative j 25 so let's simplify this so 40 times negative j 25 gives negative j 1000 divided by 40 plus negative j 25 is 40 minus j 25 now let's convert each of the values at the numerator and the denominator to the apollo forms so that gives 1000 polar negative 90 divided by 47.17 negative 32.005 now let's simplify this so we divide 1000 by 47.17 and then we have 21.20 and then polar negative 90 minus negative 32.005 and that also gives negative 57.995 so this is the combined impedance now we have this value that is 21.20 to be the impedance and then we have negative 57.995 to be the phase angle therefore to find the total supply current that is given by i equals v over z and then we have vein to be 200 so 200 divided by 21.20 and that gives 9.43 amperes so this is the value of the total supply current so that is a and also for b we have been asked to find or determine the power factor of the whole circuit so the power factor is equal to cosine of the phase angle now we have the phase angle to be negative 57.995 and then cosine of this value gives 0.53 now considering this circuit you realize that the impedance is capacitive because we have a capacitor in the circuit therefore in a capacitive circuit the current always leads the voltage therefore the power factor is going to be 0.53 leading so that's it for today's video thanks for watching and see you in my next video bye bye |
6554 | https://en.wikipedia.org/wiki/Curl_(association_football) | Curl (association football) - Wikipedia
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(Top)
1 Nomenclature
2 UsageToggle Usage subsection
2.1 Free kicks
2.2 Corners
2.3 Passing
3 Causes
4 Notable players
5 See also
6 Notes
7 References
8 External links
Curl (association football)
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From Wikipedia, the free encyclopedia
In association football (soccer), spin on the ball which will make it change direction
David Beckham (centre) scoring with a bending free kick in 2007. The ball is struck with the inside of his right foot, with his body leaning to the left to generate extra curl on the ball.
Curl or bend in association football is a definition for a spin on the ball which makes the ball move in a curved direction. When kicking the ball, the inside of the foot is often used to curl the ball, but this can also be done by using the outside of the foot. Similar to curl, the ball can also swerve in the air, without the spin on the ball which makes the ball curl.
Curling or bending the ball is especially used in free kicks, shots from outside the penalty area and crosses. Differences between balls can affect the amount of swerve and curl: traditional leather footballs were too heavy to curl without great effort, whereas lighter modern footballs curl more easily.
Nomenclature
[edit]
The deviation of a ball from the straight path in the air is known as the curl, or swerve; however, the spin on the ball that causes this is also known as the curl. Shots that curl, bend, or swerve are known as curlers, or in extreme cases, banana shots. The technique of putting curl on a ball with the outside of the foot is sometimes known as a trivela, a Portuguese term, with Ricardo Quaresma a notable user of this skill. The topspin technique of putting straight curl (instead of side curl) on a ball is known as a dip or dipping shot. Putting no spin on the ball is often used for longer distance kicks, and can cause the ball to dip, or wobble in the air unpredictably. The 1950s Brazilian star Didi is thought to have invented this technique, and used it frequently when taking free kicks, which were known as folha seca ("dry" or "dead leaf," in Portuguese) free kicks. Today it is commonly known as the knuckleball technique; this technique has also been described in the media as the "tomahawk", or even the "maledetta" ("accursed," in Italian).
Usage
[edit]
Free kicks
[edit]
Roberto Carlos' bending free kick for Brazil (yellow) against France (blue) in 1997 was struck with the outside of his left foot.
Free kick takers often curl and put spin on the ball, to curl it over or around the wall of defending players, out of the reach of the goalkeeper. Goalkeepers usually organise walls to cover one side of the goal, and then stand themselves on the other side. Thus, the free kick taker has several choices, from curling the ball around the wall with finesse, to bending the ball around the wall using power, or even going over the wall—although this last lessens the likelihood of scoring from close range.
The 1950s Brazilian star Didi is widely believed to have invented the folha seca technique; however, Italian forward Giuseppe Meazza before him is also credited with using the technique. Today, the knuckleball technique is notably used by modern-day players such as Juninho (whose technique has often been emulated), and Cristiano Ronaldo, who would strike the ball with either no or a low amount of spin, causing it to swerve unexpectedly at a point near the goal.Gareth Bale and Andrea Pirlo are also notable proponents of this technique when taking free-kicks.
Corners
[edit]
Representation of how the Magnus effect affects the ball's movement during corner kick goals
Curling can be an effective technique when taking corners. The ball gradually moves in the air towards the goal. This is referred to as an in-swinging corner. Occasionally, a corner-taker will bend the ball towards the edge of the penalty area, for an attacker to volley, or take a touch and then shoot. Rarely, a goal can be scored directly, this is called an "Olympic goal" and it requires amazing technique and a distraction of the opposing goalkeeper.
Passing
[edit]
Curling can be used in passing. Effective passes from midfield to an attacking player are often the result of a curled pass around the defender, or long cross-field passes are sometimes aided by the addition of curl or backspin. This can be done with either the inside of the foot or outside of the foot. The outside of the foot may be used when a player is facing sideways and wants to use the dominant foot to make a pass; this technique is known as the trivela.
Causes
[edit]
The fact that spin on a football makes it curl is explained by the Magnus effect. In brief, a rotating ball creates a whirlpool of air with itself at its center. Thus, the air on one side of the ball moves in the same direction the ball is travelling in, and the air on the other side moves in the opposite direction. This creates a difference in air pressure around the ball, and it is this sustained difference in pressure which causes the course of the ball to deviate.
The Magnus effect is named after German physicist Heinrich Gustav Magnus, who described the effect in 1852. In 1672, Isaac Newton had described it and correctly inferred the cause after observing tennis players in his Cambridge college.
Notable players
[edit]
Many football players are renowned for their ability to curl or bend the ball when passing or shooting at goal, either from open play or a free kick. These include: Pelé, Didi, Rivellino, Zico, Diego Maradona, Michel Platini, Roberto Baggio, Alessandro Del Piero, Gianfranco Zola, Siniša Mihajlović, Zinedine Zidane, Rivaldo, David Beckham, Roberto Carlos, Juninho, Ronald Koeman, Andrea Pirlo, Ricardo Quaresma, Gareth Bale, Ronaldinho, Thierry Henry, Neymar, Kaká, Miralem Pjanić, Rogério Ceni, Shunsuke Nakamura, Pierre van Hooijdonk, Hristo Stoichkov, Thomas Murg, Luis Chávez, Carlos Vela, Cristiano Ronaldo, Luka Modrić, Giuseppe Meazza, Ángel Di María, Kevin De Bruyne and Lionel Messi, among others.[nb 1]
See also
[edit]
Association football portal
Shooting
Free kick
Corner kick
Bend It Like Beckham (2002).
It also plays a part in the Murdoch Mysteries episode "Bend It Like Brackenreid".
Notes
[edit]
^See
References
[edit]
^Soccerpro (27 December 2019). "How to Curve a Soccer Ball". The Instep. Archived from the original on 12 October 2020. Retrieved 5 October 2020.
^"Twitter Reacts as Roberto Carlos Almost Recreates Famous Banana Free Kick". 90min.com. 11 August 2019. Archived from the original on 29 July 2022. Retrieved 29 July 2022.
^ Jump up to: ab"Bend it like a Brazilian". the Guardian. 19 May 2002. Retrieved 29 July 2022.
^Ronay, Barney (29 June 2018). "Quaresma chooses liberation over conformity with outside of his boot". The Guardian. Retrieved 13 April 2022.
^"How To Score a Dipping Shot | The Ultimate Guide To Shooting With Dip". YouTube. 6 September 2018. Archived from the original on 1 May 2020. Retrieved 30 December 2018.
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^ Jump up to: abcd"Kings of the free-kick"Archived 12 May 2015 at the Wayback Machine. FIFA.com. Retrieved 20 May 2014
^ Jump up to: abBrian Glanville (15 May 2001). "Didi". The Guardian. Retrieved 21 February 2017.
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^"Top 10 Knuckleball Goals". YouTube. 25 March 2017. Archived from the original on 22 March 2019. Retrieved 30 December 2018.
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^"Technical gifts". www.giuseppemeazza.it. Retrieved 26 October 2019.
^Hackett, Robin (13 September 2012). "Giuseppe Meazza: Champion of the world". ESPN FC. Retrieved 26 October 2019.
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^"Ranking the 16 Greatest Free-Kick Takers of All Time". Bleacher Report. Archived from the original on 6 August 2020. Retrieved 15 October 2020.
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^Wilson, Jeremy (19 June 2016). "Gareth Bale's free-kick secrets - and how he is on the brink of history at Euro 2016". The Telegraph. ISSN0307-1235. Archived from the original on 8 October 2020. Retrieved 5 October 2020.
^Patterson, Mark (21 December 2013). "Real Madrid's Gareth Bale Gives a Free Kick Masterclass". Bleacher Report. Archived from the original on 23 January 2014. Retrieved 5 October 2020.
^"Gareth Bale explains why he has altered his free-kicks in Euro 2016". Sky Sports. 20 June 2016. Archived from the original on 9 October 2020. Retrieved 5 October 2020.
^Stycer, Mauricio (11 June 2014). "Mestre de Pirlo, Juninho Pernambucano ensinou italiano a bater falta" (in Portuguese). UOL.
^ Jump up to: abG. Magnus (1852) "Über die Abweichung der Geschosse," Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, pages 1-23.
^Isaac Newton, "A letter of Mr. Isaac Newton, of the University of Cambridge, containing his new theory about light and color," Philosophical Transactions of the Royal Society, vol. 7, pages 3075-3087 (1671-1672). (Note: In this letter, Newton tried to explain the refraction of light by arguing that rotating particles of light curve as they moved through a medium just as a rotating tennis ball curves as it moves through the air.)
^Gleick, James. 2004. Isaac Newton. London: Harper Fourth Estate.
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^"Season 10 - Bend It Like Brackenreid - Murdoch Mysteries"Archived 9 February 2018 at the Wayback Machine. CBC. Retrieved 4 August 2018
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6555 | https://www.youtube.com/watch?v=Rbxb6DOjarI | Find the measure of an angle using three side lengths of a triangles - math help
Brian McLogan
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I show how to solve math problems online during live instruction in class. This is my way of providing free tutoring for the students in my class and for students anywhere in the world. Every video is a short clip that shows exaclty how to solve math problems step by step. The problems are done in real time and in front of a regular classroom. These videos are intended to help you learn how to solve math problems, review how to solve a math problems, study for a test, or finish your homework.I post all of my videos on youtube but if you are looking for other ways to interact with me and my videos you can follow me on the following pages:
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Transcript:
welcome what I want to do today is show you how to find the measure of my indicated angle of this triangle now what we see here is we have three side lengths um and we also have an angle that is what we call a right angle and it's really important that once we have a right angle we notice that we can also use our you know trigonometric properties as well so when using your trigonometric properties it's important for you to understand well what exactly do they mean so let's write them out the first one that we're going to talk about is the tangent the next one is the sign and then we'll talk about the cosine function as well so I'm also going to I wanted to write these out because through the rest of the of the video I'm going to abbreviate uh these three functions so the first one the tangent what that represents if I want to find the tangent of my angle and let's just replace this question mark with x the tangent of X is going to be equal to the ratio of two side lengths of my triangle and the two side lengths for tangent is going to be your opposite over your adjacent the S function of an angle X represents the ratio of your opposite side length over your hypotenuse and the cosine function represents the ratio of your adjacent side over your hypotenuse now you might be asking ask yourself well where are you getting uh hypotenuse adjacent and opposite well I get those by using my uh my right angle and also by using my X so when I look at this and I see my angle x one thing I need to determine is where the hypotenuse is and hopefully if you guys you know work down the Pythagorean theorem you remember the hypotenuse is always your longest longest side of your triangle it's also the side that's directly crossing my hypoten or directly crossing my 90 angle so we can label this side my hypotenuse always now your opposite and your adjacent are always going to kind of um change depending on which angle you have so here's my right angle depending on which angle I choose is going to really depend on which one is my opposite which one my adjacent so if I have this angle X the side that's adjacent is going to be the side length that's going to connect your angle with your 90° angle so therefore I can say that 65 is going to be my adjacent side therefore by proc elimination or one way to always look at it is your opposite is always the side length that's always opposite of your original angle so we can also say that 72 is going to be my opposite now for this problem we only need to choose one function that we'll work with but since they gave us all three sides I want to show you that it doesn't matter which function you choose either one you pick you're going to pick the right one as long as you make sure you plug in the correct values so I'm just going to do all of them for you because I like you so much that I really want to make sure you understand that it doesn't matter which function you use you're still going to get the exact answer as long as you use the functions correctly so the tangent of X means the opposite which is 72 over my adjacent which is 65 s of X is going to be the opposite over my hypotenuse which is 72 over 97 and therefore the cosine of x is my adjacent side 65 over my hypotenuse 97 now it'll be helpful for us to look at some of the decimals for these problems and when you're using the decimal I want you to use the full decimal in your calculator I don't want you to abbreviate but I'm just going to write down a little bit abbreviation decimal so you can understand that each one of these ratios are different so the tangent of X is going to be let's see we'll have 72 / 655 and I'm just going to abbreviate this like I said 1.1 uh 07 all right the S of X let's 16 s of X is going to equal 72 / 97 which is 74 22 and the cosine of x equal 65 / 97 which is 671 all right so notice my ratios are all going to be the same but what I want to do remember we want to find the value of x so right now we're taking the tangent of X and we get that we're taking the S of X or we're taking the cosine of x so if I want to find exactly what the value of x is I need to undo my functions I need to undo the tangent function undo the sign function undo the cosine function so how do we undo those functions well thankfully for our calculator it's what we call the inverse function so the inverse tan undoes tan the inverse cosine is the inverse function of cosine and the inverse sign is the inverse function for S so what I'll write is so if I take the inverse tan of what my value is I'm going to get my Angle now like I said I want to make sure that you guys put them in correctly because I really do not want you to abbreviate so I'm going to do 72 ID 65 and then I'm going to take the inverse tangent of that answer and what I get is 47.9 and I'm going to round that to 48° okay now when I do the inverse sign inverse sign not of X but of 74 22 so I'll do 72 / 977 and then I'm going to take inverse sign of second answer of that answer and again I get 47.9 which I'll round to 48° and lastly I'll do Co inverse cosine of 6701 and so I have 65 ID 97 and then inverse cosine of that full answer is going to give me again 47.9 so what you can I'll just round to 48° so what you'll notice ladies and gentlemen is as long as you understand what each one of these functions are and how to plug them in then to use which uh function you want to use and then use the inverse you'll understand that it doesn't matter which function you have but it will usually mentioned by your problem by what op by what values they give you but hopefully you guys can see no matter which function I chose I got the missing value from my angle was 48° |
6556 | https://www.youtube.com/watch?v=5GK5AL01TI0 | Tangents on a circle Geogebra [Tutorial]
Svijet Matematike
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Tangents on a circle Geogebra [Tutorial]
Tangente na kružnicu Geogebra [Tutorial]
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Transcript:
[음악] [음악] k [음악] [박수] [음악] [박수] [음악] k [박수] [음악] [음악] k |
6557 | https://pmc.ncbi.nlm.nih.gov/articles/PMC7020281/ | Intravesical Therapy for Non-muscle Invasive Bladder Cancer—Current and Future Options in the Age of Bacillus Calmette-Guerin Shortage - PMC
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Intravesical Therapy for Non-muscle Invasive Bladder Cancer—Current and Future Options in the Age of Bacillus Calmette-Guerin Shortage
Vishnukamal Golla
Vishnukamal Golla, MD, MPH
1 Department of Urology, The David Geffen School of Medicine at University of California, Los Angeles, Los Angeles, CA
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1, Andrew T Lenis
Andrew T Lenis, MD, MS
1 Department of Urology, The David Geffen School of Medicine at University of California, Los Angeles, Los Angeles, CA
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1, Izak Faiena
Izak Faiena, MD, MS
1 Department of Urology, The David Geffen School of Medicine at University of California, Los Angeles, Los Angeles, CA
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1, Karim Chamie
Karim Chamie, MD, MSHS
1 Department of Urology, The David Geffen School of Medicine at University of California, Los Angeles, Los Angeles, CA
2 Jonsson Comprehensive Cancer Center, University of California, Los Angeles, Los Angeles, CA
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2 Jonsson Comprehensive Cancer Center, University of California, Los Angeles, Los Angeles, CA
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PMCID: PMC7020281 PMID: 32071562
Abstract
Non-muscle invasive bladder cancer (NMIBC) is a common and burdensome malignancy. A substantial proportion of patients with intermediate- and high-risk disease will progress to invasive bladder cancer and are at a significant risk for metastasis and death. Bacillus Calmette-Guerin (BCG) therapy for selected cases has been the standard of care for nearly 40 years. Unfortunately, a world-wide shortage has made BCG challenging to obtain. Furthermore, recurrences and progressions do occur. With the US Food and Drug Administration creating a clear path to drug approval for novel treatments, many therapies have been tested, including intravesical cytotoxic chemotherapy, intravesical immunotherapy, systemic immunotherapy, and novel agents, such as gene therapy and targeted therapy. In this review, we highlight ongoing clinical trials.
Keywords: Bladder cancer, Intravesical treatment, Non-muscle invasive bladder cancer, Immunotherapy, BCG shortage
In the United States, there will be an estimated 81,000 new cases of bladder cancer in 2019, with >70% presenting as non-muscle invasive bladder (NMIBC).1,2 Although NMIBC usually carries a favorable prognosis, there is a high risk of disease recurrence and a 10% to 20% risk of progression to muscle-invasive disease.3 The common treatment for intermediate- and high-risk patients is a transurethral resection followed by intravesical therapy with bacillus Calmette-Guerin (BCG), a non-specific immunotherapy that has remained the gold standard for 40 years. This therapy, although shown to reduce tumor recurrence and progression, still has a 50% risk of failure.4 BCG failure has been variably defined in the literature and likely represents a heterogeneous group of patients. Nevertheless, these patients are at a high risk of progression to invasive and metastatic disease. The most definitive therapy with the greatest chance of cure is radical cystectomy with urinary diversion.5 However, there is a significant interest and need for organsparing treatments for this cohort of patients with limited non-invasive options.
New intravesical and systemic agents for all risk categories of NMIBC are now being evaluated in clinical trials. There has been an explosion in the number of early and late-phase clinical trials for patients who fail BCG. There are several reasons for this invigorated interest in the BCG failure space. First, only one drug, valrubicin-with admittedly low efficacy-is approved by the US Food and Drug Administration (FDA) for patients who fail BCG, making this a highly unmet medical need. Second, a worldwide shortage of BCG has limited the numbers of patients receiving adequate induction and maintenance, resulting in higher recurrence and failure rates. Finally, the FDA has underscored the urgency of this need and set a well-defined threshold for approval of novel therapies that does not require a randomized trial. Consequently, several drugs have shown promise in this space and will likely change the landscape of management of BCG failure.
Methodology
We conducted a literature search using clinicaltrials.gov, Ovid MEDLINE, and PubMed databases that included the following search terms: non-muscle invasive bladder cancer, BCG failure, intravesical therapy, immunotherapy, targeted therapy. We identified literature as recent as March 1, 2018. Recent abstracts from the American Urological Association (AUA), Society of Urologic Oncology (SUO), and the American Society of Clinical Oncology (ASCO) annual meetings were reviewed, and relevant abstracts from articles published in English were included.
Current Guideline-recommended Strategies
Initial Endoscopic Resection and Risk Stratification
The standard approach for initial treatment of NMIBC is a transurethral resection of a bladder tumor (TURBT). A visually complete TURBT provides critical staging information and surgeon experience is known to affect the rates of complete resection.6 The impact on treatment and overall prognosis is significant enough that both the AUA/SUO and European Association of Urology (EAU) guidelines recommend restaging TURBT in select patients.7 Several checklists and bladder maps are available and have been studied to help improve the quality of TURBT.7 Furthermore, a visually complete TURBT is known to aid the efficacy of future intravesical therapies. Therefore, the importance of the quality of TURBT cannot be overemphasized.
Although the definitions differ slightly, both the AUA/SUO and EAU guidelines recommend risk classifying patients into low, intermediate, and high risk at the time of each endoscopic resection in order to guide further therapy and surveillance (Table 1).7 For low-risk patients, a single dose of intravesical chemotherapy within 24 hours is recommend. Traditionally, mitomycin-C (MMC) has been used. However, adherence to this recommendation has been notoriously poor.8 Recently, intravesical gemcitabine has been shown to improve recurrence-free survival to rates like those seen with MMC, although no randomized comparison data are available.9 For intermediate-risk disease, induction with either intravesical chemotherapy (eg, MMC) or immunotherapy (eg, BCG) and 1 year of maintenance therapy is recommended. For high-risk disease, guidelines recommend induction BCG followed by 1 to 3 years of maintenance BCG (EAU) or 3 years of maintenance BCG (AUA/SUO).
TABLE 1.
AUA NMIBC Risk Stratification
| Low Risk | Intermediate Risk | High Risk |
---
| Single low-grade Ta lesion ≤3 cm | Recurrence within 1 y; low-grade Ta | Any recurrent high-grade Ta |
| Papillary urothelial neoplasm of low malignant potential | Single low-grade Ta >3 cm | High-grade T1 |
| | Low-grade Ta, multiple lesions | Carcinoma in situ |
| | High-grade Ta ≤3 cm | High-grade Ta >3 cm or multiple lesions |
| | Low-grade T1 | BCG failures in high-grade patient |
| | | Variant histology; lymphovascular invasion; high-grade prostatic urethral invasion |
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Adapted from Chang SS et al.7
AUA, American Urological Association; BCG, bacillus Calmette-Guerin; NMIBC, non-muscle invasive bladder cancer.
Adequate BCG Therapy and Definitions of Failure
BCG is debatably the most effective immunotherapy for any cancer. First used for bladder cancer over 40 years ago by Morales and colleagues, the mechanism of action is still incompletely characterized but functions via the innate and adaptive immune system to illicit a significant local inflammatory response, resulting in tumor-cell killing.10 The BCG treatment schedule consists of 6 weekly instillations, termed induction therapy, followed by 3 weekly instillations, termed maintenance therapy, at 3 and 6 months and then every 6 months thereafter for 3 years, totaling 27 instillations. BCG therapy results in complete response rates up to 66% in high-risk patients.11 However, complete courses of BCG are rare, as only 16% of patients in the pivotal SWOG study were able to complete all 27 instillations and recent estimates suggest that the real-world number is significantly lower.12 Therefore, adequate BCG therapy has been defined, which also aids in homogeneity for clinical trial purposes. Adequate BCG is at least 5 of 6 instillations of induction therapy and at least 2 of 3 maintenance instillations for at least 6 months.7
BCG failure has been variably defined in the literature and contributes to some of the heterogeneity in outcomes seen with different therapies. Currently, BCG failure is subcategorized into refractory, relapsing, unresponsive, or intolerant disease (Table 2).13,14BCG refractory disease is persistent high-grade disease at 6 months following adequate induction and maintenance, or any stage and/or grade progression at 3 months following induction. BCG relapsing disease is high-grade disease after a disease-free interval of ≥6 months after adequate BCG therapy. BCG unresponsive disease is defined as BCG refractory and BCG relapsing disease. These patients are at the highest risk of progression. Finally, BCG intolerance includes any cessation of therapy secondary to side effects. Classification of BCG failure is critically important to adequately understand the prognosis of disease and accurately characterize patient cohorts in clinical trials.
TABLE 2.
BCG Failure Definitions
| BCG Refractory Subcategories | Definition |
--- |
| BCG refractory | Persistent high-grade disease at 6 mo following adequate induction and maintenance, or any stage and/or grade progression at 3 mo following induction |
| BCG relapsing | High-grade disease after a disease-free interval of ≥6 mo after adequate BCG therapy |
| BCG unresponsive | BCG refractory and BCG relapsing disease; these patients are at the highest risk of progression |
| BCG intolerance | Includes any cessation of therapy secondary to side effects. |
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BCG, bacillus Calmette-Guerin.
Ultimately, all BCG failures should seriously consider radical cystectomy as the most effective form of salvage therapy. However, many patients will opt for bladder-sparing treatment options and will sacrifice some efficacy to preserve their native bladders. Valrubicin is currently the only drug approved by the FDA for salvage intravesical therapy in patients who fail BCG with carcinoma in situ (CIS).15 This is despite data showing a meager 20% or less of patients remaining recurrence-free at 12 months and 8% of patients after 12 months.16
BCG Shortage and Its Implications
BCG is a live attenuated strain of Mycobacterium bovis initially used to vaccinate against tuberculosis but ultimately found its way as an intravesical immunotherapy for bladder cancer in the late 1970s. Despite its gold standard status and widespread success in the treatment of intermediate- and high-risk NMIBC, several controversies have arisen specifically regarding the economics of BCG. Currently there is a critical shortage of this life-saving treatment due in part to the lengthy and costly manufacturing process, a shrinking number of FDA-approved companies producing the drug, possible financial disincentives, and increased demand.
BCG is notoriously challenging to produce. The bacteria must be fermented in approved manufacturing laboratories and the process is almost unchanged since its inception in 1921.17 Production for a single batch can take up to 3 to 4 months. Furthermore, given this multistep and often unpredictable process, irregularities can occur that affect the yield and quality of the bacteria. Finally, the sheer volume of BCG required for the treatment doses used in bladder cancer is astronomical compared with volumes used for vaccination doses.18 Compiled on top of this challenging production process, several manufacturing setbacks have created shortages over the past decade. In 2012, Sanofi Pasteur Limited discontinued production of ImmuCyst®—the Connaught substrain and the most commonly used strain of BCG in the United Kingdom and the United States—secondary to a flood in a manufacturing area in Toronto, Canada.19 Plagued by continued production issues, Sanofi had to completely shut down production of BCG by the middle of 2017. Subsequently, OncoTICE®, the Merck strain of BCG, became the most widely available form but met its own challenges with increased demand. Fast-forward to 2019 and we are on the precipice of another BCG shortage that again will have far-reaching consequences for patients with NMIBC.
In response to the current critical shortage, a joint statement from leading US-based urology associations was released in February 2019 that contained several proposed strategies to help minimize the impact of the BCG shortage on patients with NMIBC.20 The strategies are presented in Table 3. Most importantly, these strategies are at the discretion of the treating physician and should be made on a case-by-case basis, tailored to the patient and their disease.
TABLE 3.
Strategies to Minimize BCG Use During Shortages
Avoid BCG in patients with low-risk disease.
Intravesical chemotherapy is the first-line option for patients with intermediate-risk NMIBC.
Utilize alternative intravesical chemotherapy rather than BCG for patients who would normally receive BCG as a second-line therapy for intermediate-risk NMIBC.
For high-risk NMIBC, high-grade T1 and CIS patients undergoing induction therapy should be prioritized for full-strength BCG. If not available, then employ reduced 1/2 to 1/3 dose.
If the BCG stocks available for maintenance therapy for NMIBC patients, attempt 1/3 dose BCG and limit doses to 1 year.
During a BCG supply shortage, do not utilize maintenance therapy and limit induction BCG to BCG-naive patients with high-risk disease.
If no BCG is available, then physicians should consider mitomycin for induction and maintenance up to 1 year. Other options include gemcitabine, valrubicin, epirubicin, docetaxel, sequential gemcitabine/docetaxel, and/or mitomycin.
Patients with high-risk features who after shared decision making are not willing to accept alternative intravesical agents should be offered initial radical cystectomy.
When a 1/2 to 1/3 dose is used, maximize resources by treating multiple patients in the same day to avoid wasting BCG; coordinate with pharmacy prior to split dosing.
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BCG, bacillus Calmette-Guerin; CIA, carcinoma in situ; NMIBC, non-muscle invasive bladder cancer.
At the time of this writing, it is still unclear how split-vial dosing of BCG will be billed. The AUA is currently working with the Centers for Medicare & Medicaid Services (CMS) to determine the next step. Therefore, it is important for physicians to discuss with the insurance company prior to split dosing.
Adapted from American Urological Association.20
Therapeutic Options and Clinical Trials
Low-risk NMIBC
Perioperative Adjuvant Therapies.
Both AUA/SUO and EAU guidelines recommend a single perioperative dose of intravesical chemotherapy, generally within 24 hours, following endoscopic resection of presumed low-risk papillary NMIBC. Based on the meta-analyses of several clinical trials, adjuvant chemotherapy has a relative risk reduction of approximately 0.50 and an absolute risk reduction of 10% to 15%.3 Multiple institutional and population-level studies have demonstrated poor compliance with this guideline recommendation.21 Attributed to relative contraindications for its use and the potential for serious side effects, MMC has largely fallen out of favor. Recently, a randomized trial compared perioperative intravesical gemcitabine with placebo and showed a relative and absolute risk reduction of recurrence like that seen with MMC.9 Other potential adjuvant therapies included apaziquone, an alkylating prodrug, that failed to demonstrate a significant improvement in recurrence-free survival in two phase 3 randomized trials.22 Currently, another trial using apaziquone is being conducted (NCT03224182) and results are forthcoming.23
Primary Chemoablative Therapies.
An interesting novel area of research is primary chemoablation for low- and intermediate-risk NMIBC. Ablative therapies are ideal for patients who may otherwise be unable to tolerate in-office fulguration or more invasive procedures in the operating room secondary to medical comorbidities. A phase 2 (NCT02070120) and phase 3 (NCT03348969) trial are evaluating the role of MMC as a chemoablative agent. In the phase 2 CALIBER trial, preliminary results presented at the AUA Annual Meeting in 2018 demonstrate complete response in 37% of patients.24 In the phase 3 trial, investigators are evaluating biomarkers of chemotherapeutic sensitivity in patients to guide further treatment. Results are forthcoming.25 In data presented by our group at the AUA Annual Meeting in 2017, in collaboration with UroGen Pharma, a reverse thermosensitive hydrogel formulation of MMC was able to yield a complete response rate of 73% after a 6-week induction course and a 61% recurrence-free survival at 12 months without any maintenance therapy.26 Currently, the OLYMPUS Phase II study is open for patients with intermediate-risk disease.
Intermediate- and High-risk NMIBC
Therapy for BCG-naive Patients.
Currently, guidelines for intermediate-and low-risk patients for adjuvant therapy to prevent recurrence and progression clearly recommend induction BCG followed by 1 to 3 years of maintenance. Alternative strategies to curb the shortage of BCG as described above are exceptions. Nevertheless, novel therapies are needed to fill gaps left by a limited supply of BCG. First, additional strains of BCG used in other countries should be evaluated for potential approval by the FDA. To this end, SWOG S1602, led by Dr. Robert Svatek, seeks to evaluate the Japanese Tokyo strain against the TICE strain, as well as the effect of priming with intradermal BCG.27 Approval of an additional strain could significantly curb the shortage.
Several systemic agents are making their way into the BCG-naive space, although they are primarily being evaluated in the higher-risk cohort of patients who fail BCG. These clinical trials will evaluate pembrolizumab (NCT03504163,28NCT0316715129) and durvalumab (NCT0352869430). Novel agents in clinical trials for patients who fail include an intravesical typhoid vaccine call Ty21a (NCT0342123631), intravesical TSD-001 (NCT0308185832), vesigenurtacel-L (NCT0201020333), and tamoxifen citrate (NCT0219789734).
Therapy for Patients Who Have Failed BCG
Patients who have failed BCG, particularly patients in the BCG unresponsive group, have the highest rates of disease progression and death from bladder cancer. Furthermore, patients in this cohort have few additional options. Therefore, most clinical trials target this cohort of patients.
Intravesical Chemotherapy. Intravesical chemotherapy has been shown to have promising results in the treatment of NMIBC. However, combination intravesical therapy in patients who have failed BCG therapy has been shown to be even more fruitful. Initial preclinical studies used a genetically engineered mouse predisposed to progression from carcinoma in situ (CIS) to metastatic bladder cancer. In one study, they compared the results of instilling intravesical cisplatin, gemcitabine, and/or docetaxel alone or by combining two agents.35 They found that gemcitabine (pyrimidine analog that inhibits cell growth and induces apoptosis) was the single most effective agent at preventing progression to invasive disease. However, a combination of two agents, especially those that included gemcitabine, was significantly more effective at reducing tumor burden. A multi-institutional study investigating the efficacy of sequential intravesical gemcitabine and docetaxel showed recurrence free rates of 48% at 1 year and approximately 40% at 2 years.36 The successes observed in these studies was the impetus for a phase 1 trial investigating the safety profile of an intravesical triple agent salvage chemotherapy consisting of cabazitaxel (a taxane that inhibits microtubule polymerization), gemcitabine, and cisplatin. This regimen was used in patients who were BCG refractory or who had recurrent high-risk NMIBC who refused a cystectomy. Early results, published in 2017, showed a relatively benign safety profile in 8 patients with 4 patients experiencing only a grade 2 toxicity.37 Importantly, all patients were able to complete 6 weeks of induction therapy. At 6 weeks, 7 out of 8 patients had a complete response to treatment.
Systemic Immunotherapy. The realization that bladder cancer is highly sensitive to immune checkpoint inhibition (ICI) has revolutionized the treatment of advanced and metastatic disease. Sensitivity to immunotherapy, including ICI, stems from the high mutational burden of bladder cancer.38 Utilization of these therapies has moved proximal from metastatic and advanced disease and are now being evaluated in NMIBC. Briefly, programmed cell death protein-1 (PD-1) is a checkpoint protein found on T cells that acts like an off switch to prevent immune cells from attacking other cells. PD-1 is activated when it attached to program death ligand-1 (PD-L1), which is found on both normal and malignant cells.39 Therefore, malignant cells with high PD-L1 expression can evade the immune system. Specifically, in bladder cancer, higher levels of PD-L1 expression has been shown to correlate to higher grade tumors, worsened survival, and increased risk of recurrence.40,41 PD-L1 tumor cell expression was even associated with increased resistance to BCG therapy.42
There are several clinical trials investigating the role of these checkpoint inhibitors for NMIBC, four of which are investigating the role of pembrolizumab in NMIBC. The first trial combines both intravesical pembrolizumab and BCG in patients with high-risk or BCG-refractory NMIBC.43 This study is currently recruiting patients with an estimated enrollment of 27 and an estimated study completion date of February 2020. The second clinical trial compares intravenous and intravesical pembrolizumab in patients with intermediate-risk recurrent NMIBC.29 This clinical trial has an estimated enrollment of 36 patients and an estimated study completion date of August 2021. The third clinical trial, KEYNOTE 057, is a phase 2 trial for patients with high-risk NMIBC unresponsive to BCG.44 In the first cohort to yield preliminary results, patients with CIS, with and without papillary disease, had a complete response rate of 40% at 3 months and 53% of these complete responders have a median duration of response of 12.7 months. Additionally, the adverse events in this cohort were manageable with only 12% to 15% of patients having a grade 3 or 4 adverse event.45 There are currently 260 patients enrolled in this study with an estimated completion date of July 2023. Ongoing is a randomized, controlled phase 3 trial called KEYNOTE 676, which is testing pembrolizumab in combination with BCG versus BCG alone in patients refractory to BCG induction, which will further test this agent's role in high-risk BCG unresponsive NMIBC. There are currently 550 patients enrolled in this study with an estimated completion date of November 2024.46
Other checkpoint inhibitor agents in clinical trials are atezolizumab and durvalumab. Two clinical trials investigate intravenous atezolizumab either in combination with BCG or alone for patients with recurrent BCG unresponsive NMIBC.47 Only one of these trials (NCT02844816) is actively recruiting patients and has 162 patients enrolled and an estimated completion date of February 2020. The clinical trial investigating IV durvalumab is a phase 2 study for patients with BCG refractory urothelial CIS with 34 patients enrolled and a study completion date of December 2021.48 Another clinical trial will evaluate the use of intravesical durvalumab for patients with BCG-refractory NMIBC is actively recruiting patients.49 Finally, POTOMAC is a phase 2, randomized, multicenter study investigating durvalumab either with BCG or alone in high-risk, BCG-naive NMIBC with 975 patients currently enrolled and an estimated completion date of November 2024.50
ALT-803, a product of the Altor BioScience corporation, has a received Fast Track designation from the FDA. Interleukin-15 (IL-15) is a protein that induces the proliferation of natural killer cells and thereby enhances the anti-tumor immunity of cytotoxic T-cells.51,52 ALT-803 is a recombinant IL-15 superagonist complex found to have superior anti-tumor activity compared with the native IL-15 molecule.53 This drug has been found in preclinical studies to have a rapid and durable responses against multiple malignancies and virally infected cells, allowing for a potential role in targeted treatments and vaccines.52,54 As a result of these successes, ALT-803 is being investigated in a phase 1b/2 trial in combination BCG in BCG-naive patients with high-grade NMIBC. A late-breaking abstract of the phase 2 registration study at the Society of Immunotherapy for Cancer showed significant response at 3 months.55 A second phase 2 trial is studying intravesical ALT-803 in combination with BCG for BCG unresponsive NMIBC patients.56
Gene Therapy. Nadfaragene firadenovec/Syn3 (rAD-IFN/Syn3; formerly instiladrin) is another agent that was recently awarded Fast Track designation by the FDA. Interferon (IFN) has been shown in several studies to have promising results in NMIBC; however, it was ineffective due to limited contact with the bladder urothelium.57 rAD-IFN/Syn3 is a recombinant adenovirus vector that results in INF expression. This recombinant adenovirus (rAd)-mediated IFNa2b protein is co-administered with a polyamide surfactant, Syn3, that improves adenoviral transduction into the urothelial lining. Once inside the urothelial cells, the a2b gene is translated by the cells' internal DNA machinery, resulting in the secretion of high quantities of interferon a2b protein, thereby enhancing the body's natural cancer defense system. The initial phase 1 trial investigated this agent in NMIBC patients unresponsive to BCG therapy and showed that 43% of patients with IFN-α detectable in the urine had a complete remission at 3 months with durable response for 31 months.58 Additionally, patients tolerated this agent well. The phase 2 studies showed that 35% of BCG nonresponders were disease-free at 1 year.58 There is currently a phase 3 study investigating the safety and efficacy of a high dose of rAD-IFN/Syn3 in the same patient population.58
CG0070 is an oncolytic adenovirus that preferentially replicates in a deregulated retinoblastoma pathway found commonly in bladder tumor cells. The initial phase 1 study demonstrated a benign safety profile and patients who were BCG unresponsive or refused cystectomy had a complete response (CR) rate of 48.6% and a median duration of complete response of 10.4 months.4 The phase 2 study showed an overall CR rate at 6 months of 50% for patients with CIS.4 Additional interim analysis showed a CR of 30% at 12 months.59
Targeted Therapy. Vicinium is recombinant fusion protein consisting of an epithelial cell adhesion molecule (EpCAM)-specific anti Targeted Therapy. Vicinium is recombinant fusion protein consisting of an epithelial cell adhesion molecule (EpCAM)-specific antibody fragment fused to a pseudomonas exotoxin. This drug targets the EpCAM antigen found to be overexpressed in urothelial tumor cells.60 Once vicinium binds to the EpCAM antigen on the tumor cells, the drug is internalized and irreversibly blocks proteins synthesis, inducing cell death.61 This drug was evaluated for patients with high-grade, BCG-unresponsive NMIBC, in phase 1 and 2 studies and had an excellent safety profile.62 In CIS patients, the complete response rate at 3 months was 40% in a phase 2 trial.62 This trial is currently in phase 3 of evaluation with interim results showing a 42% complete response in CIS patients and only 4% of patients with a grade 3 or higher adverse event.63 There is also a phase 1 clinical trial that is investigating the combination of durvalumab and vicinium for NMIBC patients intolerant or unresponsive to BCG.64 Preclinical work with an earlier form of vincinium demonstrated an abscopal effect and synergy when used with an ICI, which was the impetus for this particular trial. There are currently 40 patients in this trial with an estimated study completion date of July 1, 2021.64
Conclusions
NMIBC is a common and potentially lethal cancer. Unfortunately, the mainstay of treatment, BCG is in short supply with a dwindling number of producers. Several novel agents have made it to Fast Track status from the FDA and are pending approval. Novel agents include various other combinations of different cytotoxic therapies, systemic ICI, intravesical ICI, and other agents, such as gene therapy and targeted therapy. In the future, ideal treatments would be tailored to the specific somatic mutations present within the tumor.
Main Points.
The current mainstay of treatment for non-muscle invasive bladder cancer, bacillus Calmette-Guerin (BCG), is in short supply with a dwindling number of producers. It is paramount to explore alternative therapies.
Novel agents include various other combinations of different cytotoxic therapies, systemic immune checkpoint inhibition (ICI), intravesical ICI, and other agents, such as gene therapy and targeted therapy.
In the future, ideal treatments would be tailored to the specific somatic mutations present within the tumor.
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Tracking inheritance of human mitochondrial DNA
Mother’s age, a bottleneck, and selection affect how disease-causing mutations persist
November 18, 2019
By Gail McCormick
UNIVERSITY PARK, Pa. — New insight into how genetic information stored in human mitochondria is passed from one generation to the next could have important implications for genetic counseling of women planning pregnancies, according to a study by researchers at Penn State and the University of California, Berkeley. The findings are especially relevant to women carrying disease-causing mutations within their mitochondrial DNA and those suffering from mitochondrial diseases.
Mitochondria are cellular subunits that produce energy, and mitochondrial diseases are among the most common genetic disorders. They occur in one in 5,000 people and can cause a wide range of health problems, ranging from fatigue to developmental disabilities to seizures to compromised heart function and poor vision.
One in eight women in the general population is a carrier of mitochondrial disease. These women carry copies of their mitochondrial DNA that contain disease-causing mutations that can be transmitted to their offspring, but do not suffer from the disease themselves. Predicting the chances that female carriers will transmit disease-causing mitochondrial DNA to their children has been challenging.
Mitochondrial DNA is separate from the rest of the cell’s DNA, which is located in the nucleus. Unlike the nuclear DNA that is inherited from both parents, mitochondrial DNA is inherited from the mother. Each mitochondrion carries two to 10 copies of the mitochondrial DNA, which may vary slightly due to mutations — replacements, insertions or deletions of DNA letters — which in some cases lead to mitochondrial disease. Whether or not a mutation results in a mitochondrial disease, and the severity of the disease, often depends on the proportion of an individual’s mitochondrial DNA that carries the mutation.
The new study, which appears online the week of Nov. 18 in the journal Proceedings of the National Academies of Sciences, reveals how mitochondrial DNA is transmitted between generations. The study also found that children born to older mothers carry more mitochondrial mutations than children born to younger mothers. This could have important implications for inheritance of disease-related mutations.
“Some tissues with high energetic demands — such as muscle and brain — have hundreds to thousands of mitochondria and thus thousands of mitochondrial DNA copies,” said Kateryna Makova, Pentz Professor of Biology at Penn State and the co-corresponding author of the study. “Mitochondrial DNA is inherited from the mother, but only a small number of these copies actually get passed on in each egg. That means that at some point, there is a reduction in the number of mitochondrial DNA between generations, causing what biologists call a bottleneck, which can reduce the amount of genetic diversity passed on from mother to child. Understanding this bottleneck, when it happens, and how many mitochondrial DNA get through, could improve our understanding of how disease-related mutations get passed on.”
The research team found that there is a drastic bottleneck of mitochondrial DNA that takes place during oogenesis, the process by which egg cells are created, reducing thousands of copies of mitochondrial DNA down to only about seven to 10. Like shaking colored beads out of a bottle with a narrow neck, where the few beads that get through might not reflect the overall composition of the bottle, the mitochondrial DNA that gets passed on to the offspring may not have the proportion of disease-related mutations present in the mother.
“The size of the bottleneck affects the probability that a child inherits a disease-causing mutation from her or his mother,” said Peter Wilton, postdoctoral researcher at the University of California, Berkeley, and a co-first author of the study. “We found that, because of the narrow bottleneck, mutations in maternal mitochondrial DNA that were not very frequent were often lost within one generation — they were not present in the children. But those that were present at a higher proportion in the mother were more likely to get passed on, and in some cases may persist for up to five to 10 generations.”
To accomplish this study, the research team relied on a collaboration with co-author Dr. Ian Paul, a pediatrician at the Penn State Milton S. Hershey Medical Center, who oversaw sample collection from 345 individuals in 96 multi-generational families. Families included at least two siblings, their mother, and in some cases the mother’s mother and additional generations.
Because the team studied mothers and multiple children in each family, they were able to infer when during oogenesis this bottleneck occurs. If the bottleneck occurs before cells split into separate lineages that will eventually become the eggs that make each child, then siblings should have a similar composition of mitochondrial DNA — the same seven to 10 copies of mitochondrial DNA that made it through the bottleneck. If the bottleneck instead occurs after these lineages have separated — with each lineage independently reducing the amount of DNA during the bottleneck — then siblings should have very different compositions of mitochondrial DNA, which is what the researchers found.
“Knowing the size and timing of the bottleneck has important implications for genetic counseling,” said Makova. “Because the bottleneck occurs separately in each child’s cell lineage, a mother that is a carrier for a mitochondrial disease could pass on a disease-related mutation to one child but not the other. Now that we know when the bottleneck occurs, we hope to investigate oogenesis at a molecular level in animal models to get a better understanding of the process.”
The study also suggests that egg precursor cells with high proportions of disease-causing mutations might not make it to the next generation because of the evolutionary process called natural selection.
“Selection against disease-causing mutations likely occurs during oogenesis, and might be facilitated by the bottleneck,” said Arslan Zaidi, postdoctoral researcher at Penn State at the time of the study and the co-first author. The third co-first author with Wilton and Zaidi was Marcia Shu-Wei Su, postdoctoral researcher at Penn State at the time of the study.
Importantly, children born to mothers that gave birth later in life had more mutation-carrying copies of their mitochondrial DNA as compared to younger mothers.
“All of the eggs are essentially fully developed when a girl is born, so they sit dormant for a long time,” said Zaidi. “Our results tell us that something is happening within these dormant eggs, for example their mitochondrial DNA might be dividing, which provides an opportunity to acquire mutations. Age may also play a role in the probability that a child might inherit a disease-causing mutation.”
In addition to Makova, Wilton, Paul, Zaidi and Su, the research team includes Barbara Arbeithuber, Kate Anthony and Anton Nekrutenko at Penn State, and Rasmus Nielsen at the University of California, Berkeley. Funding for the work was provided by the National Institutes of Health. Additional funding was provided by the Office of Science Engagement, the Eberly College of Science, the Huck Institute of the Life Sciences and the Institute for CyberScience at Penn State, as well as, in part, under grants from the Pennsylvania Department of Health using Tobacco Settlement and CURE Funds. The department specifically disclaims any responsibility for any analyses, responsibility or conclusions.
Last Updated June 6, 2021
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6559 | https://miro.com/templates/venn-diagram-for-learning-and-education/ | Venn Diagram for Learning and Education
A Venn diagram is a graphical representation that shows similarities and differences between items, concepts or data.
Overview
A Venn diagram is a graphical representation that shows similarities and differences between items, concepts or data. They utilize shapes, often circles, to illustrate relationships and compare data.
Venn diagrams are certainly one of the most well-known types of diagrams. This design, featuring a series of simple circles or ellipses or rectangles that overlap is ubiquitous. The overlapping region of a Venn diagram indicates similarities while others showcase differences. Although the Venn diagram was created to describe mathematically logical relationships, itâs now used to show all types of relationships.
Uses of Venn Diagrams in Education and Learning
Venn diagrams are powerful visual tools used in education and learning to represent relationships between different sets of information. They consist of overlapping circles, each representing a set, with the overlaps illustrating the commonalities between the sets. Here are several key uses of Venn diagrams in the educational context:
1. Concept Comparison and Contrast
One of the primary uses of Venn diagrams is to help students compare and contrast concepts, ideas, or objects. By placing different characteristics in separate circles and common attributes in the overlapping areas, students can easily visualize similarities and differences. This method enhances critical thinking and helps in organizing thoughts more systematically.
2. Classifying Information
Venn diagrams are useful for classifying information into distinct categories. This is particularly helpful in subjects like science, where students need to categorize living organisms, chemical elements, or geological formations based on specific characteristics.
3. Analyzing Relationships
They assist in analyzing complex relationships between multiple sets of data. This analysis can be crucial in subjects such as mathematics, where students learn about set theory and probability.
4. Problem Solving
In problem-solving scenarios, Venn diagrams help break down problems into manageable parts, making it easier to identify solutions. They allow students to visually organize information, which can lead to a better understanding and resolution of the problem.
5. Curriculum Development
Educators use Venn diagrams to develop curricula by mapping out overlaps between different subject areas. This helps in creating interdisciplinary lessons that integrate concepts from multiple fields of study.
6. Enhancing Memory and Recall
Visual aids like Venn diagrams enhance memory retention and recall by providing a clear and concise way to display information. The visual format makes it easier for students to remember relationships between different pieces of information.
7. Facilitating Discussions and Debates
Venn diagrams serve as effective tools for facilitating classroom discussions and debates. They provide a structured way for students to present and organize their ideas and arguments.
8. Promoting Collaborative Learning
In group activities, Venn diagrams encourage collaborative learning by allowing students to collectively brainstorm and organize their thoughts. This fosters teamwork and enhances communication skills.
Three examples from the above areas have been created to help you understand the application of the Venn diagram in the Education and Learning domain.
Good Luck
Khawaja Rizwan
Get started with this template right now.
Target Audience Template
Works best for:
Marketing, Desk Research, Prioritization
Understanding your target audience is vital to business success. How can you market yourself effectively if you donât know who youâre targeting? Using the Target Audience template, you can review valuable data about who your customers are and what they want from your product or service.
Gantt Chart for Project Scheduling
Works best for:
Planning, Strategy
Gantt Chart for Project Scheduling helps you plan and visualize project timelines. It enables you to manage tasks, deadlines, and dependencies, ensuring efficient project execution. Ideal for project managers looking to streamline their scheduling process.
Fishbone Diagram for Marketing
Works best for:
Fishbone diagram
Identifying the root causes of marketing challenges is essential for effective strategy development. The Fishbone Diagram for Marketing template helps you systematically explore factors affecting your marketing efforts. Categorize potential causes into areas such as market research, product positioning, promotional strategies, and distribution channels. This structured analysis enables your team to pinpoint and address issues, optimizing marketing campaigns.
Strategy Map Template
Works best for:
Leadership, Strategic Planning, Mapping
How do your individual or team goals relate to an organizationâs overall strategy? A Strategy Map is a stylized picture of your organizationâs strategy and objectives. Itâs powerful because it provides a clear visual guide to how these various elements work together. Strategy Maps can help align various different team goals with the overall strategy and mission. With the Strategy Map in place, teams can create set actionable, relevant KPIs. Strategy mapping is often considered part of the balanced scorecard (BSC) methodology, which is a strategic planning tool for setting overall team goals.
3-Circle Venn Diagram
Works best for:
Education, Diagrams, Brainstorming
Venn diagrams have been a staple of business meetings and presentations since the 1800s, and thereâs a good reason why. Venn diagrams provide a clear, effective way to visually showcase relationships between datasets. They serve as a helpful visual aid in brainstorming sessions, meetings, and presentations. You start by drawing a circle containing one concept, and then draw an overlapping circle containing another concept. In the space where the circles overlap, you can make note of the conceptsâ similarities. In the space where they do not, you can make note of their differences.
Value Chain Analysis Template
Works best for:
Leadership, Strategic Planning, Workflows
First coined by Harvard Business School professor Michael Porter, the value chain analysis helps your team evaluate your business activities so you can find ways to improve your competitive advantage. A value chain is a set of activities that a company performs in order to deliver a valuable product from start to finish. The analysis itself allows your team to visualize all the business activities involved in creating the productâand helps you identify inefficiencies, bottlenecks, and miscommunication within the process.
Product
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Plans and Pricing |
6560 | https://scienceandjoe.com/2019/03/01/chem-03-stoichiometry-part-1b-determining-masses-of-reactants-and-products-of-a-chemical-reaction-with-conversion-factors/ | Science and Joe
Determining the Masses of Reactants and Products of a Chemical Reaction (with Conversion Factors)
The following tutorial will cover how to calculate the masses of chemical substances involved in a chemical reaction via stoichiometric calculations using conversion factors to cancel out units.
If you would like to see how these calculations can be done without the use of conversion factors to cancel out units, please refer to ‘Determining the Masses of Reactants and Products of a Chemical Reaction (without Conversion Factors)’. Also, if you would like to see how chemical equations are balanced using stoichiometric coefficients and an explanation of some of the key terms and phrases of stoichiometry (e.g. stoichiometry coefficients and mole ratio) please also refer to ‘Determining the Masses of Reactants and Products of a Chemical Reaction (without Conversion Factors)’.
Reminder: A conversion factor is a fraction that consists of two equivalent values with different units of measurement. The main purpose of a conversion factor is to cancel out the unit we are converting from and only leave the unit we are converting to for an answer. In stoichiometry, that would initially be from grams to moles for the substance we know the mass of, but then from moles to grams for the substance we are working out the mass of.
This method first involves using the mass and molar mass of the substance we know the mass of to work out its amount in moles. We then use mole ratio to work out the amount in moles of the substance we want to know the mass of and then use its amount in moles and its molar mass to work out its mass.
Question 1
Gaseous
ammonia reacts with copper (II) oxide (when heated) to produce
copper, nitrogen gas and water.
If
14.50 grams of copper was produced, what was the mass of ammonia gas?
Step 1: Write out the reaction as an equation with the correct chemical formulas.
Step 2: Balance the equation with stoichiometric coefficients.
Step 3 :Work out the molar mass of the substance you know the mass of – which is copper.
M = 63.5 g mol-1
(It’s just the relative atomic mass of copper.)
Step 4: Work
out the molar mass of the substance you want to know the mass of –
which is ammonia.
M = 14.0 + (3 x 1.01) = 17.03 g mol-1
Step 5: Identify the mole ratio of ammonia to copper using the balanced equation.
The balanced equation indicates that the mole ratio of ammonia to copper is 2 moles to 3 moles.
Step 6: Arrange the calculated molar masses and the mole ratio into conversion factors in a calculation and cancel units.
Note:
Instead of including mol-1 in the unit for molar mass, we
replace it with the chemical formula of the substance when we include
the value in a conversion factor.
The
arrangement of three conversion factors in the above calculation
allows us to cancel out the units we don’t want for our answer.
The
first variable in the numerator is the mass of copper which is the
substance we know the mass of. This is followed by a conversion
factor consisting of the molar mass of copper and the equivalent
value of 1 mole of copper. The second conversion factor indicates the
mole ratio of ammonia to copper which is 2 moles to 3 moles. The
third conversion factor consists of the molar mass of ammonia and the
equivalent value of 1 mole of ammonia.
The arrangement of these values allows us to cancel out units by having the same units aligned to one another (diagonally) in the numerator and the denominator.
The units of g Cu cancel; the units of mol Cu cancel and the units of mol NH3 cancel.
This
then leaves only with the unit of g NH3 which is the unit
we want to use to express the mass of ammonia gas acting as a
reactant.
Step
7: Carry out the calculation with only the unit we want for the
answer.
We are going to round our answer to four significant figures because the mass of the other substance we are given in the question is given in four significant figures.
So we have worked out that 2.592 grams of ammonia gas acted as a reactant to produce 14.50 grams of copper.
Question 2
(This is an example
of a reversible reaction.)
Nitrogen gas reacts with hydrogen gas to produce ammonia gas.
What mass of hydrogen gas is needed to produce 94.0 grams of ammonia gas?
Step 1: Write out the equation.
Step 2: Balance the equation with stoichiometric coefficients.
Step
3: Work out the molar mass of the substance you know the mass of
– which is ammonia.
M
= 14.0 + (3x 1.01) = 17.03 g mol-1
Step
4: Work out the molar mass of the substance you want to know the
mass of – which is hydrogen.
(Hydrogen
exists as a diatomic molecule – meaning there is 2 atoms
rather than 1.)
M
= 2 x 1.01 = 2.02
Step 5: Identify the mole ratio of hydrogen to ammonia using the balanced chemical equation.
The balanced equation indicates that the mole ratio of hydrogen to ammonia is 3 moles to 2 moles.
Step 6: Arrange the calculated molar masses and the mole ratio into conversion factors in a calculation and cancel out units.
Step 7: Carry out the calculation with only the unit we want for the answer.
We are going to round our answer to four significant figures because the mass of the other substance we are given in the question is given in four significant figures.
So
we have worked out that 16.72 grams of hydrogen gas is involved as a
reactant to produce 94.00 grams of ammonia gas.
So there we go we have worked out the masses of chemical substances involved in a chemical reaction using a balanced chemical equation and conversion factors to cancel out units. .
Over the next few tutorials on stoichiometry, we’ll be going through how to work out the number of particles involved either as a reactant or a product in a chemical reaction. We’ll also go over some more mole ratio problems.
Bye for now.
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6561 | https://www.k5learning.com/blog/lines-of-symmetry | Reading & Math for K-5
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Line of Symmetry Explained
Line of Symmetry Explained
What is a line of symmetry?
A line of symmetry is a line which divides a shape into two parts that match exactly. One way to easily work out if a shape has a line of symmetry is to fold it. When the folded part sits perfectly on top with all edges matching the fold line is a line of symmetry.
Let’s look at a few shapes and their lines of symmetry.
Triangles line of symmetry
A triangle can have three, one or zero lines of symmetry:
An equilateral triangle has three lines of symmetry.
An isosceles triangle has one line of symmetry.
A scalene triangle has no lines of symmetry.
Quadrilaterals line of symmetry
A square has four lines of symmetry.
A rectangle has two lines of symmetry.
An irregular quadrilateral has no lines of symmetry.
Polygons line of symmetry
Polygons are shapes with all equal sides and all equal angles.
The equilateral triangle and the square are both polygons, with three and four lines of symmetry respectively.
A regular pentagon has five lines of symmetry.
A regular hexagon has six lines of symmetry.
A regular heptagon has seven lines of symmetry.
Lines of symmetry worksheets
In our grade 4 geometry section, we have a set of worksheets for students to practice identifying and drawing lines of symmetry.
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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-.
StatPearls [Internet].
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Treasure Island (FL): StatPearls Publishing; 2025 Jan-.
Labyrinthitis
David Barkwill; Ryan Winters; Rubeena Arora.
Author Information and Affiliations
Authors
David Barkwill1; Ryan Winters2; Rubeena Arora3.
Affiliations
1 Norwich Medical School, Poole General Hospital
2 Ochsner Health System
3 Lady Harding E Medical College
Last Update: July 23, 2025.
Continuing Education Activity
Labyrinthitis is an inflammatory condition of the inner ear that can cause sudden vertigo and often mimics a stroke. Symptoms typically include dizziness, nausea, vomiting, hearing loss, and tinnitus. While most cases are viral in origin, bacterial infections, autoimmune disorders, or systemic diseases may also be responsible. Some patients recover fully, while others may experience lasting impairments to their hearing or balance (vestibular function). The differential diagnosis includes more serious conditions, making expeditious evaluation and accurate diagnosis central to optimizing patient outcomes.
This activity explores the etiology, pathophysiology, differential diagnosis, and management of labyrinthitis, with particular emphasis on distinguishing it from other causes of vertigo and hearing loss. Clinicians will gain a deeper understanding of the diagnostic criteria, differential diagnosis, treatment, and incorporation of multidisciplinary care for patients with labyrinthitis. Interprofessional strategies, in collaboration with key professionals such as audiologists, vestibular therapists, and pharmacists, are emphasized to support patient safety, effective medication management, and long-term recovery.
Objectives:
Differentiate labyrinthitis from other causes of vertigo, such as vestibular neuritis, Meniere disease, and cerebrovascular accidents.
Assess patient history and physical exam findings to guide diagnostic decision-making.
Apply evidence-based treatment strategies based on underlying etiology and symptom severity.
Coordinate with the interprofessional team to enhance follow-up care and referrals for patients with labyrinthitis.
Access free multiple choice questions on this topic.
Introduction
Labyrinthitis is an inflammation of the membranous labyrinth of the inner ear, presenting with symptoms such as vertigo, nausea, vomiting, tinnitus, and/or hearing loss. Epidemiological data on labyrinthitis are limited, but the incidence increases with age. Notably, labyrinthitis can mimic other conditions, such as a cerebrovascular accident (CVA), necessitating a thorough evaluation before a final diagnosis. Bacterial or viral infections of the ear usually cause labyrinthitis, but it can be a manifestation of systemic disease due to autoimmunity, medication toxicity, or infection such as human immunodeficiency virus (HIV) or syphilis. The etiology determines the appropriate treatment and symptom control. Although some patients recover completely, others experience residual balance or hearing deficits.
Etiology
Labyrinthitis is an inflammation of the membranous labyrinth caused by viruses, bacteria, or systemic diseases. In rare cases, it can cause labyrinthitis ossificans, a disease in which pathological new bone formation occurs within the membranous labyrinth. Vestibular neuritis (also referred to as vestibular neuronitis) is often used interchangeably with labyrinthitis, as the symptoms and clinical picture are similar. However, true vestibular neuritis is characterized by inflammation confined to the vestibular nerve without the involvement of the membranous labyrinth. For the remainder of this article, we will focus exclusively on labyrinthitis.
Viral Labyrinthitis
The most common cause of labyrinthitis is a viral infection of the upper respiratory tract. Local inflammatory mediators are produced in response to the viral infection and enter the cochlea through either the round or oval window, leading to inflammation. Less commonly, viruses can directly enter and infect the cochlea, possibly causing hearing impairment. For instance, labyrinthitis secondary to maternal rubella or cytomegalovirus (CMV) infections is one of the most common causes of congenital deafness. In the post-natal period, mumps and measles are the leading causes of viral hearing loss. Ramsay-Hunt syndrome, also known as herpes zoster oticus, is due to a latent varicella-zoster virus infection reactivating, often years after the primary viral infection. This syndrome typically produces a vesicular rash, usually on the oral mucosa or ear, accompanied by peripheral facial nerve paralysis. Additionally, the virus can infect the vestibular and cochlear nerves in 25% of cases. Moreover, CMV-mediated inflammation can be a cause of sudden sensorineural hearing loss, although the cause is usually unknown.
Bacterial Labyrinthitis
Bacterial labyrinthitis typically arises from either bacterial meningitis or bacterial acute otitis media. Similarly, 20% of children with bacterial meningitis will develop auditory or vestibular symptoms. Inflammation occurs through 2 different mechanisms in serous and suppurative labyrinthitis. In serous labyrinthitis, the inflammation is secondary to bacterial toxins and/or host cytokines and inflammatory mediators passing into the membranous labyrinth via the round or oval window. Suppurative labyrinthitis is inflammation caused directly by a bacterial infection. Bacteria enter the inner ear from the middle ear via the round window (most common) or the oval window. They can also enter the central nervous system via the cochlear aqueduct or auditory canal. Congenital or acquired defects in the bony labyrinth can predispose to further bacterial infection.
Autoimmune Labyrinthitis
Both polyarteritis nodosa and granulomatosis with polyangiitis are rare autoimmune conditions that are associated with labyrinthitis.
HIV/ Syphilis
Both syphilis and HIV infections are associated with labyrinthitis. However, it is unclear whether the inflammation is due to opportunistic infections or if HIV causes the inflammation.
Epidemiology
There is limited research on the incidence and prevalence of labyrinthitis, as most epidemiologic studies focus on broader categories of vestibular dysfunction or vertigo rather than labyrinthitis alone. In a prospective study of 200 patients presenting with sudden sensorineural hearing loss, labyrinthitis was identified in 3% of cases based on MRI findings.While vestibular dysfunction and balance abnormalities increase with age, the precise prevalence of labyrinthitis is difficult to determine, although it is considered rare.
Otogenic suppurative labyrinthitis can occur at any age and is usually associated with a cholesteatoma or secondary to untreated, longstanding otitis media. Suppurative bacterial labyrinthitis, a complication of bacterial meningitis, is the most common cause of deafness in children under age 2. Fortunately, it is exceedingly rare in the post-antibiotic era.
Pathophysiology
Anatomy
The inner ear consists of 2 structures: the bony labyrinth and the membranous labyrinth. The bony labyrinth is a collection of bony cavities situated within the temporal bone of the skull, comprising three main parts: the vestibule, the cochlea, and the 3 semicircular canals. All three of these structures contain a substance called perilymph, also known as Cotunnius' liquid. Perilymph is similar in composition to cerebrospinal fluid, with sodium as the primary cation and a lower concentration of potassium than intracellular fluid.
The membranous labyrinth is situated within the bony labyrinth and comprises four structures: the saccule, utricle, semicircular ducts, and cochlear duct. Endolymph fills the membranous labyrinth. It has a principal cation of potassium and a lower concentration of sodium, similar to intracellular fluid. The gradient across Reissner's membrane, created by the opposing concentrations of endolymph and perilymph, is responsible for generating the electrical impulses of the hair cells, known as the endocochlear potential.
The inner ear has 2 connections with the middle ear and 2 connections with the central nervous system. The oval window connects the vestibule to the middle ear, and the round window connects the cochlear duct to the middle ear. The internal auditory canal and cochlear aqueduct connect the inner ear and central nervous system (CNS).
History and Physical
Labyrinthitis is a diagnosis of exclusion. When evaluating a patient with suggested labyrinthitis, it is important to screen for risk factors. Helpful clues in the history include recent viral upper respiratory tract infections, cholesteatoma, a history of ear surgery, a history of temporal bone fracture, meningitis, or acute or chronic otitis media.
Nausea, vomiting, and severe ‘room spinning’ vertigo are the most commonly reported symptoms of labyrinthitis. It is also essential to clarify the number of episodes of actual ‘room spinning' or 'whirling' vertigo, as opposed to more generalized sensations of imbalance or lightheadedness, which would not be indicative of labyrinthitis. The peak of vertigo rarely lasts longer than 72 hours, but ataxia and occasional brief episodes may persist for several weeks. The clinical course can be more prolonged if the patient has comorbid anxiety or depression.
The differential diagnosis of labyrinthitis includes vestibular neuritis, stroke, and Meniere disease. Vestibular neuritis is distinguished by the presence of hearing loss or tinnitus, symptoms that are absent in labyrinthitis. The presence of vertigo with cranial nerve deficits, crossed or uncrossed cortical tract findings, and numbness could indicate a CVA affecting the brain stem. Meniere disease is characterized by fluctuating aural fullness, tinnitus, and hearing loss, often accompanied by vertiginous episodes.
On physical examination, nystagmus is often present, and the fast phase typically moves away from the affected ear. Patients can also experience gait and balance abnormalities. Therefore, Romberg sign, tandem gait, cerebellar, and meningeal testing are essential components of the neurologic examination. Notably, Rinne and Weber hearing tests will likely demonstrate sensorineural hearing loss in the affected ear; therefore, a formal audiological evaluation is required. Otoscopy may provide clues to the etiology of the disease, such as otitis media or cholesteatoma. Finally, if bacterial meningitis is a consideration, skin examination may reveal a non-blanching rash.
Evaluation
Audiometry is useful to assess the presence or extent of sensorineural hearing loss. Specific investigations into the vestibular system (eg, evoked myogenic potentials, electronystagmography, and rotary chair tests) are not indicated in the acute phase of the disease. However, they can be useful in assessing long-term compensation and residual deficits, as well as in differentiating inner ear pathology from other etiologies.
The laboratory tests are tailored to the patient's symptoms and differential diagnoses. Severe vomiting should prompt a basal metabolic panel to assess electrolyte abnormalities and hydration status. Cerebrospinal fluid analysis is performed if bacterial meningitis is suggested. Subsequently, HIV and syphilis serology are helpful in high-risk individuals or atypical presentations. Finally, testing for autoimmune disease is considered in patients with systemic findings or atypical presentations, even if serological titers are negative.
Imaging studies, such as magnetic resonance imaging (MRI) and computed tomography (CT) scans, can be useful for identifying alternative pathologies. Furthermore, expert consensus recommends imaging of the cerebello-pontine angle if the difference between the ears exceeds 15 dB, because 13% of acoustic neuromas present with sudden hearing loss. Gadolinium-enhanced magnetic resonance imaging (Gd-MRI) predicts the development of hearing loss following bacterial meningitis, a complication that occurs in approximately 14% of cases.
Treatment / Management
Treatment is primarily supportive and is tailored to the etiology and symptoms.
Viral labyrinthitis treatment typically involves hydration and bed rest in an outpatient setting. However, patients must be counseled to seek further medical intervention if their condition worsens or they develop neurological symptoms (e.g., weakness/numbness, diplopia, slurred speech, and gait disturbance). Notably, antiviral medications and steroids are not supported by current evidence.
The treatment of bacterial labyrinthitis depends on the source of infection. Oral antibiotics are the first-line treatment for acute otitis media with an intact tympanic membrane. Intravenous antibiotics are recommended for recalcitrant infections. If findings are suggestive of bacterial meningitis, intravenous antibiotics are commenced immediately, while confirmatory CSF and imaging testing are performed concurrently.
Autoimmune labyrinthitis is treated initially with corticosteroids. If the disease is refractory to corticosteroid therapy, other immunomodulators are considered, such as azathioprine, etanercept, or cyclophosphamide. Compared with corticosteroids, these medications are preferable for chronic cases due to their favorable safety profile and fewer adverse effects. Treatment with immunosuppressants requires specialist referral.
If serology testing is consistent with syphilis or HIV infection, appropriate treatment and referral to specialist care are warranted.
During the initial phase of vertigo, patients often lie motionless with their eyes closed due to dizziness and nausea. However, early mobilization can promote vestibular compensation and improve prognosis.
Benzodiazepines and antihistamines are useful initially. However, treatment with these medications should not last more than 72 hours because they can inhibit vestibular compensation. Some experts advise against these vestibular suppressant medications for all but intractable symptoms. Antiemetics, such as prochlorperazine or ondansetron, may help control nausea and vomiting. Patients with sudden hearing loss should receive high-dose corticosteroids and specialist referral.
A small minority of patients may have residual tinnitus following sensorineural hearing loss. Reactive depression may coexist with prolonged tinnitus, and early intervention with tinnitus re-training, tinnitus maskers, hearing aids, and/or biofeedback can aid recovery.
Surgical intervention is required in a minority of cases. For example, mastoidectomy is required in patients with cholesteatoma or severe mastoiditis. Occasionally, patients may require drainage of effusions or myringotomy if labyrinthitis is secondary to otitis media.
Once the acute labyrinthitis has resolved, patients with persistent vestibular symptoms should be referred for vestibular rehabilitation.
Differential Diagnosis
Vestibular neuritis presents similarly to labyrinthitis but without accompanying hearing loss.
Meniere disease causes hearing loss and vertigo, but the episodes are usually intermittent.
Benign positional vertigo causes dizziness but not hearing loss, and patients typically exhibit an abnormal Dix-Hallpike test.
Posterior fossa CVA typically presents with an abnormal neurological examination, including ataxia, hoarseness, dysarthria, or dysphagia. A CT or MRI of the head can aid in diagnosis.
Acoustic neuromas/ vestibular schwannomas are diagnosed with a GdMRI.
Inner ear malformationstypically present with progressive or congenital hearing loss and can be diagnosed with CT or MRI imaging.
Temporal bone fracture is suggested with a recent history of head trauma, and is diagnosed with CT imaging.
Inner ear hemorrhage is commonly associated with trauma and is visible on MRI.
Temporal bone neoplasm usually presents with cranial nerve deficits or facial paralysis and is apparent on MRI and/or CT.
Multiple sclerosis often presents with additional systemic findings, such as spasticity, optic neuritis, or other cranial nerve deficits.
Pertinent Studies and Ongoing Trials
Several trials are studying the effects of novel treatments for labyrinthitis. Superoxide dismutase limits hearing loss and prevents labyrinthitis ossificans in gerbils with bacterial meningitis. Additionally, TNF-alpha inhibitors reduce postmeningitic cochlear injury and hearing loss. Some evidence suggests corticosteroid therapy for pneumococcal meningitis may prevent labyrinthitis ossificans. Furthermore, research into intra-tympanic corticosteroids as a superior therapeutic administration route shows promising results. Finally, cochlear microperfusion and antioxidant therapy have potential as adjuvant therapies.
Prognosis
The acute vertigo of labyrinthitis usually resolves after 48 to 72 hours. However, mild symptoms may persist for several weeks. The prognosis is generally favorable if there are no serious neurological sequelae. Conversely, neurological complications may require further interventions and portend a more guarded prognosis. For example, ventriculoperitoneal shunts are necessary in patients with hydrocephalus from bacterial meningitis. Furthermore, a prolonged course of benzodiazepines and/or antihistamines for vertigo can delay vestibular recovery. Additionally, suppurative labyrinthitis can cause permanent hearing impairment.
Complications
After a severe case of labyrinthitis, some patients may develop persistent hearing loss or tinnitus, which may require the use of hearing aids or tinnitus-specific therapies.
Bilateral vestibular hypofunction is a debilitating complication associated with bilateral labyrinthitis, typically resulting from bacterial meningitis. It can cause visual impairment (oscillopsia) and impaired spatial awareness.
Complete deafness is a rare complication of bilateral labyrinthitis associated with bacterial meningitis. In addition, labyrinthitis ossificans is a complication of suppurative labyrinthitis. Moreover, undertreated bacterial labyrinthitis can result in mastoiditis. This condition typically responds to intravenous antibiotics but can require mastoidectomy with tympanoplasty in severe cases. Rarely, a labyrinthectomy may be necessary to treat labyrinthitis caused by cholesteatoma.
Deterrence and Patient Education
Labyrinthitis is often a complication of infection, such as otitis media or meningitis. Therefore, early diagnosis and effective management of infections can minimize the risk of labyrinthitis or long-term complications. Furthermore, vaccinations against measles, mumps, and rubella can protect against labyrinthitis in vulnerable populations.
Patients with vertigo should be encouraged to mobilize as soon as possible to improve vestibular compensation and prognosis. Patients who receive a prolonged course of benzodiazepines and/or antihistamines for vertigo may have delayed vestibular recovery.
Enhancing Healthcare Team Outcomes
The interprofessional team is crucial in delivering optimal care to patients with labyrinthitis. First, primary care clinicians should be familiar with the signs, symptoms, and differential diagnosis of labyrinthitis. These clinicians are the first line of defense and can administer prompt and effective management, thereby reducing potentially life-threatening conditions and complications of labyrinthitis.
Both nurses and pharmacists are invaluable in the initial management of a patient with labyrinthitis and must be familiar with the disease. For example, nursing staff can assist patients at high risk of falls due to vertigo. Pharmacists can recommend the most appropriate medications for nausea and vertigo. For example, pharmacists' knowledge of the role of benzodiazepines and antihistamines in short-term management can enable them to collaborate with physicians. They also know medications that may cause symptoms similar to labyrinthitis, and identify adverse effects or interactions.
Patients sometimes experience residual vestibular symptoms, which can significantly impact their quality of life. Both occupational and physical therapists can treat persistent vertigo and ataxia. Occupational therapists can also conduct assessments of the home environment and implement changes to enhance patient safety. Audiologists perform ongoing monitoring of hearing impairments to ensure optimal patient care. In summary, an interprofessional team approach can improve patient outcomes and quality of life.
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: Hermann R, Ionescu EC, Dumas O, Tringali S, Truy E, Tilikete C. Bilateral Vestibulopathy: Vestibular Function, Dynamic Visual Acuity and Functional Impact. Front Neurol. 2018;9:555. [PMC free article: PMC6048872] [PubMed: 30042725]
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: Bloch SL, McKenna MJ, Adams J, Friis M. Labyrinthitis Ossificans: On the Mechanism of Perilabyrinthine Bone Remodeling. Ann Otol Rhinol Laryngol. 2015 Aug;124(8):649-54. [PubMed: 25757630]
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: Laulajainen Hongisto A, Jero J, Markkola A, Saat R, Aarnisalo AA. Severe Acute Otitis Media and Acute Mastoiditis in Adults. J Int Adv Otol. 2016 Dec;12(3):224-230. [PubMed: 27895000]
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6563 | https://www.youtube.com/watch?v=ORyhjp3-HYI | Subtracting Mixed Numbers (Unlike Denominators) | Math with Mr. J
Math with Mr. J
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Posted: 6 Jan 2020
Welcome to Subtracting Mixed Numbers with Mr. J! Need help with how to subtract mixed numbers? You're in the right place!
Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for how to subtract mixed numbers with unlike denominators. Mr. J will go through subtracting mixed numbers examples and explain the subtracting mixed numbers steps.
About Math with Mr. J: This channel offers instructional videos that are directly aligned with math standards. Teachers, parents/guardians, and students from around the world have used this channel to help with math content in many different ways. All material is absolutely free.
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Hopefully this video is what you're looking for when it comes to subtracting mixed numbers.
Have a great rest of your day and thanks again for watching!
1700 comments
Transcript:
welcome to math with mr. J in this video we're going to discuss how to subtract mixed numbers and as you can see there are two problems on your screen now we're going to go through in order to get this down so let's jump right in to number one and for number one we have five and three-fourths minus two and a half so the first thing I like to do when subtracting or adding mixed numbers I like to rewrite the problem vertically up and down so I'm going to underneath my original problem rewrite 5 and 3/4 minus 2 and 1/2 now notice I have my fractional part of the mixed numbers lined up and the whole number part of the mixed numbers lined up now what you need to do we need to subtract the fractional part first now just like when we add or subtract fractions we need a common denominator and obviously a denominator of four and two we don't have the same denominator so we're going to need to rename both these mixed numbers the fractional part of the mixed numbers with a common denominator so let's think what's going to be our common denominator for a 4 and a 2 hopefully you're thinking 4 so let's rename these mixed numbers well 5 and 3/4 already has a denominator of 4 so we can keep it as is now 2 and 1/2 though we're going to need to rename so let's think 2 times 2 gives me that denominator of 4 so I need to do the same thing to the top in order to properly rename so we would do 1 times 2 is 2 now we have our problem rewritten with a common denominator now subtract the fractional parts and then the whole number part so 3/4 minus 2/4 well 3 minus 2 is 1 keep our denominator of 4 the same and then do 5 minus 2 which gives us 3 always look to see if the answer can be simplified the fractional part 1/4 1/4 is in lowest terms so we are done the solution or answer to number 1 is 3 and 1/4 let's take a look at number 2 let's rewrite the problem vertically this means which means up and down 7 and 1/3 minus 3 and 5/6 all right let's do the fractional part first we do not have a common denominator so we will need to rename with a common denominator so a 3 & a 6 let's think hopefully we're thinking a common denominator of 6 so 7 and how many sixths well 3 times 2 is 6 1 times 2 is 2 minus 3 5 6 already has a denominator of 6 so there is our rewritten problem so let's subtract the fractional part 2 6 minus 5/6 well we can't do that because we cannot do 2 minus 5 so just like a whole number subtraction problem we need to borrow to be able to do the fractional part of this problem so how you do that with mixed numbers we're going to borrow 1 from this 7 and it's going to turn to a 6 and we're going to add one whole to that 2/6 now I'm gonna write one whole as six over six so I can add it to that fractional part they need a common denominator in order to add that whole there so again six over six is one whole I just put it with a denominator of six so now we have six holes right and then two plus six is eight keep my denominator of six subtract three and five six now we could do 8 minus 5 is 3 keep my denominator of 6 and then we have 6 minus 3 is 3 check to see if we can simplify the fractional part and we can a common factor between 3 & 6 is 3 so I can divide by 3 to get the fractional part in lowest terms and this actually turns out to be 3 and a half that would be our final answer there so there you have it there's how you subtract mixed numbers rewrite the problem vertically see if you need a common denominator rename if you do then go ahead and subtract the fractional part then the whole number part thanks so much for watching until next time peace |
6564 | https://artofproblemsolving.com/wiki/index.php/1988_AHSME_Problems?srsltid=AfmBOorrn5mO3Q4780bqlM9NYD2uGxY4GZ1Vki69g8ScqCbwPiz4jOBx | Art of Problem Solving
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1988 AHSME Problems
1988 AHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF
Instructions
This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.
No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
Figures are not necessarily drawn to scale.
You will have 90 minutes working time to complete the test.
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•26•27•28•29•30
Contents
[hide]
1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8
9 Problem 9
10 Problem 10
11 Problem 11
12 Problem 12
13 Problem 13
14 Problem 14
15 Problem 15
16 Problem 16
17 Problem 17
18 Problem 18
19 Problem 19
20 Problem 20
21 Problem 21
22 Problem 22
23 Problem 23
24 Problem 24
25 Problem 25
26 Problem 26
27 Problem 27
28 Problem 28
29 Problem 29
30 Problem 30
31 See also
Problem 1
Solution
Problem 2
Triangles and are similar, with corresponding to and to . If , and , then is:
Solution
Problem 3
Four rectangular paper strips of length and width are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?
Solution
Problem 4
The slope of the line is
Solution
Problem 5
If and are constants and , then is
Solution
Problem 6
A figure is an equiangular parallelogram if and only if it is a
Solution
Problem 7
Estimate the time it takes to send blocks of data over a communications channel if each block consists of "chunks" and the channel can transmit chunks per second.
Solution
Problem 8
If and , what is the ratio of to ?
Solution
Problem 9
An table sits in the corner of a square room, as in Figure below. The owners desire to move the table to the position shown in Figure . The side of the room is feet. What is the smallest integer value of for which the table can be moved as desired without tilting it or taking it apart?
Solution
Problem 10
In an experiment, a scientific constant is determined to be with an error of at most . The experimenter wishes to announce a value for in which every digit is significant. That is, whatever is, the announced value must be the correct result when is rounded to that number of digits. The most accurate value the experimenter can announce for is
Solution
Problem 11
On each horizontal line in the figure below, the five large dots indicate the populations of cities and in the year indicated. Which city had the greatest percentage increase in population from to ?
Solution
Problem 12
Each integer through is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?
Solution
Problem 13
If then what is ?
Solution
Problem 14
For any real number a and positive integer k, define
What is
?
Solution
Problem 15
If and are integers such that is a factor of , then is
Solution
Problem 16
and are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side and side is the altitude of . The ratio of the area of to the area of is
Solution
Problem 17
If and , find
Solution
Problem 18
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives th prize and the winner bowls #3 in another game. The loser of this game receives th prize and the winner bowls #2. The loser of this game receives rd prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes?
Solution
Problem 19
Simplify
Solution
Problem 20
In one of the adjoining figures a square of side is dissected into four pieces so that and are the midpoints of opposite sides and is perpendicular to . These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, , in this rectangle is
Solution
Problem 21
The complex number satisfies . What is ? Note: if , then .
Solution
Problem 22
For how many integers does a triangle with side lengths and have all its angles acute?
Solution
Problem 23
The six edges of a tetrahedron measure and units. If the length of edge is , then the length of edge is
Solution
Problem 24
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is , and one of the base angles is . Find the area of the trapezoid.
Solution
Problem 25
and are pairwise disjoint sets of people. The average ages of people in the sets and are and respectively. Find the average age of the people in set .
Solution
Problem 26
Suppose that and are positive numbers for which
What is the value of ?
Solution
Problem 27
In the figure, , and is tangent to the circle with center and diameter . In which one of the following cases is the area of an integer?
Solution
Problem 28
An unfair coin has probability of coming up heads on a single toss. Let be the probability that, in independent toss of this coin, heads come up exactly times. If , then
Solution
Problem 29
You plot weight against height for three of your friends and obtain the points . If and , which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.
Solution
Problem 30
Let . Give , consider the sequence defined by for all . For how many real numbers will the sequence take on only a finite number of different values?
Solution
See also
AMC 12 Problems and Solutions
Mathematics competition resources
1988 AHSME (Problems • Answer Key • Resources)
Preceded by
1987 AHSMEFollowed by
1989 AHSME
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All AHSME Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
Retrieved from "
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6565 | https://www.youtube.com/watch?v=QBZhZaXHA20 | Construct sinusoidal functions : Khan Academy
Khan Man Math
1200 subscribers
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Posted: 21 Feb 2023
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Transcript:
this will be for the con exercise construct sinusoidal functions now you need to be familiar with the basic uh sine and cosine functions all right just a quick review here the graph in red is y equals sine of x the one in blue is y equals negative sine of x all right so take note that the Y equals sine of x starts at the midline and heads to the max whereas negative sine of x starts at the midline and heads to the minimum all right and here we have in red the cosine of x and in blue the negative cosine of x notice that cosine starts at a Max and then heads to the midline whereas negative cosine of x starts at a minimum and heads to the midline okay so the graph of a sinusoidal function intersects its midline at 0 2 and then has a minimum at three negative six write the formula of the function X is in radians all right I highly recommend you drew you draw a quick sketch all right so it doesn't have to be perfect but I suggest you do this on your paper all right so it intersects the midline at x equals zero y equals two and then it has a minimum point at x equals three y equals negative six all right this is three negative six so um let's see here if you remember from the sine and negative sine graphs the one that starts at the midline and heads to the minimum is going to be a negative sign all right and as far as the amplitude is concerned going from midline to minimum the amplitude travels from y equals 2 to y equals negative six all right so the amplitude is just going to be the absolute value of 2 minus negative six which of course is just eight okay again midline to minimum would be the distance for the amplitude so that's eight I'm sorry let me give you the equation here we're looking at a times sine of VX plus c okay so a the absolute value of a is the amplitude but um amplitude is always positive but we know this is negative because it went from midline to minimum all right and for the period we know that equals 2 pi over B absolute value of B all right and we can see from here we're going from midline to minimum so that's only going to be a fourth of the period so that horizontal distance from here to here is only going to be one-fourth of the period okay so the period distance one-fourth of the period it runs from zero to three so a fourth of the period is three so the period is just going to be three times four all right so 2 pi over B equals 12. all right and multiplying both sides by B it cancels here and then dividing both sides by 12 it cancels here and we have 2 over 12 pi which is just pi over 6. so the value for B is just pi over six and that's times x and finally for the midline which is the value for C the midline is at zero comma two okay so that's just going to be a plus two so that is your final answer all right so remember uh a has to deal with amplitude B has to deal with period and C is the midline all right negative eight sine pi over six plus two negative eight sine parentheses pi over six and don't forget the X I think I forgot the X plus two even for the ones that you get correct I would check how they answered the question just to go over what you just did and have see how it matches up with what they did all right the graph of a sine function has a maximum point at zero five and then a minimum at this point right the formula all right quick sketch here it says it starts at the maximum okay so it starts here at zero five and then it goes to the minimum at two Pi negative five so basically goes like that and that's going to be your minimum 2 pi comma negative five all right let's do the amplitude first we see that um this goes from a Max to A Min it goes from five to negative five so the amplitude is just going to be 5 minus negative five which of course is 10. all right I'll save that for later and then for the value of B let's see the period if you go from a Max to A Min that's going to be half the period okay the rest the rest of the period would be Min to Max right so going from here to here that's going to be half the period all right and we went from this is the horizontal distance so we went from x equals zero to x equals two Pi all right 2 pi minus zero that's going to be 2 pi so half the period is 2 pi that means the period is just going to be 2 pi times 2. which of course is 4 pi all right finding the value of B we know that the period equals 2 pi over the absolute value of B which we said was 4 pi okay multiplying both sides by B you can cross multiply if you want same thing the B's cancel solving for B divide both sides by 4 pi cancels here 2 pi over 4 Pi the pies cancel two over four is one half all right so we got that and finally the value for C is the midline okay the midline is going to be in between the Max and the min we see the max is at five and the minimum is at negative five what number is in between five and negative five zero okay we don't even have to do any math there all right so C equals zero the only thing I didn't do now is if the state if it were sine or cosine okay you should have your sine and cosine graphs handy while you're doing this these problems but the one that starts at a Max and then heads to the midline and to the men that is a positive cosine all right so the formula we're looking at is a cosine of BX plus c all right same formula If This Were the sign all right so positive cosine we said the amplitude was 10 and this is positive cosine so it's just going to be a 10 okay B times x it's going to be one half X no Pi this time and then plus C well C was just zero okay because the midline is exactly in between five and negative five so that's it 10 cosine of one half X okay ten cosine parentheses one slash two hit the right arrow key X 10 cosine of one half X okay let's see what I did wrong okay I made a mistake on the amplitude it's actually five cosine of one-half X big mistake because what I did was I went from uh Max to Min I went from five to negative five I was supposed to take half of that number remember the um amplitude if you go from Max to min the amplitude is only going to be half of that okay so I made a mistake and I went I went all the way so I got 10 instead of five all right so that is a five very easy to make mistakes on these all right the graph of assigned function intersects midline at zero five and then has a max at this point all right quick sketch the midline is at zero comma five and then the max is at 0.75 comma seven so it heads up to the max okay and that point is going to be 0.75 comma seven all right now if you look at your sine and cosine functions and negative sign negative cosine functions the one that goes from midline to maximum that's going to be a positive sine function okay so we're just doing a times sine BX plus c all right let's look at a the amplitude amplitude goes from midline to Max okay we're going from from seven to five all right so it's just seven minus five and that equals two all right this is a positive sine function so we have two times the sine and for the period if you go to midline midline to Max that's only one-fourth of the period all right so from here to here that's only one-fourth of the period the horizontal distance it goes from zero to 0.75 so 0.75 minus zero is just 0.75 but that is only one-fourth of the period so we have to multiply this by four and that's going to equal three now we use our formula for period we know period is 2 pi over B absolute value and that's going to equal three all right multiplying both sides by B the B's cancel dividing both sides by three the threes cancel 2 pi over 3 equals B all right so BX is going to be 2 pi 2 pi over three times x finally for the midline the plus C that determines how high or low the graph is shifted the midline is at the midline is at five you look at the y coordinate okay if this graph kept going is this midline this is Max it would be like that and that's the midline right there so it's at x equals five two sine 2 pi over three X plus five two sine 2 pi over three times x outside parentheses plus five all right last one minimum point at zero negative three midline at one one all right let's see minimums at zero negative three so that's down here zero negative three and it goes to the midline at 1-1 so it starts to head up and that's one comma one all right if this were to keep going this is only the midline here so this would go like that and then like that all right because this is the minimum if you look at your graphs for the sine cosine negative sine negative cosine this is clearly a negative cosine okay now we need to do the amplitude Okay the amplitude is the horizontal distance from the midline to the minimum okay so that's 1 minus negative three that's going to be 1 minus negative three I don't need the absolute value because I put the bigger number first so that's just going to be a four so that goes here now for the period that is the horizontal distance okay so this is going from a minimum to the midline all right minimum to midline that's only one fourth of the period so it goes from zero to one so that is just one unit so the period if one-fourth of the period is one the period is going to be one times four which is four all right the formula for the period is 2 pi over the absolute value of B that's going to equal four all right times B on both sides cancels here divided by 4 on both sides cancels there two over four is a half times pi so one half pi is your value for B all right so that's going to be one half pi times x don't forget the X please all right remember we're using a times cosine of BX plus c finally the midline the y-coordinate is at y equals one all right see this here that's your midline right there that's that y equals one so it's plus one all right so that's it you can instead of one half pi you can just make it pi over two if you want that's the same thing anyhow negative four cosine pi over two X plus one all right negative four cosine parentheses Pi slash two X and the plus one is outside the parentheses like I said you might want to even if you get the answer correct you might want to click on their answer and just see what they did all right but it's the same it's the same thing that I did in this case |
6566 | https://openstax.org/books/principles-microeconomics-3e/pages/chapter-3 | Answer Key Chapter 3 - Principles of Microeconomics 3e | OpenStax
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Principles of Microeconomics 3e
Chapter 3
Principles of Microeconomics 3eChapter 3
Contents Contents
Highlights
Table of contents
Preface
1 Welcome to Economics!
2 Choice in a World of Scarcity
3 Demand and Supply
4 Labor and Financial Markets
5 Elasticity
6 Consumer Choices
7 Production, Costs, and Industry Structure
8 Perfect Competition
9 Monopoly
10 Monopolistic Competition and Oligopoly
11 Monopoly and Antitrust Policy
12 Environmental Protection and Negative Externalities
13 Positive Externalities and Public Goods
14 Labor Markets and Income
15 Poverty and Economic Inequality
16 Information, Risk, and Insurance
17 Financial Markets
18 Public Economy
19 International Trade
20 Globalization and Protectionism
A | The Use of Mathematics in Principles of Economics
B | Indifference Curves
C | Present Discounted Value
Answer Key
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
References
Index
Search for key terms or text.
Close
1.
Since $1.60 per gallon is above the equilibrium price, the quantity demanded would be lower at 550 gallons and the quantity supplied would be higher at 640 gallons. (These results are due to the laws of demand and supply, respectively.) The outcome of lower Qd and higher Qs would be a surplus in the gasoline market of 640 – 550 = 90 gallons.
2.
To make it easier to analyze complex problems. Ceteris paribus allows you to look at the effect of one factor at a time on what it is you are trying to analyze. When you have analyzed all the factors individually, you add the results together to get the final answer.
3.
An improvement in technology that reduces the cost of production will cause an increase in supply. Alternatively, you can think of this as a reduction in price necessary for firms to supply any quantity. Either way, this can be shown as a rightward (or downward) shift in the supply curve.
An improvement in product quality is treated as an increase in tastes or preferences, meaning consumers demand more paint at any price level, so demand increases or shifts to the right. If this seems counterintuitive, note that demand in the future for the longer-lasting paint will fall, since consumers are essentially shifting demand from the future to the present.
An increase in need causes an increase in demand or a rightward shift in the demand curve.
Factory damage means that firms are unable to supply as much in the present. Technically, this is an increase in the cost of production. Either way you look at it, the supply curve shifts to the left.
4.
More fuel-efficient cars means there is less need for gasoline. This causes a leftward shift in the demand for gasoline and thus oil. Since the demand curve is shifting down the supply curve, the equilibrium price and quantity both fall.
Cold weather increases the need for heating oil. This causes a rightward shift in the demand for heating oil and thus oil. Since the demand curve is shifting up the supply curve, the equilibrium price and quantity both rise.
A discovery of new oil will make oil more abundant. This can be shown as a rightward shift in the supply curve, which will cause a decrease in the equilibrium price along with an increase in the equilibrium quantity. (The supply curve shifts down the demand curve so price and quantity follow the law of demand. If price goes down, then the quantity goes up.)
When an economy slows down, it produces less output and demands less input, including energy, which is used in the production of virtually everything. A decrease in demand for energy will be reflected as a decrease in the demand for oil, or a leftward shift in demand for oil. Since the demand curve is shifting down the supply curve, both the equilibrium price and quantity of oil will fall.
Disruption of oil pumping will reduce the supply of oil. This leftward shift in the supply curve will show a movement up the demand curve, resulting in an increase in the equilibrium price of oil and a decrease in the equilibrium quantity.
Increased insulation will decrease the demand for heating. This leftward shift in the demand for oil causes a movement down the supply curve, resulting in a decrease in the equilibrium price and quantity of oil.
Solar energy is a substitute for oil-based energy. So if solar energy becomes cheaper, the demand for oil will decrease as consumers switch from oil to solar. The decrease in demand for oil will be shown as a leftward shift in the demand curve. As the demand curve shifts down the supply curve, both equilibrium price and quantity for oil will fall.
A new, popular kind of plastic will increase the demand for oil. The increase in demand will be shown as a rightward shift in demand, raising the equilibrium price and quantity of oil.
5.
Step 1. Draw the graph with the initial supply and demand curves. Label the initial equilibrium price and quantity.
Step 2. Did the economic event affect supply or demand? Jet fuel is a cost of producing air travel, so an increase in jet fuel price affects supply.
Step 3. An increase in the price of jet fuel caused an increase in the cost of air travel. We show this as an upward or leftward shift in supply.
Step 4. A leftward shift in supply causes a movement up the demand curve, increasing the equilibrium price of air travel and decreasing the equilibrium quantity.
6.
Step 1. Draw the graph with the initial supply and demand curves. Label the initial equilibrium price and quantity.
Step 2. Did the economic event affect supply or demand? A tariff is treated like a cost of production, so this affects supply.
Step 3. A tariff reduction is equivalent to a decrease in the cost of production, which we can show as a rightward (or downward) shift in supply.
Step 4. A rightward shift in supply causes a movement down the demand curve, lowering the equilibrium price and raising the equilibrium quantity.
7.
A price ceiling (which is below the equilibrium price) will cause the quantity demanded to rise and the quantity supplied to fall. This is why a price ceiling creates a shortage.
8.
A price ceiling is just a legal restriction. Equilibrium is an economic condition. People may or may not obey the price ceiling, so the actual price may be at or above the price ceiling, but the price ceiling does not change the equilibrium price.
9.
A price ceiling is a legal maximum price, but a price floor is a legal minimum price and, consequently, it would leave room for the price to rise to its equilibrium level. In other words, a price floor below equilibrium will not be binding and will have no effect.
10.
Assuming that people obey the price ceiling, the market price will be below equilibrium, which means that Qd will be more than Qs. Buyers can only buy what is offered for sale, so the number of transactions will fall to Qs. This is easy to see graphically. By analogous reasoning, with a price floor the market price will be above the equilibrium price, so Qd will be less than Qs. Since the limit on transactions here is demand, the number of transactions will fall to Qd. Note that because both price floors and price ceilings reduce the number of transactions, social surplus is less.
11.
Because the losses to consumers are greater than the benefits to producers, so the net effect is negative. Since the lost consumer surplus is greater than the additional producer surplus, social surplus falls.
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Book title: Principles of Microeconomics 3e
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6567 | https://www.splashlearn.com/math-vocabulary/distance-between-point-and-plane | Distance Between Point and Plane: Definition, Formula, Examples
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Distance Between Point and Plane: Definition, Formula, Examples, FAQs
Home » Math Vocabulary » Distance Between Point and Plane: Definition, Formula, Examples, FAQs
What Is the Distance between Point and Plane?
Formula for Distance between a Point and a Plane
How to Find the Distance from the Point to the Plane
Solved Examples on Distance Between Point and Plane
Practice Problems on Distance between Point and Plane
Frequently Asked Questions about the Distance between Point and Plane
What Is the Distance between Point and Plane?
The distance between a point and a plane is equal to the length of the perpendicular drawn to the plane from the given point.
You can draw an infinite number of line segments from a given point to a plane. However, the distance between a point and a plane is the shortest distance possible between the point and the plane.
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Formula for Distance between a Point and a Plane
The distance between the point P(x 0,y 0,z 0) and the plane with the equation
A x+B y+C z+D=0 is given by the formula:
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
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How to Find the Distance from the Point to the Plane
Now that we know the formula to calculate the shortest distance from point to plane, we simply substitute the values in the formula and simplify.
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
Step 1: Write down the values of x 0,y 0,z 0 from the coordinates of the point.
Step 2: Write down the values of A, B,C, and D from the equation of plane.
Step 3: To find the value in the numerator, replace the values of x, y, and z in the equation of the plane by by x 0,y 0, and z 0 respectively. Write the absolute value of the final answer.
Step 4:To find the value in the denominator, find the square root of the sum of squares of A, B, and C.
Step 5:Simplify.
How to Apply the Distance from Point to Plane Formula
Let’s understand how to find the distance between a point and a plane with the help of an example.
Example: Determine the distance between the pointP=(3,1,2)and the plane3 x+4 y+z+3=0.
Step 1:Comparing the coordinates of the point P=(3,1,2) with P=(x 0,y 0,z 0), we get
x 0=3,y 0=1,z 0=2
Step 2:Comparing the equation of the plane 3 x+4 y+z+3=0 with
A x+B y+C z+D=0, we get
A=3,B=4,C=1,D=3
Step 3: Substitute the values in the formula and simplify.
We know that the formula for distance between point and plane is
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
Substituting the values in the formula, we get
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
d=|3×3+4×1+1×2+3|(9+16+1)
d=|9+4+2+3|9+16+1
d=18 26
d=9 26 13 units
Facts about Distance between Point and Plane
If the given point lies on the given plane, then the distance (D) between the point and the plane is equal to zero.
The distance between origin and plane A x+B y+C z+D=0 is given by
d=|D|A 2+B 2+C 2
Distance between the point P=(x 0,y 0) and the line A x+B y+C=0 is given by
d=|A x 0+B y 0+C|A 2+B 2
Distance between the point (x, y, z) and the xy-plane is |z|.
Distance between the point (x, y, z) and the yz-plane is |x|.
Distance between the point (x, y, z) and the xz-plane is |y|.
Conclusion
In this article, we learned how to find the distance between a point and the plane, the formula, and application of the formula. Now, let’s solve some practice problems and examples for better understanding.
Solved Examples on Distance Between Point and Plane
1. Find the distance between the point(−5,−8,−6)and the plane−2 𝑥+𝑦+2 𝑧=7.
Solution:
We know that the formula for distance between point and plane is:
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
Here, A=−2,B=1,C=2,D=−7,
x 0=−5,y 0=−8,z 0=−6
Substituting the values in the formula, we get
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
d=|−2×−5+1×−8+2×−6+−7|(4+1+4)
d=|(10)+(−8)+(−1 2)+−7|9
d=|−17|9
d=173 units
The distance between the point (−5,−8,−6) and the plane −2 𝑥+𝑦+2 𝑧=7 is 173 units.
2. Find the distance between the point(2,−1,3)and the plane−2 𝑥+2 𝑦+𝑧=3.
Solution:
We know that the formula for distance between point and plane is:
d(P,π)=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
(x 0,y 0,z 0)=(2,−1,3),
𝑎=−2,𝑏=2,𝑐=1, and 𝑑=−3
Substituting these values, we get
𝑑=(−2)(2)+(2)(−1)+(1)(3)−3(4+4+1)
𝑑=|(−4)+(−2)+(3)+−3|9
𝑑=|−6|9
𝑑=6 3
𝑑=2 units
3. Find the distance from the point (1, 2, 0) to the plane3 x−4 y−5 z−2=0.
Solution:
We know that the formula for distance between point and plane is:
d(P,π)=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
Here, A=3,B=−4,C=−5,D=−2,x 0=1,y 0=2,z 0=0
Substituting the values in the formula, we get
d=|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
d=|3×1+(−4)×2+(−5)×0−2|3 2+(−4)2+(−5)2
d=|(3)+(−8)+(0)−2|9+16+25
d=7 50
d=7 5 2 units
The distance from the point (1, 2, 0) to the plane 3 x−4 y−5 z−2=0 is 7 5 2 units
Practice Problems on Distance between Point and Plane
Distance Between Point and Plane: Definition, Formula, Examples, FAQs
Attend this quiz & Test your knowledge.
1
The shortest distance from a point to a plane is the length of the ____ to the plane drawn from the point.
perpendicular
Line segment
adjacent
None of the above
Correct Incorrect
Correct answer is: perpendicular
The shortest distance from a point to a plane is the length of a perpendicular to the plane.
2
If the given point lies on the given plane, then the distance between the point and the plane is equal to _.
2
1
0
None of the above
Correct Incorrect
Correct answer is: 2
If the given point lies on the given plane, then the distance between the point and the plane is equal to zero.
3
Distance between the point (1, 2, 3) and xy - plane is _
1
2
3
0
Correct Incorrect
Correct answer is: 3
Distance between the point (x, y, z) and xy - plane is | z | = 3 units.
4
The formula for distance between origin and plane A x+B y+C z+D=0 is
|D|A 2+B 2+C 2
|A x 0+B y 0+C z 0+D|A 2+B 2+C 2
|D|A 2−B 2−C 2
|D|
Correct Incorrect
Correct answer is: |D|A 2+B 2+C 2
The distance between point and plane is calculated using the formula:
d=|D|A 2+B 2+C 2
Frequently Asked Questions about the Distance between Point and Plane
What is the formula for finding the distance between two points?
The distance between two points with coordinates (x 1,y 1,z 1) and (x 2,y 2,z 2) is defined by the distance formula d = \sqrt{(x_{2}\;-\;x_{1})^{2} + (y_{2}\;-\;y_{1})^{2} + (z_{2}\;-\;z_{1})^{2}}$.
What is the distance between a point and the XZ plane?
The distance between a point (x 0,y 0,z 0) and x – zplane is given by the absolute value of the y-coordinate, |y 0|.
What is the distance between point and plane when the point lies on the given plane?
If the given point lies on the given plane, it satisfies the equation of the plane, and thus the distance between the plane and the point on the plane is zero.
What is the shortest distance between a point to a plane?
The shortest distance from a point to a plane is the length of the perpendicular drawn from the point to the plane.
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6568 | https://juniperfamilyhealth.com/blog/2023/6/20/contraindications-to-hormone-replacement-therapy-by-dr-meghan-van-drimmelen-nd | Contraindications to Hormone Replacement Therapy - By Dr. Meghan van Drimmelen, ND
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Contraindications to Hormone Replacement Therapy - By Dr. Meghan van Drimmelen, ND
June 20, 2023Meghan van Drimmelen
As a naturopathic doctor who regularly prescribes hormone replacement therapy (HRT) to women going through perimenopause and menopause, I understand the importance of individualizing treatment plans to maximize benefits and minimize risks. While HRT remains the most effective treatment for vasomotor symptoms (VMS) and the genitourinary syndrome of menopause, it is important to be aware of the contraindications for this therapy. Women who have contraindications for HRT may still benefit from other therapies, such as non-hormonal treatments for VMS or low-dose vaginal estrogen therapy for genitourinary syndrome of menopause.
Contraindications to HRT
Based on the North American Menopause Society’s (NAMS) guidelines, HRT (which includes Health Canada approved bioidentical hormone therapy) is contraindicated in the following situations:
1. History of breast cancer
Women who have a history of breast cancer should not use HRT. This is because estrogen can stimulate the growth of breast cancer cells.
2. History of endometrial cancer
Women who have a history of endometrial cancer should not use HRT. This is because estrogen can increase the risk of endometrial cancer. However, women who have had a hysterectomy (removal of the uterus) can use HRT without an increased risk of endometrial cancer.
3. History of blood clots
Women who have a history of blood clots (deep vein thrombosis or pulmonary embolism) or who have a personal history of a high risk blood clotting disorder should not use HRT. This is because estrogen can increase the risk of blood clots. Women who have a history of stroke or heart attack should also avoid HRT.
4. Liver disease
Women who have liver disease should not use HRT. This is because the liver plays a key role in metabolizing estrogen, and liver disease can affect the way the body processes estrogen.
5. Abnormal vaginal bleeding
Women who have unexplained vaginal bleeding should not use HRT until the cause of the bleeding has been determined. This is because HRT can mask the symptoms of certain gynecological conditions, such as endometrial cancer.
HRT can be an effective treatment for menopausal symptoms, but it is not suitable for everyone. Women who have a history of breast cancer, endometrial cancer, blood clots, a blood clotting disorder, liver disease, or abnormal vaginal bleeding should not use HRT. If you are considering HRT, it is important to discuss the risks and benefits with your naturopathic doctor or other health care provider trained in HRT prescribing. They can help you determine if HRT is right for you and, if so, what type and dose of HRT is appropriate.
Book in for an initial naturopathic consultation to learn more about your options. We are here to help!
In health and happiness,
Dr. Meghan van Drimmelen, ND | Naturopath Victoria BC
Reference:
NAMs Position Statement: The 2022 hormone therapy position statement of The North American Menopause Society
InWomen's HealthTagsMenopause, Perimenopause, Postmenopause, Hormones, Hormone Replacement Therapy, Bioidentical Hormones, Naturopath Victoria BC, Naturopathic Medicine
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← Understanding the Benefits & Risks of Menopausal Hormone Therapy - By Dr. Meghan van Drimmelen, ND3 Nutrients Worth Testing for Fertility - By Dr. Meghan van Drimmelen, ND →
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Mechanical equivalent of heat
Mechanical energy can be converted into heat, and heat can be converted into some mechanical energy. This important physical observation is known as the mechanical equivalent of heat. This means one can change the internal energy of a system by either doing work to the system, or adding heat to the system. This concept is fundamental to thermodynamics which applies the ideas of heat and work in order to create useful systems such as engines, power plants, and refrigerators. This equivalence of heat and motion was tested in a classic experiment by James Joule in 1843, who used the change in potential energy of falling masses to stir water. The water increased in temperature, much like it would when put over a flame. This showed that the downward motion of the masses which caused the water to be stirred (a form of mechanical motion) could in fact be equated to an increase in water temperature—or an increase in the heat,
This idea of work and heat equivalence is stated in the First law of thermodynamics, which says that the change in internal energy of a system is the sum of the work done and the heat added to any system. From this, if a system is observed at any state, it is impossible to tell whether it reached this state from an input of work, an input of heat, or a combination of the two. This is shown in Figure 2 below.
For Further Reading
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References
↑ Wikimedia Commons [Online], Available:
↑ Hyperphysics, Mechanical equivalence of heat [Online], Available:
↑ M. Zemansky, The Use and Misuse of the Word "Heat" in Physics Teaching [PDF], Available:
↑ Hyperphysics, Heat and Work Example [Online], Available:
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6570 | https://www.vitalsource.com/products/a-textbook-of-fluid-mechanics-and-hydraulic-rajput-r-k-v9789385401374?srsltid=AfmBOorKqw4a1glFuWdDLyW1tiv9NXKGqIy7LrQWEuOf_eVRgPk8ubdh | A Textbook of Fluid Mechanics and Hydraulic Machines (In SI Units)
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6571 | https://www.wordhippo.com/what-is/the-opposite-of/disjointed.html | What is the opposite of disjointed?
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What is the opposite of disjointed? ======================================= Need antonyms for disjointed? Here's a list of opposite words from our thesaurus that you can use instead. ====================================================================================================================== Contexts ▼▲ Adjective Opposite of lacking a coherent sequence or connection Opposite of consisting of disorganized, untidy, or incomplete parts Opposite of not clearly expressed or pronounced Opposite of standing on its own apart from the others Opposite of in a state of confusion Opposite of lacking a coherent sequence or connection Opposite of not flowing smoothly, especially of speech or writing Opposite of having breaks or interruptions Opposite of having a deformed shape, especially one that is unsightly Opposite of made up of abruptly disconnected parts or sounds Opposite of not having all the necessary or appropriate parts Opposite of forming or viewed as a unit apart or by itself Verb Opposite of to have thrown into disorder Opposite of past tense for to force or break apart … more ▼▲ Adjective ▲ Opposite of lacking a coherent sequence or connection coherent connected contiguous jointed ordered united intelligible relevant logical meaningful continuous understandable Adjective ▲ Opposite of consisting of disorganized, untidy, or incomplete parts uniform complete coherent all-embracing unified comprehensive cohesive Adjective ▲ Opposite of not clearly expressed or pronounced articulate clear coherent comprehensible fluent intelligible well-spoken communicative eloquent understandable silver-tongued perfect natural fluid smooth voluble well-versed expressive lucid comprehendible grandiloquent graspable logical rational fathomable illuminating magniloquent unambiguous perspicuous pellucid discernable apprehensible cogent transpicuous smooth-tongued clear-cut smooth-spoken clearly expressed easy to understand Adjective ▲ Opposite of standing on its own apart from the others together attached combined connected sympathetic unified united whole joined entwined linked tied associated coupled integrated bound merged conjoined conjugated touching affixed inseparable banded cemented fused fixed interfused welded Adjective ▲ Opposite of in a state of confusion ordered neat orderly organisedUK organizedUS shipshape tidy bandbox crisp kempt neatened snug tidied trim uncluttered well-ordered systematic straight well-organized methodical structured systematizedUS standardizedUS coordinated methodized logical methodic controlled catalogedUS arranged efficient formulated well thought-out tabulated regulated well-thought-out standardisedUK well-defined cataloguedUK meticulous fastidious systematisedUK Adjective ▲ Opposite of lacking a coherent sequence or connection clear exact lucid precise Adjective ▲ Opposite of not flowing smoothly, especially of speech or writing flowing smooth Adjective ▲ Opposite of having breaks or interruptions continuous constant habitual periodic regular repeated steady connected unbroken permanent lasting Adjective ▲ Opposite of having a deformed shape, especially one that is unsightly undeformed shapely well-built well-proportioned Adjective ▲ Opposite of made up of abruptly disconnected parts or sounds legato Adjective ▲ Opposite of not having all the necessary or appropriate parts accomplished complete developed finished full perfect unified whole adequate ample enough sufficient total from A to Z Adjective ▲ Opposite of forming or viewed as a unit apart or by itself dependent connected interdependent appurtenant conditional contingent determined by provisory reliant subject to “The characteristic flavor of any coffee is dependent on its origin and the degree of roast.” Verb ▲ Opposite of to have thrown into disorder arranged organizedUS ordered aligned arrayed assembled classified filed structured tidied catalogedUS categorizedUS classed grouped systematizedUS coordinated marshaledUS marshalledUK methodized ranged regulated standardizedUS tabulated tidied up cataloguedUK categorisedUK neatened organisedUK put in order sorted out spruced up standardisedUK straightened straightened up systematisedUK fixed up alined lined up systemized disposed drew up drawn up “Our documents were in a mess and we arranged them to restore some organization into our filing system.” Verb ▲ Opposite of past tense for to force or break apart joined linked unified united retained Find more words! Use for blank tiles (max 2)Advanced SearchAdvanced Search Use for blank spacesAdvanced Search Advanced Word Finder Related Words and Phrases ------------------------- disjoint disjointedness disjointly disjointedly disjoints disjointing disjointednesses See Also -------- What is another word for disjointed? Sentences with the word disjointed Words that rhyme with disjointed What is the adverb for disjointed? What is the adjective for disjointed? What is the noun for disjointed? What is the opposite of disjoint? Use our Antonym Finder Nearby Words disjointedly disjointedness disjointednesses disjointing disjointly disjoints disjoint disjoins disjoining disjoined disjoin disir 10-letter Words Starting With d di dis disj disjo disjoi disjoin disjoint disjointeFind Antonyms go
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6572 | https://artofproblemsolving.com/community/c2437h1033423s3_polynomial_game?srsltid=AfmBOopHBvd1kHF84U-OpW5F-XKj93Fjc4Zf2EdEik_JpQN2Wj0ChCxy | An Excursion through Mathematical problems : Polynomial game
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An Excursion through Mathematical problems ==========================================
Polynomial game
by boywholived, Oct 6, 2012, 12:02 PM
A polynomial has three undetermined coefficients denoted by stars. The players A and B move alternately, replacing a star by a real number until all stars are replaced. A wins if all zeros of the polynomial are complex.B wins if at least one zero is real. Show that B can win in spite of his only second move.
This problem is not yet unsolved(on AOPS) as on Oct 06 2012
The above is the link where I have posted. Check it to find any recent improvement
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I am sad to call me a dictator since I dictate this shout box.. Please somebody else fight..
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Hi everyone just leave a shout as you wish by scrolling down and typing in the box and click on submit
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Leave a shout by just scorlling down and typing in the box..
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@Magnolia, Thanks.
But that's not mine all credits to those who made the CSS.
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Like your CSS!
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@pi_Plus_45x23,. Now you are a contributer.
Eagerly waiting to see your first post.
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Hey all, I updated my theme. How is it?>
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Hurray.
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Please feel free to express your opinions and solutions to problems.
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@mathpassion.. I will send a private message with a breif solution.
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Complex number Summation is a tough question, Please post solution too.
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@gh6r Thanks for your shout.. Hoping more shouts and your opinion to the posts in future
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Please leave your comments about the problems
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Hello shouting is fun. like being the fifth. But who is sixth ??
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hello I am the fourth one to shout...well Who is the fifth?
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6573 | https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-38/issue-4/Integer-Solutions-to-the-Equation-y2xx2pm-pk/10.1216/RMJ-2008-38-4-1285.pdf |
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6574 | https://codes.findlaw.com/ga/title-10-commerce-and-trade/ga-code-sect-10-6a-3/ | Georgia Code Title 10. Commerce and Trade § 10-6A-3 | FindLaw
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FindLaw/
Codes/
Georgia/
Title 10. Commerce and Trade/
§ 10-6A-3
Georgia Code Title 10. Commerce and Trade § 10-6A-3
Current as of January 01, 2024 | Updated byFindLaw Staff
As used in this chapter, the term:
(1) “Agency” means every relationship in which a real estate broker acts for or represents another as a client by the latter's written authority in a real property transaction.
(2) “Broker” means any individual or entity issued a broker's real estate license by the Georgia Real Estate Commission pursuant to Chapter 40 of Title 43. The term “broker” includes the broker's affiliated licensees except where the context would otherwise indicate.
(3) “Brokerage” means the business or occupation of a real estate broker.
(4) “Brokerage engagement” means a written contract wherein the seller, buyer, landlord, or tenant becomes the client of the broker and promises to pay the broker a valuable consideration or agrees that the broker may receive a valuable consideration from another in consideration of the broker producing a seller, buyer, tenant, or landlord ready, able, and willing to sell, buy, or rent the property or performing other brokerage services.
(5) “Brokerage relationship” means the agency and nonagency relationships which may be formed between the broker and the broker's clients and customers, as described in this chapter.
(6) “Client” means a person who is being represented by a real estate broker in an agency capacity pursuant to a brokerage engagement.
(7) “Common source information companies” means any person, firm, or corporation that is a source, compiler, or supplier of information regarding real estate for sale or lease and other data and includes but is not limited to multiple listing services.
(8) “Customer” means a person who is not being represented by a real estate broker in an agency capacity pursuant to a brokerage engagement but for whom a broker may perform ministerial acts in a real estate transaction pursuant to either a verbal or written agreement.
(9) “Designated agent” means one or more licensees affiliated with a broker who are assigned by the broker to represent solely one client to the exclusion of all other clients in the same transaction and to the exclusion of all other licensees affiliated with the broker.
(10) “Dual agent” means a broker who simultaneously has a client relationship with both seller and buyer or both landlord and tenant in the same real estate transaction.
(11) “Material facts” means those facts that a party does not know, could not reasonably discover, and would reasonably want to know.
(12) “Ministerial acts” means those acts described in Code Section 10-6A-14 and such other acts which do not require the exercise of the broker's or the broker's affiliated licensee's professional judgment or skill.
(12.1) “Option to enter into a brokerage engagement” means a written contract wherein a broker pays valuable consideration or agrees that a seller, buyer, landlord, or tenant may receive a valuable consideration from another in consideration of such seller, buyer, landlord, or tenant granting such broker an exclusive right to enter into a brokerage engagement with such seller, buyer, landlord, or tenant during the term of the written contract.
(13) “Timely” means a reasonable time under the particular circumstances.
(14) “Transaction broker” means a broker who has not entered into a client relationship with any of the parties to a particular real estate transaction and who performs only ministerial acts on behalf of one or more of the parties, but who is paid valuable consideration by one or more parties to the transaction pursuant to a verbal or written agreement for performing brokerage services.
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6575 | https://www.youtube.com/watch?v=pEW9j_Q2fIo | If the equation ax^2 + 2x + a = 0 has two distinct roots, if
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If the equation ax^2 + 2x + a = 0 has two distinct roots, if
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फाइंड देखिए क्या का रहा है आईएफ डी इक्वेशन ax² + 2x + ए = 0 हे तू डिस्टिंगू तू आईएफ अच्छा इसके दो डिस्टिंग्विश में से किसको सरफेस फी करेगा यहां पर देखिए दिया हुआ है ax² + 2x + 8 = यहां पे क्या का रहा है किधर इक्वेशन और डी जिवन इक्वेशन है तू डिस्टिंक्ट [संगीत] रूट इसके दो डिस्टिंग्विश है तो दो डिस्टेंस [संगीत] 17 के बीच होगा आप माइंस वन सेवन के बीच का कोई भी वैल्यू रखेंगे तो यह वन से ली दें होगा उसका स्क्वायर [संगीत] माइंस वन से वन के बीच में हो रहा है अब ऑप्शन कौन सा देखिए ए का वैल्यू प्लस माइंस वन नहीं हो सकता है ये सही है सही ना सही है जीरो आप जीरो हो सकता है तो यहां इनके बीच में लाइक कर रहा है ना जी लेकिन बहुत सारे रियल करेगा यह भी नहीं करेगा [संगीत] |
6576 | https://www.coursehero.com/file/170320746/Chp09-Homeworkpdf/ | Velocity Profiles and Flow Calculations in Hydraulic Systems | Course Hero
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1 of 3 By Thabang Mafokoane (May2020)CHAPTER 9: VELOCITY PROFILES FOR CIRCULAR SECTIONS AND FLOW IN NONCIRCULAR SECTIONS HOMEWORK PROBLEMS
9.4. Compute points on the velocity profile from the tube wall to the centerline of a standard hydraulic steel tube, 50 mm OD × 1.5 mm wall, if the volume flow rate of SAE 30 oil (sg =0.89) at 110C is 25 L/min. Use increments of 4.0 mm and include the velocity at the centerline.
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9.10.A large pipeline with a 1.200-m inside diameter carries oil similar to SAE 10 at 40C (sg= 0.8). Compute the volume flow rate required to produce a Reynolds number of 3.60 × 10 4.Then, if the pipe is clean steel, compute several points of the velocity profile and plot the data in a manner similar to that shown in Fig. 9.7.
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9.20.Figure 9.14 shows a heat exchanger in which each of two DN 150 Schedule 40 pipes carries 450 L/min of water. The pipes are inside a rectangular duct whose inside dimensions are 200 mm by 400 mm. Compute the velocity of flow in the pipes. Then, compute the required volume flow rate of water in the duct to obtain the same average velocity.
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9.22.Air with a specific weight of 12.5 N/m3 and a dynamic viscosity of 2.0 × 10-5 Pa.s flows through the shaded portion of the duct shown in Fig. 9.16 at the rate of 150 m 3/h. Calculate the Reynolds number of the flow.
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9.26.Refer to Fig. 9.14, which shows two DN 150 Schedule 40 pipes inside a rectangular duct.Each pipe carries 450 L/min of water at 20˚C. Compute the Reynolds number for the flow of water. Then, for benzene (sg = 0.862) at 70˚C flowing inside the duct, compute the volume flow rate required to produce the same Reynolds number.
2 of 3
By Thabang Mafokoane (May2020)9.28.Water at 10˚C is flowing in the shell shown in Fig. 9.18 at the rate of 850 L/min. The shell is a 50 mm OD × 1.5 mm wall copper tube and the inside tubes are 15 mm OD × 1.2 mm wall copper tubes. Compute the Reynolds number for the flow.9.30.Figure 9.20 shows a liquid-to-air heat exchanger in which air flows at 50 m 3/h inside a rectangular passage and around a set of five vertical tubes. Each tube is a standard hydraulic steel tube, 15 mm OD × 1.2 mm wall. The air has a density of 1.15 kg/m3 and a dynamic viscosity of 1.63 × 10-5 Pa.s. Compute the Reynolds number for the air flow.
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3 of 3 By Thabang Mafokoane (May2020)9.39.For the system described in Problem 9.26, compute the pressure drop for both fluids between two points 3.80 m apart if the duct is horizontal. Use the roughness for steel pipe for all surfaces.9.41.For the shell-and-tube heat exchanger described in Problem 9.28, compute the pressure drop for the flow of water in the shell. Use the roughness for copper for all surfaces. The length is 3.60 m.9.43.For the glycerin described in Problem 9.31, compute the pressure drop for a horizontal duct 22.6 m long. All surfaces are copper.Source:Mott. R.L and Untener, J.A. (2017). Applied Fluid Mechanics (seventh edition, global edition). Boston: Pearson Education Limited.
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NRICH topics: Properties of numbers Modular arithmetic
Resources tagged with: Modular arithmetic
There are 51 NRICH Mathematical resources connected to Modular arithmetic, you may find related items under Properties of numbers.
Broad Topics > Properties of numbers > Modular arithmetic
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How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Age 11 to 14
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Age 11 to 14
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On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.
Age 14 to 16
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Guesswork
Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.
Age 14 to 16
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Take Three from Five
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Age 11 to 16
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Where Can We Visit?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Age 11 to 14
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Differences
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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Prime AP
What can you say about the common difference of an AP where every term is prime?
Age 16 to 18
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Days and Dates
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Age 11 to 14
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Happy birthDay
Can you interpret this algorithm to determine the day on which you were born?
Age 16 to 18
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Zeller's Birthday
What day of the week were you born on? Do you know? Here's a way to find out.
Age 14 to 16
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Elevens
Add powers of 3 and powers of 7 and get multiples of 11.
Age 16 to 18
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The Chinese Remainder Theorem
In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."
Age 14 to 18
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An Introduction to Modular Arithmetic
An introduction to the notation and uses of modular arithmetic
Age 14 to 18
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The Knapsack Problem and Public Key Cryptography
An example of a simple Public Key code, called the Knapsack Code is described in this article, alongside some information on its origins. A knowledge of modular arithmetic is useful.
Age 16 to 18
Problem Primary curriculum Secondary curriculum
Knapsack
You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.
Age 14 to 16
Challenge Level
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The Public Key
Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.
Age 16 to 18
Challenge Level
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Double Time
Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.
Age 16 to 18
Challenge Level
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Readme
Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.
Age 16 to 18
Challenge Level
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Modular Fractions
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
Age 16 to 18
Challenge Level
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Transposition Fix
Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?
Age 14 to 16
Challenge Level
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Check Code Sensitivity
You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.
Age 14 to 16
Challenge Level
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Check Codes
Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?
Age 14 to 16
Challenge Level
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Obviously?
Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.
Age 14 to 18
Challenge Level
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Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.
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Find the remainder when 3^{2001} is divided by 7.
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6578 | https://www.coursehero.com/file/75177034/chapter-3-lecture-notespdf/ | chapter 3 lecture notes.pdf - Engineering Electromagnetics W.H. Hayt Jr. and J. A. Buck Chapter 3: Electric Flux Density Gauss' Law and | Course Hero
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Engineering Electromagnetics W.H. Hayt Jr. and J. A. Buck Chapter 3:Electric Flux Density, Gauss’ Law,and Divergence
Faraday Experiment He started with a pair of metal spheres of different sizes; the larger one consisted of two hemispheres that could be assembled around the smaller sphere
+Q Faraday Apparatus, Before Grounding The inner charge, Q, induces an equal and opposite charge, -Q,on the inside surface of the outer sphere, by attracting free electrons in the outer material toward the positive charge. This means that before the outer sphere is grounded, charge +Q resides on the outside surface of the outer conductor.
Faraday Apparatus, After Grounding q = 0 ground attached Attaching the ground connects the outer surface to an unlimited supply of free electrons, which then neutralize the positive charge layer. The net charge on the outer sphere is then the charge on the inner layer, or -Q.
Interpretation of the Faraday Experiment q = 0 Faraday concluded that there occurred a charge “displacement” from the inner sphere to the outer sphere.Displacement involves a flow or flux, Ψ,existing within the dielectric, and whose magnitude is equivalent to the amount of “displaced” charge.Specifically:
Electric Flux Density q = 0 The density of flux at the inner sphere surface is equivalent to the density of charge there (in Coul/m 2)
Vector Field Description of Flux Density q = 0
A vector field is established which points in the direction of the “flow”or displacement. In this case, the direction is the outward radial direction in spherical coordinates. At each surface,we would have:
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6579 | https://www.sciencedirect.com/science/article/pii/S2210833513000282 | Malignant otitis externa: An emerging scourge - ScienceDirect
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Journal of Clinical Gerontology and Geriatrics
Volume 4, Issue 4, December 2013, Pages 128-131
Case report
Malignant otitis externa: An emerging scourge
Author links open overlay panelS. Prasanna Kumar, A. Ravikumar MS, DNB, DLORCS, MAMS, L. Somu MS, Nazrin Mohd Ismail MBBS
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Abstract
Malignant otitis externa is an emerging problem in developing countries. We are seeing an increase in this condition in elderly people with diabetes. This paper reports on a retrospective chart review of 18 patients with malignant otitis externa in a tertiary care center. We have comprehensively discussed the clinical features, diagnosis, and management of this disease. Malignant otitis externa is suspected in high-risk patients, i.e., elderly individuals with diabetes, immunocompromised, and immunosuppressed patients. Early and prolonged use of appropriate antimicrobials will reduce the morbidity and mortality associated with this condition. The role of surgery is limited to biopsy, the draining of microabscesses, or the removal of sequestrum.
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6580 | https://www.scribd.com/document/447093252/Pigeonhole-Principle-Real-Life-Applicati-pdf | Pigeonhole Principle Real Life Applicati PDF | PDF | Combinatorics | Logic
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Pigeonhole Principle Real Life Applicati PDF
This document discusses the pigeonhole principle, a mathematical concept stating that if n items are put into m containers where n > m, then at least one container must contain more than one…
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Pigeonhole Principle: the Real Life Applications and Mathematical Investigation
Pengsheng GUO, Qing YU, Yang WANG, Yiwei GONG March 2013
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Contents
1 Bac kgr oun d an d Hi sto ry22 Re al Li fe Ca se s2
2.1 Bir thda y Pa rado x. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Ap pl e Pr obl em. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 F und ame nt al P roof3
3.1 Fir st Pig eonh ole Pri nci ple. . . . . . . . . . . . . . . . . . . . . . . 33.2 Seco nd Pig eonh ole Prin cip le. . . . . . . . . . . . . . . . . . . . . . 3
4 Ex te ns io n of th e The or y4
4.1 Ave rage V alue Princip le, A VP. . . . . . . . . . . . . . . . . . . . . 44.2 Ram se y Th eo rem a nd i ts P roof. . . . . . . . . . . . . . . . . . . . 54.3 Prin cip le of Inc lus ion a nd Ex clu sio n. . . . . . . . . . . . . . . . . . 6
5 Appl icat ions o f Pige onhol e Pri ncip le7
5.1 Decisi on Mak ing St rategy for Opt imum Outcome Using Pigeonh olePrinciple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.1.1 Si mp le Ca se s. . . . . . . . . . . . . . . . . . . . . . . . . . 75.1.2 Adv ance d Cas e: Av erag e V alu e Prin cip le. . . . . . . . . . . 85.1.3 Cas e of the P rinc ipl e of In clu sio n and Ex clu sio n. . . . . . . 95.2 Qu est io n 6 f rom 20t h IM O. . . . . . . . . . . . . . . . . . . . . . . 9
6 Rai sed Que sti ons11
6.1 Ram se y Nu mb er. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 16.2 Gen eral ise d I.E.P. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
7 Co nc lu si on12
1
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O
ver hundr e ds of thousa nds of ye ars, mathe matic ians had disc over e d the Pi-geonhole Principle and applied this law into real life applications. Pigeonhole Principle presents the most essential and basic part in the mathematics of count-ing and sorting. This research paper introduces the topic of Pigeonhole Principle,including theorems born from this basis and discusses several cases related to the prin cip le. Thi s p ap er will als o pr ese nt the fundam ent al pr o of of the the ory and some advanced questions in abstract mathematics which are related to the theory of sets and graphs.
1 Ba c kg ro un d a nd Hi st or y
The first statement of the Pigeonhole Principle was made by Johann Peter Gustav Lejeune Dirichlet(1805-1859) in 1834 under the name
Schubfachprinzip
(“drawer principle” or “shelf principle”). For this reason it is also commonly called Dirich-let’s box principle, Dirichlet’s drawer principle or simply “Dirichlet principle”. The phrase “Pigeonhole Principle” was firstly used in a serious mathematics journal by mathematician Raphael M. Robinson in the year of 1940.The original “drawer” name is still in use in French (“principe des tiroirs”),Polish (“zasada szufladkowa”), Italian (“principio dei cassetti”), German (“Schub-fachp rinzip”), and Chinese(“
”).This simple assertion has been used in many applications that range from computer data compression to problems that involve infinite sets that cannot be put into one-to-one correspondence. It has been generalised to probabilistic appli-cations as well
1
.
2 R e a l Li f e C a s e s
2.1 Bi rt hd a y P ar ad o x
2
On one hot summer day, a primary school is holding a big birthday celebrating party for the old headmaster, Professor Edison. On the party, Mark, a-grade-two pupil, asked his mother a question:”Mom, how many people are there on the party of Professor Edison?”“367.” His mother answered.“Is there anyone sharing the same birthday?”
1
If
n
pigeons are
randomly
placed in
m
pigeonholes with uniform probability of 1
/m
, then at least one pigeonhole will hold more than one pigeon with probability of
1
−
m
!(
m
−
n
)!
m
n
2
this is the Pigeonhole Principle part of birthday paradox, for the full paradox please refer to”en.wikipedia.org/wiki/Birthday paradox”
2
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“No, I don’t think so, everyone’s birthday is different.”“You must have counted wrongly.” Mark smiled.“No it couldn’t be. I counted it twice.” Mark’s mother argued.“In the way of mathematics, the situation that 367 students in the house sharing the same birthday can’t happen.”...(This story is an example of the First Pigeonhole Principle)
2.2 Ap pl e Pr ob le m
Professor Francois de Ciel, who is teaching at a primary school, wants to teach his class about simple calculating this morning. He intends to use apples as bonus when the kids are getting correct answers.When he is holding those fresh apples from bracket in his hands, an idea came to his mind: can he distribute those 19 apples to 9 children with everyone got two apples and no apple is left? If he does not eat any apple.The answer must be impossible, but what’s the reason behind it?...(This story is an example of the Second Pigeonhole Principle)
3 F un da me n ta l Pr oof
3.1 Fi rst Pi ge on ho le Pr in ci pl e
If
n
items are put into
m
pigeonholes with
n >m
(
m,n
∈
N
∗
), then at least one pigeonhole must contain more than one item.
Proof.
Assume all the pigeonholes contains at most one item, then the maximum number of the total items that the
n
pigeonholes can contain is
n
, which is less than
m
.By contradiction, there must be at least one pigeonhole contains more than one item.Proved.
3.2 Se co nd Pi ge on ho le Pr in ci pl e
If (
mn
−
1) items ate put into n pigeonholes where
m,n
∈
N
∗
, then there will be at least one pigeonhole must contain less than
m
item(s).3
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6581 | https://www.youtube.com/watch?v=bjsYdM2oym8 | 16 - Simplify Logarithms - Part 1 (Log Bases, Calculate Logarithms & More)
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In this lesson, you will learn how to simplify logarithms in expressions and in equations. We will learn that a logarithm (log) is really the inverse of an exponential. Because of this, the log expression can be written in exponential form. Writing a logarithm as an exponential function is the easiest way to calculate logarithms.
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Transcript:
hello welcome back the title of this lesson is called simplifying logarithms this is part one of several up until this point we have covered the concept of the exponential function you should have already watched a lesson on what an exponential function is it's a number raised to a power of a variable such as like for instance 2 to the X 3 to the X those are exponential functions we've also covered in great detail the concept of a logarithm the logarithm is the inverse function of the exponential function so you also should have learned in the past lessons on what an inverse function is so logarithm an exponential function go together like peanut butter and jelly they're inverses of one another so there's two things I want you to remember in the back your mind as we go and we start simplifying logarithms and really applying the definition of the logarithm the first thing is I want you to always remember that a logarithm is the inverse function it's kind of the opposite function of an exponential and likewise the exponential is an opposite function of a logarithm so later on down the road we're going to use an Exponential's and logarithms to undo each other we're going to use them to solve equations because we can undo those those functions with the inverse function right and the second thing is I want you to understand that a logarithm when you calculate what it's doing it's giving you back an exponent we talked about that in the last lesson when we introduce logarithms when you calculate logarithm of 10 a logarithm base 2 of 15 or logarithm base 5 of 9 or whatever the calculation you get back is it's returning the exponent back the exponent required to calculate that number that you're taking the logarithm of so here we go simplifying some logs I want you to remember both of those things which we've covered in the past so here's our first little problem logarithm base 5 of 125 now all of these logarithm problems are going to have a base involved just like all Exponential's they have to have a base to define the exponential two to the power of X 10 to the power of X 5 to the power of X those numbers are the bases of the exponential and because a logarithm is the inverse of an exponential logarithms have a base as well the base here is 5 now what you need to do to translate this to solve this to simplify this is you need to translate it as follows the base which is 5 raised to the power of something which what this whole thing is gonna equal to I'm gonna call it X it's gonna be equal to 125 this is gonna be how we solve every one of these logarithm problems when you see a logarithm what it's telling you is there's some exponent that is going to end up being this value of X that's gonna be such that 5 the base raised to that exponent gives you what you're taking the logarithm of so that's why I'm telling you logarithms give you back they calculate for you the exponent required to do that remember that logarithms give you the exponent back the exponent here we're calling X so how do we calculate what X is equal to well you can do it a couple of different ways but let's write it like this on the right hand side we know that 125 can be written as 5 to the power of 3 5 times 5 is 25 25 times again 5 is 125 now we have the base is the same on the both sides of the equal sign so the exponents then must be equal x must be equal to 3 and so for your final answer what I want you to write down is the following thing I want you to say logarithm base 5 of the number 125 is equal to 3 this is what you would circle on your test if I say simplify this what I want to know is what is it's equal to what is it equal to if you dump it in a calculator or computer what do you get but more intuitively than that I want you to remember the logarithm gives me the exponent back in this case is giving me that exponent back of a 3 telling me that if I take this base 2 raised to this exponent I'm gonna get back whatever I'm calculating the logarithm of because the logarithm gives me the exponent back I've said that about 10 times I'll stop saying it I want you to remember it because a lot of times students kind of forget what logarithms are after you start using them for a while you kind of forget what you're doing after a while once you get a little more comfortable but as you get deeper in math you're going to need to remember what exactly they're doing all right next problem what if I have longer them base 3 of the number 81 again I'm gonna calculate this it's going to give me the exponent back what exponent I take 3 just whatever exponent this log is gonna calculate has got to be equal to the number I'm taking the logarithm of it's returning the exponent back now what exponent makes this work the easiest thing to do is try to get the right-hand side and put it in terms of this same base but you know that 9 times 9 is 81 so I can write that as 9 squared and then you remember that 9 can be written as 3 squared squared right so this is 9 squared and because I have an exponent raised to an exponent I can write this as 3 to the 4th power now if you just happen to know that 3 to the 4th power is 81 then you could skip all this stuff here but most of us don't remember that 3 to the 4th power is 81 so you go step by step you say well 9 times 9 81 oh I can write 9 it's 3 times 3 so now I have the same base and because I have the same base I now know that then X must be equal to 4 now generally there's no X in this problem anywhere so what I really want you to write down is the following logarithm base 3 of 81 is equal to 4 4 is the exponent returned by the logarithm function such that the base raised to this exponent is equal to what I'm taking the logarithm of all right let's go off to a much much easier problem let's go logarithm base 3 of the number 1 how do you calculate this anytime you see these things you're gonna do it the same way you're gonna translate this and say 3 raised to the power of something which is what this log is gonna calculate it's gonna be equal to whatever I'm taking the log of and you might say well in this problem I could get the basis to be the same so I can find the exponent I could get the basis to look the same so I can find this exponent here but here there's no way that I can get these bases the same but then you start thinking about it further and you say well wait a minute though 3 to the power of something is 1 anything to the 0 power is equal to 1 so if I put a 0 here 3 to the 0 is 1 it's something you're gonna have to remember when an exponent is 0 then it always evaluates and calculates to be the number 1 so log base 3 log of the number 1 is equal to 0 and that's the final answer there in fact a log of any base of the number one base whatever base whatever base you have logarithm of the number one is always going to be 0 if this were a base 10 then 10 to the power of something is 1 ok the X is still 0 if it was base 1717 to the power of thing is one something the exponent is always zero so logarithm no matter what base you have of the number one always gets you zero that's something you might want to remember but if you forget it you just calculate it again you know when you need to all right now for some reason fractions really trip up students so let's do one with a fraction logarithm base 5 of the number one over 25 so probably a better way to write it is to wrap this in parenthesis to tell you it's a base 5 logarithm of the number which is a fraction 1 over 25 some people just get very confused by that you cannot let fractions trip you up it's the same thing this logarithm is going to spit back and exponent you it's 5 raised to that exponent is when an equal to this number here 1 over 25 so you have to figure out what exponent does this so you really want to try to get them into the B of the same base so 5 so the power of X you can write this as 1 over 5 squared that's pretty close because you have a 5 there but this fraction makes it not quite right so how do you handle that then you remember oh wait a minute I can write this 1 over 5 squared as 5 to the negative 2 power now I have everything in the same form as before base is the same on both sides so then I can equate the exponents X is negative 2 but again there's no X in this problem so what I want you to really do is say the logarithm base 5 of the number 1 over 25 is equal to negative 2 that is the simplification here so it's returning an exponent back the reason it's a negative logarithm is because 5 raised to the negative 2 will be 1 over 5 squared which is exactly what you have here so if you thought about it long enough you probably could just figure out what the exponent was by thinking you know and kind of staring at it but the problem is when the problems get more complicated staring at the problem without doing anything is really dangerous because maybe you know you never figure out what to do so you have to try to get these bases the same that's the proper way to go all right so what if we have not just a fraction or a number but what if we have a radical in their logarithm base six of the number six times the square root of six now this six times the square root of six is is what the logarithm is is working on so it's this whole number here how do you proceed same exact thing six the base here to the power of some exponent what the logarithm is going to give me back is equal to this number here six square root of six now this is where most students will just stop they don't know what to do you understand it when you have numbers you can understand it when you have fractions you start throwing radicals in there they freeze they don't know what to do it's because in algebra you have to use what you already know you know that this radical can be written as an exponent so you have to write it as an exponent six multiplied by what would this be six to the power of one half that is what this radical is equal to the square root then you have the same base here and they're multiplied so this is six to the power of one and this is six to the power of one half so just like x times x squared you can add those exponents to give you X cubed here the bases are the same so I can add these exponents so what I'm going to have is 1 plus 1/2 now you may not remember what that is I remember what it is but a lot of people don't so if you have to add the number 1 plus 1/2 you would write this as 2 over 2 to get a common denominator and so what you're gonna get is 3 halves so when you add 1 plus 1/2 you get 3 halves so 6 to the power of X is 6 to the power of 3 halves adding these exponents together so you see you have it now where the bases are the same X then B has to be equal to 3 halves again there's no X in this problem I kind of use that to figure out the answer so really what you want to do is you want to say the logarithm base 6 logarithm of 6 times the square root of 6 is equal to 3 halves if you go and grab a calculator and dump it in there that's where you're gonna get 3 halves all right let's do another problem with radicals it seems to be one of the things that trip people up what if you have a logarithm a base 4 logarithm simply of the square root whoops wrote down the wrong number in here square root of 2 that's what you're taking the logarithm of you take the base to the power of something that this logarithm is going to give you back is equal to the square root of two but now you know that this square root you need to write it as an exponent two to the one halves one half like this now this base is a four in this base is a two they're not quite the same so I in order to figure this out I really need to change this for write it as two squared to the power of X all I did was take the four and make it two squared because I want to have the same base on both sides now I have a exponent raised to an exponent so really this is two to the power of two times X because two to M times X there and you get two to the one-half now what do I have same base on both sides of the equal sign so because of that I'd go over here and say that 2x this exponent is equal to one-half now I know a lot of you can do this in your head but if you can't this is what you do you say you want to divide by two so you have one half on the right hand side divided by two because I'm dividing by 2 to get X by itself but really two is really two over one so I'm gonna take this fraction division and change it to multiplication this fraction division becomes a multiplication I take the denominator flip it over so I have one half and then you have X is 1/4 all right and well you know I circled this but really what I want you to write is that the logarithm the base for logarithm of the square root of 2 whoops I can write 2 correctly is 1/4 base for logarithm of the square root of 2 is equal to 1/4 and you should check this out I mean if you go back over here and put it into this guy here the fourth root of this guy when you calculate it is going to give you the square root of 2 go and do a factor tree and you'll see that that's the case or another way to do it is to just go through here and you can plug in the 1/4 here and then 1/4 times 2 is gonna give you 1/2 which is exactly what you have on the right hand side so there's lots of different ways to verify what we're getting but never ever ever forget that these logarithms are returning back to you exponents because it's the opposite it's the backwards function of an exponential function that's what it is last problem just to get a little practice what if you have the logarithm it's a base seven logarithm of the following thing forty-nine but it's the cube root of 49 how do we calculate this alright well what you want to do is you want to say 7 to the power of something that this logarithm is going to return back to you is going to be equal to the cube root of 49 but you know that these cube roots can always be written as exponents so you always want to do that 49 so the 1/3 power because it's a cube root and you know that you have a 7 as a base 49 rings a bell with 7 so I know that I can write this 49 as 7 times 7 7 squared again raised to the 1/3 power but then I have a power raised to a power so what I'm gonna have 2 times 1/3 is 2/3 whoops let me write the 7 first it's going to be 2/3 when you multiply these guys together so now I have the same base and I can equate the exponents X is 2/3 and so what you want to say is logarithm base 7 of the cube root of 49 is equal to 2/3 and this is the final answer all right this is really important stuff you know because what happens a lot of times is we introduce a concept of a logarithm we teach you what it is we teach you kind of why we use it and then if you don't master these kind of essential core concepts of logarithms then when you try to do any kind of problem later on down the road like in calculus this is used extensively and you don't really know where the logarithm is or how to manipulate it then you get stuck and you start making silly mistakes because you don't really know what it is never ever ever forget logarithms give you back they calculate for you an exponent they're giving you the exponent back the exponent required to get this number when you take the base raised to that exponent to give you that number and then the second thing is never forget logarithms are just the inverse function of Exponential's we're gonna use that extensively going forward follow me on to the next lesson after you can solve all of these yourself we'll get some more practice with simplifying these logarithms |
6582 | https://dl.icdst.org/pdfs/files4/c469d51a20d35bdc3d8578eb2b9fe7ca.pdf | ZOOLOGY Integrated Principles of Fourteenth Edition Hickman Roberts Keen Larson I’Anson Eisenhour Cleveland P. Hickman, Jr. WASHINGTON AND LEE UNIVERSITY Larry S. Roberts FLORIDA INTERNATIONAL UNIVERSITY Susan L. Keen UNIVERSITY OF CALIFORNIA AT DAVIS Allan Larson WASHINGTON UNIVERSITY Helen I’Anson WASHINGTON AND LEE UNIVERSITY David J. Eisenhour MOREHEAD STATE UNIVERSITY Original Artwork by WILLIAM C. OBER, M.D. Washington and Lee University and Shoals Marine Laboratory and CLAIRE W. GARRISON, B.A. Shoals Marine Laboratory, Cornell University INTEGRATED PRINCIPLES OF ZOOLOGY, FOURTEENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2006, 2004, 2001, and 1997. 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 consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.
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1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 9 8 7 ISBN 978–0–07–297004–3 MHID 0–07–297004–9 Publisher: Janice Roerig-Blong Executive Editor: Patrick E. Reidy Developmental Editor: Debra A. Henricks Senior Marketing Manager: Tami Petsche Project Manager: April R. Southwood Senior Production Supervisor: Laura Fuller Lead Media Project Manager: Jodi K. Banowetz Media Producer: Daniel M. Wallace Associate Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri Senior Photo Research Coordinator: John C. Leland Photo Research: Mary Reeg Supplement Producer: Melissa M. Leick Compositor: Laserwords Private Limited Typeface: 10/12 Garamond Printer: R. R. Donnelley Willard, OH Front cover image: Polar Bear; © Digital Vision Back cover images: Polar bear paws; © Creatas/PunchStock, Polar bear standing on the ice; © Geostock/ Getty Images The credits section for this book begins on page 880 and is considered an extension of the copyright page.
Library of Congress Cataloging-in-Publication Data Integrated principles of zoology / Cleveland P. Hickman, Jr. ... [et al.]. – 14th ed. p. cm. Includes index. ISBN 978–0–07–297004–3 — ISBN 0–07–297004–9 (hard copy : alk. paper) 1. Zoology. I. Hickman, Cleveland P. QL47.2.H54 2008 590–dc22 2007024506 www.mhhe.com iii iii C O N T E N T S I N B R I E F 20 Crustaceans 420 21 Hexapods 441 22 Chaetognaths, Echinoderms, and Hemichordates 469 23 Chordates 496 24 Fishes 514 25 Early Tetrapods and Modern Amphibians 543 26 Amniote Origins and Nonavian Reptiles 563 27 Birds 585 28 Mammals 612 P A R T F O U R Activity of Life 29 Support, Protection, and Movement 644 30 Homeostasis: Osmotic Regulation, Excretion, and Temperature Regulation 666 31 Internal Fluids and Respiration 686 32 Digestion and Nutrition 708 33 Nervous Coordination: Nervous System and Sense Organs 726 34 Chemical Coordination: Endocrine System 753 35 Immunity 771 36 Animal Behavior 785 P A R T F I V E Animals and Their Environments 37 The Biosphere and Animal Distribution 806 38 Animal Ecology 825 Glossary 843 Credits 880 Index 883 About the Authors ix Preface xi P A R T O N E Introduction to Living Animals 1 Life: Biological Principles and the Science of Zoology 2 2 The Origin and Chemistry of Life 21 3 Cells as Units of Life 37 4 Cellular Metabolism 58 P A R T T W O Continuity and Evolution of Animal Life 5 Genetics: A Review 76 6 Organic Evolution 104 7 The Reproductive Process 137 8 Principles of Development 158 P A R T T H R E E Diversity of Animal Life 9 Architectural Pattern of an Animal 185 10 Taxonomy and Phylogeny of Animals 199 11 Protozoan Groups 217 12 Sponges and Placozoans 246 13 Radiate Animals 260 14 Flatworms, Mesozoans, and Ribbon Worms 289 15 Gnathiferans and Smaller Lophotrochozoans 313 16 Molluscs 331 17 Annelids and Allied Taxa 362 18 Smaller Ecdysozoans 384 19 Trilobites, Chelicerates, and Myriapods 402 iv CHAPTER 4 Cellular Metabolism 58 Energy and the Laws of Thermodynamics 59 The Role of Enzymes 60 Chemical Energy Transfer by ATP 62 Cellular Respiration 63 Metabolism of Lipids 70 Metabolism of Proteins 71 Management of Metabolism 72 Summary 73 P A R T T W O Continuity and Evolution of Animal Life CHAPTER 5 Genetics: A Review 76 Mendel’s Investigations 77 Chromosomal Basis of Inheritance 77 Mendelian Laws of Inheritance 81 Gene Theory 90 Storage and Transfer of Genetic Information 91 Genetic Sources of Phenotypic Variation 100 Molecular Genetics of Cancer 101 Summary 101 iv T A B L E O F C O N T E N T S About the Authors ix Preface xi P A R T O N E Introduction to Living Animals CHAPTER 1 Life: Biological Principles and the Science of Zoology 2 Fundamental Properties of Life 4 Zoology as a Part of Biology 10 Principles of Science 11 Theories of Evolution and Heredity 15 Summary 19 CHAPTER 2 The Origin and Chemistry of Life 21 Water and Life 22 Organic Molecular Structure of Living Systems 24 Chemical Evolution 28 Origin of Living Systems 31 Precambrian Life 32 Summary 35 CHAPTER 3 Cells as Units of Life 37 Cell Concept 38 Organization of Cells 40 Mitosis and Cell Division 52 Summary 56 www . m hhe . com /hi ckmani pz14e Table of Contents v CHAPTER 6 Organic Evolution 104 Origins of Darwinian Evolutionary Theory 105 Darwinian Evolutionary Theory: The Evidence 108 Revisions of Darwin’s Theory 126 Microevolution: Genetic Variation and Change Within Species 126 Macroevolution: Major Evolutionary Events 132 Summary 134 CHAPTER 7 The Reproductive Process 137 Nature of the Reproductive Process 138 The Origin and Maturation of Germ Cells 142 Reproductive Patterns 146 Structure of Reproductive Systems 147 Endocrine Events That Orchestrate Reproduction 149 Summary 156 CHAPTER 8 Principles of Development 158 Early Concepts: Preformation Versus Epigenesis 159 Fertilization 160 Cleavage and Early Development 162 An Overview of Development Following Cleavage 164 Suites of Developmental Characters 166 Mechanisms of Development 170 Gene Expression During Development 172 Vertebrate Development 175 Development of Systems and Organs 179 Summary 182 P A R T T H R E E Diversity of Animal Life CHAPTER 9 Architectural Pattern of an Animal 185 Hierarchical Organization of Animal Complexity 186 Animal Body Plans 187 Components of Metazoan Bodies 190 Complexity and Body Size 193 Summary 195 CHAPTER 10 Taxonomy and Phylogeny of Animals 199 Linnaeus and Taxonomy 200 Species 201 Taxonomic Characters and Phylogenetic Reconstruction 205 Theories of Taxonomy 207 Major Divisions of Life 212 Major Subdivisions of the Animal Kingdom 213 Summary 215 CHAPTER 11 Protozoan Groups 217 How Do We Defi ne Protozoan Groups? 218 Form and Function 221 Major Protozoan Taxa 228 Phylogeny and Adaptive Diversifi cation 243 Summary 244 CHAPTER 12 Sponges and Placozoans 246 Origin of Metazoa 247 Phylum Porifera: Sponges 248 Phylum Placozoa 257 Summary 258 CHAPTER 13 Radiate Animals 260 Phylum Cnidaria 261 Phylum Ctenophora 282 Phylogeny and Adaptive Diversifi cation 285 Summary 287 vi Table of Contents CHAPTER 14 Flatworms, Mesozoans and Ribbon Worms 289 Phylum Acoelomorpha 290 Clades Within Protostomia 291 Phylum Platyhelminthes 292 Phylum Mesozoa 307 Phylum Nemertea (Rhynchocoela) 307 Summary 311 CHAPTER 15 Gnathiferans and Smaller Lophotrochozoans 313 Clade Gnathifera 314 Phylum Gnathostomulida 314 Phylum Micrognathozoa 316 Phylum Rotifera 316 Phylum Acanthocephala 319 Phylum Cycliophora 321 Phylum Gastrotricha 321 Phylum Entoprocta 323 Lophophorates 324 Phylum Ectoprocta (Bryozoa) 325 Phylum Brachiopoda 326 Phylum Phoronida 327 Phylogeny 328 Summary 329 CHAPTER 16 Molluscs 331 Molluscs 332 Form and Function 333 Classes of Molluscs 336 Phylogeny and Adaptive Diversifi cation 357 Summary 360 CHAPTER 17 Annelids and Allied Taxa 362 Phylum Annelida, Including Pogonophorans (Siboglinids) 364 Phylum Echiura 379 Phylum Sipuncula 380 Evolutionary Signifi cance of Metamerism 381 Phylogeny and Adaptive Diversifi cation 381 Summary 382 CHAPTER 18 Smaller Ecdysozoans 384 Phylum Nematoda: Roundworms 386 Phylum Nematomorpha 393 Phylum Kinorhyncha 394 Phylum Priapulida 394 Phylum Loricifera 395 Clade Panarthropoda 396 Phylogeny 399 Summary 400 CHAPTER 19 Trilobites, Chelicerates, and Myriapods 402 Phylum Arthropoda 403 Subphylum Trilobita 406 Subphylum Chelicerata 407 Subphylum Myriapoda 414 Phylogeny and Adaptive Diversifi cation 416 Summary 418 CHAPTER 20 Crustaceans 420 Subphylum Crustacea 422 A Brief Survey of Crustaceans 430 Phylogeny and Adaptive Diversifi cation 439 Summary 439 CHAPTER 21 Hexapods 441 Class Insecta 443 Insects and Human Welfare 459 Phylogeny and Adaptive Diversifi cation 465 Summary 466 CHAPTER 22 Chaetognaths, Echinoderms, and Hemichordates 469 Phylum Chaetognatha 471 Clade Ambulacraria 472 Phylum Echinodermata 472 Phylogeny and Adaptive Diversifi cation 488 Phylum Hemichordata 490 Phylogeny and Adaptive Diversifi cation 493 Summary 494 CHAPTER 23 Chordates 496 The Chordates 497 Five Chordate Hallmarks 500 Ancestry and Evolution 501 Subphylum Urochordata (Tunicata) 502 Subphylum Cephalochordata 504 Subphylum Vertebrata (Craniata) 505 Summary 512 www . m hhe . com /hi ckmani pz14e Table of Contents vii CHAPTER 24 Fishes 514 Ancestry and Relationships of Major Groups of Fishes 515 Living Jawless Fishes 515 Class Chondrichthyes: Cartilaginous Fishes 520 Osteichthyes: Bony Fishes 525 Structural and Functional Adaptations of Fishes 529 Summary 541 CHAPTER 25 Early Tetrapods and Modern Amphibians 543 Movement onto Land 544 Early Evolution of Terrestrial Vertebrates 544 Modern Amphibians 548 Summary 561 CHAPTER 26 Amniote Origins and Nonavian Reptiles 563 Origin and Early Evolution of Amniotes 564 Characteristics of Nonavian Reptiles That Distinguish Them from Amphibians 568 Characteristics and Natural History of Reptilian Orders 570 Summary 583 CHAPTER 27 Birds 585 Origin and Relationships 586 Structural and Functional Adaptations for Flight 587 Flight 598 Migration and Navigation 601 Social Behavior and Reproduction 603 Bird Populations 606 Summary 610 CHAPTER 28 Mammals 612 Origin and Evolution of Mammals 613 Structural and Functional Adaptations of Mammals 617 Humans and Mammals 631 Human Evolution 632 Summary 640 P A R T F O U R Activity of Life CHAPTER 29 Support, Protection, and Movement 644 Integument 645 Skeletal Systems 648 Animal Movement 654 Summary 663 CHAPTER 30 Homeostasis: Osmotic Regulation, Excretion, and Temperature Regulation 666 Water and Osmotic Regulation 667 Invertebrate Excretory Structures 671 Vertebrate Kidney 673 Temperature Regulation 679 Summary 684 CHAPTER 31 Internal Fluids and Respiration 686 Internal Fluid Environment 687 Composition of Blood 688 Circulation 690 Respiration 698 Summary 706 viii Table of Contents CHAPTER 32 Digestion and Nutrition 708 Feeding Mechanisms 709 Digestion 712 Organization and Regional Function of Alimentary Canals 714 Regulation of Food Intake 720 Nutritional Requirements 722 Summary 724 CHAPTER 33 Nervous Coordination: Nervous System and Sense Organs 726 Neurons: Functional Units of Nervous Systems 727 Synapses: Junctions Between Nerves 730 Evolution of Nervous Systems 733 Sense Organs 740 Summary 751 CHAPTER 34 Chemical Coordination: Endocrine System 753 Mechanisms of Hormone Action 754 Invertebrate Hormones 756 Vertebrate Endocrine Glands and Hormones 758 Summary 769 CHAPTER 35 Immunity 771 Susceptibility and Resistance 772 Innate Defense Mechanisms 772 Immunity in Invertebrates 774 Acquired Immune Response in Vertebrates 775 Blood Group Antigens 782 Summary 783 CHAPTER 36 Animal Behavior 785 Describing Behavior: Principles of Classical Ethology 787 Control of Behavior 788 Social Behavior 792 Summary 802 P A R T F I V E Animals and Their Environments CHAPTER 37 The Biosphere and Animal Distribution 806 Distribution of Life on Earth 807 Animal Distribution (Zoogeography) 817 Summary 823 CHAPTER 38 Animal Ecology 825 The Hierarchy of Ecology 826 Extinction and Biodiversity 839 Summary 841 Glossary 843 Credits 880 Index 883 ix A B O U T T H E A U T H O R S CLEVELAND P. HICKMAN, JR. Cleveland P. Hickman, Jr., Professor Emeritus of Biology at Washington and Lee University in Lexington, Virginia, has taught zoology and animal physiology for more than 30 years. He received his Ph.D. in comparative physiology from the Uni-versity of British Columbia, Vancouver, B.C., in 1958 and taught animal physiol-ogy at the University of Alberta before moving to Washington and Lee Univer-sity in 1967. He has published numer-ous articles and research papers in fi sh physiology, in addition to co-authoring these highly successful texts: Integrated Principles of Zoology, Biology of Animals, Animal Diversity, Laboratory Studies in Animal Diversity, and Laboratory Studies in Integrated Principles of Zoology. Over the years Dr. Hickman has led many fi eld trips to the Galápagos Islands. His current research is on intertidal zona-tion and marine invertebrate systematics in the Galápagos. He has published three field guides in the Galápagos Marine Life Series for the identifi cation of echi-noderms, marine molluscs, and marine crustaceans. His interests include scuba diving, woodworking, and participating in cham-ber music ensembles. Dr. Hickman can be contacted at: hickmanc@wlu.edu LARRY S. ROBERTS Larry S. Roberts, Professor Emeritus of Biology at Texas Tech University and an adjunct professor at Florida International University, has extensive experience teaching invertebrate zoology, marine biology, parasitology, and developmental biology. He received his Sc.D. in parasitol-ogy at the Johns Hopkins University and is the lead author of Schmidt and Roberts’s Foundations of Parasitology, sixth edition. Dr. Roberts is also co-author of Integrated Principles of Zoology, Biology of Animals, and Animal Diversity, and is author of The Underwater World of Sport Diving. Dr. Roberts has published many research articles and reviews. He has served as President of the American Society of Parasitologists, Southwestern Association of Parasitologists, and South-eastern Society of Parasitologists, and is a member of numerous other professional societies. Dr. Roberts also serves on the Editorial Board of the journal, Parasitology Research. His hobbies include scuba div-ing, underwater photography, and tropical horticulture. Dr. Roberts can be contacted at: Lroberts1@compuserve.com SUSAN KEEN Susan Keen is a lecturer in the Section of Evolution and Ecology at the University of California at Davis. She received her Ph.D. in zoology from the University of Califor-nia at Davis, following a M.Sc. from the University of Michigan at Ann Arbor. She is a native of Canada and obtained her undergraduate education at the University of British Columbia in Vancouver. Dr. Keen is an invertebrate zoologist fascinated with jellyfi sh life histories. She has a particular interest in life cycles where both asexual and sexual phases of organ-isms are present, as they are in most jel-lyfi shes. Her other research has included work on sessile marine invertebrate com-munities, spider populations, and Andean potato evolution. Dr. Keen has been teaching evolution and animal diversity within the Introduc-tory Biology series for 13 years. She enjoys all facets of the teaching process, from lectures and discussions to the design of effective laboratory exercises. In addition to her work with introductory biology, she offers seminars for the Davis Honors Challenge program, and for undergraduate and graduate students interested in teach-ing methods for biology. She was given an Excellence in Education Award from the Associated Students group at Davis in 2004. She attended the National Acad-emies Summer Institute on Undergraduate Education in Biology in 2005, and was a National Academies Education Fellow in the Life Sciences for 2005–2006. Her inter-ests include weight training, horseback rid-ing, gardening, travel, and mystery novels. Dr. Keen can be contacted at: slkeen@ucdavis.edu ALLAN LARSON Allan Larson is a professor at Washington University, St. Louis, MO. He received his Ph.D. in genetics at the University of Cali-fornia, Berkeley. His fi elds of specializa-tion include evolutionary biology, molec-ular population genetics and systematics, and amphibian systematics. He teaches courses in introductory genetics, zoology, macroevolution, molecular evolution, and the history of evolutionary theory, and has organized and taught a special course in evolutionary biology for high-school teachers. Dr. Larson has an active research labo-ratory that uses DNA sequences to exam-ine evolutionary relationships among ver-tebrate species, especially in salamanders and lizards. The students in Dr. Larson’s laboratory have participated in zoological fi eld studies around the world, including projects in Africa, Asia, Australia, Mada-gascar, North America, South America, the Indo-Pacific Ocean, and the Carib-bean Islands. Dr. Larson has authored numerous scientific publications, and has edited for the journals The American Naturalist, Evolution, Journal of Experi-mental Zoology, Molecular Phylogenetics and Evolution, and Systematic Biology. Dr. Larson serves as an academic advisor to undergraduate students and supervises the undergraduate biology curriculum at Washington University. Dr. Larson can be contacted at: larson@wustl.edu HELEN I’ANSON Helen I’Anson, a native of England, is professor of biology at Washington and Lee University in Lexington, Virginia. She received her Ph.D. in physiology at the x About the Authors University of Kentucky, Lexington, KY, and postdoctoral training at the University of Michigan, Ann Arbor, MI. She teaches courses in animal physiology, microanat-omy, neuroendocrinology, general biol-ogy, and reproductive physiology. She has an active research program that focuses on the neural regulation of reproductive development. In particular, she is inter-ested in how energy is partitioned in the developing animal, how signals from food and food storage depots are monitored by the brain, and how such signals are trans-duced to regulate reproductive activity at the onset of puberty in mammals. Her interests include gardening, hiking, fi shing, aromatherapy, music, and partici-pating in choral ensembles. Dr. I’Anson can be contacted at: iansonh@wlu.edu DAVID J. EISENHOUR David J. Eisenhour is an associate profes-sor of biology at Morehead State Univer-sity in Morehead, Kentucky. He received his Ph.D. in zoology from Southern Illi-nois University, Carbondale. He teaches courses in environmental science, human anatomy, general zoology, comparative anatomy, ichthyology, and vertebrate zoology. David has an active research program that focuses on systematics, con-servation biology, and natural history of North American freshwater fi shes. He has a particular interest in the diversity of Kentucky’s fi shes and is writing a book about that subject. He and his graduate students have authored several publica-tions. David serves as an academic advisor to prepharmacy students. His interests include fi shing, landscap-ing, home remodeling, and entertaining his three young children, who, along with his wife, are enthusiastic participants in fi eldwork. Dr. Eisenhour can be contacted at: d.eisenhour@morehead-st.edu xi I ntegrated Principles of Zoology is a college text designed for an introductory course in zoology. This fourteenth edition, as with previous editions, describes the diversity of animal life and the fascinating adaptations that enable animals to inhabit so many ecological niches. We retain in this revision the basic organization of the thir-teenth edition and its distinctive features, especially emphasis on the principles of evolution and zoological science. Also retained are several pedagogical features that have made previous edi-tions easily accessible to students: opening chapter dialogues drawn from the chapter’s theme; chapter summaries and review questions to aid student comprehension and study; concise and visually appealing illustrations; in-text derivations of generic names; chapter notes and essays that enhance the text by offer-ing interesting sidelights to the narrative; literature citations; and an extensive glossary providing pronunciation, derivation, and defi nition of terms used in the text. NEW TO THE FOURTEENTH EDITION The authors welcome to the fourteenth edition Susan Keen, who supervised this revision. Many improvements are the direct result of Susan’s new perspectives and those of many zoology instruc-tors who submitted reviews of the thirteenth edition. We revised all chapters to streamline the writing and to incorporate new dis-coveries and literature citations. Our largest formal revision is to include a cladogram of animal phyla on the inside front cover of the book, and to reorder chapter contents in Part Three (Diver-sity of Animal Life) to match the arrangement of phyla on the cladogram. Each chapter in Part Three begins with a small image of the zoological cladogram highlighting the phylum or phyla covered in the chapter, followed by an expanded cladogram of the contents of each major phylum. We place stronger empha-sis on phylogenetic perspectives throughout the book. Material formerly presented separately as “biological contributions” and “characteristics” of phyla is consolidated in a boxed list of phy-lum “characteristics” for each chapter in Part Three. New photo-graphs are added to illustrate animal diversity in many phyla. Material new to the fourteenth edition expands and updates our coverage of eight major principles: (1) scientifi c process and the role of theory, (2) cellular systems and metabolism, (3) endo-symbiotic theory of eukaryotic origins, (4) physiological and ecological systems, (5) populational processes and conservation, (6) evolutionary developmental biology, (7) phylogenetic tests of morphological homologies, and (8) taxonomy. Exciting new fossil discoveries and molecular phylogenies contribute impor-tant changes to the last three principles. The primary changes to each major principle are summarized here with references to the relevant chapters. P R E F A C E Scientifi c Process and the Role of Theory Many changes throughout the book increase the integration of hypothetico-deductive methodology in discussing new dis-coveries and controversies. We begin in Chapter 1 with a more detailed explanation of the hypothetico-deductive method of science and the important contrast between the comparative method versus experimental biology as complementary means of testing hypotheses. The role of theory in science is illus-trated explicitly using Darwin’s theory of common descent in Chapter 6. Uses of Darwin’s theory of common descent to test evolutionary hypotheses and to construct taxonomies get expanded treatment in Chapter 10, including a new conceptual distinction between classifi cation and systematization and cov-erage of DNA barcoding in species identifi cation. Cellular Systems and Metabolism We expand in Chapter 3 our coverage of the components of eukaryotic cells, the biological roles of subcellular structures, and specializations of cellular surfaces. Expanded molecular topics include pH (Chapter 2), prions as diseases of protein conforma-tion (Chapter 2), lipid metabolism (Chapter 4), and accumulation of “junk” or “parasitic” DNA in animal genomes (Chapter 5). In Chapter 7, a new boxed essay reports the discovery of actively dividing germ cells in adult female mammals, and a revised boxed essay updates applications of cell biology to contraceptive medicine. Endosymbiotic Theory of Eukaryotic Origins The history of the endosymbiotic theory is presented in more detail, including the empirical testing of its original claims and its more recent expansion to cover a broader evolutionary domain (Chapter 2). Important molecular phylogenetic evidence for sepa-rate evolutionary origins of nuclear, mitochondrial, and chloroplast genomes is presented in the form of a new global “tree of life” relating prokaryotic and eukaryotic genomes (Chapter 10). The role of endosymbiosis in diversifi cation of unicellular eukaryotes gets new coverage in Chapter 11, and evolutionary loss of mito-chondria from some infectious unicellular eukaryotes is added to Chapter 2. Physiological and Ecological Systems Numerous revisions address organismal physiology and its eco-logical consequences, beginning with the addition of “move-ment” as a general characteristic of life in Chapter 1. We add xii Preface new results on tracheal respiration in insects (Chapter 21), respi-ratory gas transport in terrestrial arthropods and in vertebrates (Chapter 31), and lung ventilation in vertebrates (Chapter 31). Also revised are the plans of vertebrate circulatory systems, coro-nary circulation, and excitation and control of the heart (Chap-ter 31). New material appears on regulation of food intake and of digestion (Chapter 32), digestive processes in the vertebrate small intestine (Chapter 32), and foregut fermentation in rumi-nant mammals (Chapter 28). Evolution of centralized nervous systems, chemoreception, mechanoreception and photorecep-tion in invertebrates gets new coverage in Chapter 33, with expanded explanation of synapses and conduction of action potentials. Endocrinology of invertebrates is expanded, and ver-tebrate endocrinology is updated to include discussion of white adipose tissue as an endocrine organ, the pancreatic polypeptide (PP) hormone, and controversies regarding medicinal uses of anabolic steroids (Chapter 34). Invertebrate excretory systems, especially arthropod kidneys, get expanded coverage in Chap-ter 30. We cover regional endothermy in fi shes (Chapter 24), and add new explanatory material on the importance of water and osmotic regulation, especially in marine fi shes (Chapter 30). Revision of Chapter 35 updates our knowledge of susceptibility and resistance to disease, including acquired immune defi ciency. We add a new section on cetacean echolocation (Chapter 28), greater explanation of frog mating systems (Chapter 25), and avian reproductive strategies, including extra-pair copulations (Chapter 27). We provide greater coverage of scientifi c contro-versies regarding bee communication, eusociality, and genet-ics of animal behavior (Chapter 36). Concepts of food chains and food webs are now distinguished, and quantitative data are added to illustrate them using ecological pyramids (Chapter 38). Populational Processes and Conservation Modes of speciation receive expanded coverage and explanation (Chapter 6), as do concepts of fi tness and inclusive fi tness (Chap-ter 6), and costs and benefi ts of sexual versus asexual reproduc-tion (Chapter 7). Conservation of natural populations is updated, especially in fi shes (Chapter 24), mammals (Chapter 36), and tuataras (Chapter 26). Historical biogeographic processes are illustrated with expanded coverage of explanations for Wallace’s Line, the geographic contact between evolutionarily disparate faunas (Chapter 37). Evolutionary Developmental Biology This rapidly growing discipline gets updated coverage both in concept and application. New concepts of developmental mod-ularity and evolvability join our general coverage of evolution-ary biology in Chapter 6. We discuss in Chapter 8 new evidence that some sponges have two germ layers. Cnidarian develop-ment and life cycles get expanded coverage, and the diplo-blastic status of cnidarians and ctenophores is reconsidered in light of new phylogenetic results (Chapter 13). We provide molecular-genetic interpretations of the diploblast-triploblast distinction (Chapter 13) and updated details of triploblastic development (Chapter 14). Insights from genomic and devel-opmental studies offer new interpretations of metazoan ori-gins (Chapter 12) and suggest that changes in the expression of a single gene underlie alternative developmental pathways of arthropod limbs (uniramous versus biramous; Chapter 19). Developmental differences among chaetognaths, protostomes, and deuterostomes are reevaluated in light of new phylogenetic evidence (Chapter 22). We restructure our general coverage of body plans (Chapter 9) and provide greater explanation of the complex development of gastropod torsion (Chapter 16). Phylogenetic Tests of Morphological Homologies New molecular phylogenies and fossil discoveries revise our interpretations of many homologies and reveal independent evo-lution of similar characters in different groups. In light of these issues, we expand our coverage of the concept of homoplasy in Chapter 10. Chapter 16 incorporates new evidence challenging homology of metamerism in annelids and molluscs and illus-trating the scientifi c process in action. Evidence from Hox gene expression is used in Chapter 19 to homologize the cephalo-thorax of spiders with heads of other arthropods and to sup-port phylogenetic evidence for multiple origins of uniramous limbs from biramous ones in phylum Arthropoda. Homology of diffuse epidermal nervous systems and tripartite coeloms of echinoderms and hemichordates and nonhomology of dorsal hollow nerve chords of hemichordates and chordates change our favored hypotheses for relationships among these groups (Chapter 22). Developmental comparisons demonstrate non-homology of coelomic compartmentalization of lophophorates with that of echinoderms and hemichordates (Chapter 22). New data and interpretations revise inferred characteristics of the most recent chordate ancestor (Chapter 23), origin and diversifi cation of amniotes and their adaptations for terrestrial life (Chapter 26), evolution of the mammalian middle ear (Chapter 28), and details of hominid morphological evolution (Chapter 28). Taxonomy New molecular-phylogenetic and fossil data reject some fami-liar taxa and suggest new ones. We discuss evidence for a sister-group relationship of choanofl agellates and metazoans in Chapter 12. Chapter 14 discusses new phylogenetic results that underlie recognition of phylum Acoelomorpha and a revised phylogenetic hypothesis for nemertine worms. Acan-thocephalans now appear to descend from a rotiferan ancestor (Chapter 15). Clade Clitellata (oligochaetes and leeches), pogo-nophorans, and vestimentiferans descend from polychaete annelids according to new phylogenetic data, making poly-chaetes paraphyletic (Chapter 17). Chapter 18 presents new evi-dence for clade Panarthropoda (Onychophora, Tardigrada, and Arthropoda). Chapters 19 and 21 present evidence supporting www . m hhe . com /hi ckmani pz14e Preface xiii recognition of clade Pancrustacea (crustaceans and hexapods) and rejection of arthropod subphylum Uniramia. Chapter 20 includes new evidence that hexapods derive from a crustacean ancestor, and Pentastomida is subsumed in Crustacea. Ento-gnatha and Insecta form separate clades within subphylum Hexapoda (Chapter 21). We update recognition of insect orders in Chapter 21. We introduce in Chapter 22 clade Ambulacraria (Echinodermata and Hemichordata), which is likely the sister group of chordates (Chapter 23). The fossil genus Haikouella gets increased coverage and illustration as the likely sister taxon to craniates (Chapter 23). Changes to fi sh taxonomy include using the clade name Petromyzontida for lampreys and remov-ing bichirs from chondrosteans (Chapter 24). Early tetrapod evo-lution is extensively revised with reference to new fossil discov-eries, including the genus Tiktaalik (Chapter 25). We replace the traditional use of “Reptilia” with one including the traditional reptiles, birds, and all descendants of their most recent com-mon ancestor (Chapter 26). Phylogenetic results place turtles in the clade Diapsida (Chapter 26), contrary to earlier hypotheses. Amphisbaenians are now included within lizards according to their phylogenetic position, and the section on relationships of snakes to lizards is expanded (Chapter 26). Chapter 27 includes fairly extensive revisions of avian taxonomy based upon phylo-genetic results from DNA sequence data. TEACHING AND LEARNING AIDS To help students in vocabulary development, key words are boldfaced and derivations of technical and zoological terms are provided, along with generic names of animals where they fi rst appear in the text. In this way students gradually become familiar with the more common roots that form many technical terms. An extensive glossary provides pronunciation, derivation, and defi -nition of each term. Many new terms were added to the glossary or rewritten for this edition. A distinctive feature of this text is a prologue for each chap-ter that highlights a theme or fact relating to the chapter. Some prologues present biological, particularly evolutionary, princi-ples; those in Part Three on animal diversity illuminate distin-guishing characteristics of the group presented in the chapter. Chapter notes, which appear throughout the book, aug-ment the text material and offer interesting sidelights without interrupting the narrative. We prepared many new notes for this edition and revised several existing notes. To assist students in chapter review, each chapter ends with a concise summary, a list of review questions, and annotated selected references. The review questions enable a student to self-test retention and understanding of the more important chapter material. Again, William C. Ober and Claire W. Garrison have strength-ened the art program for this text with many new full-color paintings that replace older art, or that illustrate new material. Bill’s artistic skills, knowledge of biology, and experience gained from an earlier career as a practicing physician have enriched this text through ten of its editions. Claire practiced pediatric and obstetric nursing before turning to scientifi c illustration as a full-time career. Texts illustrated by Bill and Claire have received national recognition and won awards from the Association of Medical Illustrators, American Institute of Graphic Arts, Chicago Book Clinic, Printing Industries of America, and Bookbuilders West. They are also recipients of the Art Directors Award. SUPPLEMENTS For Students Online Learning Center The Online Learning Center for Integrated Principles of Zoology is a great place to review chapter material and to enhance your study routine. Visit www.mhhe.com/hickmanipz14e for access to the following online study tools: • Chapter quizzing • Key term fl ash cards • Web links • And more! For Instructors Online Learning Center The Integrated Principles of Zoology Online Learning Cen-ter ( www.mhhe.com/hickmanipz14e ) offers a wealth of teaching and learning aids for instructors and students. Instruc-tors will appreciate: • A password-protected Instructor’s Manual • Access to the new online Presentation Center including illustrations, photographs, and tables from the text • PowerPoint lecture outlines • Links to professional resources • Online activities for students such as chapter quizzing, key-term fl ash cards, web links, and more! NEW! Presentation Center www.mhhe.com/hickmanipz14e Build Instructional Materials Wherever, Whenever, and However You Want! Accessed through the Online Learning Center, the Presentation Center is an online digital library containing assets such as art, photos, tables, PowerPoint image slides, and other media types that can be used to create customized lectures, visually enhanced tests and quizzes, compelling course websites, or attractive printed support materials. Access to Your Book, Access to All Books! The Pre-sentation Center library includes thousands of assets from many McGraw-Hill titles. This ever-growing resource gives instructors the power to utilize assets specifi c to an adopted textbook as well as content from all other books in the library. xiv Preface Nothing Could Be Easier! Presentation Center’s dynamic search engine allows you to explore by discipline, course, textbook chapter, asset type, or keyword. Simply browse, select, and download the fi les you need to build engaging course materials. All assets are copy-righted by McGraw-Hill Higher Education but can be used by instructors for classroom purposes. Instructor’s Manual This helpful ancillary provides chapter outlines, lecture enrich-ment suggestions, lesson plans, a list of changes from the pre v-ious edition, and source materials.
Test Bank A computerized test bank utilizing testing software to create customized exams is available with this text. The user-friendly software allows instructors to search for questions by topic or format, edit existing questions or add new ones, and scramble questions to create multiple versions of the same test. Word fi les of the test bank questions are provided for those instructors who prefer to work outside the test-generator software. Course Management Systems Online content is available for a variety of course management systems including BlackBoard and WebCT. Laboratory Studies in Integrated Principles of Zoology by Cleveland Hickman, Jr., and Lee B. Kats Now in its fourteenth edition, this lab manual was written to accompany Integrated Principles of Zoology, and can be easily adapted to fi t a variety of course plans. Biology Digitized Video Clips McGraw-Hill is pleased to offer digitized biology video clips on DVD! Licensed from some of the highest-quality science video producers in the world, these brief segments range from about fi ve seconds to just under three minutes in length and cover all areas of general biology from cells to ecosystems. Engaging and informative, McGraw-Hill’s digitized biology videos will help cap-ture students’ interest while illustrating key biological concepts and processes. Includes video clips on mitosis, Darwin’s fi nches, cichlid mouth brooding, sponge reproduction, and much more! ACKNOWLEDGMENTS The authors extend their sincere thanks to the faculty reviewers whose numerous suggestions for improvement were of the great-est value in the revision process. Their experience with students of varying backgrounds, and their interest in and knowledge of the subject, helped to shape the text into its fi nal form. Kenneth Andrews, East Central University Patricia Biesiot, University of Southern Mississippi Mark Blackmore, Valdosta State University Raymond Bogiatto, California State University Roger Choate, Oklahoma City Community College Jerry Cook, Sam Houston State University Tamara Cook, Sam Houston State University Michael Harvey, Broward Community College Robert Hoyt, Western Kentucky University Timothy Judd, Southeast Missouri State University Robert L. Koenig, Southwest Texas Jr. College William Kroll, Loyola University of Chicago Sharyn Marks, Humboldt State University R. Patrick Randolf, University of California–Davis Anthony Stancampiano, Oklahoma City Community College Rodney Thomas, University of South Carolina Michael Toliver, Eureka College Stim Wilcox, Binghamton University The authors express their appreciation to the editors and support staff at McGraw-Hill Higher Education who made this project possible. Special thanks are due Patrick Reidy, Executive Editor, and Debra Henricks, Developmental Editor, who were the driving forces in piloting this text throughout its development. April Southwood, Project Manager, somehow kept authors, text, art, and production programs on schedule. John Leland oversaw the extensive photographic program and Brenda Rolwes man-aged the book’s interior and cover design. We are indebted to them for their talents and dedication. Although we make every effort to bring to you an error-free text, errors of many kinds inevitably fi nd their way into a textbook of this scope and complexity. We will be grateful to readers who have comments or suggestions concerning content to send their remarks to Debra Henricks, Developmental Editor at debra_henricks@mcgraw-hill.com . Cleveland P. Hickman, Jr. Larry S. Roberts Susan Keen Allan Larson Helen I’Anson David J. Eisenhour 1 P A R T O N E Life: Biological Principles and the Science of Zoology The Origin and Chemistry of Life Cells as Units of Life Cellular Metabolism 1 2 3 4 Introduction to Living Animals A tube anemone (cerianthid, Botruanthus benedini ) from the eastern Pacifi c . C H A P T E R 1 Life: Biological Principles and the Science of Zoology The Uses of Principles We gain knowledge of the animal world by actively applying impor-tant guiding principles to our investigations. Just as the exploration of outer space is both guided and limited by available technologies, exploration of the animal world depends critically on our questions, methods, and principles. The body of knowledge that we call zool-ogy makes sense only when the principles that we use to construct it are clear. The principles of modern zoology have a long history and many sources. Some principles derive from laws of physics and chemistry, which all living systems obey. Others derive from the scientifi c method, which tells us that our hypotheses regarding the animal world are useless unless they guide us to gather data that potentially can refute them. Many important principles derive from previous studies of the living world, of which animals are one part. Principles of heredity, variation, and organic evolution guide the study of life from the simplest unicellular forms to the most complex animals, fungi, and plants. Because life shares a common evolution-ary origin, principles learned from the study of one group often per-tain to other groups as well. By tracing the origins of our operating principles, we see that zoologists are not an island unto themselves but part of a larger scientifi c community. We begin our study of zoology not by focusing narrowly within the animal world, but by searching broadly for our most basic prin-ciples and their diverse sources. These principles simultaneously guide our studies of animals and integrate those studies into the broader context of human knowledge. Zoologist studying the behavior of yellow baboons ( Papio cynocephalus ) in the Amboseli Reserve, Kenya. www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 3 Z oology, the scientifi c study of animal life, builds on centu-ries of human inquiry into the animal world. Mythologies of nearly every human culture attempt to solve the mys-teries of animal life and its origin. Zoologists now confront these same mysteries with the most advanced methods and technologies developed by all branches of science. We start by documenting the diversity of animal life and organizing it in a systematic way. This complex and exciting process builds on the contributions of thousands of zoologists working in all dimensions of the biosphere ( Figure 1.1 ). We strive through this work to understand how ani-mal diversity originated and how animals perform the basic pro-cesses of life that permit them to occupy diverse environments. This chapter introduces the fundamental properties of ani-mal life, the methodological principles on which their study is based, and two important theories that guide our research: (1) the theory of evolution, which is the central organizing Figure 1.1 A few of the many dimensions of zoological research. A, Observing moray eels in Maui, Hawaii. B, Working with tranquilized polar bears. C, Banding mallard ducks. D, Observing Daphnia pulex (150) microscopically. E, Separating growth stages of crab larvae at a marine laboratory. A B C D E 4 PART ONE Introduction to Living Animals principle of biology, and (2) the chromosomal theory of inheri-tance, which guides our study of heredity and variation in ani-mals. These theories unify our knowledge of the animal world. FUNDAMENTAL PROPERTIES OF LIFE Does Life Have Defi ning Properties? We begin with the diffi cult question, What is life? Although many attempts have been made to defi ne life, simple defi nitions are doomed to failure. When we try to give life a simple defi nition, we look for fi xed properties maintained throughout life’s his-tory. However, the properties that life exhibits today (pp. 4–9) are very different from those present at its origin. The history of life shows extensive and ongoing change, which we call evolu-tion. As the genealogy of life progressed and branched from the earliest living form to the millions of species alive today, new properties evolved and passed from parents to their offspring. Through this process, living systems have generated many rare and spectacular features that have no counterparts in the non-living world. Unexpected properties emerge on many different lineages in life’s evolutionary history, producing the great organ-ismal diversity observed today. We might try to defi ne life by universal properties evident at its origin. Replication of molecules, for example, can be traced to life’s origin and represents one of life’s universal properties. Defi ning life in this manner faces the major problem that these are the properties most likely to be shared by some nonliving forms. To study the origin of life, we must ask how organic molecules acquired the ability for precise replication. But where do we draw the line between those replicative processes that character-ize life and those that are merely general chemical features of the matter from which life arose? Replication of complex crystalline structures in nonliving chemical assemblages might be confused, for example, with the replicative molecular properties associated with life. If we defi ne life using only the most advanced char-acteristics of the highly evolved living systems observed today, the nonliving world would not intrude on our defi nition, but we would eliminate the early forms of life from which all others descended and which give life its historical unity. Ultimately our defi nition of life must be based on the com-mon history of life on earth. Life’s history of descent with modi-fi cation gives it an identity and continuity that separates it from the nonliving world. We can trace this common history back-ward through time from the diverse forms observed today and in the fossil record to their common ancestor that arose in the atmosphere of the primitive earth (see Chapter 2). All organisms forming part of this long history of hereditary descent from life’s common ancestor are included in our concept of life. We do not force life into a simple defi nition, but we can read-ily identify the living world through its history of common evolu-tionary descent. Many remarkable properties have arisen during life’s history and are observed in various combinations among liv-ing forms. These properties, discussed in the next section, clearly identify their possessors as part of the unifi ed historical entity called life. All such features occur in the most highly evolved forms of life, such as those that compose the animal kingdom. Because they are so important for maintenance and functioning of living forms that possess them, these properties should persist through life’s future evolutionary history. General Properties of Living Systems The most outstanding general features in life’s history include chemical uniqueness; complexity and hierarchical organization; reproduction (heredity and variation); possession of a genetic program; metabolism; development; environmental interaction; and movement. 1. Chemical uniqueness. Living systems demonstrate a uni-que and complex molecular organization. Living systems assemble large molecules, known as macromolecules, that are far more complex than the small molecules of nonliving matter. These macromolecules are composed of the same kinds of atoms and chemical bonds that occur in nonliv-ing matter and they obey all fundamental laws of chemis-try; it is only the complex organizational structure of these macromolecules that makes them unique. We recognize four major categories of biological macromolecules: nucleic acids, proteins, carbohydrates, and lipids (see Chapter 2). These categories differ in the structures of their component parts, the kinds of chemical bonds that link their subunits together, and their functions in living systems. The general structures of these macromolecules evolved and stabilized early in the history of life. With some modi-fi cations, these same general structures are found in every form of life today. Proteins, for example, contain about 20 specifi c kinds of amino acid subunits linked together by peptide bonds in a linear sequence ( Figure 1.2 ). Additional bonds occurring between amino acids that are not adjacent to each other in the protein chain give the protein a com-plex, three-dimensional structure (see Figures 1.2 and 2.15). A typical protein contains several hundred amino acid sub-units. Despite the stability of this basic protein structure, the ordering of the different amino acids in the protein mole-cule is subject to enormous variation. This variation under-lies much of the diversity that we observe among different kinds of living forms. The nucleic acids, carbohydrates, and lipids likewise contain characteristic bonds that link variable subunits (see Chapter 2). This organization gives living sys-tems both a biochemical unity and great potential diversity. 2. Complexity and hierarchical organization. Living sys-tems demonstrate a unique and complex hierarchical orga-nization. Nonliving matter is organized at least into atoms and molecules and often has a higher degree of organi-zation as well. However, atoms and molecules are com-bined into patterns in the living world that do not exist in the nonliving world. In living systems, we fi nd a hierarchy of levels that includes, in ascending order of complexity, macromolecules, cells, organisms, populations, and spe-cies ( Figure 1.3 ). Each level builds on the level below it www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 5 Figure 1.2 A computer simulation of the three-dimensional structure of the lysozyme protein (A), which is used by animals to destroy bacteria. The protein is a linear string of molecular subunits called amino acids, connected as shown in B, which fold in a three-dimensional pattern to form the active protein. The white balls correspond to carbon atoms, the red balls to oxygen, the blue balls to nitrogen, the yellow balls to sulfur, the green balls to hydrogen, and the black balls (B) to molecular groups formed by various combinations of carbon, oxygen, nitrogen, hydrogen, and sulfur atoms that differ among amino acids. Hydrogen atoms are not shown in A. The purple molecule in A is a structure from the bacterial cell wall that is broken by lysozyme. A B Amino acid Amino acid Amino acid and has its own internal structure, which is also often hier-archical. Within the cell, for example, macromolecules are compounded into structures such as ribosomes, chromo-somes, and membranes, and these are likewise combined in various ways to form even more complex subcellu-lar structures called organelles, such as mitochondria (see Chapters 3 and 4). The organismal level also has a hierarchical substructure; cells combine to form tissues, which combine to form organs, which likewise combine to form organ systems (see Chapter 9). Cells ( Figure 1.4 ) are the smallest units of the biologi-cal hierarchy that are semiautonomous in their ability to conduct basic functions, including reproduction. Repli-cation of molecules and subcellular components occurs only within a cellular context, not independently. Cells Figure 1.4 Electron micrograph of ciliated epithelial cells and mucus-secreting cells (see pp. 192–195). Cells are the basic building blocks of living organisms. Figure 1.3 Volvox globator (see pp. 229–231) is a multicellular chlorophytan that illustrates three different levels of the biological hierarchy: cellular, organismal, and populational. Each individual spheroid (organism) contains cells embedded in a gelatinous matrix. The larger cells function in reproduction, and the smaller ones perform the general metabolic functions of the organism. The individual spheroids together form a population. 6 PART ONE Introduction to Living Animals are therefore considered the basic units of living systems (see Chapter 3). We can isolate cells from an organism and cause them to grow and to multiply under laboratory conditions in the presence of nutrients alone. This semi-autonomous replication is not possible for any individual molecules or subcellular components, which require addi-tional cellular constituents for their reproduction. Each successively higher level of the biological hierar-chy is composed of units of the preceding lower level in the hierarchy. An important characteristic of this hierarchy is that the properties of any given level cannot be inferred even from the most complete knowledge of the proper-ties of its component parts. A physiological feature, such as blood pressure, is a property of the organismal level; it is impossible to predict someone’s blood pressure simply by knowing the physical characteristics of individual cells of the body. Likewise, systems of social interaction, as observed in bees, occur at the populational level; it is not possible to infer properties of this social system by studying individual bees in isolation. The appearance of new characteristics at a given level of organization is called emergence, and these character-istics are called emergent properties. These properties arise from interactions among the component parts of a system. For this reason, we must study all levels directly, each one being the focus of a different subfi eld of biol-ogy (molecular biology; cell biology; organismal anatomy, physiology and genetics; population biology; Table 1.1 ). Emergent properties expressed at a particular level of the biological hierarchy are certainly infl uenced and restricted by properties of the lower-level components. For exam-ple, it would be impossible for a population of organisms that lack hearing to develop a spoken language. Nonethe-less, properties of parts of a living system do not rigidly determine properties of the whole. Many different spoken languages have emerged in human culture from the same basic anatomical structures that permit hearing and speech. The freedom of the parts to interact in different ways makes possible a great diversity of potential emergent properties at each level of the biological hierarchy. Different levels of the biological hierarchy and their particular emergent properties are built by evolution. Before multicellular organisms evolved, there was no distinction between the organismal and cellular levels, and this distinc-tion is still absent from single-celled organisms (see Chap-ter 11). The diversity of emergent properties that we see at all levels of the biological hierarchy contributes to the dif-fi culty of giving life a simple defi nition or description. 3. Reproduction. Living systems can reproduce themselves. Life does not arise spontaneously but comes only from prior life, through reproduction. Although life certainly ori-ginated from nonliving matter at least once (see Chapter 2), this origin featured enormously long periods of time and conditions very different from the current biosphere. At each level of the biological hierarchy, living forms repro-duce to generate others like themselves ( Figure 1.5 ). Genes are replicated to produce new genes. Cells divide to pro-duce new cells. Organisms reproduce, sexually or asexually, to produce new organisms (see Chapter 7). Populations may become fragmented to produce new populations, and species may split to produce new species through a pro-cess called speciation. Reproduction at any hierarchical level usually features an increase in numbers. Individual genes, cells, organisms, populations, or species may fail to reproduce themselves, but reproduction is nonetheless an expected property of these individuals. Reproduction at each of these levels shows the com-plementary, and yet apparently contradictory, phenomena T A B L E 1 . 1 Different Hierarchical Levels of Biological Complexity That Display Reproduction, Variation, and Heredity Level Timescale of Reproduction Fields of Study Methods of Study Some Emergent Properties Cell Hours (mammalian cell ~16 hours) Cell biology Microscopy (light, electron), biochemistry Chromosomal replication (meiosis, mitosis), synthesis of macromolecules (DNA, RNA, proteins, lipids, polysaccharides) Organism Hours to days (unicellular); days to years (multicellular) Organismal anatomy, physiology, genetics Dissection, genetic crosses, clinical studies, physiological experimentation Structure, functions and coordination of tissues, organs and organ systems (blood pressure, body temperature, sensory perception, feeding) Population Up to thousands of years Population biology, population genetics, ecology Statistical analysis of variation, abundance, geographical distribution Social structures, systems of mating, age distribution of organisms, levels of variation, action of natural selection Species Thousands to millions of years Systematics and evolutionary biology, community ecology Study of reproductive barriers, phylogeny, paleontology, ecological interactions Method of reproduction, reproductive barriers www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 7 of heredity and variation. Heredity is the faithful transmis-sion of traits from parents to offspring, usually (but not nec-essarily) observed at the organismal level. Variation is the production of differences among the traits of different indi-viduals. In a reproductive process, properties of descendants resemble those of their parents to varying degrees but usu-ally are not identical to them. Replication of deoxyribonu-cleic acid (DNA) occurs with high fi delity, but errors occur at repeatable rates. Cell division is exceptionally precise, espe-cially with regard to the nuclear material, but chromosomal changes occur nonetheless at measurable rates. Organis-mal reproduction likewise demo nstrates both heredity and variation, the latter most obvious in sexually reproducing forms. Production of new populations and species also dem-onstrates conservation of some properties and changes of others. Two closely related frog species may have similar mating calls but differ in the rhythm of repeated sounds. Interaction of heredity and variation in the repro ductive process is the basis for organic evolution (see Chapter 6). If heredity were perfect, living systems would never change; if variation were uncontrolled by heredity, biological sys-tems would lack the stability that allows them to persist through time. 4. Possession of a genetic program. A genetic program pro-vides fi delity of inheritance ( Figure 1.6 ). Structures of the protein molecules needed for organismal development and functioning are encoded in nucleic acids (see Chapter 5). For animals and most other organisms, genetic informa-tion is contained in DNA. DNA is a very long, linear chain of subunits called nucleotides, each of which contains a sugar phosphate (deoxyribose phosphate) and one of four nitrogenous bases (adenine, cytosine, guanine, or thymine, abbreviated A, C, G, and T, respectively). The sequence of nucleotide bases contains a code for the order of amino acids in the protein specifi ed by the DNA molecule. The correspondence between the sequence of bases in DNA and the sequence of amino acids in a protein is called the genetic code. A B C Figure 1.5 Reproductive processes observed at four different levels of biological complexity. A, Molecular level—electron micrograph of a replicating DNA molecule. B, Cellular level—micrograph of cell division at mitotic telophase. C, Organismal level—a king snake hatching. D, Species level—formation of new species in the sea urchin (Eucidaris) after geographic separation of Caribbean (E. tribuloides) and Pacifi c (E. thouarsi) populations by the formation of a land bridge.
Pacific Ocean Central America Caribbean Sea Gulf of Mexico D 8 PART ONE Introduction to Living Animals The genetic code arose early in the evolutionary his-tory of life, and the same code occurs in bacteria and in the nuclear genomes of almost all animals and plants. The near constancy of this code among living forms provides strong evidence for a single origin of life. The genetic code has undergone very little evolutionary change since its origin because an alteration would disrupt the structure of nearly every protein, which would in turn severely disrupt cellular functions that require very specifi c protein structures. Only in the rare instance that the altered protein structures main-tain their cellular functions would such a change be able to survive and be reproduced. Evolutionary change in the genetic code has occurred in the DNA contained in animal mitochondria, the organelles that regulate cellular energy. The genetic code in animal mitochondrial DNA therefore is slightly different from the standard code of nuclear and bacterial DNA. Because mitochondrial DNA specifi es far fewer proteins than nuclear DNA, the likelihood of getting a change in the code that maintains cellular functions is greater there than in the nucleus. 5. Metabolism. Living organisms maintain themselves by ac-q uiring nutrients from their environments ( Figure 1.7 ). The nutrients are used to obtain chemical energy and molecular components for building and maintaining the living system (see Chapter 4). We call these essential chemical processes metabolism. They include digestion, acquisition of energy (respiration), and synthesis of molecules and structures. Metabolism is often viewed as an interaction of destruc-tive (catabolic) and constructive (anabolic) reactions. The most fundamental anabolic and catabolic chemical pro-cesses used by living systems arose early in the evolution-ary history of life, and all living forms share them. These reactions include synthesis of carbohydrates, lipids, nucleic acids, and proteins and their constituent parts and cleavage of chemical bonds to recover energy stored in them. In ani-mals, many fundamental metabolic reactions occur at the cellular level, often in specifi c organelles found throughout the animal kingdom. Cellular respiration occurs, for exam-ple, in mitochondria. Cellular and nuclear membranes regu-late metabolism by controlling the movement of molecules across the cellular and nuclear boundaries, respectively. The study of complex metabolic functions is called physi-ology. We devote a large portion of this book to describ-ing and comparing the diverse tissues, organs, and organ systems that different groups of animals have evolved to perform the basic physiological functions of life (see Chapters 11 through 36). 6. Development. All organisms pass through a characteristic life cycle. Development describes the characteristic changes that an organism undergoes from its origin (usually the fer-tilization of an egg by sperm) to its fi nal adult form (see Chapter 8). Development usually features changes in size and shape, and differentiation of structures within an organ-ism. Even the simplest one-celled organisms grow in size and replicate their component parts until they divide into two or more cells. Multicellular organisms undergo more dramatic changes during their lives. Different developmental Figure 1.6 James Watson and Francis Crick with a model of the DNA double helix (A). Genetic information is coded in the nucleotide base sequence inside the DNA molecule. Genetic variation is shown (B) in DNA molecules that are similar in base sequence but differ from each other at four positions. Such differences can encode alternative traits, such as different eye colors.
T A T A C G G G C T A G C T A C G T A T A C G G C G C T A G C T A C G T A T A T A T A T A C G C T A G C T A C G C G T A T A G C G C T A G C G C C G T A T A 5 3 5 3 5 3 5 3 A B www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 9 stages of some multicellular forms are so dissimilar that they are hardly recognizable as belonging to the same species. Embryos are dis tinctly different from juvenile and adult forms into which they develop. Even postembryonic development of some organisms includes stages dramati-cally different from each other. The trans formation that occurs from one stage to another is called metamorpho-sis. There is little resemblance, for example, among the egg, larval, pupal, and adult stages of metamorphic insects ( Figure 1.8 ). Among animals, early stages of development are often more similar among organisms of related species than are later deve lopmental stages. In our survey of ani-mal diversity, we describe all stages of observed life histo-ries but concentrate on adult stages in which diversity tends to be greatest. 7. Environmental interaction. All animals interact with their environments. The study of organismal interaction with an environment is called ecology. Of special interest are the factors that infl uence geographic distribution and abundance of animals (see Chapters 37 and 38). The sci-ence of ecology reveals how an organism perceives envi-ronmental stimuli and responds in appropriate ways by adjusting its metabolism and physiology ( Figure 1.9 ). All organisms respond to environmental stimuli, a property called irritability. The stimulus and response may be simple, such as a unicellular organism moving from or toward a light source or away from a noxious substance, or it may be quite complex, such as a bird responding to a compli-cated series of signals in a mating ritual (see Chapter 36). Life and environment are inseparable. We cannot isolate the evolutionary history of a lineage of organisms from the environments in which it occurred. 8. Movement. Living systems and their parts show precise and controlled movements arising from within the system. The energy that living systems extract from their environments permits them to initiate controlled movements. Such move-ments at the cellular level are essential for reproduction, growth, and many responses to stimuli in all living forms and for development in multicellular ones. Autonomous movement reaches great diversity in animals, and much of this book comprises descriptions of animal movement and the many adaptations that animals have evolved for loco-motion. On a larger scale, entire populations or species may disperse from one geographic location to another one over time through their powers of movement. Movement characteristic of nonliving matter, such as that of particles in solution, radioactive decay of nuclei, and eruption of volcanoes is not precisely controlled by the moving objects themselves and often involves forces entirely external to them. The adaptive and often purposeful movements initi-ated by living systems are absent from the nonliving world. Figure 1.7 Feeding processes illustrated by (A) an ameba surrounding food and (B) a chameleon capturing insect prey with its projectile tongue.
A B Figure 1.8 A, Adult monarch butterfl y emerging from its pupal case. B, Fully formed adult monarch butterfl y.
A B 10 PART ONE Introduction to Living Animals Life Obeys Physical Laws To untrained observers, these eight properties of life may appear to violate basic laws of physics. Vitalism, the idea that life is endowed with a mystical vital force that violates physical and chemical laws, was once widely advocated. Biological research has consistently rejected vitalism, showing instead that all living systems obey basic laws of physics and chemis-try. Laws governing energy and its transformations (thermody-namics) are particularly important for understanding life (see Chapter 4). The fi rst law of thermodynamics is the law of conservation of energy. Energy is neither created nor destroyed but can be transformed from one form to another. All aspects of life require energy and its transformation. The energy to support life on earth fl ows from the fusion reactions in our sun and reaches the earth as light and heat. Sunlight captured by green plants and cyanobacteria is transformed by photo-synthesis into chemical bonds. Energy in chemical bonds is a form of potential energy released when the bond is broken; the energy is used to perform numerous cellular tasks. Energy transformed and stored in plants is then used by animals that eat the plants, and these animals may in turn provide energy for other animals that eat them. The second law of thermodynamics states that physical systems tend to proceed toward a state of greater disorder, or entropy. Energy obtained and stored by plants is subsequently released by various mechanisms and fi nally dissipated as heat. The complex molecular organization in living cells is attained and maintained only as long as energy fuels the organization. The ultimate fate of materials in the cells is degradation and dissipation of their chemical bond energy as heat. The process of evolution whereby organismal complexity can increase over time may appear at fi rst to violate the second law of thermody-namics, but it does not. Organismal complexity is achieved and maintained only by the constant use and dissipation of energy flowing into the biosphere from the sun. Survival, growth, and reproduction of animals require energy that comes from breaking complex food molecules into simple organic waste. The processes by which animals acquire energy through nutri-tion and respiration reveal themselves to us through the many physiological sciences. ZOOLOGY AS A PART OF BIOLOGY Animals form a distinct branch on the evolutionary tree of life. It is a large and old branch that originated in the Precambrian seas over 600 million years ago. Animals form part of an even larger limb known as eukaryotes, organisms whose cells contain mem-brane-enclosed nuclei. This larger limb includes plants, fungi and numerous unicellular forms. Perhaps the most distinctive char-acteristic of the animals as a group is their means of nutrition, which consists of eating other organisms. Evolution has elabo-rated this basic way of life through diverse systems for capturing and processing a wide array of food items and for locomotion. Animals are distinguished also by the absence of character-istics that have evolved in other eukaryotes. Plants, for example, use light energy to produce organic compounds (photosynthe-sis), and they have evolved rigid cell walls that surround their cell membranes; photosynthesis and cell walls are absent from animals. Fungi acquire nutrition by absorption of small organic molecules from their environments, and their body plan contains tubular fi laments called hyphae; these structures are absent from the animal kingdom. Figure 1.9 A lizard regulates its body temperature by choosing different locations (microhabitats) at different times of day.
Late afternoon Morning Midday www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 11 Some organisms combine properties of animals and plants. For example, Euglena ( Figure 1.10 ) is a motile, single-celled organism that resembles plants in being photosynthetic, but it resembles animals in its ability to eat food particles. Euglena is part of a separate eukaryotic lineage that diverged from those of plants and animals early in the evolutionary history of eukary-otes. Euglena and other unicellular eukaryotes are sometimes grouped as the kingdom Protista, although this kingdom is an arbitrary grouping of unrelated lineages that violates taxonomic principles (see Chapter 10). The fundamental structural and developmental features evo lved by the animal kingdom are presented in Chapters 8 and 9. PRINCIPLES OF SCIENCE Nature of Science We stated in the fi rst sentence of this chapter that zoology is the scientifi c study of animals. A basic understanding of zoology therefore requires an understanding of what science is, what it is not, and how knowledge is gained using the scientifi c method. Science is a way of asking questions about the natural world and sometimes obtaining precise answers to them. Although sci-ence, in the modern sense, has arisen recently in human history (within the last 200 years or so), the tradition of asking questions about the natural world is an ancient one. In this section, we examine the methodology that zoology shares with science as a whole. These features distinguish sciences from activities that we exclude from the realm of science, such as art and religion. Despite an enormous impact of science on our lives, many people have only a minimal understanding of the nature of sci-ence. For example, on March 19, 1981, the governor of Arkan-sas signed into law the Balanced Treatment for Creation-Science and Evolution-Science Act (Act 590 of 1981). This act falsely pre-sented “creation-science” as a valid scientifi c endeavor. “Creation-science” is actually a religious position advocated by a minority of the American religious community, and it does not qualify as science. The enactment of this law led to a historic lawsuit tried in December 1981 in the court of Judge William R. Overton, U.S. District Court, Eastern District of Arkansas. The suit was brought by the American Civil Liberties Union on behalf of 23 plain-tiffs, including religious leaders and groups representing several denominations, individual parents, and educational associations. The plaintiffs contended that the law was a violation of the First Amendment to the U.S. Constitution, which prohibits “estab-lishment of religion” by government. This prohibition includes passing a law that would aid one religion or prefer one religion over another. On January 5, 1982, Judge Overton permanently enjoined the State of Arkansas from enforcing Act 590. Considerable testimony during the trial dealt with the nature of science. Some witnesses defi ned science simply, if not very informatively, as “what is accepted by the scientifi c community” and “what scientists do.” However, on the basis of other testi-mony by scientists, Judge Overton was able to state explicitly these essential characteristics of science: 1. It is guided by natural law. 2. It has to be explanatory by reference to natural law. 3. It is testable against the observable world. 4. Its conclusions are tentative and therefore not necessarily the fi nal word. 5. It is falsifi able. Pursuit of scientifi c knowledge must be guided by the physical and chemical laws that govern the state of existence. Scientifi c knowledge must explain what is observed by reference to natu-ral law without requiring intervention of a supernatural being or force. We must be able to observe events in the real world, directly or indirectly, to test hypotheses about nature. If we draw a conclusion relative to some event, we must be ready always to discard or to modify our conclusion if further observations contradict it. As Judge Overton stated, “While anybody is free to approach a scientifi c inquiry in any fashion they choose, they cannot properly describe the methodology used as scientifi c if they start with a conclusion and refuse to change it regardless of the evidence developed during the course of the investigation.” Science is separate from religion, and the results of science do not favor one religious position over another. Unfortunately, the religious position formerly called “creation-science” has reappeared in American politics with the name “intelli-gent-design theory.” We are forced once again to defend the teach-ing of science against this scientifi cally meaningless dogma. Scientifi c Method These essential criteria of science form the hypothetico-deductive method. The fi rst step of this method is the gen-eration of hypotheses or potential answers to the question being asked. These hypotheses are usually based on prior observations of nature or derived from theories based on such observations. Scientifi c hypotheses often constitute general statements about nature that may explain a large number of diverse observations. Darwin’s hypothesis of natural selection, for example, explains the observations that many different species have properties that adapt them to their environments. Figure 1.10 Some organisms, such as the single-celled Euglena (shown here) and Volvox (see Figure 1.3), combine properties that distinguish animals (locomotion) and plants (photosynthetic ability).
C (photo 12 PART ONE Introduction to Living Animals On the basis of the hypothesis, a scientist must make a predic-tion about future observations. The scientist must say, “If my hypothesis is a valid explanation of past observations, then future observations ought to have certain characteristics.” The best hypotheses are those that make many predictions which, if found erroneous, will lead to rejection, or falsifi cation, of the hypothesis. The scientifi c method may be summarized as a series of steps: 1. Observation 2. Question 3. Hypothesis 4. Empirical test 5. Conclusions 6. Publication Observations illustrated in Figure 1.1A-E form a critical fi rst step in evaluating the life histories of natural populations. For example, observations of crab larvae shown in Figure 1.1E might cause the observer to question whether rate of larval growth is higher in undisturbed populations than in ones exposed to a chemical pollutant. A null hypothesis is then generated to permit an empirical test. A null hypothesis is one worded in a way that would permit data to reject it if it is false. In this case, the null hypothesis is that larval growth rates for crabs in undisturbed habitats are the same as those in polluted habitats. The investiga-tor then performs an empirical test by gathering data on larval growth rates in a set of undisturbed crab populations and a set of populations subjected to the chemical pollutant. Ideally, the undisturbed populations and the chemically treated populations are equivalent for all conditions except presence of the chemical in question. If measurements show consistent differences in growth rate between the two sets of populations, the null hypoth-esis is rejected. One then concludes that the chemical pollutant does alter larval growth rates. A statistical test is usually needed to ensure that the differences between the two groups are greater than would be expected from chance fl uctuations alone. If the null hypothesis cannot be rejected, one concludes that the data do not show any effect of the chemical treatment. The results of the study are then published to communicate fi ndings to other researchers, who may repeat the results, perhaps using additional populations of the same or a different species. Conclusions of the initial study then serve as the observations for further questions and hypotheses to reiterate the scientifi c process. Note that a null hypothesis cannot be proved correct using the scientifi c method. If the available data are compatible with it, the hypothesis serves as a guide for collecting additional data that potentially might reject it. Our most successful hypotheses are the ones that make specifi c predictions confi rmed by large numbers of empirical tests. The hypothesis of natural selection was invoked to explain variation observed in British moth populations ( Figure 1.11 ). In industrial areas of England having heavy air pollution, many pop-ulations of moths contain primarily darkly pigmented (melanic) individuals, whereas moth populations inhabiting clean forests show a much higher frequency of lightly pigmented individuals. The hypothesis suggests that moths can survive most effectively by matching their surroundings, thereby remaining invisible to birds that seek to eat them. Experimental studies have shown that, consistent with this hypothesis, birds are able to locate and then to eat moths that do not match their surroundings. Birds in the same area frequently fail to fi nd moths that match their sur-roundings, leaving them to reproduce and to increase their num-bers relative to conspicuous moths. Another testable prediction of the hypothesis of natural selection is that when polluted areas are cleaned, the moth populations should demonstrate an increase in frequency of lightly pigmented individuals. Observations of such populations confi rmed the result predicted by natural selection. If a hypothesis is very powerful in explaining a wide variety of related phenomena, it attains the status of a theory. Natural selection is a good example. Our example of the use of natu-ral selection to explain observed pigmentation patterns in moth populations is only one of many phenomena to which natural selection applies. Natural selection provides a potential explana-tion for the occurrence of many different traits distributed among virtually all animal species. Each of these instances constitutes a Figure 1.11 Light and melanic forms of the peppered moth, Biston betularia on A, a lichen-covered tree in unpolluted countryside and B, a soot-covered tree near industrial Birmingham, England. These color variants have a simple genetic basis. C, Recent decline in the frequency of the melanic form of the peppered moth with falling air pollution in industrial areas of England. The frequency of the melanic form still exceeded 90% in 1960, when smoke and sulfur dioxide emissions were still high. Later, as emissions fell and light-colored lichens began to grow again on the tree trunks, the melanic form became more conspicuous to predators. By 1986, only 50% of the moths were still of the melanic form, the rest having been replaced by the light form.
Smoke pollution level 300 200 100 0 100 90 80 70 60 50 40 1960 1970 1980 Winter smoke, µg/m3 Percent frequency of melanic peppered moths A B C Number of melanic moths www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 13 specifi c hypothesis generated from the theory of natural selec-tion. Note, however, that falsifi cation of a specifi c hypothesis does not necessarily lead to rejection of the theory as a whole. Natural selection may fail to explain origins of human behavior, for example, but it provides an excellent explanation for many structural modifi cations of the pentadactyl (fi ve-fi ngered) verte-brate limb for diverse functions. Scientists test many subsidiary hypotheses of their major theories to ask whether their theories are generally applicable. Most useful are theories that explain the largest array of different natural phenomena. We emphasize that the meaning of the word “theory,” when used by scientists, is not “speculation” as it is in ordinary English usage. Failure to make this distinction has been prominent in creationist challenges to evolution. The creationists have spoken The use of animals to serve human needs raises challenging ethical questions. Most controversial is the issue of animal use in biomedical and behavioral research and in testing commercial products.
Congress has passed a series of amendments to the Federal Ani-mal Welfare Act, a body of laws covering care of vertebrate animals in laboratories and other facilities. These amendments are known as the three R’s: Reduction in the number of animals needed for research; Refi nement of techniques that might cause stress or suf-fering; Replacement of live animals with simulations or cell cultures whenever possible. As a result, the total number of animals used each year in research and in testing of commercial products has declined. Developments in cellular and molecular biology also have contributed to a decreased use of animals for research and testing. An animal rights movement has created an awareness of the needs of animals used in research and has stimulated researchers to discover more humane alternatives.
Computers and culturing of cells can substitute for experiments on animals only when the basic principles involved are well known. When the principles themselves are being scrutinized and tested, com-puter modeling is not suffi cient. The National Research Council con-cedes that although the search for alternatives to animals in research and testing will continue, “the chance that alternatives will completely replace animals in the foreseeable future is nil.” Realistic immediate goals, however, are reduction in number of animals used, replace-ment of mammals with other vertebrates, and refi nement of experi-mental procedures to reduce discomfort of the animals being tested.
Medical and veterinary progress depends on research using ani-mals. Every drug and vaccine developed to improve the human con-dition has been tested fi rst on animals. Research using animals has enabled medical science to eliminate smallpox and polio from at least some parts of the world, and to immunize against diseases pre-viously common and often deadly, including diphtheria, mumps, and rubella. It also has helped to create treatments for cancer, diabetes, heart disease, and depression, and to develop surgical procedures including heart surgery, blood transfusions, and cataract removal. AIDS research is wholly dependent on studies using animals. The similarity of simian AIDS, identifi ed in rhesus monkeys, to human AIDS has permitted the disease in monkeys to serve as a model for the human disease. Recent work indicates that cats, too, may be use-ful models for development of an AIDS vaccine. Skin grafting experi-ments, fi rst done with cattle and later with other animals, opened a new era in immunological research with vast contributions to treat-ment of disease in humans and other animals.
Research using animals also has benefi ted other animals through the development of veterinary cures. The vaccines for feline leuke-mia and canine parvovirus were fi rst introduced to other cats and dogs. Many other vaccinations for serious diseases of animals were developed through research on animals—for example, rabies, distem-per, anthrax, hepatitis, and tetanus. No endangered species is used in general research (except to protect that species from total extinc-tion). Thus, research using animals has provided enormous benefi ts to humans and other animals. Still, much remains to be learned about treatment of diseases such as cancer, AIDS, diabetes, and heart dis-ease, and research with animals will be required for this purpose.
Despite the remarkable benefi ts produced by research on ani-mals, advocates of animal rights consider the harm done to animals in some research unethical. The most extreme animal-rights activ-ists advocate total abolition of all forms of research using animals. The scientifi c community is deeply concerned about the impact of such attacks on the ability of scientists to conduct important experi-ments that will benefi t people and animals. If we are justifi ed to use animals for food and fi ber and as pets, are we not justifi ed in experimentation to benefi t human welfare when these studies are conducted humanely and ethically?
The Association for Assessment and Accreditation of Laboratory Animal Care International supports the use of animals to advance medicine and science when nonanimal alternatives are not avail-able and when animals are treated in an ethical and humane way. Accreditation by this organization allows research institutions to demonstrate excellence in their standards of animal care. Nearly all major institutions receiving funding from the National Institutes of Health have sought and received this accreditation. See the website at www.aaalac.org for more information on accreditation of labora-tory animal care.
References on Animal-Research Ethics Commission on Life Sciences, National Research Council. 1988. Use of laboratory animals in biomedical and behavioral research. Washington, D.C., National Academy Press. Statement of national policy on guidelines for use of animals in biomedical research. Includes a chapter on benefi ts derived from use of animals.
Groves, J. M. 1997. Hearts and minds: the controversy over laboratory animals. Philadelphia, Pennsylvania, Temple University Press. Thoughtful review of the controversy by an activist who conducted extensive interviews with activists and animal-research supporters.
Paul, E. F., and J. Paul, eds. 2001. Why animal experimentation matters: the use of animals in medical research. New Brunswick, New Jersey, Social Philosophy and Policy Foundation, and Transaction Publishers. Essays by scientists, historians, and philosophers that express a defense of animal experimentation, demonstrating its moral acceptability and historical importance.
Ethics in Animal Research 14 PART ONE Introduction to Living Animals of evolution as “only a theory,” as if it were little better than a guess. In fact, the theory of evolution is supported by such mas-sive evidence that biologists view repudiation of evolution as tan-tamount to repudiation of reason. Nonetheless, evolution, along with all other theories in science, is not proven in a mathemati-cal sense, but it is testable, tentative, and falsifi able. Powerful theories that guide extensive research are called paradigms. The history of science shows that even major paradigms are subject to refutation and replacement when they fail to account for our observations of the natural world. They are then replaced by new paradigms in a process called a scientifi c revolution. For example, prior to the 1800s, animal species were studied as if they were specially created entities whose essential properties remained unchanged through time. Darwin’s theories led to a scientifi c revolution that replaced these views with the evolu-tionary paradigm. The evolutionary paradigm has guided bio-logical research for more than 140 years, and to date there is no scientifi c evidence that falsifi es it; it has strong explanatory power and continues to guide active inquiry into the natural world. Evolutionary theory is generally accepted as the corner-stone of biology. Chemists and physicists often use the term “law” to denote highly corroborated theories that appear to apply without exception to the physical world. Such laws are considered uniform throughout time and space. Because the biological world is temporally and spatially bounded, and because evolutionary change has produced an enor-mous diversity of forms with different emergent properties at mul-tiple levels ( Table 1.1 ), biologists now avoid using the term law for their theories. Nearly all of the biological laws proposed in the past have been found to apply only to some of life’s diverse forms and not to all. Mendel’s laws of inheritance, for example, do not apply to bacteria and often are violated even in animal and plant species that usually follow them. Darwin’s theories of perpetual change and common descent of living forms (p. 15) are perhaps the only state-ments that one meaningfully might call laws of biology. Experimental versus Evolutionary Sciences The many questions asked about the animal world since Aristo-tle can be grouped into two major categories. The fi rst category seeks to understand the proximate or immediate causes that underlie the functioning of biological systems at a particular time and place. These include the problems of explaining how animals perform their metabolic, physiological, and behavioral functions at the molecular, cellular, organismal, and even populational lev-els. For example, how is genetic information expressed to guide the synthesis of proteins? What causes cells to divide to produce new cells? How does population density affect the physiology and behavior of organisms? Mayr, E. 1985. Chapter 25 in D. Kohn, ed. The Darwinian Heritage. Princeton, Princeton University Press. The biological sciences that investigate proximate causes are called experimental sciences, and they proceed using the experimental method. Our goal is to test our understanding of a biological system. We predict the results of an experimental disturbance of the system based on our current understanding of it. If our understanding is correct, then the predicted outcome should occur. If, after the experimental disturbance, we see an unexpected outcome, we then discover that our understanding is incorrect or incomplete. Experimental conditions are repeated to eliminate chance occurrences that might produce erroneous con-clusions. Controls —repetitions of the experimental procedure that lack the disturbance—are established to eliminate unknown factors that might bias the outcome of the experiment. The pro-cesses by which animals maintain a body temperature under different environmental conditions, digest their food, migrate to new habitats, or store energy are some additional examples of physiological phenomena studied by experiment (see Chapters 29 through 36). Subfi elds of biology that constitute experimental sciences include molecular biology, cell biology, endocrinology, developmental biology, and community ecology. In contrast to questions concerning the proximate causes of biological systems are questions of the ultimate causes that have produced these systems and their distinctive characteristics through evolutionary time. For example, what are the evolutionary factors that caused some birds to acquire complex patterns of seasonal migration between temperate and tropical areas? Why do differ-ent species of animals have different numbers of chromosomes in their cells? Why do some animal species maintain complex social systems, whereas other species have solitary individuals? The biological sciences that address questions of ultimate cause are called evolutionary sciences, and they proceed largely using the comparative method rather than experimen-tation. Characteristics of molecular biology, cell biology, organ-ismal structure, development, and ecology are compared among related species to identify their patterns of variation. The pat-terns of similarity and dissimilarity are then used to test hypoth-eses of relatedness, and thereby to reconstruct the evolutionary tree that relates the species being studied. Recent advances in DNA sequencing technology permit detailed tests of relation-ships among all animal species. The evolutionary tree is then used to examine hypotheses of the evolutionary origins of the diverse molecular, cellular, organismal, and populational prop-erties observed in the animal world. Clearly, evolutionary sci-ences rely on results of experimental sciences as a starting point. Evolutionary sciences include comparative biochemistry, molec-ular evolution, comparative cell biology, comparative anatomy, comparative physiology, and phylogenetic systematics. A scientist’s use of the phrase “ultimate cause,” unlike Aristotle’s usage, does not imply a preconceived goal for natural phenomena. An argument that nature has a predetermined goal, such as evolution of the human mind, is termed teleological. Teleology is the mistaken notion that evolution of living organisms is guided by purpose toward an optimal design. A major success of Darwinian evolutionary theory is its rejection of teleology in explaining biological diversifi cation. www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 15 THEORIES OF EVOLUTION AND HEREDITY We turn now to a specifi c consideration of the two major para-digms that guide zoological research today: Darwin’s theory of evolution and the chromosomal theory of inheritance. Darwin’s Theory of Evolution Darwin’s theory of evolution is now over 140 years old (see Chapter 6). Darwin articulated the complete theory when he published his famous book On the Origin of Species by Means of Natural Selection in England in 1859 ( Figure 1.12 ). Biologists today are frequently asked, “What is Darwinism?” and “Do biolo-gists still accept Darwin’s theory of evolution?” These questions cannot be given simple answers, because Darwinism encom-passes several different, although mutually compatible, theories. Professor Ernst Mayr of Harvard University argued that Darwin-ism should be viewed as fi ve major theories. These fi ve theories have somewhat different origins and different fates and cannot be treated as only a single statement. The theories are (1) perpet-ual change, (2) common descent, (3) multiplication of species, (4) gradualism, and (5) natural selection. The fi rst three theories are generally accepted as having universal application through-out the living world. The theories of gradualism and natural selection are controversial among evolutionists, although both are strongly advocated by a large portion of the evolutionary community and are important components of the Darwinian evo-lutionary paradigm. Gradualism and natural selection are clearly part of the evolutionary process, but their explanatory power might not be as widespread as Darwin intended. Legitimate sci-entifi c controversies regarding gradualism and natural selection often are misrepresented by creationists as challenges to the fi rst three theories listed, although the validity of those fi rst three theories is strongly supported by all relevant observations. 1. Perpetual change. This is the basic theory of evolution on which the others are based. It states that the living world is neither constant nor perpetually cycling, but is always chang-ing. The properties of organisms undergo transformation across generations throughout time. This theory originated in antiquity but did not gain widespread acceptance until Darwin advocated it in the context of his other four theo-ries. “Perpetual change” is documented by the fossil record, which clearly refutes creationists’ claims for a recent origin of all living forms. Because it has withstood repeated testing and is supported by an overwhelming number of observa-tions, we now regard “perpetual change” as a scientifi c fact. 2. Common descent. The second Darwinian theory, “com-mon descent,” states that all forms of life descended from a common ancestor through a branching of lineages ( Fig-ure 1.13 ). The opposing argument, that the different forms of life arose independently and descended to the present Figure 1.12 Modern evolutionary theory is strongly identifi ed with Charles Robert Darwin, who, with Alfred Russel Wallace, provided the fi rst credible explanation of evolution. This photograph of Darwin was taken in 1854 when he was 45 years old. His most famous book, On the Origin of Species, appeared fi ve years later.
Figure 1.13 An early tree of life drawn in 1874 by the German biologist, Ernst Haeckel, who was strongly infl uenced by Darwin’s theory of common descent. Many of the phylogenetic hypotheses shown in this tree, including the unilateral progression of evolution toward humans ( Menschen, top), have been refuted.
16 PART ONE Introduction to Living Animals in linear, unbranched genealogies, has been refuted by comparative studies of organismal form, cell structure, and macromolecular structures (including those of the genetic material, DNA). All of these studies confi rm the theory that life’s history has the structure of a branching evolutionary tree, called a phylogeny. Species that share recent common ancestry have more similar features at all levels than do spe-cies whose most recent common ancestor is an ancient one. Much current research is guided by Darwin’s theory of com-mon descent toward reconstructing life’s phylogeny using the patterns of similarity and dissimilarity observed among species. The resulting phylogeny serves as the basis for our taxonomic classifi cation of animals (see Chapter 10). 3. Multiplication of species. Darwin’s third theory states that the evolutionary process produces new species by splitting and transforming older ones. Species are now generally viewed as reproductively distinct populations of organisms that usually but not always differ from each other in organismal form. Once species are fully formed, interbreeding among members of different species does not occur or is too restricted to permit the species’ lin-eages to merge. Evolutionists generally agree that the split-ting and transformation of lineages produces new species, although there is still much controversy concerning details of this process (see Chapter 6) and the precise meaning of the term “species” (see Chapter 10). Much active scientifi c research examines historical processes that generate new species. 4. Gradualism. Gradualism states that the large differences in anatomical traits that characterize diverse species origi-nate through the accumulation of many small incremen-tal changes over very long periods of time. This theory is important because genetic changes having very large effects on organismal form are usually harmful to an organism. It is possible, however, that some genetic variants that have large effects are nonetheless suffi ciently benefi cial to be favored by natural selection. Therefore, although gradual evolu-tion is known to occur, it may not explain the origins of all structural differences that we observe among species ( Fig-ure 1.14 ). Scientists are still actively studying this question. 5. Natural selection. Natural selection, Darwin’s most famous theory, rests on three propositions. First, there is varia-tion among organisms (within populations) for anatomical, behavioral, and physiological traits. Second, the variation is at least partly heritable so that offspring tend to resemble their parents. Third, organisms with different variant forms are expected to leave different numbers of offspring to future generations. Variants that permit their possessors most effec-tively to exploit their environments will preferentially survive to be transmitted to future generations. Over many genera-tions, favorable new traits will spread throughout a popula-tion. Accumulation of such changes leads, over long periods of time, to production of new organismal characteristics and new species. Natural selection is therefore a creative process that generates novel forms from the small individual varia-tions that occur among organisms within a population. Figure 1.14 Gradualism provides a plausible explanation for the origins of different bill shapes in the Hawaiian honeycreepers shown here. This theory has been challenged, however, as an explanation of the evolution of such structures as vertebrate scales, feathers, and hair from a common ancestral structure. The geneticist Richard Goldschmidt viewed the latter forms as unbridgeable by any gradual transformation series.
Akiapolaau Nukupuu Kauai akialoa Maui parrotbill Ou Kona finch Laysan finch Palila Anianiau Amakihi Akepa Alauwahio Crested honeycreeper Ula-ai-hawane Mamos Apapane Iiwi www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 17 Natural selection explains why organisms are constructed to meet the demands of their environments, a phenomenon called adaptation ( Figure 1.15 ). Adaptation is the expected result of a process that accumulates the most favorable variants occurring in a population throughout long periods of evolutionary time. Adaptation was viewed previously as strong evidence against evolution, and Darwin’s theory of natural selection was therefore important for convincing people that a natural process, capable of being studied scientifi cally, could produce new species. The demonstration that natural processes could produce adaptation was important to the eventual acceptance of all fi ve Darwinian theories. Darwin’s theory of natural selection faced a major obstacle when it was fi rst proposed: it lacked a successful theory of hered-ity. People assumed incorrectly that heredity was a blending process, and that any favorable new variant appearing in a pop-ulation therefore would be lost. The new variant arises initially in a single organism, and that organism therefore must mate with one lacking the favorable new trait. Under blending inheri-tance, the organism’s offspring would then have only a diluted form of the favorable trait. These offspring likewise would mate with others that lack the favorable trait. With its effects diluted by half each generation, the trait eventually would cease to exist. Natural selection would be completely ineffective in this situation. Darwin was never able to counter this criticism success-fully. It did not occur to Darwin that hereditary factors could be discrete and nonblending and that a new genetic variant therefore could persist unaltered from one generation to the next. This principle is called particulate inheritance. It was established after 1900 with the discovery of Gregor Mendel’s genetic experiments, and it was eventually incorporated into what we now call the chromosomal theory of inheritance. We use the term neo-Darwinism to describe Darwin’s theories as modifi ed by incorporating this theory of inheritance. Mendelian Heredity and the Chromosomal Theory of Inheritance The chromosomal theory of inheritance is the foundation for cur-rent studies of genetics and evolution in animals (see Chapters 5 and 6). This theory comes from the consolidation of research done in the fi elds of genetics, which was founded by the experi-mental work of Gregor Mendel ( Figure 1.16 ), and cell biology. Genetic Approach The genetic approach consists of mating or “crossing” populations of organisms that are true-breeding for alternative traits, and then following hereditary transmission of those traits through subse-quent generations. “True-breeding” means that a population main-tains across generations only one of the alternative traits when propagated in isolation from other populations. For example, most populations of fruit fl ies produce only red-eyed individu-als, generation after generation, regardless of the environments in which they are raised; such strains are true-breeding for red eyes. Some laboratory strains of fruit fl ies produce only white-eyed indi-viduals and are therefore true-breeding for white eyes (p. 88). Gregor Mendel studied the transmission of seven variable features in garden peas, crossing populations that were true-breeding for alternative traits (for example, tall versus short plants). In the fi rst generation (called the F 1 generation, for “fi l-ial”), only one of the alternative parental traits was observed; there was no indication of blending of the parental traits. In the example, the offspring (called F 1 hybrids because they represent a cross between two different forms) formed by crossing the tall and short plants were tall, regardless of whether the tall trait was inherited from the male or the female parent. These F 1 hybrids were allowed to self-pollinate, and both parental traits were found among their offspring (called the F 2 generation), although the trait observed in the F 1 hybrids (tall plants in this example) was Figure 1.15 According to Darwinian evolutionary theory, the different forms of these vertebrate forelimbs were molded by natural selection to adapt them for different functions. We show in later chapters that, despite these adaptive differences, these limbs share basic structural similarities.
Frog Hawk Gecko Bat Whale Human 18 PART ONE Introduction to Living Animals three times more common than the other trait. Again, there was no indication of blending of the parental traits ( Figure 1.17 ). Mendel’s experiments showed that the effects of a genetic factor can be masked in a hybrid individual, but that these fac-tors are not physically altered during the transmission process. He postulated that variable traits are specifi ed by paired heredi-tary factors, which we now call “genes.” When gametes (eggs or sperm) are produced, the two genes controlling a particular feature are segregated from each other and each gamete receives only one of them. Fertilization restores the paired condition. If an organism possesses different forms of the paired genes for a feature, only one of them is expressed in its appearance, but both genes nonetheless are transmitted unaltered in equal num-bers to the gametes produced. Transmission of these genes is particulate, not blending. Mendel observed that inheritance of one pair of traits is independent of inheritance of other paired Figure 1.16 A, Gregor Johann Mendel. B, The monastery in Brno, Czech Republic, now a museum, where Mendel performed his experiments with garden peas.
A B Figure 1.17 Different predictions of particulate versus blending inheritance regarding the outcome of Mendel’s crosses of tall and short plants. The prediction of particulate inheritance is upheld and the prediction of blending inheritance is falsifi ed by the results of the experiments. The reciprocal experiments (crossing short female parents with tall male parents) produced similar results. (P1 parental generation; F1 fi rst fi lial generation; F2 second fi lial generation.) PARTICULATE INHERITANCE (observed) BLENDING INHERITANCE (not observed) P1 F1 F2 Tall females Short males All intermediate All intermediate P1 F1 F2 Tall females Short males Tall and short (3:1 ratio) All tall www . m hhe . com /hi ckmani pz14e CHAPTER 1 Life: Biological Principles and the Science of Zoology 19 traits. We now know, however, that not all pairs of traits are inherited independently of each other; different traits that tend to be inherited together are said to be genetically linked (p. 88). Numerous studies, particularly of the fruit fl y, Drosophila mela-nogaster, have shown that principles of inheritance discovered initially in plants apply also to animals. Contributions of Cell Biology Improvements in microscopes during the 1800s permitted cytol-ogists to study the production of gametes by direct observa-tion of reproductive tissues. Interpreting the observations was Figure 1.18 An early nineteenth-century micrographic drawing of sperm from (1) guinea pig, (2) white mouse, (3) hedgehog, (4) horse, (5) cat, (6) ram, and (7) dog. Some biologists initially interpreted these as parasitic worms in the semen, but in 1824, Jean Prévost and Jean Dumas correctly identifi ed their role in egg fertilization.
1 5 4 3 6 7 2 initially diffi cult, however. Some prominent biologists hypoth-esized, for example, that sperm were parasitic worms in semen ( Figure 1.18 ). This hypothesis was soon falsifi ed, and the true nature of gametes was clarifi ed. As the precursors of gametes prepare to divide early in gamete production, the nuclear mate-rial condenses to reveal discrete, elongate structures called chro-mosomes. Chromosomes occur in pairs that are usually similar but not identical in appearance and informational content. The number of chromosomal pairs varies among species. One mem-ber of each pair is derived from the female parent and the other from the male parent. Paired chromosomes are physically associ-ated and then segregated into different daughter cells during cell division prior to gamete formation ( Figure 1.19 ). Each resulting gamete receives one chromosome from each pair. Different pairs of chromosomes are sorted into gametes independently of each other. Because the behavior of chromosomal material during gamete formation parallels that postulated for Mendel’s genes, Sutton and Boveri in 1903 through 1904 hypothesized that chro-mosomes were the physical bearers of genetic material. This hypothesis met with extreme skepticism when fi rst proposed. A long series of tests designed to falsify it nonetheless showed that its predictions were upheld. The chromosomal theory of inheri-tance is now well established. Figure 1.19 Paired chromosomes being separated before nuclear division in the process of forming gametes. S U M M A R Y Zoology is the scientifi c study of animals, and it is part of biol-ogy, the scientifi c study of life. Animals and life in general can be identifi ed by attributes that they have acquired over their long evo-lutionary histories. The most outstanding attributes of life include chemical uniqueness, complexity and hierarchical organization, reproduction, possession of a genetic program, metabolism, devel-opment, interaction with the environment, and movement. Biologi-cal systems comprise a hierarchy of integrative levels (molecular, 20 PART ONE Introduction to Living Animals cellular, organismal, populational, and species levels), each of which demonstrates a number of specifi c emergent properties. Science is characterized by the acquisition of knowledge by constructing and then testing hypotheses through observations of the natural world. Science is guided by natural law, and its hypoth-eses are testable, tentative, and falsifi able. Zoological sciences can be subdivided into two categories, experimental sciences and evolu-tionary sciences. Experimental sciences use the experimental method to ask how animals perform their basic metabolic, developmental, behavioral, and reproductive functions, including investigations of their molecular, cellular, and populational systems. Evolutionary sciences use the comparative method to reconstruct the history of life, and then use that history to understand how diverse species and their molecular, cellular, organismal, and populational prop-erties arose through evolutionary time. Hypotheses that withstand repeated testing and therefore explain many diverse phenomena gain the status of a theory. Powerful theories that guide extensive research are called “paradigms.” The major paradigms that guide the study of zoology are Darwin’s theory of evolution and the chromo-somal theory of inheritance. The principles given in this chapter illustrate the unity of bio-logical science. All components of biological systems are guided by natural laws and are constrained by those laws. Living organisms arise only from other living organisms, just as new cells can be pro-duced only from preexisting cells. Reproductive processes occur at all levels of the biological hierarchy and demonstrate both heredity and variation. Interaction of heredity and variation at all levels of the biological hierarchy produces evolutionary change and has gener-ated the great diversity of animal life documented throughout this book. R E V I E W Q U E S T I O N S Why is life diffi cult to defi ne? What are the basic chemical differences that distinguish living from nonliving systems? Describe the hierarchical organization of life. How does this organization lead to the emergence of new properties at different levels of biological complexity? What is the relationship between heredity and variation in reproducing biological systems? Describe how evolution of complex organisms is compatible with the second law of thermodynamics. What are the essential characteristics of science? Describe how evolutionary studies fi t these characteristics whereas “scientifi c creationism” or “intelligent-design theory” does not. Use studies of natural selection in British moth populations to illustrate the hypothetico-deductive method of science. 1.
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7. How do we distinguish the terms hypothesis, theory, paradigm, and scientifi c fact? How do biologists distinguish experimental and evolutionary sciences? What are Darwin’s fi ve theories of evolution (as identifi ed by Ernst Mayr)? Which are accepted as fact and which continue to stir controversy among biologists? What major obstacle confronted Darwin’s theory of natural selection when it was fi rst proposed? How was this obstacle overcome? How does neo-Darwinism differ from Darwinism? Describe the respective contributions of the genetic approach and cell biology to formulating the chromosomal theory of inheritance. 8.
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13. S E L E C T E D R E F E R E N C E S Futuyma, D. J. 1995. Science on trial: the case for evolution. Sunderland, Massachusetts, Sinauer Associates, Inc. A defense of evolutionary biology as the exclusive scientifi c approach to the study of life’s diversity. Kitcher, P. 1982. Abusing science: the case against creationism. Cambridge, Massachusetts, MIT Press. A treatise on how knowledge is gained in science and why creationism does not qualify as science. Note that the position refuted as “scientifi c creationism” in this book is equivalent in content to the position more recently termed “intelligent-design theory.” Kuhn, T. S. 1970. The structure of scientifi c revolutions, ed. 2, enlarged. Chicago, University of Chicago Press. An infl uential and controversial commentary on the process of science. Mayr, E. 1982. The growth of biological thought: diversity, evolution and inheritance. Cambridge, Massachusetts, The Belknap Press of Harvard University Press. An interpretive history of biology with special reference to genetics and evolution. Medawar, P. B. 1989. Induction and intuition in scientifi c thought. London, Methuen & Company. A commentary on the basic philosophy and methodology of science. Moore, J. A. 1993. Science as a way of knowing: the foundations of modern biology. Cambridge, Massachusetts, Harvard University Press. A lively, wide-ranging account of the history of biological thought and the workings of life. Perutz, M. F. 1989. Is science necessary? Essays on science and scientists. New York, E. P. Dutton. A general discussion of the utility of science. Pigliucci, M. 2002. Denying evolution: creationism, scientism, and the nature of science. Sunderland, Massachusetts, Sinauer Associates, Inc. A critique of science education and the public perception of science. Rennie, J. 2002. 15 answers to creationist nonsense. Sci. Am. 287:78–85 (July). A guide to the most common arguments used by creationists against evolutionary biology, with concise explanations of the scientifi c fl aws of creationists’ claims. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! C H A P T E R 2 The Origin and Chemistry of Life Spontaneous Generation of Life? From ancient times, people commonly thought that life arose repeat-edly by spontaneous generation from nonliving material in addition to parental reproduction. For example, frogs appeared to arise from damp earth, mice from putrefi ed matter, insects from dew, and mag-gots from decaying meat. Warmth, moisture, sunlight, and even star-light often were mentioned as factors that encouraged spontaneous generation of living organisms. Among the efforts to synthesize organisms in the laboratory is a recipe for making mice, given by the Belgian plant nutritionist Jean Baptiste van Helmont (1648). “If you press a piece of under-wear soiled with sweat together with some wheat in an open jar, after about 21 days the odor changes and the ferment . . . changes the wheat into mice. But what is more remarkable is that the mice which came out of the wheat and underwear were not small mice, not even miniature adults or aborted mice, but adult mice emerge!” In 1861, the great French scientist Louis Pasteur convinced scientists that living organisms cannot arise spontaneously from nonliving matter. In his famous experiments, Pasteur introduced fermentable material into a fl ask with a long S-shaped neck that was open to air. The fl ask and its contents were then boiled for a long time to kill any microorganisms that might be present. After-ward the fl ask was cooled and left undisturbed. No fermentation occurred because all organisms that entered the open end were deposited in the neck and did not reach the fermentable material. When the neck of the fl ask was removed, microorganisms in the air promptly entered the fermentable material and proliferated. Pasteur concluded that life could not originate in the absence of previously existing organisms and their reproductive elements, such as eggs and spores. Announcing his results to the French Academy, Pasteur proclaimed, “Never will the doctrine of spontaneous generation arise from this mortal blow.” All living organisms share a common ancestor, most likely a population of colonial microorganisms that lived almost 4 billion years ago. This common ancestor was itself the product of a long period of prebiotic assembly of nonliving matter, including organic molecules and water, to form self-replicating units. All living organ-isms retain a fundamental chemical composition inherited from their ancient common ancestor. Earth’s abundant supply of water was critical for the origin of life. 22 PART ONE Introduction to Living Animals A ccording to the big-bang model, the universe originated from a primeval fi reball and has been expanding and cooling since its inception 10 to 20 billion years ago. The sun and planets formed approximately 4.6 billion years ago from a spherical cloud of cosmic dust and gases. The cloud col-lapsed under the infl uence of its own gravity into a rotating disc. As material in the central part of the disc condensed to form the sun, gravitational energy was released as radiation. The pressure of this outwardly directed radiation prevented a collapse of the nebula into the sun. The material left behind cooled and eventu-ally produced the planets, including earth ( Figure 2.1 ). In the 1920s, Russian biochemist Alexander I. Oparin and British biologist J. B. S. Haldane independently proposed that life originated on earth after an inconceivably long period of “abiogenic molecular evolution.” Rather than arguing that the first living organisms miraculously originated all at once, a notion that formerly discouraged scientifi c inquiry, Oparin and Haldane argued that the simplest form of life arose gradually by the progressive assembly of small molecules into more complex organic molecules. Molecules capable of self-replication eventu-ally would be produced, ultimately leading to assembly of living microorganisms. WATER AND LIFE The origin and maintenance of life on earth depend critically upon water. Water is the most abundant of all compounds in cells, forming 60% to 90% of most living organisms. Water has several extraordinary properties that explain its essential role in living systems and their origin. These properties result largely from hydrogen bonds that form between its molecules ( Figure 2.2 ). Figure 2.1 Solar system showing narrow range of thermal conditions suitable for life. TOO HOT TOO COLD Neptune Pluto Jupiter Saturn Uranus Sun Mercury Earth Mars Venus Figure 2.2 Geometry of water molecules. Each water molecule is linked by hydrogen bonds ( dashed lines ) to four other molecules. If imaginary lines connect the water molecules as shown, a tetrahedron is obtained. O O O O O H H H H H H H H H H Hydrogen bond Water molecule www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 23 Water has a high specifi c heat capacity: 1 calorie is required to elevate the temperature of 1 g of water 1° C, a higher thermal capacity than any other liquid except ammonia. Much of this heat energy is used to rupture some hydrogen bonds in addition to increasing the kinetic energy (molecular movement), and thus the temperature, of the water. Water’s high thermal capacity greatly moderates environmental temperature changes, thereby protect-ing living organisms from extreme thermal fl uctuation. Water also has a high heat of vaporization, requiring more than 500 calo-ries to convert 1 g of liquid water to water vapor. All hydrogen bonds between a water molecule and its neighbors must be rup-tured before that water molecule can escape the surface and enter the air. For terrestrial animals (and plants), cooling produced by evaporation of water is important for expelling excess heat. Another property of water important for life is its unique density behavior during changes of temperature. Most liquids become denser with decreasing temperature. Water, however, reaches its maximum density at 4° C while still a liquid, then becomes less dense with further cooling ( Figure 2.3 ). Therefore, ice fl oats rather than sinking to the bottoms of lakes and ponds. If A calorie is defi ned as the amount of heat required to heat 1 g of water from 14.5° to 15.5° C. Although the calorie is the traditional unit of heat widely used in publications and tables, it is not part of the International System of Units (the SI system) which uses the joule (J) as the energy unit (1 cal 4.184 J). ice were denser than liquid water, bodies of water would freeze solid from the bottom upward in winter and might not melt com-pletely in summer. Such conditions would severely limit aquatic life. In ice, water molecules form an extensive, open, crystal-like network supported by hydrogen bonds that connect all mole-cules. The molecules in this lattice are farther apart, and thus less dense, than in liquid water at 4° C. Water has high surface tension, exceeding that of any other liquid but mercury. Hydrogen bonding among water molecules produces a cohesiveness important for maintaining protoplasmic form and movement. The resulting surface ten-sion creates an ecological niche (see p. 826) for insects, such as water striders and whirligig beetles, that skate on the surfaces of ponds ( Figure 2.4 ). Despite its high surface tension, water has low viscosity, permitting movement of blood through minute capillaries and of cytoplasm inside cellular boundaries. Water is an excellent solvent. Salts dissolve more extensively in water than in any other solvent. This property results from the dipolar nature of water, which causes it to orient around charged particles dissolved in it. When, for example, crystalline NaCl dis-solves in water, the Na and Cl ions separate ( Figure 2.5 ). The negative zones of the water dipoles attract the Na ions while the positive zones attract the Cl ions. This orientation keeps the ions separated, promoting their dissociation. Solvents lack-ing this dipolar character are less effective at keeping the ions separated. Binding of water to dissolved protein molecules is essential to the proper functioning of many proteins. Water also participates in many chemical reactions in living organisms. Many compounds are split into smaller pieces by the addition of a molecule of water, a process called hydrolysis. Likewise, larger compounds may be synthesized from smaller components by the reverse of hydrolysis, called condensation reactions. R R H O Hydrolysis R OH H R R OH H R Con 2 — — — — — → densation R R H O 2 → — Figure 2.3 When water freezes at 0° C, the four partial charges of each atom in the molecule interact with the opposite charges of atoms in other water molecules. The hydrogen bonds between all the molecules form a crystal-like lattice structure, and the molecules are farther apart (and thus less dense) than when some of the molecules have not formed hydrogen bonds at 4° C. Water molecules Hydrogen bonds Figure 2.4 Because of hydrogen bonds between water molecules at the water-air interface, the water molecules cling together and create a high surface tension. Thus some insects, such as this water strider, can literally walk on water. 24 PART ONE Introduction to Living Animals ORGANIC MOLECULAR STRUCTURE OF LIVING SYSTEMS Chemical evolution in the prebiotic environment produced simple organic compounds that ultimately formed the build-ing blocks of living cells. The term “organic” refers broadly to compounds that contain carbon. Many also contain hydrogen, oxygen, nitrogen, sulfur, phosphorus, salts, and other elements. Carbon has a great ability to bond with other carbon atoms in chains of varying lengths and confi gurations. Carbon-to-carbon combinations introduce the possibility of enormous complexity and variety into molecular structure. More than a million organic compounds are known. We review the kinds of organic molecules found in living systems, followed by further discussion of their origins in earth’s primitive reducing atmosphere. Carbohydrates: Nature’s Most Abundant Organic Substance Carbohydrates are compounds of carbon, hydrogen, and oxy-gen. These elements usually occur in the ratio of 1 C: 2 H: 1 O and are grouped as HCOH. Carbohydrates function in protoplasm mainly as structural elements and as a source of chemical energy. Glucose is the most important of these energy-storing carbohydrates. Familiar examples of carbohydrates include sugars, starches, and cellulose (the woody structure of plants). Cellulose occurs on earth in greater quantities than all other organic materials combined. Carbohydrates are synthe-sized by green plants from water and carbon dioxide, with the aid of solar energy. This process, called photosynthesis, is a reaction upon which all life depends, for it is the starting point in the formation of food. Figure 2.5 When a crystal of sodium chloride dissolves in water, the negative ends of the dipolar molecules of water surround the Na ions, while the positive ends of water molecules face the Cl ions. The ions are thus separated and do not reenter the salt lattice. + – – + + – Na Cl Salt crystal Hydration shells In pure liquid water ( distilled water), a small fraction of the water molecules split into ions of hydrogen (H ) and hydroxide (OH ); the concentration of both ions is 10 7 moles/liter. An acidic substance, when dissolved in water, contributes H ions to solution, thereby increasing their concentration and causing an excess of H ions over OH ions in solution. A basic substance does the reverse, contribut-ing OH ions to the solution and making OH ions more common than H ions. The degree to which a solution is acidic or basic is criti-cal for most cellular processes and requires precise quantifi cation and control; the structure and function of dissolved proteins, for example, depend critically on the concentration of H in the solution. The pH scale quantifi es the degree to which a solution is acidic or basic. The scale ranges from 0 to 14 and represents the addi-tive inverse of the logarithm (base 10) of the H concentration (in moles/liter) of the solution. Pure liquid water therefore has a pH of 7 (H concentration 10 7 moles/liter). A solution with pH 6.0 has an H concentration ten times higher than that of pure water and is acidic, whereas a solution with pH 8.0 has an H concentration ten times lower than pure water and is basic. A concentrated strong acid, such as hydrochloric acid (HCl, known commercially as “muriatic acid” used to clean masonry) has an H concentration of ~1 10 0 mole/liter, giving a pH of 0 (a concentration of H 10,000,000 times that of pure water). A concentrated base, such as sodium hydroxide (NaOH, used commercially in liquid drain cleaners) has an H con-centration of approximately 10 14 mole/liter, giving a pH of 14. A buffer is a dissolved substance (solute) that causes a solution to resist changes in pH because the buffer can remove added H and OH ions from solution by binding them into compounds. Dissolved carbon dioxide in the form of bicarbonate (HCO 3 ) is a buffer that helps to protect human blood (pH 7.3 to 7.5) from changes in pH. H ions are removed from solution when they react with bicarbonate ions to form carbonic acid, which then dissociates into carbon dioxide and water. The excess carbon dioxide is removed during exhalation (p. 703). OH ions are removed from solution when this reaction is reversed, forming bicarbonate and hydrogen ions. The excess bicar-bonate ions are secreted in the urine (p. 676), and the hydrogen ions serve to increase blood pH back to normal levels. Severe health prob-lems occur if the pH of blood drops to 7 or rises to 7.8. pH of Water Solutions Because water is critical to the support of life, the continu-ing search for extraterrestrial life usually begins with a search for water. Plans for a human outpost on the moon likewise depend upon fi nding water there. As we write, NASA is planning to crash a space probe into the moon in 2009 in a search for ice; the moon’s south pole is a prime candidate for a human outpost if ice is found there. www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 25 Carbohydrates are usually grouped into the following three classes: (1) monosaccharides, or simple sugars; (2) disaccha-rides, or double sugars; and (3) polysaccharides, or complex sugars. Simple sugars have a single carbon chain containing 4 carbons (tetroses), 5 carbons (pentoses), or 6 carbons (hex-oses). Other simple sugars have up to 10 carbons, but these sugars are not biologically important. Simple sugars, such as glucose, galactose, and fructose, all contain a free sugar group, C C H OH O in which the double-bonded O may be attached to the terminal or nonterminal carbons of a chain. The hexose glucose (also called dextrose) is particularly important to the living world. Glu-cose is often shown as a straight chain ( Figure 2.6A ), but in water it forms a cyclic compound ( Figure 2.6B ). The “chair” diagram ( Figure 2.7 ) of glucose best represents its true confi guration, but all forms of glucose, however represented, are chemically equiv-alent. Other hexoses of biological signifi cance include galactose and fructose, which are compared with glucose in Figure 2.8 . Disaccharides are double sugars formed by bonding two simple sugars. An example is maltose (malt sugar), composed of two glucose molecules. As shown in Figure 2.9 , the two glu-cose molecules are joined by removing a molecule of water, causing the sharing of an oxygen atom by the two sugars. All disaccharides are formed in this manner. Two other common disaccharides are sucrose (ordinary cane, or table, sugar), formed by the linkage of glucose and fructose, and lactose (milk sugar), composed of glucose and galactose. Polysaccharides are composed of many molecules of simple sugars (usually glucose) linked in long chains called polymers. Their empirical formula is usually written (C 6 H 10 O 5 ) n , where n Figure 2.6 Two ways of depicting the simple sugar glucose. In A, the carbon atoms are shown in open-chain form. When dissolved in water, glucose tends to assume a ring form as in B. In this ring model the carbon atoms located at each turn in the ring are usually not shown. H OH O C H O H C H O H C H O H C H CH2OH H OH H OH H OH H H O H O H C H C O H A B Figure 2.7 “Chair” representation of a glucose molecule. OH OH HO OH H H H H H H H C C C C C O C OH Figure 2.8 These three hexoses are the most common monosaccharides. Figure 2.9 Formation of a double sugar (disaccharide maltose) from two glucose molecules with the removal of one molecule of water. OH O CH2OH OH HO OH HO Glucose Maltose Glucose OH HO OH O OH O CH2OH O CH2OH O CH2OH OH OH OH OH OH 26 PART ONE Introduction to Living Animals designates the number of simple-sugar subunits in the polymer. Starch is the common polymer in which sugar is stored in most plants and is an important food for animals. Chitin is an impor-tant structural polysaccharide in the exoskeletons of insects and other arthropods (p. 404). Glycogen is an important polymer for storing sugar in animals. It is stored mainly in liver and muscle cells in vertebrates. When needed, glycogen is converted to glu-cose and delivered by blood to the tissues. Another polymer is cellulose, the principal structural carbohydrate of plants. Lipids: Fuel Storage and Building Material Lipids are fats and fatlike substances. They are molecules of low polarity; consequently, they are virtually insoluble in water but are soluble in organic solvents, such as acetone and ether. The three principal groups of lipids are neutral fats, phospholipids, and steroids. Neutral Fats The neutral or “true” fats are major fuels of animals. Stored fat is derived either directly from dietary fat or indirectly from dietary carbohydrates that the body has converted to fat for storage. Fats are oxidized and released into the bloodstream as needed to meet tissue demands, especially those of active muscle. Neutral fats include triglycerides, which contain glycerol and three molecules of fatty acids. Neutral fats are therefore esters, a combination of an alcohol (glycerol) and an acid. Fatty acids in triglycerides are simply long-chain monocarboxylic acids; they vary in size but are commonly 14 to 24 carbons long. Production of a typical fat by the union of glycerol and stearic acid is shown in Figure 2.10A . In this reaction, three fatty acid molecules are seen to have united with OH groups of glycerol to form stearin (a neutral fat) plus three molecules of water. Most triglycerides contain two or three different fatty acids attached to glycerol, and bear ponderous names such as myris-toyl stearoyl glycerol ( Figure 2.10B ). The fatty acids in this tri-glyceride are saturated; every carbon within the chain holds two hydrogen atoms. Saturated fats, more common in animals than in plants, are usually solid at room temperature. Unsatu-rated fatty acids, typical of plant oils, have two or more carbon atoms joined by double bonds; the carbons are not “saturated” with hydrogen atoms and are available to form bonds with other atoms. Two common unsaturated fatty acids are oleic acid and linoleic acid ( Figure 2.11 ). Plant fats, such as peanut oil and corn oil, tend to be liquid at room temperature. Phospholipids Unlike fats that are fuels and serve no structural roles in the cell, phospholipids are important components of the molecular organization of tissues, especially membranes. They resemble triglycerides in structure, except that one of the three fatty acids is replaced by phosphoric acid and an organic base. An exam-ple is lecithin, an important phospholipid of nerve membranes ( Figure 2.12 ). Because the phosphate group on phospholipids is charged and polar and therefore soluble in water, and the remainder of the molecule is nonpolar, phospholipids can bridge two environments and bind water-soluble molecules, such as proteins, to water-insoluble materials. Steroids Steroids are complex alcohols. Although they are structurally unlike fats, they have fatlike properties. The steroids are a large group of biologically important molecules, including cholesterol ( Figure 2.13 ), vitamin D3, many adrenocortical hormones, and sex hormones. Amino Acids and Proteins Proteins are large, complex molecules composed of 20 kinds of amino acids ( Figure 2.14 ). The amino acids are linked by pep-tide bonds to form long, chainlike polymers. In the formation of a peptide bond, the carboxyl group of one amino acid is linked by a covalent bond to the amino group of another, with elimina-tion of water, as shown here: Figure 2.10 Neutral fats. A, Formation of a neutral fat from three molecules of stearic acid (a fatty acid) and glycerol. B, A neutral fat bearing three different fatty acids. Stearin (1 mol) Glycerol (1 mol) Stearic acid (3 mol) C17H35CO OH H O—CH2 C17H35CO OH + H O—CH C17H35CO OH H O—CH2 C17H35COO—CH2 C17H35COO—CH + 3H2O C17H35COO—CH2 CH2—O—C—(CH2)12—CH3 C—H O CH2—O—C—(CH2)16—CH3 H3C—(CH2)14—C—O— O O A B R H2O OH H CH C N O H R OH H CH C N O O H R OH CH C C N O R H CH N H H + + Amino group Carboxyl group Amino group Carboxyl group Peptide bond www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 27 Figure 2.11 Unsaturated fatty acids. Oleic acid has one double bond and linoleic acid has two double bonds. The remainder of both acids is saturated. CH3—(CH2)7—CH CH—(CH2)7 — COOH Oleic acid CH3—(CH2)4—CH CH—CH2—CH CH—(CH2)7 — COOH Linoleic acid recognize four levels of protein organization called primary, secondary, tertiary, and quaternary structures. The primary structure of a protein is the sequence of amino acids composing the polypeptide chain. Because bonds between the amino acids in the chain can form only a limited number of stable angles, certain recurring structural patterns are assumed by the chain. These bond angles generate the second-ary structure, such as the alpha-helix, which makes helical turns in a clockwise direction like a screw ( Figure 2.15 ). The spirals of the chains are stabilized by hydrogen bonds, usually between a hydrogen atom of one amino acid and the peptide-bond oxygen of another amino acid from an adjacent turn of the helix. The helical and other confi gurations formed by the poly-peptide chain bend and fold, giving the protein its complex, yet stable, three-dimensional tertiary structure ( Figure 2.15 ). The folded chains are stabilized by chemical bonds between pairs of amino acids from different parts of the polypeptide chain. These bonds form between “side groups,” parts of the amino acid not involved in a peptide bond. An example is the disulfi de bond, a covalent bond between the sulfur atoms in two cysteine amino acids that are brought together by folds in the polypeptide chain. Also stabilizing the tertiary structure of proteins are hydrogen bonds, ionic bonds, and hydrophobic bonds. Figure 2.12 Lecithin (phosphatidyl choline), an important phospholipid of nerve membranes.
Choline group Oleoyl group Palmitoyl group Water-soluble end Fat-soluble end H2C O C H C O CH3 N—CH3 CH2 CH2 O P —O—CH2—C—CH2—O O O H3C— O— CH2 H2C CH2 H2C CH2 H2C H2C CH2 H2C CH2 H2C CH2 CH2 CH2 CH2 CH2 CH3 H2C H2C H2C H2C CH2 H2C CH2 CH2 H2C H2C CH3 H2C H H C C + – Figure 2.13 Cholesterol, a steroid. All steroids have a basic skeleton of four rings (three 6-carbon rings and one 5-carbon ring) with various side groups attached.
HO CH3 CH3 CH2 CH3 CH CH2 Cholesterol CH2 CH H3C H3C Figure 2.14 Five of the twenty kinds of amino acids.
Tryptophan C C NH2 H H N H CH CH2 CH2 CH2 O OH C C NH2 H CH2 HS O OH C O OH Glycine Proline Cysteine C H C C NH2 H CH2 CH2 O OH C C NH2 H CH2 O OH O OH Glutamic acid C N CH The combination of two amino acids by a peptide bond forms a dipeptide having a free amino group on one end and a free carboxyl group on the other; therefore, additional amino acids can be joined until a long chain is produced. The 20 dif-ferent kinds of amino acids can be arranged in an enormous variety of sequences of up to several hundred amino acid units, accounting for the large diversity of proteins found among living organisms. A protein is not just a long string of amino acids; it is a highly organized molecule. For convenience, biochemists 28 PART ONE Introduction to Living Animals The term quaternary structure describes proteins that con-tain more than one polypeptide chain. For example, hemoglobin (the oxygen-carrying substance in blood) of higher vertebrates is composed of four polypeptide subunits held together in a single protein molecule ( Figure 2.15 ). Proteins perform many functions in living organisms. They form the structural framework of protoplasm and many cellular components. Many proteins function as enzymes, the biological catalysts required for almost every reaction in the body. Enzymes lower the activation energy required for specifi c reactions and enable life processes to proceed at moderate temperatures rather than requiring high temperatures. Enzymes control the reac-tions by which food is digested, absorbed, and metabolized. They promote the synthesis of structural materials for growth and to replace those lost by wear. They determine the release of energy used in respiration, growth, muscle contraction, physical and mental activities, and many other activities. Enzyme action is described in Chapter 4 (p. 60). A prion is an infectious protein particle in which a pro-tein of the host organism is contorted into an abnormal three-dimensional structure. Upon infection, the prion causes its host’s normal copies of the protein to be refolded into the abnormal form, with pathological results. In “mad cow disease,” a prion infection severely damages brain tissues and is fatal. Fatal neuro-logical diseases associated with transmissible prions occur also in people (for example, kuru), and in sheep and goats (scrapie). Nucleic Acids Nucleic acids are complex polymeric molecules whose sequence of nitrogenous bases encodes the genetic information necessary for biological inheritance. They store directions for the synthesis of enzymes and other proteins, and are the only molecules that can (with the help of the right enzymes) replicate themselves. The two kinds of nucleic acids in cells are deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). They are polymers of repeated units called nucleotides, each of which contains a sugar, a nitrogenous base, and a phosphate group. In addition to their role in nucleic acids, nucleotides have an important role as transporters of chemical energy in cellular metabolism (p. 62). Because the structure of nucleic acids is crucial to the mecha-nism of inheritance and protein synthesis, detailed information on nucleic acids is presented in Chapter 5 (p. 91). CHEMICAL EVOLUTION Both Haldane and Oparin proposed that earth’s primitive atmo-sphere consisted of simple compounds such as water, molecu-lar hydrogen, methane, and ammonia, but lacked oxygen gas Figure 2.15 Structure of proteins. The amino acid sequence of a protein (primary structure) encourages the formation of hydrogen bonds between nearby amino acids, producing coils and foldbacks (the secondary structure). Bends and helices cause the chain to fold back on itself in a complex manner (tertiary structure). Individual polypeptide chains of some proteins aggregate to form a functional molecule composed of several subunits (quaternary structure).
Primary structure Secondary structure Tertiary structure Quaternary structure Pleated sheet Helix H H H HO H H H H H H H H H H H H H H H H H C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C N N N N N N N N N N N N N N N N N N N O O O O O O O O O O O O O O O O O O O www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 29 (O 2 , also called “molecular oxygen”). The nature of the prime-val atmosphere is critical for understanding life’s origin. The organic compounds that compose living organisms are neither synthesized outside cells nor stable in the presence of molecular oxygen, which is abundant in the atmosphere today. The best evidence indicates, however, that the primitive atmosphere con-tained not more than a trace of molecular oxygen. The primeval atmosphere therefore was a reducing one, consisting primarily of molecules in which hydrogen exceeds oxygen; methane (CH 4 ) and ammonia (NH 3 ), for example, constitute fully reduced com-pounds. Such compounds are called “reducing” because they tend to donate electrons to other compounds, thereby “reduc-ing” those compounds (p. 64). During this time, the earth was bombarded by large (100 km diameter) comets and meteorites, generating heat that repeatedly vaporized its oceans. This reducing atmosphere was conducive to the prebiotic synthesis that led to life’s beginnings, although totally unsuited for the organisms alive today. Haldane and Oparin proposed that ultraviolet radiation of such a gas mixture caused many organic substances, such as sugars and amino acids, to form. Haldane proposed that the early organic molecules accumulated in the primitive oceans to form a “hot dilute soup.” In this primordial broth, carbohydrates, fats, proteins, and nucleic acids could have assembled to form the earliest structures capable of guiding their own replication. If the simple gaseous compounds present in the early atmosphere are mixed with methane and ammonia in a closed glass system and kept at room temperature, they never react chemically with each other. To produce a chemical reaction, a continuous source of free energy sufficient to overcome reaction-activation barriers must be supplied. Ultraviolet light from the sun must have been intense on earth before the accu-mulation of atmospheric oxygen; ozone, a three-atom form of oxygen located high in the atmosphere, now blocks much of the ultraviolet radiation from reaching the earth’s surface. Electrical discharges could have provided further energy for chemical evo-lution. Although the total amount of electrical energy released by lightning is small compared with solar energy, nearly all of the energy of lightning is effective in synthesizing organic com-pounds in a reducing atmosphere. A single fl ash of lightning through a reducing atmosphere generates a large amount of organic matter. Thunderstorms may have been one of the most important sources of energy for organic synthesis. Widespread volcanic activity is another possible source of energy. One hypothesis maintains, for example, that life did not originate on the surface of the earth, but deep beneath the sea in or around hydrothermal vents (p. 816). Hydrothermal vents are submarine hot springs; seawater seeps through cracks in the seafl oor until the water comes close to hot magma. The water is then superheated and expelled forcibly, carrying various dis-solved molecules from the superheated rocks. These molecules include hydrogen sulfi de, methane, iron ions, and sulfi de ions. Hydrothermal vents have been discovered in several locations beneath the deep sea, and they would have been much more widely prevalent on the early earth. Interestingly, many heat- and sulfur-loving bacteria grow in hot springs today. Prebiotic Synthesis of Small Organic Molecules The Oparin-Haldane hypothesis stimulated experimental work to test the hypothesis that organic compounds characteristic of life could be formed from the simpler molecules present in the prebiotic environment. In 1953, Stanley Miller and Harold Urey in Chicago successfully simulated the conditions thought to pre-vail on the primitive earth. Miller built an apparatus designed to circulate a mixture of methane, hydrogen, ammonia, and water past an electric spark ( Figure 2.16 ). Water in the fl ask was boiled to produce steam that helped to circulate the gases. The prod-ucts formed in the electrical discharge (representing lightning) were condensed in the condenser and collected in the U-tube and small fl ask (representing an ocean). After a week of continuous sparking, approximately 15% of the carbon from the reducing “atmosphere” had been converted into organic compounds that collected in the “ocean.” The most Figure 2.16 Dr. S. L. Miller with a replica of the apparatus used in his 1953 experiment on the synthesis of amino acids with an electric spark in a strongly reducing atmosphere.
Tungsten electrodes 5-L flask Condenser Water containing synthesized organic compounds To vacuum 500-ml flask with boiling water Stopcocks for removing samples Water Methane Ammonia Hydrogen Gases 30 PART ONE Introduction to Living Animals striking fi nding was that many compounds related to life were synthesized. These compounds included four of the amino acids commonly found in proteins, urea, and several simple fatty acids. We can appreciate the astonishing nature of this synthesis when we consider that there are thousands of known organic com-pounds with structures no more complex than those of the amino acids formed. Yet in Miller’s synthesis, most of the relatively few substances formed were compounds found in living organisms. This result was surely no coincidence, and it suggests that pre-biotic synthesis on the primitive earth may have occurred under conditions not greatly different from those that Miller simulated. Miller’s experiments have been criticized in light of current opinion that the early atmosphere on earth was quite different from Miller’s strongly reducing simulated atmosphere. Neverthe-less, Miller’s work stimulated many other investigators to repeat and to extend his experiment. Amino acids were found to be synthesized in many different kinds of gas mixtures that were heated (volcanic heat), irradiated with ultraviolet light (solar radiation), or subjected to electrical discharge (lightning). The only conditions required to produce amino acids were that the gas mixture be reducing and that it be subjected violently to a source of energy. In other experiments, electrical discharges were passed through mixtures of carbon monoxide, nitrogen, and water, yielding amino acids and nitrogenous bases. Although reaction rates were much slower than in atmospheres contain-ing methane and ammonia, and yields were poor in compari-son, these experiments support the hypothesis that the chemical beginnings of life can occur in atmospheres that are only mildly reducing. The need for methane and ammonia, however, led to proposals that these substances might have been introduced by comets or meteorites, or that they were synthesized near the hydrothermal vents. Thus the experiments of many scientists have shown that highly reactive intermediate molecules such as hydrogen cyanide, formaldehyde, and cyanoacetylene are formed when a reducing mixture of gases is subjected to a violent energy source. These molecules react with water and ammonia or nitrogen to form more complex organic molecules, including amino acids, fatty acids, urea, aldehydes, sugars, and nitrogenous bases (purines and pyrimidines), all of the building blocks required for the syn-thesis of the most complex organic compounds of living matter. Further evidence for the natural abiotic synthesis of amino acids comes from fi nding amino acids in meteorites, such as the Mur-chison meteorite that landed in Australia in 1969. Formation of Polymers The next stage in chemical evolution involved the joining of amino acids, nitrogenous bases, and sugars to yield larger mol-ecules, such as proteins and nucleic acids. Such synthesis does not occur easily in dilute solutions, because excess water drives reactions toward decomposition (hydrolysis). Although the prim-itive ocean might have been called a “primordial soup,” it was probably a rather dilute one containing organic material that was approximately one-tenth to one-third as concentrated as chicken bouillon. Need for Concentration Prebiotic synthesis must have occurred in restricted areas where concentrations of the reactants were high. Violent weather on the primitive earth would have created enormous dust storms; impacts of meteorites would have lofted great amounts of dust into the atmosphere. The dust particles could have become foci of water droplets. Salt concentration in the particles could have been high and provided a concentrated medium for chemical reactions. Alternatively, perhaps the surface of the earth was too warm to have oceans but not too hot for a damp surface. This condition would have resulted from constant rain and rapid evaporation. Thus, the earth’s surface could have become coated with organic molecules, an “incredible scum.” Prebiotic molecules might have been concentrated by adsorption on the surface of clay and other minerals. Clay can concentrate and condense large amounts of organic molecules. The surface of iron pyrite (FeS 2 ) also has been suggested as a site for the evo-lution of biochemical pathways. The positively charged surface of pyrite would attract a variety of negative ions, which would bind to its surface. Furthermore, pyrite is abundant around hydrothermal vents, compatible with the hydrothermal-vent hypothesis. Thermal Condensations Most biological polymerizations are condensation (dehydra-tion) reactions, in which monomers are linked together by the removal of water (p. 23). In living systems, condensa-tion reactions always occur in an aqueous (cellular) envi-ronment containing appropriate enzymes. Without enzymes and energy supplied by ATP, macromolecules (proteins and nucleic acids) of living systems soon decompose into their constituent monomers. Dehydration reactions could have occurred without enzymes in primitive earth conditions by thermal condensation. The simplest dehydration is accomplished by driving water from solids by direct heating. For example, heating a mixture of all 20 amino acids to 180° C produces a good yield of poly-peptides. The thermal synthesis of polypeptides to form “protein-oids” has been studied extensively by the American scientist Sidney Fox. He showed that heating dry mixtures of amino acids and then mixing the resulting polymers with water forms small spherical bodies. These proteinoid microspheres ( Figure 2.17 ) possess certain characteristics of living systems. Each is not more than 2 µm in diameter, comparable in size and shape to spherical bacteria. The outer walls of the micro-spheres appear to have a double layer, and they show osmotic properties and selective diffusion. They may grow by accretion or proliferate by budding like bacteria. Proteinoids might have been used to assemble the fi rst cells from macromolecular pre-cursors. Formation of these polymers requires conditions likely to have occurred only in volcanoes. Organic polymers might have condensed on or in volcanoes and then, wetted by rain or dew, reacted further in solution to form polypeptides or polynucleotides. www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 31 Figure 2.17 Electron micrograph of proteinoid microspheres. These proteinlike bodies can be produced in the laboratory from polyamino acids and may represent precellular forms. They have defi nite internal ultrastructure. (1700) ORIGIN OF LIVING SYSTEMS The fossil record reveals that life existed 3.8 billion years ago; therefore, the origin of the earliest form of life can be estimated at approximately 4 billion years BP . The fi rst living organisms were protocells, autonomous membrane-bound units with a complex functional organization that permitted the essential activity of self-reproduction. The primitive chemical systems that we have described lack this essential property. The principal problem in understanding the origin of life is explaining how primitive chemical systems could have become organized into living, autonomous, self-reproducing cells. As we have seen, a lengthy chemical evolution on the primi-tive earth produced several molecular components of living forms. In a later stage of evolution, nucleic acids (DNA and RNA) began to behave as simple genetic systems that directed the syn-thesis of proteins, especially enzymes. However, this conclusion leads to a troublesome chicken-egg paradox: (1) How could nucleic acids have appeared without the enzymes that catalyze their synthesis? (2) How could enzymes have evolved without nucleic acids to encode their amino acid sequence? These ques-tions come from a long-accepted notion that only proteins could act as enzymes. Startling evidence presented in the 1980s indi-cates that RNA in some instances has catalytic activity. Catalytic RNA (ribozymes) can mediate processing of mes-senger RNA (removal of introns, p. 95), and can catalyze forma-tion of peptide bonds. Strong evidence suggests that translation of mRNA by ribosomes (p. 95) is catalyzed by their RNA, not protein, content. Therefore the earliest enzymes could have been RNA, and the earliest self-replicating molecules could have been RNA. Investi-gators are now calling this stage the “RNA world.” Nonetheless, proteins have several important advantages over RNA as catalysts, and DNA is a more stable carrier of genetic information than RNA. The fi rst protocells containing protein enzymes and DNA should have been selectively favored over those with only RNA. Once this protocellular stage of organization was reached, natural selection (pp. 124–126) would have acted on these prim-itive self-replicating systems. This stage was critical. Before this stage, biogenesis was shaped by the favorable environmental conditions on the primitive earth and by the nature of the react-ing elements themselves. When self-replicating systems became responsive to natural selection, they began to evolve. The more rapidly replicating and more successful systems were favored, thereby gradually evolving effi cient replicators. Evolution of the genetic code and fully directed protein synthesis followed. The system now meets the requirements for being the common ancestor of all living organisms. Origin of Metabolism Living cells today are organized systems with complex and highly ordered sequences of enzyme-mediated reactions. How did such vastly complex metabolic schemes develop? The exact history of this phase of life’s evolution is unknown. We present here a model of the simplest sequence of events that could explain the origin of observed metabolic properties of living systems. Organisms that can synthesize their food from inorganic sources using light or another source of energy are called auto-trophs (Gr. autos, self, trophos, feeder) ( Figure 2.18 ). Organisms Figure 2.18 Koala, a heterotroph, feeding on a eucalyptus tree, an autotroph. All heterotrophs depend for their nutrients directly or indirectly on autotrophs, which capture the sun’s energy to synthesize their own nutrients.
32 PART ONE Introduction to Living Animals lacking this ability must obtain their food supplies directly from the environment and are called heterotrophs (Gr. heteros, another, trophos, feeder). The earliest postulated microor-ganisms are sometimes called primary heterotrophs because they relied on environmental sources for their food and existed prior to the evolution of any autotrophs. They were probably anaerobic organisms similar to bacteria of the genus Clostrid-ium. Because chemical evolution had supplied generous stores of organic nutrients in the prebiotic soup, the earliest organisms would not have been required to synthesize their own food. Autotrophs would have had a tremendous selective advantage over the primary heterotrophs in areas where organic nutrients became depleted. Evolution of autotrophic organisms most likely required acquisition of enzymatic activities to catalyze conversion of inorganic molecules to more complex ones, such as carbohy-drates. The numerous enzymes of cellular metabolism appeared when cells started to utilize proteins for catalytic functions. Carl Woese challenges the traditional view that the fi rst organisms were primary heterotrophs. He fi nds it easier to visualize membrane -associated molecular aggregates that absorbed visible light and converted it with some effi ciency into chemical energy. Thus the fi rst organisms would have been autotrophs. Woese also suggests that the earliest “metabolism” comprised numerous chemical reac-tions catalyzed by nonprotein cofactors (substances necessary for the function of many of the protein enzymes in living cells). These cofactors would have been associated with membranes. Appearance of Photosynthesis and Oxidative Metabolism Autotrophy evolved in the form of photosynthesis. In photo-synthesis, hydrogen atoms obtained from water react with car-bon dioxide obtained from the atmosphere to generate sugars and molecular oxygen. Energy is stored in the form of covalent bonds between carbon atoms in the sugar molecule. Sugars pro-vide nutrition to the organism and molecular oxygen is released into the atmosphere.
6 6 6 CO H O light C H O O 2 2 6 12 6 2 → This equation summarizes the many reactions now known to occur in photosynthesis. Undoubtedly these reactions did not appear simultaneously, and other reduced compounds, such as hydrogen sulfi de (H 2 S), probably were the early sources of hydrogen. Gradually, oxygen produced by photosynthesis accumu-lated in the atmosphere. When atmospheric oxygen reached approximately 1% of its current level, ozone began to accumu-late and to absorb ultraviolet radiation, thereby greatly restricting the amount of ultraviolet light that reached the earth. Land and surface waters then were occupied by photosynthetic organisms, thereby increasing oxygen production. Accumulation of atmospheric oxygen would interfere with anaerobic cellular metabolism that had evolved in the primitive reducing atmosphere. As the atmosphere slowly accumulated oxygen gas (O 2 ), a new and highly effi cient kind of metabolism appeared: oxidative (aerobic) metabolism. By using available oxygen as a terminal electron acceptor (p. 68) and completely oxidizing glucose to carbon dioxide and water, much of the bond energy stored by photosynthesis could be recovered. Most living forms became completely dependent upon oxidative metabolism. Our atmosphere today is strongly oxidizing. It contains 78% molecular nitrogen, approximately 21% free oxygen, 1% argon, and 0.03% carbon dioxide. Although the time course for produc-tion of atmospheric oxygen is much debated, the most important source of oxygen is photosynthesis. Almost all oxygen currently produced comes from cyanobacteria (blue-green algae), eukary-otic algae, and plants. Each day these organisms combine approx-imately 400 million tons of carbon dioxide with 70 million tons of hydrogen to produce 1.1 billion tons of oxygen. Oceans are a major source of oxygen. Almost all oxygen produced today is consumed by organisms for respiration; otherwise, the amount of oxygen in the atmosphere would double in approximately 3000 years. Because Precambrian fossil cyanobacteria resemble mod-ern cyanobacteria, it is reasonable to suppose that oxygen enter-ing the early atmosphere came from their photosynthesis. PRECAMBRIAN LIFE The Precambrian period covers the geological time before the beginning of the Cambrian period some 570 to 600 million years BP . Most major animal phyla appear in the fossil record within a few million years at the beginning of the Cambrian period. This appearance has been called the “Cambrian explosion” because before this time, fossil deposits are mostly devoid of any organ-isms more complex than single-celled bacteria. Comparative molecular studies (p. 206) now suggest that the rarity of Precam-brian fossils may represent poor fossilization rather than absence of animal diversity from the Precambrian period. Nonetheless, animals make a relatively late appearance in the history of life on earth. What were the early forms of life that generated both the oxidizing atmosphere critical for animal evolution and the evolutionary lineage from which animals would arise? Prokaryotes and the Age of Cyanobacteria (Blue-Green Algae) The earliest bacterium-like organisms proliferated, giving rise to a great variety of forms, some of which were capable of photo-synthesis. From these arose the oxygen-producing cyanobacte-ria approximately 3 billion years ago. Bacteria are called prokaryotes, meaning literally “before the nucleus.” They contain a single, large molecule of DNA not located in a membrane-bound nucleus, but found in a nuclear region, or nucleoid. The DNA is not complexed with histone proteins, and prokaryotes lack membranous organelles such as mitochondria, plastids, Golgi apparatus, and endoplasmic reticu-lum (see Chapter 3). During cell division, the nucleoid divides and replicates of the cell’s DNA are distributed to the daughter cells. Prokaryotes lack the chromosomal organization and chro-mosomal (mitotic) division seen in animals, fungi, and plants. www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 33 The name “algae” is misleading because it suggests a relationship to eukaryotic algae, and many scientists prefer the alternative name “cyanobacteria” rather than “blue-green algae.” These organisms were responsible for producing an oxygen-rich atmosphere that replaced earth’s primitive reducing atmosphere. Studies of biochemical reac-tions in extant cyanobacteria suggest that they evolved in a time of fl uctuating oxygen concentration. For example, although they can tolerate atmospheric concentrations of oxygen (21%), the optimum concentration for many of their metabolic reactions is only 10%. Bacteria and especially cyanobacteria ruled earth’s oceans unchallenged for 1 to 2 billion years. The cyanobacteria reached the zenith of their success approximately 1 billion years BP , when fi lamentous forms produced great fl oating mats on the oceans’ surfaces. This long period of cyanobacterial dominance, encom-passing approximately two-thirds of the history of life, has been called with justifi cation the “age of blue-green algae.” Bacteria and cyanobacteria are so completely different from forms of life that evolved later that they were placed in a separate taxonomic kingdom, Monera. Carl Woese and his colleagues at the University of Illinois discovered that the prokaryotes actually comprise at least two distinct lines of descent: the Eubacteria (“true” bacteria) and the Archaebacteria also called Archaea (p. 213). Although these two groups of bacteria look very much alike when viewed with the electron microscope, they are biochemically distinct. Archaebac-teria differ fundamentally from bacteria in cellular metabolism, and their cell walls lack muramic acid, which is present in the cell walls of all Eubacteria. The most compelling evidence for differentiating these two groups comes from the use of one of the newest and most powerful tools at the disposal of the evolu-tionist, sequencing of nucleic acids (see note). Woese found that Archaebacteria differ fundamentally from other bacteria in the sequence of bases in ribosomal RNA (p. 95). Woese considers the Archaebacteria so distinct from the true bacteria that they should be considered a separate taxonomic kingdom, Archaea. The Monera then comprise only the true bacteria (see pp. 212–213 for further discussion and criticism of this taxonomy). Appearance of Eukaryotes Eukaryotes (“true nucleus”; Figure 2.19 ) have cells with mem-brane-bound nuclei containing chromosomes composed of chromatin. Constituents of eukaryotic chromatin include proteins called histones and RNA, in addition to DNA. Some nonhistone proteins are found associated with both prokaryotic DNA and eukaryotic chromosomes. Eukaryotes are generally larger than pro-karyotes and contain much more DNA. Cellular division usually is by some form of mitosis. Within their cells are numerous membra-nous organelles, including mitochondria, in which the enzymes for oxidative metabolism are packaged. Eukaryotes include animals, fungi, plants, and numerous single-celled forms formerly called “protozoans” or “protists.” Fossil evidence suggests that single-celled eukaryotes arose at least 1.5 billion years ago ( Figure 2.20 ). Molecular sequencing has emerged as a very successful approach to unraveling the ancient genealogies of living forms. The sequences of nucleotides in the DNA of an organism’s genes are a record of evolutionary relationships, because every gene that exists today is an evolved copy of a gene that existed millions or even billions of years ago. Genes become altered by mutations through the course of time, but vestiges of the original gene usually per-sist. With modern techniques, one can determine the sequence of nucleotides in an entire molecule of DNA or in short segments of the molecule. When corresponding genes are compared between two different organisms, the extent to which the genes differ can be correlated with the time elapsed since the two organisms diverged from a common ancestor. Similar comparisons are made with RNA and proteins. These methods also permit scientists to synthesize long-extinct genes and proteins and to measure biochemical prop-erties of the extinct proteins. Figure 2.19 Comparison of prokaryotic and eukaryotic cells. Prokaryotic cells are about one-tenth the size of eukaryotic cells.
Mitochondrion Golgi apparatus Nucleus Cell membrane Cytoplasm with ribosomes DNA Nucleoid Eukaryote Prokaryote Cell wall Cell membrane Lysosome Endoplasmic reticulum 34 PART ONE Introduction to Living Animals Because the organizational complexity of eukaryotes is much greater than that of prokaryotes, it is diffi cult to visualize how a eukaryote could have arisen from any known prokaryote. The American biologist Lynn Margulis ( Figure 2.21 ) and others have proposed that eukaryotes did not arise from any single pro-karyote but were derived from a symbiosis (“life together”) of two or more types of bacteria. Mitochondria and plastids (pho-tosynthetic organelles found only in plant cells), for example, each contain their own DNA (apart from the nucleus of the cell), which has some prokaryotic characteristics. Nuclei, plastids, and mitochondria each contain genes encoding ribosomal RNA. Comparisons of the sequences of bases of these genes show that the nuclear, plastid, and mito-chondrial DNAs represent distinct evolutionary lineages. Plastid and mitochondrial DNAs are closer in their evolutionary history to bacterial DNAs than to the eukaryotic nuclear DNA. Plastids are closest evolutionarily to cyanobacteria, and mitochondria are closest to another group of bacteria (purple bacteria), consistent with the symbiotic hypothesis of eukaryotic origins. Mitochon-dria contain the enzymes of oxidative metabolism, and plastids (a plastid with chlorophyll is a chloroplast) conduct photosyn-thesis. It is easy to see how a host cell able to accommodate such guests in its cytoplasm would have gained enormous evo-lutionary success. The endosymbiotic theory proposes that a population ances-tral to eukaryotic cells, derived from and resembling anaerobic (lacking oxidative metabolism) bacteria, evolved a nucleus and other intracellular membranes (p. 41) from infoldings of the cell membrane. Cells of this population acquired, by ingestion or parasitism, aerobic bacteria that avoided digestion and came to reside in the host cell’s cytoplasm (p. 41). The endosymbi-otic aerobic bacteria would have metabolized oxygen, which is toxic for their anaerobic host, and the anaerobic host cell would have given its aerobic residents food and physical protection. This mutually benefi cial relationship would produce selection for the host cells and their residents to evolve a means of mak-ing their relationship a permanent one. Among the evolutionary outcomes of this selection would be compactness of the endo-symbiont and its loss of genes redundant with those of its host (or the reverse). Data collected to test this proposed mechanism show that its conditions are reasonable ones. Fossil data show that both aerobic and anaerobic bacteria were well established by 2.5 billion years ago, and that cells containing nuclei and internal membranes fi rst appeared at this time. Some anaerobic, nucle-ated forms that lack mitochondria are alive today, including the human parasite Giardia intestinalis, although these forms prob-ably represent descendants of lineages that formerly had mito-chondria and lost them rather than lineages whose ancestry never Figure 2.20 The clock of biological time. A billion seconds ago it was 1961, and most students using this text had not yet been born. A billion minutes ago the Roman empire was at its zenith. A billion hours ago Neanderthals were alive. A billion days ago the fi rst bipedal hominids walked the earth. A billion months ago the dinosaurs were at the climax of their radiation. A billion years ago no animal had ever walked on the surface of the earth.
Invasion of land by animals Humans appear Invasion of land by plants Oldest multicellular organisms 1 BILLION YEARS 2 BILLION YEARS Oldest eukaryote fossils 4 BILLION YEARS Oldest known rocks Earliest isotopic traces of life Oldest prokaryote fossils 3 BILLION YEARS First evidence of photosynthesis Midnight Noon Afternoon Morning Figure 2.21 Dr. Lynn Margulis, whose endosymbiotic theory of the origins of mitochondria and chloroplasts is strongly supported by molecular evolutionary studies.
www . m hhe . com /hi ckmani pz14e CHAPTER 2 The Origin and Chemistry of Life 35 featured mitochondria. Eukaryotic cells containing mitochondria are evident approximately 1.2 billion years ago. Bacteria have been introduced experimentally into single-celled eukaryotes and propagated as a symbiotic unit for many generations. Such experiments have shown further that the host cell can become dependent upon its resident bacteria for proteins whose func-tions formerly were performed by the host population prior to the experimental endosymbiosis. In addition to claiming that mitochondria and plastids originated as bacterial symbionts, Lynn Margulis argues that eukaryote fl agella, cilia (locomotory structures), and even the spindle of mitosis came from a kind of bacterium like a spirochete. Indeed, she suggests that this association (the spirochete with its new host cell) made evolution of mitosis possible. Margulis’s evidence that organelles are former partners of an ancestral cell is now accepted by most biologists. Such merging of disparate organisms to produce evolu-tionarily novel forms is called symbiogenesis. The first eukaryotes were undoubtedly unicellular, and many were photosynthetic autotrophs. Some of these forms lost their photosynthetic ability and became heterotrophs, feeding on eukaryotic autotrophs and prokaryotes. As cyanobacteria were cropped, their dense fi lamentous mats began to thin, providing space for other organisms. Carnivores appeared and fed on her-bivores. Soon a balanced ecosystem of carnivores, herbivores, and primary producers appeared. By freeing space, cropping herbivores encouraged a greater diversity of producers, which in turn promoted evolution of new and more specialized croppers. An ecological pyramid developed with carnivores at the top of the food chain (p. 834). The burst of evolutionary activity that followed at the end of the Precambrian period and beginning of the Cambrian period was unprecedented. Some investigators hypothesize that the explanation for the “Cambrian explosion” lies in the accumula-tion of oxygen in the atmosphere to a critical threshold level. Larger, multicellular animals required the increased effi ciency of oxidative metabolism; these pathways could not be supported under conditions of limiting oxygen concentration. S U M M A R Y Living organisms show a remarkable uniformity in their chemical constituents and metabolism, refl ecting their common descent from an ancient ancestor. Life on earth could not have appeared without water, the pri-mary component of living cells. The unique structure of water and its ability to form hydrogen bonds between adjacent water mole-cules are responsible for its special properties: solvency, high heat capacity, boiling point, surface tension, and lower density as a solid than as a liquid. Life also depends critically on the chemistry of carbon. Carbon is especially versatile in bonding with itself and with other atoms, and it is the only element capable of forming the large molecules found in living organisms. Carbohydrates are composed primarily of carbon, hydrogen, and oxygen grouped as HCOH. The simplest carbohy-drates are sugars, which serve as immediate sources of energy in liv-ing systems. Monosaccharides, or simple sugars, may bond together to form disaccharides or polysaccharides, which serve as storage forms of sugar or perform structural roles. Lipids constitute another class of large molecules featuring chains of carbon compounds; fats exist principally as neutral fats, phospholipids, and steroids. Proteins are large molecules composed of amino acids linked by peptide bonds. Many proteins function as enzymes that catalyze biological reactions. Each kind of protein has a characteristic primary, second-ary, tertiary, and often, quaternary structure critical for its functioning. Nucleic acids are polymers of nucleotide units, each composed of a sugar, a nitrogenous base, and a phosphate group. They contain the material of inheritance and function in protein synthesis. Experiments by Louis Pasteur in the 1860s convinced scien-tists that organisms do not arise repeatedly from inorganic matter. About 60 years later, A. I. Oparin and J. B. S. Haldane provided an explanation for how a common ancestor of all living forms could have arisen from nonliving matter almost 4 billion years ago. The origin of life followed a long period of “abiogenic molecular evo-lution” on earth in which organic molecules slowly accumulated in a “primordial soup.” The atmosphere of the primitive earth was reducing, with little or no free oxygen present. Ultraviolet radiation, electrical discharges of lightning, or energy from hydrothermal vents could have provided energy for early formation of organic mole-cules. Stanley Miller and Harold Urey demonstrated the plausibility of the Oparin-Haldane hypothesis by simple but ingenious experi-ments. The concentration of reactants necessary for early synthesis of organic molecules might have been provided by damp surfaces, clay particles, iron pyrite, or other conditions. RNA might have been the primordial biomolecule, performing the functions of both genetic coding of information and catalysis. When self-replicating systems became established, evolution by natural selection could have increased their diversity and complexity. The fi rst organisms are hypothesized to have been primary heterotrophs, living on energy stored in molecules dissolved in a primordial soup. Later evolution produced autotrophic organisms, which can synthesize their own organic nutrients (carbohydrates) from inorganic materials. Autotrophs are better protected than het-erotrophs from depletion of organic compounds from their environ-ments. Molecular oxygen began to accumulate in the atmosphere as an end product of photosynthesis, an autotrophic process that produces sugars and oxygen by reacting water and carbon dioxide. Cyanobacteria appear to be primarily responsible for generation of atmospheric oxygen early in life’s history. All bacteria are prokaryotes, organisms that lack a membrane-bound nucleus and other organelles in their cytoplasm. The pro-karyotes consist of two genetically distinct groups, Archaebacteria and Monera. The eukaryotes apparently arose from symbiotic unions of two or more types of prokaryotes. The genetic material (DNA) of eukary-otes is borne in a membrane-bound nucleus, and also in mitochon-dria and sometimes plastids. Mitochondria and plastids have resem-blances to bacteria, and their DNA is more closely allied to that of certain bacteria than to eukaryotic nuclear genomes. 36 PART ONE Introduction to Living Animals R E V I E W Q U E S T I O N S Explain each of these properties of water, and describe how each is conferred by the dipolar nature of a water molecule: high specifi c heat capacity; high heat of vaporization; unique density behavior; high surface tension; good solvent for ions of salts. What was the composition of the earth’s atmosphere at the time of the origin of life, and how did it differ from the atmosphere of today? Regarding the experiments of Miller and Urey described in this chapter, explain what constituted the following in each case: observations, hypothesis, deduction, prediction, data, control. (The scientifi c method is described on pp. 11–14.) Explain the signifi cance of the Miller-Urey experiments. Name three different sources of energy that could have powered reactions on early earth to form organic compounds. By what mechanism might organic molecules have been concentrated in a prebiotic world so that further reactions could occur? 1.
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6. Name two simple carbohydrates, two storage carbohydrates, and a structural carbohydrate. What characteristic differences in molecular structure distinguish lipids and carbohydrates? Explain the difference between the primary, secondary, tertiary, and quaternary structures of a protein. What are the important nucleic acids in a cell, and of what units are they constructed? Distinguish among the following: primary heterotroph, autotroph, secondary heterotroph. What is the source of oxygen in the present-day atmosphere, and what is its metabolic signifi cance to most organisms living today? Distinguish prokaryotes from eukaryotes. Describe Margulis’s view on the origin of eukaryotes from prokaryotes. What was the “Cambrian explosion” and how might you explain it? 7.
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15. S E L E C T E D R E F E R E N C E S Berg, J. M., T. L. Tymoczko, and L. Stryer. 2007. Biochemistry, ed. 6. New York, W. H. Freeman. A current and thorough textbook of biochemistry. Conway, Morris, S. 1993. The fossil record and the early evolution of the Metazoa. Nature 361: 219–225. An important summary correlating fossil and molecular evidence. Fenchel, T. 2002. Origin and early evolution of life. Oxford, Oxford Univ. Press. A review of current theories of the origin and early diversifi cation of living forms. Gesteland, R. F., and J. F. Atkins, editors. 1999. The RNA world. Cold Spring Harbor, New York, Cold Spring Harbor Laboratory Press. Evidence that there was a period when RNA served in both catalysis and transmission of genetic information. Kasting, J. F. 1993. Earth’s early atmosphere. Science 259: 920–926. Most investigators agree that there was little or no oxygen in the atmosphere of early earth and that there was a signifi cant increase about 2 billion years ago. Knoll, A. H. 1991. End of the Proterozoic Eon. Sci. Am. 265: 64–73 (Oct.). Multicellular animals probably originated only after oxygen in the atmosphere accumulated to a critical level. Lehninger, A. L., D. L. Nelson, and M. M. Cox. 2005. Lehninger principles of biochemistry, ed. 4. New York, Worth Publishers, Inc. Clearly presented advanced textbook in biochemistry. Lodish, H., A. Berk, S. L. Zipursky, P. Matsudira, D. Baltimore, and J. Darnell. 2004. Molecular cell biology, ed. 5. New York, W. H. Freeman & Co. Thorough treatment; begins with fundamentals such as energy, chemical reactions, bonds, pH, and biomolecules, then proceeds to advanced molecular biology. Margulis, L. 1998. Symbiotic planet: a new look at evolution. New York, Basic Books. An important discussion of symbiogenesis in evolution. Orgel, L. E. 1994. The origin of life on the earth. Sci. Am. 271: 77–83 (Oct.). There is growing evidence for an RNA world, but diffi cult questions are unanswered. Rand, R. P. 1992. Raising water to new heights. Science 256: 618. Cites some ways that water can affect function of protein molecules. Wainright, P. O., G. Hinkle, M. L. Sogin, and S. L. Stickel. 1993. Monophyletic origins of the Metazoa: an evolutionary link with fungi. Science 260: 940–942. Molecular evidence that multicellular animals share a more recent common ancestor with fungi than with plants or other eukaryotes. Waldrop, M. M. 1992. Finding RNA makes proteins gives “RNA world” a big boost. Science 256: 1396–1397. Reports on the signifi cance of Noller et al. (1992, Science 256: 1416), who found that ribosomal RNA can catalyze the formation of peptide bonds and hence proteins. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! C H A P T E R transformed by photosynthesis into chemical bond energy. Chemical bond energy is a form of potential energy that can be released when the bond is broken; the energy is used to per-form electrical, mechanical, and osmotic tasks in the cell. Ulti-mately, all energy is dissipated into heat. This is in accord with the second law of thermodynamics, which states that there is a tendency in nature to proceed toward a state of greater molec-ular disorder, or entropy. Thus the high degree of molecular organization in living cells is attained and maintained only as long as energy fuels the organization. The Fabric of Life It is a remarkable fact that living forms, from amebas and unicellular algae to whales and giant redwood trees, are formed from a single type of building unit: cells. All animals and plants are composed of cells and cell products. Thus the cell theory is another great unifying concept of biology. New cells come from division of preexisting cells, and the activity of a multicellular organism as a whole is the sum of activities and interactions of its constituent cells. Energy is required to support life’s activities, and virtually all of it flows from sunlight that is captured by green plants and algae and 3 Cells as Units of Life A humpback whale, Megaptera novaeangliae, leaps from the water. 38 PART ONE Introduction to Living Animals CELL CONCEPT More than 300 years ago the English scientist and inventor Robert Hooke, using a primitive compound microscope, observed boxlike cavities in slices of cork and leaves. He called these compartments “little boxes or cells.” In the years that followed Hooke’s fi rst demonstration of the remarkable powers of the microscope to the Royal Society of London in 1663, biologists gradually began to realize that cells were far more than simple containers fi lled with “juices.” Cells are the fabric of life ( Figure 3.1 ). Even the most primi-tive cells are enormously complex structures that form the basic units of all living organisms. All tissues and organs are composed of cells. In a human an estimated 60 trillion cells interact, each performing its specialized role in an organized partnership. In single-celled organisms all functions of life are performed within the confi nes of one microscopic package. There is no life without cells. The idea that a cell represents the basic structural and func-tional unit of life is an important unifying concept of biology. With the exception of some eggs, which are the largest cells (in volume) known, cells are small and mostly invisible to the unaided eye. Consequently, our understanding of cells paral-leled technical advances in the resolving power of microscopes. The Dutch microscopist Antoni van Leeuwenhoek sent letters to the Royal Society of London containing detailed descriptions of the numerous organisms he had observed using high-quality single lenses that he had made (1673 to 1723). In the early nineteenth century, improved design of microscopes permitted biologists to see separate objects only 1 µm apart. This advance was quickly followed by new discoveries that laid the ground-work for the cell theory —a theory stating that all living organ-isms are composed of cells. In 1838 Matthias Schleiden, a German botanist, announced that all plant tissue was composed of cells. A year later one of his countrymen, Theodor Schwann, described animal cells as being similar to plant cells, an understanding that had been long delayed because animal cells are bounded only by a nearly invis-ible plasma membrane rather than a distinct cell wall characteristic Figure 3.1 Biological organization from simple atoms to complex organisms. Atoms from molecules and macromolecules are assembled into organelles within each cell. Cells are grouped into tissues, organs, and organ systems to form a complex multicellular organism. CELLULAR LEVEL www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 39 of plant cells. Schleiden and Schwann are thus credited with the unifying cell theory that ushered in a new era of productive explo-ration in cell biology. Another German, Rudolf Virchow, recog-nized that all cells came from preexisting cells (1858). In 1840 J. Purkinje introduced the term protoplasm to describe cell contents. Protoplasm was at fi rst thought to be a granular, gel-like mixture with special and elusive life properties of its own; cells were viewed as bags of thick soup containing a nucleus. Later the interior of cells became increasingly visible as microscopes were improved and better tissue-sectioning and staining techniques were introduced. Rather than being a uni-form granular soup, a cell’s interior is composed of numerous cellular organelles, each performing a specifi c function in the life of a cell. Today we realize that the components of a cell are so highly organized, structurally and functionally, that describing its contents as “protoplasm” is like describing the contents of an automobile engine as “autoplasm.” How Cells Are Studied Light microscopes, with all their variations and modifi cations, have contributed more to biological investigation than any other instrument. They have been powerful exploratory tools for 300 years, and they continue to be so more than 50 years after inven-tion of the electron microscope. However, electron microscopy has vastly enhanced our appreciation of the delicate internal organization of cells, and modern biochemical, immunological, physical, and molecular techniques have contributed enormously to our understanding of cell structure and function. Electron microscopes employ high voltages to direct a beam of electrons through or at the surface of objects examined. The wavelength of the electron beam is approximately 0.00001 that of ordinary white light, thus permitting far greater magnifi cation and resolution. In preparation for viewing using the transmission electron microscope, specimens are cut into extremely thin sections (10 nm to 100 nm thick) and treated with “electron stains” (ions of elements such as osmium, lead, and uranium) to increase con-trast between different structures. Electrons pass through a speci-men, and images are seen on a fl uorescent screen and photo-graphed ( Figure 3.2 ). In contrast, specimens prepared for scanning electron microscopy are not sectioned, and electrons do not pass through them. The whole specimen is coated with an electron-dense material and bombarded with electrons, causing some electrons to be refl ected back and secondary electrons to be emitted. An apparent three-dimensional image is recorded in the photograph. Although the magnifi cation capability of scanning instruments is not as great as transmission microscopes, much has been learned about surface features of organisms and cells, as well as inter-nal, membrane-bound structures. Examples of scanning electron micrographs are shown on pp. 145, 163, and 686. A still greater level of resolution can be achieved with X-ray crystallography and nuclear magnetic resonance (NMR) spectros-copy. These techniques reveal the shapes of biomolecules and the relationships among atoms within them. Both techniques are laborious, but NMR spectroscopy does not require purifi ca-tion and crystallization of a substance, and molecules can be observed in solution. Advances in techniques of cell study (cytology) are not limited to improvements in microscopes but include new meth-ods of tissue preparation, staining for microscopic study, and the great contributions of modern biochemistry and molecu-lar biology. For example, the various organelles of cells have differing, characteristic densities. Cells can be disrupted with most of the organelles remaining intact, then centrifuged in a density gradient ( Figure 3.3 ), and relatively pure preparations of each organelle may be recovered. Thus the biochemical functions of various organelles may be studied separately. DNA and various types of RNA can be extracted and studied. Many enzymes can be purifi ed and their characteristics determined. We use radioactive isotopes to study many metabolic reactions and pathways in cells. Modern chromatographic techniques can separate chemically similar intermediates and products. A particular cellular protein can be extracted and purifi ed, and specifi c antibodies (see p. 775) can be prepared against the protein. When the antibody is complexed with a fl uorescent substance and the complex is used to “stain” cells, the complex binds to the protein of interest, and its precise location in cells can be determined. Figure 3.2 Comparison of optical paths of light and transmission electron microscopes. To facilitate comparison, the scheme of the light microscope has been inverted from its usual orientation with light source below and image above. In an electron microscope the lenses are magnets to focus the beam of electrons. Light microscope Transmission electron microscope Lamp Condenser lens Specimen Objective lens Eyepiece Image at eye or photographic plate Electron source Condenser lens Specimen Objective lens Projector lens Image at fluorescent screen or photographic plate 40 PART ONE Introduction to Living Animals ORGANIZATION OF CELLS If we were to restrict our study of cells to fi xed and sectioned tis-sues, we would be left with the erroneous impression that cells are static, quiescent, rigid structures. In fact, a cell’s interior is in a constant state of upheaval. Most cells are continually changing shape; their organelles twist and regroup in a cytoplasm teeming with starch granules, fat droplets, and vesicles of various sorts. This description is derived from studies of living cell cultures with time-lapse photography and video. If we could see the swift shut-tling of molecular traffi c through gates in the cell membrane and the metabolic energy transformations within cell organelles, we would have an even stronger impression of internal turmoil. How-ever, cells are anything but bundles of disorganized activity. There is order and harmony in cell functioning. Studying this dynamic phenomenon through a microscope, we realize that, as we gradu-ally comprehend more and more about these units of life, we are gaining a greater understanding of the nature of life itself. Prokaryotic and Eukaryotic Cells We already described the radically different cell plan of prokary-otes and eukaryotes (p. 32). A fundamental distinction, expressed in their names, is that prokaryotes lack the membrane-bound nucleus present in all eukaryotic cells. Among other differences, eukaryotic cells have many membranous organelles (special-ized structures that perform particular functions within cells) ( Table 3.1 ). Despite these differences, which are of paramount impor-tance in cell studies, prokaryotes and eukaryotes have much in common. Both have DNA, use the same genetic code, and syn-thesize proteins. Many specifi c molecules such as ATP perform similar roles in both. These fundamental similarities imply com-mon ancestry. The upcoming discussion is restricted to eukary-otic cells, of which all animals are composed. T A B L E 3 . 1 Comparison of Prokaryotic and Eukaryotic Cells Characteristic Prokaryotic Cell Eukaryotic Cell Cell size Mostly small (1–10 µm) Mostly large (10–100 µm) Genetic system DNA with some DNA-binding protein; simple, circular DNA molecule in nucleoid; nucleoid is not membrane bound DNA complexed with DNA-binding proteins in complex linear chromosomes within nucleus with membranous envelope; circular mitochondrial and chloroplast DNA Cell division Direct by binary fi ssion or budding; no mitosis Some form of mitosis; centrioles in many; mitotic spindle present Sexual system Absent in most; highly modifi ed if present Present in most; male and female partners; gametes that fuse to form zygote Nutrition Absorption by most; photosynthesis by some Absorption, ingestion, photosynthesis by some Energy metabolism No mitochondria; oxidative enzymes bound to cell membrane, not packaged separately; great variation in metabolic pattern Mitochondria present; oxidative enzymes packaged therein; more unifi ed pattern of oxidative metabolism Intracellular movement None Cytoplasmic streaming, phagocytosis, pinocytosis Flagella/cilia If present, not with “9 2” microtubular pattern With “9 2” microtubular pattern Cell wall Contains disaccharide chains cross-linked with peptides If present, not with disaccharide polymers linked with peptides Figure 3.3 Separation of cell organelles in a density gradient by ultracentrifugation. The gradient is formed by layering sucrose solutions in a centrifuge tube, then carefully placing a preparation of mixed organelles on top. The tube is centrifuged at about 40,000 revolutions per minute for several hours, and organelles become separated down the tube according to their density.
Plasma membrane Golgi vesicles Rough endoplasmic reticulum Organelle fraction Density of sucrose (g/cm3) 1.10 1.12 1.14 1.15 1.16 1.20 Centrifuged at 40,000 rpm www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 41 Components of Eukaryotic Cells and Their Functions Typically, eukaryotic cells are enclosed within a thin, selectively permeable plasma membrane ( Figure 3.4 ). The most promi-nent organelle is a spherical or ovoid nucleus, enclosed within two membranes to form a double-layered nuclear envelope ( Figure 3.4 ). Cellular material located between the cell mem-brane and nuclear envelope is collectively called cytoplasm. Within the cytoplasm are many organelles, such as mitochon-dria, Golgi complexes, centrioles, and endoplasmic reticulum. Plant cells typically contain plastids, some of which are photo-synthetic organelles, and plant cells bear a cell wall containing cellulose outside the cell membrane. The fl uid-mosaic model is the currently accepted concept describing plasma membrane structure. By electron microscopy, a plasma membrane appears as two dark lines, each approxi-mately 3 nm thick, at each side of a light zone ( Figure 3.5 ). The entire membrane is 8 to 10 nm thick. This image is the result of a phospholipid bilayer, two layers of phospholipid molecules, all oriented with their water-soluble (hydrophilic) ends toward the outside and their fat-soluble portions (hydrophobic) toward the inside of the membrane ( Figure 3.6 ). An important charac-teristic of the phospholipid bilayer is that it is fl uidlike, giving the membrane fl exibility and allowing the phospholipid mol-ecules to move sideways freely within their own monolayer. Molecules of cholesterol are interspersed in the lipid portion of the bilayer ( Figure 3.6 ). They make the membrane even less permeable to water-soluble ions and molecules and decrease membrane fl exibility. Glycoproteins (proteins with carbohydrates attached) are essential components of plasma membranes ( Figure 3.6 ). Some of these proteins transport substances such as charged ions across the membrane (see p. 46, membrane function). Oth-ers act as specifi c receptors for various molecules or as highly specifi c cell markers. For example, the self/nonself recognition that enables the immune system to react to invaders (see Chap-ter 35) is based on proteins of this type. Some aggregations of protein molecules form pores or channels through which small polar molecules may enter (see gap junctions, p. 46, and channels, p. 49). Like the phospholipid molecules, most glyco-proteins can move laterally in the membrane, although more slowly. Nuclear envelopes contain less cholesterol than cell mem-branes, and pores in the envelope ( Figure 3.7 ) allow molecules to move between nucleus and cytoplasm. Nuclei contain linear chromosomes suspended in nucleoplasm. The chromosomes are normally loosely condensed, fl exible strands of chromatin, composed of a complex of DNA, and DNA-binding proteins. Chromosomal DNA carries the genetic information encoding cellular RNA and protein molecules (see Chapter 5). The linear chromosomes only become condensed and visible as discrete structures during cell division (see p. 52 for mitosis and p. 77 for meiosis). Nucleoli are specialized parts of certain chromosomes that stain in a characteristically dark manner. They carry multi-ple copies of the DNA information used to synthesize ribosomal Figure 3.4 Generalized cell with principal organelles, as might be seen with the electron microscope.
Centriole Lysosome Mitochondrion Ribosomes Free ribosomes Rough endoplasmic reticulum Smooth endoplasmic reticulum Cytoplasm Nucleus Nucleolus Nuclear envelope Smooth endoplasmic reticulum Cytoskeleton Golgi apparatus Plasma membrane Microvilli Figure 3.5 Plasma membranes of two adjacent cells. Each membrane (between arrows) shows a typical dark-light-dark staining pattern. (325,000) 42 PART ONE Introduction to Living Animals RNA. After transcription from DNA, ribosomal RNA combines with protein to form the two subunits of ribosomes, which leave the nucleolus and pass to the cytoplasm through pores in the nuclear enve-lope. Ribosomes are sites of polypeptide or protein synthesis. They perform this function free, within the cytoplasm, when manufacturing polypeptides for use in the cytoplasm or nucleus. Alternatively, they become attached to endoplasmic reticu-lum (ER) when manufacturing polypeptides des-tined for the plasma membrane, lysosomes, or for export. The outer membrane of the nuclear envelope is continuous with a cytoplasmic membranous sys-tem called endoplasmic reticulum (ER) ( Figures 3.7 and 3.8 ). The space between the membranes of the nuclear envelope communicates with the space between the ER membranes (cisterna, pl., cis-ternae). The ER membranes may be covered on their outer surfaces with ribosomes and are thus designated rough ER, or they may lack ribosomal covering and be called smooth ER. Ribosomes on rough ER synthesize polypeptides that enter the ER cisternae or membrane and are destined for incor-poration into the plasma membrane ( Figure 3.9 ), for export from the cell, or they are bound for the lyso-somes. Smooth ER functions in synthesis of lipids and phospholipids. Figure 3.6 Diagram illustrating fl uid-mosaic model of a plasma membrane.
Figure 3.6 Diagram illustrating fl uid-mosaic model of a plasma membrane.
Oligosaccharide side chain Hydrophobic ends of phospholipids Hydrophilic heads of phospholipids Cholesterol Peripheral protein Inside of cell Plasma membrane Outside of cell Glycolipid Transmembrane proteins (Glycoproteins) Nucleolus Endoplasmic reticulum Glycogen particles Nucleus Mitochondrion Figure 3.7 Electron micrograph of part of hepatic cell of rat showing portion of nucleus (left) and surrounding cytoplasm. Endoplasmic reticulum and mitochondria are visible in cytoplasm, and pores (arrows) can be seen in nuclear envelope. (14,000) www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 43 The Golgi complex ( Figures 3.9 and 3.10 ) is composed of a stack of membranous vesicles that function in storage, modi-fi cation, and packaging of polypeptide and protein products produced by rough ER. The vesicles do not synthesize polypep-tide or protein but may add complex carbohydrates to the mol-ecules. Small vesicles of ER membrane containing polypeptide or protein detach and then fuse with sacs on the cis or “forming face” of a Golgi complex. After modifi cation, the polypeptides or proteins become incorporated into vesicles that detach from the trans or “maturing face” of the complex ( Figures 3.9 and 3.10 ). The contents of some of these vesicles may be expelled to the outside of the cell, as secretory products such as from a glandular cell. Some may carry integral polypeptides or pro-teins for incorporation into the plasma membrane, such as receptors or carrier proteins. Others may contain enzymes that remain in the same cell that produces them. Such vesicles are called lysosomes (literally “loosening body,” a body capable of causing lysis, or disintegration). Enzymes that they contain are involved in the breakdown of foreign material, including bacte-ria engulfed by a cell. Lysosomes also are capable of breaking down injured or diseased cells and worn-out cellular compo-nents. Their enzymes are so powerful that they kill the cell that formed them if enough of the lysosome membranes rupture. In normal cells the enzymes remain safely enclosed within the protective vesicle membranes. Lysosomal vesicles may pour their enzymes into a larger membrane-bound body containing an ingested food particle, a food vacuole or phagosome (see Figure 3.21 ) .
Figure 3.8 Endoplasmic reticulum. A, Endoplasmic reticulum is continuous with the nuclear envelope. It may have associated ribosomes (rough endoplasmic reticulum) or not (smooth endoplasmic reticulum). B, Electron micrograph showing rough endoplasmic reticulum. (28,000) Ribosomes Rough endoplasmic reticulum Smooth endoplasmic reticulum A B Figure 3.9 Golgi complex (Golgi body, Golgi apparatus). A, The smooth cisternae of the Golgi complex have enzymes that modify polypeptides or proteins synthesized by the rough endoplasmic reticulum. B, Electron micrograph of a Golgi complex. (46,000) Secretory vesicles Transport vesicles A B cis face trans face 44 PART ONE Introduction to Living Animals Mitochondria (sing., mitochondrion ) ( Figure 3.11 ) are conspicuous organelles present in nearly all eukaryotic cells. They are diverse in size, number, and shape; some are rodlike, and others are nearly spherical. They may be scattered uni-formly throughout the cytoplasm or localized near cell surfaces and other regions of high metabolic activity. A mitochondrion is composed of a double membrane. The outer membrane is smooth, whereas the inner membrane is folded into numerous platelike or fi ngerlike projections called cristae (sing., crista; Figure 3.11 ), which increase the internal surface area where chemical reactions occur. These characteristic features make mitochondria easy to identify among organelles. Mitochon-dria are often called “powerhouses of cells,” because enzymes located on the cristae catalyze the energy-yielding steps of aero-bic metabolism (see Figure 4.14, p. 64). ATP (adenosine triphos-phate), the most important energy-transfer molecule of all cells, is produced in this organelle. Mitochondria are self-replicating. They have a tiny, circular genome, much like genomes of pro-karyotes except much smaller, which contains DNA specifying some, but not all, proteins of a mitochondrion. Eukaryotic cells characteristically have a sys-tem of tubules and fi laments that form a cyto-skeleton ( Figures 3.12 and 3.13 ). These provide support and maintain the form of cells, and in many cells, they provide a means of locomo-tion and translocation of macromolecules and organelles within the cell. The cytoskeleton is composed of microfi laments, microtubules, and intermediate filaments. Microfilaments are thin, linear structures, fi rst observed distinctly in muscle cells, where they are responsible for the ability of a cell to contract. They are made of a protein called actin. Several dozen other pro-teins are known that bind with actin and deter-mine its confi guration and behavior in particular cells. One of these is myosin, whose interac-tion with actin causes contraction in muscle and other cells (p. 658). Actin microfi laments provide a means for movement of molecules and organ-elles through the cytoplasm, as well as movement of messenger RNA (p. 93) from the nucleus to particular positions within the cytoplasm. Actin and actin-binding proteins are also important in movement of vesicles between the ER, Golgi complex, and plasma membrane or lysosomes. Micro-tubules, somewhat larger than microfi laments, are tubular struc-tures composed of a protein called tubulin ( Figure 3.13 ). Each tubulin molecule is actually a doublet composed of two globu-lar proteins. The molecules are attached head-to-tail to form a strand, and 13 strands aggregate to form a microtubule. Because the tubulin subunits in a microtubule are always attached head-to-tail, the ends of the microtubule differ chemically and function-ally. One end (called the plus end) both adds and deletes tubulin subunits more rapidly than the other end (the minus end). Micro-tubules play a vital role in moving chromosomes toward daugh-ter cells during cell division (p. 52), and they are important in intracellular architecture, organization, and transport. In addition, microtubules form essential parts of the structures of cilia and fl a-gella (see next section). Microtubules radiate from a microtubule organizing center, the centrosome, near the nucleus. Centro-somes are not membrane bound. Within centrosomes are found a pair of centrioles ( Figures 3.4 and 3.14 ), which are themselves Figure 3.10 System for assembling, isolating, and secreting polypeptides and proteins for export in a eukaryotic cell, for lysosomes, or for incorporation into plasma membrane.
Golgi complex Lysosome Proteins and polypeptides for export Proteins and polypeptides for export Secretory vesicle Rough endoplasmic reticulum Cytoplasm Plasma membrane Vesicle incorporation in plasma membrane Transport vesicle Smooth endoplasmic reticulum Vesicle containing transmembrane proteins/polypeptides Figure 3.11 Mitochondria. A, Structure of a typical mitochondrion. B, Electron micrograph of mitochondria in cross and longitudinal section. (30,000) Inner membrane Outer membrane Matrix Crista A B www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 45 composed of microtubules. Each centriole of a pair lies at right angles to the other and is a short cylinder of nine triplets of microtubules. They replicate before cell division. Although cells of higher plants do not have centrioles, a microtubule organiz-ing center is present. Intermediate fi laments are larger than microfilaments but smaller than microtubules. There are five biochemically distinct types of intermediate fi laments, and their composition and arrangement depend on the cell type in which they occur. The type of intermediate fi lament is often determined in cancerous cells so that the original cell type can be identifi ed. Knowing the particular cell type can often help in determining treatment options. Surfaces of Cells and Their Specializations The free surface of epithelial cells (cells that cover the surface of a structure or line a tube or cavity; see p. 195) sometimes bears either cilia or fl agella (sing., cilium, fl agellum ). These are motile extensions of the cell surface that sweep materials past the cell. In many single-celled organisms and some small multicellular forms, they propel the entire organism through a liquid medium (see pp. 239, 282). Flagella provide the means of locomotion for male reproductive cells (sperm) of most animals (see p. 146) and many plants. Cilia and fl agella have different beating patterns (see p. 656), but their internal structure is the same. With few exceptions, the internal structures of locomotory cilia and fl agella are com-posed of a long cylinder of nine pairs of microtubules enclosing a central pair (see Figure 29.11 ). At the base of each cilium or fl agellum is a basal body (kinetosome), which is identical in structure to a single centriole. Many cells move neither by cilia nor fl agella but by ame-boid movement using pseudopodia. Some groups of uni-cellular eukaryotes (p. 224), migrating cells in embryos of multicellular animals, and some cells of adult multicellular animals, such as white blood cells, show ameboid movement. Cytoplasmic streaming through the action of actin microfi la-ments extends a lobe (pseudopodium) outward from the sur-face of the cell. Continued streaming in the direction of a pseudopodium brings cytoplasmic organelles into the lobe and accomplishes movement of the entire cell. Some special-ized pseudopodia have cores of microtubules (p. 227), and Figure 3.13 A, The microtubules in kidney cells of a baby hamster have been rendered visible by treatment with a preparation of fl uorescent proteins that specifi cally bind to tubulin. B, A microtubule is composed of 13 strands of tubulin molecules, and each molecule is a dimer. Tubulin dimers are added to and removed from the () end of the microtubule more rapidly than at the () end.
Tubulin dimer (+) end (–) end A B Figure 3.12 Cytoskeleton of a cell, showing its complex nature. Three visible cytoskeletal elements, in order of increasing diameter, are microfi laments, intermediate fi laments, and microtubules. (66,600) Microfilaments Microtubules Intermediate filaments 46 PART ONE Introduction to Living Animals movement is effected by assembly and disassembly of the tubulin subunits. Cells covering the surface of a structure (epithelial cells) or cells packed together in a tissue may have specialized junctional complexes between them. Nearest the free surface, the mem-branes of two cells next to each other appear to fuse, forming a tight junction ( Figure 3.15 ). They are formed from rows of transmembrane proteins that bind tightly between adjacent cells. Tight junctions function as seals to prevent the passage of mol-ecules between cells from one side of a layer of cells to another, because there is usually a space of about 20 nm between the plasma membranes of adjacent cells. The number of rows of transmembrane proteins in the tight junction determines how closely adjacent cells are sealed to each other. Tight junctions between intestinal cells, for example, force molecules from the intestinal contents to pass through epithelial cells during absorption, rather than between them. Adhesion junctions ( Figure 3.15 ) occur just beneath tight junctions. These anchor-ing junctions are similar to tight junctions in that they encircle the cell. They are different from tight junctions in that they do not seal adjacent cells to each other. Rather, the transmem-brane proteins link together across a small intercellular space. Inside adjacent cells, the transmembrane proteins are attached to actin microfi laments and thus attach the cytoskeletons of adja-cent cells to each other. Modifi ed adhesion junctions are found between cardiac muscle cells, and these hold the cells together as the heart beats throughout the life of an organism (p. 694). At various points beneath tight junctions and adhesion junctions, in epithelial cells, small ellipsoid discs occur, within the plasma membrane in each cell. These appear to act as “spot-welds” and are called desmosomes ( Figure 3.15 ) . From each desmosome a tuft of intermediate fi laments extends into the cytoplasm, and transmembrane linker proteins extend through the plasma mem-brane into the intercellular space to bind the discs of adjacent cells together. Desmosomes are not seals but seem to increase the strength of the tissue. Many are found between the cells of the skin in vertebrates (p. 645). Hemidesmosomes ( Figure 3.15 ) are found at the base of cells and anchor them to underlying connective tissue layers. Gap junctions ( Figure 3.15 ), rather than serving as points of attachment, provide a means of inter-cellular communication. They form tiny canals between cells, so that their cytoplasm becomes continuous, and small molecules and ions can pass from one cell to the other. Gap junctions may occur between cells of epithelial, nervous, and muscle tissues. Another specialization of cell surfaces is the “lacing together” of adjacent cell surfaces where plasma membranes of the cells infold and interdigitate very much like a zipper. These infoldings are especially common in epithelial cells of kidney tubules and serve to increase the surface area of the cells for absorption or secretion. The distal or apical boundaries of some epithelial cells, as seen by electron microscopy, show regularly arranged micro-villi (sing., microvillus ). They are small, fi ngerlike projections consisting of tubelike evaginations of the plasma membrane with a core of cytoplasm containing bundles of actin microfi la-ments ( Figures 3.15 and 3.16 ). They are seen clearly in the lining of the intestine where they greatly increase the absorptive and digestive surface. Such specializations appear as brush borders by light microscopy. Membrane Function The incredibly thin, yet sturdy, plasma membrane that encloses every cell is vitally important in maintaining cellular integrity. Once erroneously considered to be a rather static entity that defi ned cell boundaries and kept cell contents in place, plasma membranes (also called the plasmalemma) form dynamic struc-tures having remarkable activity and selectivity. They are a per-meability barrier that separates the interior from the external environment. They regulate the fl ow of molecules into and out of the cell, and provide many of the unique functional properties of specialized cells. Membranes inside a cell surround a variety of organelles. Indeed, a cell is a system of membranes that divide it into numerous compartments. It has been estimated that if all mem-branes present in one gram of liver tissue were spread out fl at, they would cover 30 square meters! Internal membranes share Figure 3.14 The centrosome. A, Each centrosome contains a pair of centrioles and each centriole is composed of nine triplets of microtubules arranged as a cylinder. B, Electron micrograph of a pair of centrioles, one in longitudinal (right) and one in cross section (left). The normal orientation of centrioles is at right angles to each other.
Microtubule triplet A B www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 47 many structural features of plasma membranes and are the site for many of a cell’s enzymatic reactions. A plasma membrane acts as a selective gatekeeper for entrance and exit of many substances involved in cell metabo-lism. Some substances can pass through with ease, others enter slowly and with diffi culty, and still others cannot enter at all. Because conditions outside a cell are different from and more variable than conditions within a cell, it is necessary that passage of substances across the membrane be rigorously controlled. We recognize three principal ways that a substance may enter across a cell membrane: (1) by diffusion along a concen-tration gradient; (2) by a mediated transport system, in which the substance binds to a specifi c site on a transmembrane pro-tein that assists it across the membrane; and (3) by endocytosis, in which the substance is enclosed within a vesicle that forms from the membrane surface and detaches inside the cell. Diffusion and Osmosis Diffusion is a movement of particles from an area of higher con-centration to an area of lower concentration of the particles or molecules, thus tending to equalize the concentration throughout the area of diffusion. If a living cell surrounded by a membrane is immersed in a solution having a higher concentration of solute Figure 3.15 Junction types and locations are shown in columnar epithelial cells. Actin microfi laments (shown in green) and intermediate fi laments (shown in orange) attach adhesion junctions and desmosomes, respectively, to the cytoskeleton.
Microvilli Tight junction Adhesion junction Junctional complex Intermediate filament Desmosome (anchoring junction) Gap junction (communicating junction) Hemidesmosome (anchoring junction) Basal lamina 48 PART ONE Introduction to Living Animals molecules than the fl uid inside the cell, a concentration gradi-ent instantly exists between the two fl uids across the membrane. Assuming that the membrane is permeable to the solute, there is a net movement of solute toward the inside, the side having the lower concentration. The solute diffuses “downhill” across the membrane until its concentrations on each side are equal. Most cell membranes are selectively permeable, that is, permeable to water but variably permeable or impermeable to solutes. In free diffusion it is this selectiveness that regulates molecular traffi c. As a rule, gases (such as oxygen and carbon dioxide), urea, and lipid-soluble solutes (such as fats, fatlike substances, and alcohol; see p. 26) are the only solutes that can diffuse through biological membranes with any degree of freedom. Because many water-soluble molecules readily pass through membranes, such movements cannot be explained by simple diffusion. Sugars, many electrolytes, and macromolecules are moved across membranes by carrier-mediated processes that are described in the next section. If we place a membrane between two unequal concentrations of solutes to which the membrane is impermeable, water fl ows through the membrane from the more dilute to the more concen-trated solution. The water molecules move across the membrane down a concentration gradient from an area where the water mol-ecules are more concentrated to an area on the other side of the membrane where they are less concentrated. This is osmosis. We can demonstrate osmosis in a simple experiment by tying a selectively permeable membrane tightly over the end of a funnel. The funnel is fi lled with a salt solution and placed in a beaker of pure water so that the water levels inside and outside the funnel are equal. In a short time the water level in the glass tube of the funnel rises, indicating a net movement of water through the membrane into the salt solution ( Figure 3.17 ). Inside the funnel are salt molecules, as well as water mole-cules, while the beaker contains only water molecules. Thus the concentration of water is less inside the funnel because some of the available space is occupied by the nondiffusible salt ions. A concentration gradient exists for water molecules in the system. Water diffuses from the region of greater concentration of water (pure water in the beaker) to the region of lesser water concen-tration (salt solution inside the funnel). As water enters the salt solution, the fl uid level in the funnel rises. Eventually the pressure produced by the increasing weight of solution in the funnel pushes water molecules out as fast as they enter. The level in the funnel becomes stationary and the Figure 3.16 Electron micrograph of microvilli. (59,000) Figure 3.17 Simple membrane osmometer. A, The end of a tube containing a salt solution is closed at one end by a selectively permeable membrane. The membrane is permeable to water but not to salt. B, When the tube is immersed in pure water, water molecules diffuse through the membrane into the tube. Water molecules are in higher concentration in the beaker because they are diluted inside the tube by salt ions. Because the salt cannot diffuse out through the membrane, the volume of fl uid inside the tube increases, and the level rises. C, When the weight of the column of water inside the tube exerts a downward force (hydrostatic pressure) causing water molecules to leave through the membrane in equal number to those that enter (osmotic pressure), the volume of fl uid inside the tube stops rising. At this point the hydrostatic pressure is equivalent to the osmotic pressure.
3% salt solution Selectively permeable membrane Distilled water Salt solution rising Solution stops rising when weight of column (hydrostatic pressure) equals osmotic pressure Water A B C www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 49 system is in equilibrium. The osmotic pressure of the solution is equivalent to the hydrostatic pressure necessary to prevent further net entry of water. The concept of osmotic pressure is not without problems. A solution reveals an osmotic “pressure” only when it is separated from solvent by a selectively permeable membrane. It can be disconcerting to think of an isolated bottle of salt solution as having “pressure” much as compressed gas in a bottle ( hydro-static pressure) would have. Furthermore, the osmotic pressure is really the hydrostatic pressure that must be applied to a solu-tion to keep it from gaining water if the solution were separated from pure water by a selectively permeable membrane. Conse-quently, biologists frequently use the term osmotic potential rather than osmotic pressure. However, since the term “osmotic pressure” is so fi rmly fi xed in our vocabulary, it is necessary to understand the usage despite its potential confusion. The concept of osmosis is very important in understanding how animals control their internal fl uid and solute environment (see Chapter 30). For example, marine bony fi shes maintain a solute concentration in their blood about one-third of that in sea-water; they are hypoosmotic to seawater. If a fi sh swims into a river mouth and then up a freshwater stream, as salmon do, it would pass through a region where its blood solutes were equal in concentration to those in its environment (isosmotic), then enter freshwater, where its blood solutes were hyperosmotic to those in its environment. It must have physiological mechanisms to avoid net loss of water in the sea and gain of water in the river. Diffusion Through Channels Water and dissolved ions, since they are charged, cannot dif-fuse through the phospholipid component of the plasma mem-brane. Instead, they pass through specialized pores or channels created by transmembrane proteins. Ions and water move through these channels by diffusion. Ion channels allow spe-cifi c ions of a certain size and charge to diffuse through them. They may allow ion diffusion at all times or they may be gated channels, requiring a signal to open or close them. Gated ion channels may open or close when a signaling molecule binds to a specifi c binding site on the transmembrane protein ( chemically-gated ion channels; Figure 3.18A ) or when the ionic charge across a plasma membrane changes ( voltage-gated ion channels; Figure 3.18B ). Ion diffusion through channels is the basis of signaling mechanisms in the nervous system (see Chapter 33, p. 729) and in muscles (see Chapter 29, p. 660). Water channels are called aquaporins, and several different types have been discovered. They are especially important in the digestive system for absorption of water from food (see Chapter 32, p. 720), and in the kidney for water reabsorption during urine formation (see Chapter 30, p. 679). Carrier-Mediated Transport We have seen that a plasma membrane is an effective bar-rier to free diffusion of most molecules of biological signifi -cance, yet it is essential that such materials enter and leave a cell. Nutrients such as sugars and materials for growth such as amino acids must enter a cell, and wastes of metabolism must leave. Such molecules are moved across a membrane by spe-cial transmembrane proteins called transporters, or carriers. Transporters enable solute molecules to cross the phospholipid bilayer ( Figure 3.19A ). Transporters are usually quite specifi c, recognizing and transporting only a limited group of chemical substances or perhaps even a single substance. At high concentrations of solute, mediated transport sys-tems show a saturation effect. This means simply that the rate of infl ux reaches a plateau beyond which increasing the solute concentration has no further effect on infl ux rate ( Figure 3.19B ). This is evidence that the number of transporters available in a membrane is limited. When all transporters become occupied by solutes, the rate of transport is at a maximum and it cannot be increased. Simple diffusion shows no such limitation; the greater the difference in solute concentrations on the two sides of the membrane, the faster the infl ux. Two distinctly different kinds of mediated transport mech-anisms are recognized: (1) facilitated diffusion, in which a transporter assists a molecule to diffuse through the membrane that it cannot otherwise penetrate, and (2) active transport, in which energy is supplied to the transporter system to transport molecules in the direction opposite to a concentration gradient ( Figure 3.20 ). Facilitated diffusion therefore differs from active Figure 3.18 Gated channels require a signal to open (or close) them. A, Chemically-gated ion channels open (or close) when a signaling molecule binds to a specifi c binding site on the transmembrane protein. B, Voltage-gated ion channels open (or close) when the ionic charge across the membrane changes.
Gate Channel closed Channel open A Chemically-gated ion channel B Voltage-gated ion channel Signal Ions + – + + 50 PART ONE Introduction to Living Animals transport in that it sponsors movement only in a downhill direc-tion (in the direction of a concentration gradient) and requires no metabolic energy to drive the transport system. In many animals facilitated diffusion aids in transport of glu-cose (blood sugar) into body cells that oxidize it as a principal energy source for the synthesis of ATP. The concentration of glu-cose is greater in blood than in the cells that consume it, favor-ing inward diffusion, but glucose is a water-soluble molecule that does not, by itself, penetrate cell membranes rapidly enough to support the metabolism of many cells; the carrier-mediated transport system increases the inward fl ow of glucose. In active transport, molecules are moved uphill against the forces of passive diffusion. Active transport always involves an expenditure of energy (from ATP) because materials are trans-ported against a concentration gradient. Among the most impor-tant active-transport systems in all animals are those that maintain sodium and potassium ion gradients between cells and the sur-rounding extracellular fl uid or external environment. Most animal cells require a high internal concentration of potassium ions for protein synthesis at the ribosome and for certain enzymatic func-tions. The potassium ion concentration may be 20 to 50 times greater inside a cell than outside. Sodium ions, on the other hand, may be 10 times more concentrated outside a cell than inside. This sodium gradient forms the basis for electrical signal genera-tion within the nervous system of animals (see Chapter 33, p. 728). Both of these ionic gradients are maintained by active transport of potassium ions into and sodium ions out of the cell. In many cells outward transport of sodium is linked to inward transport of potassium; this same transporter molecule does both. As much as 10% to 40% of all energy produced by cells is consumed by the sodium-potassium exchange pump ( Figure 3.20 ). Figure 3.19 Facilitated transport. A, A transporter protein molecule binds with a molecule to be transported (substrate) on one side of a plasma membrane, changes shape, and releases the molecule on the other side. Facilitated transport takes place in the direction of a concentration gradient. B, Rate of transport increases with increasing substrate concentration until all transporter molecules are occupied.
Inside of cell Outside of cell Transporter molecule B A Rate of influx Extracellular concentration All carrier molecules occupied Figure 3.20 Sodium-potassium pump, powered by bond energy of ATP, maintains the normal gradients of these ions across the cell membrane. The pump works by a series of conformational changes in the transporter: Step 1. Three ions of Na bind to the interior end of the transporter, producing a conformational (shape) change in the protein complex. Step 2. The complex binds a molecule of ATP, cleaves it, and phosphate binds to the complex. Step 3. The binding of the phosphate group induces a second conformational change, passing the three Na ions across the membrane, where they are now positioned facing the exterior. This new conformation has a very low affi nity for the Na ions, which dissociate and diffuse away, but it has a high affi nity for K ions and binds two of them as soon as it is free of the Na ions. Step 4. Binding of the K ions leads to another conformational change in the complex, this time leading to dissociation of the bound phosphate. Freed of the phosphate, the complex reverts to its original conformation, with the two K ions exposed on the interior side of the membrane. This conformation has a low affi nity for K ions so that they are now released, and the complex has the conformation it started with, having a high affi nity for Na ions.
3 Na+ 3 Na+ 2 K+ 2 K+ 2 K+ ATP ADP P P P STEP 1: Three ions of Na+ bind to the interior end of the transporter.
STEP 2: The complex binds a molecule of ATP and cleaves it.
STEP 3: The 3 Na+ ions are released to the exterior, and 2 K+ ions bind to the transporter.
STEP 4: The phosphate is released and the 2 K+ ions move into the cell.
Inside cell Transporter www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 51 Endocytosis Endocytosis, the ingestion of material by cells, is a collective term that describes three similar processes: phagocytosis, pinocytosis, and receptor-mediated endocytosis ( Figure 3.21 ). They are path-ways for specifi cally internalizing solid particles, small molecules and ions, and macromolecules, respectively. All require energy and thus may be considered forms of active transport. Phagocytosis, which literally means “cell eating,” is a com-mon method of feeding among protozoa and lower metazoa. It is also the way in which white blood cells (leukocytes) engulf cellular debris and uninvited microbes or other pathogens in the blood. By phagocytosis, an area of the plasma membrane, coated externally with specifi c receptors and internally with actin and actin-binding proteins, forms a pocket that engulfs the solid material. The membrane-enclosed vesicle, a food vacuole or phagosome, then detaches from the cell surface and moves into the cytoplasm where it fuses with lysosomes, and its con-tents are digested by lysosomal enzymes. Pinocytosis is similar to phagocytosis except that small areas of the surface membrane are invaginated into cells to form tiny vesicles. The invaginated pits and vesicles are called caveo-lae (ka-vee´o-lee). Specifi c binding receptors for the molecule or ion to be internalized are concentrated on the cell surface of caveolae. Pinocytosis apparently functions for intake of at least some vitamins, and similar mechanisms may be important in translocating substances from one side of a cell to the other (see “exocytosis,” next heading) and internalizing signal molecules, such as some hormones or growth factors. Receptor-mediated endocytosis is a specifi c mechanism for bringing large molecules within the cell. Proteins of the plasma membrane specifi cally bind particular molecules (termed ligands in this process), which may be present in the extra-cellular fl uid in very low concentrations. The invaginations of the cell surface that bear the receptors are coated within the cell with a protein called clathrin; hence, they are described as clathrin-coated pits. As a clathrin-coated pit with its receptor-bound ligand invaginates and is brought within the cell, it is uncoated, the receptor and the ligand are dissociated, and the receptor and membrane material are recycled back to the sur-face membrane. Lysosomes fuse with the remaining vesicle, now called an endosome, and its contents are digested and absorbed into the cytoplasm. Some important proteins, peptide hormones, and cholesterol are brought into cells in this manner. In phagocytosis, pinocytosis, and receptor-mediated endo-cytosis some amount of extracellular fl uid is necessarily trapped in the vesicle and nonspecifi cally brought within the cell. We describe this as bulk-phase endocytosis. Exocytosis Just as materials can be brought into a cell by invagination and formation of a vesicle, the membrane of a vesicle can fuse with the plasma membrane and extrude its contents to the surrounding medium. This is the process of exocytosis. This process occurs in various cells to remove undigestible residues of substances brought in by endocytosis, to secrete substances such as hormones ( Figure 3.10 ), and to transport a substance completely across a cellular barrier (transcytosis), as we just mentioned. For example, a substance may be picked up on one side of the wall of a blood vessel by pinocytosis, moved across the cell, and released by exocytosis. Figure 3.21 Three types of endocytosis. In phagocytosis the cell membrane binds to a large particle and extends to engulf it, forming a membrane-enclosed vesicle, a food vacuole or phagosome. In pinocytosis small areas of cell membrane, bearing specifi c receptors for a small molecule or ion, invaginate to form caveolae. Receptor-mediated endocytosis is a mechanism for selective uptake of large molecules in clathrin-coated pits. Binding of the ligand to the receptor on the surface membrane stimulates invagination of pits.
Phagocytosis Pinocytosis Receptor-mediated Endocytosis Ligands Clathrin-coated pit Clathrin Lysosomal enzymes Lysosomal enzymes Membrane-enclosed vesicle Microbe Small molecule or ion Receptors Translocation to opposite side of cell (transcytosis) Receptors and ligands are dissociated Release or Caveolae Vesicle is uncoated Receptors and membranes are recycled 52 PART ONE Introduction to Living Animals Actin and actin-binding proteins are now to be essential cytoskeletal components in the processes of endocytosis and exocytosis. MITOSIS AND CELL DIVISION All cells arise from the division of preexisting cells. All cells found in most multicellular organisms originated from the divi-sion of a single cell, a zygote, which is the product of union (fertilization) of an egg and a sperm (gametes). Cell division provides the basis for one form of growth, for both sexual and asexual reproduction, and for transmission of hereditary qualities from one cell generation to another cell generation. In the formation of body cells (somatic cells) the process of nuclear division is mitosis. By mitosis each “daughter cell” is ensured a complete set of genetic instructions. Mitosis is a delivery system for distributing the chromosomes and the DNA they contain to continuing cell generations. Thus, a single zygote divides by mitosis to produce a multicellular organism, and dam-aged cells are replaced by mitosis during wound healing. As an animal grows, its somatic cells differentiate and assume different functions and appearances because of differential gene action. Although most of the genes in specialized cells remain silent and unexpressed throughout the lives of those cells, every cell pos-sesses a complete genetic complement. Mitosis ensures equal-ity of genetic potential; later, other processes direct the orderly expression of genes during embryonic development by select-ing from the genetic instructions that each cell contains. (These fundamental properties of cells of multicellular organisms are discussed further in Chapter 8.) In animals that reproduce asexually (see Chapter 7), mito-sis is the only mechanism for the transfer of genetic information from parent to progeny, and thus the progeny are genetically identical to the parents in this case. In animals that reproduce sexually (see Chapter 7), the parents must produce sex cells (gametes or germ cells) that contain only half the usual number of chromosomes, so that progeny formed by the union of gam-etes will not contain double the parental number of chromo-somes. This requires a special type of reductional division called meiosis, described in Chapter 5 (p. 77). Structure of Chromosomes As mentioned on page 41, DNA in eukaryotic cells occurs in chromatin, a complex of DNA with associated protein. Chroma-tin is organized into a number of discrete linear bodies called chromosomes (color bodies), so named because they stain deeply with certain biological dyes. In cells that are not dividing, chromatin is loosely organized and dispersed, so that individual chromosomes cannot be distinguished by light microscopy (see Figure 3.24 , Interphase). Before division the chromatin becomes more compact, and chromosomes can be recognized and their individual morphological characteristics determined. They are of varied lengths and shapes, some bent and some rodlike. Their number is constant for a species, and every body cell (but not the germ cells) has the same number of chromosomes regardless of a cell’s function. A human, for example, has 46 chromosomes in each somatic cell. During mitosis (nuclear division) chromosomes shorten fur-ther and become increasingly condensed and distinct, and each assumes a shape partly characterized by the position of a con-striction, the centromere ( Figure 3.22 ). The centromere is the location of the kinetochore, a disc of proteins that binds with microtubules of the spindle that forms during mitosis. When chromosomes become condensed, the DNA is inac-cessible, such that transcription (see Chapter 5, p. 93) cannot occur. Chromosomal condensation does, however, enable a cell to distribute chromosomal material effi ciently and equally to daughter cells during cell division. Phases in Mitosis There are two distinct stages of cell division: division of nuclear chromosomes (mitosis) and division of cytoplasm (cytokine-sis). Mitosis (that is, chromosomal segregation) is certainly the most obvious and complex part of cell division and that of great-est interest to cytologists. Cytokinesis normally immediately fol-lows mitosis, although occasionally the nucleus may divide a number of times without a corresponding division of the cyto-plasm. In such a case the resulting mass of protoplasm containing many nuclei is called a multinucleate cell. An example is the giant resorptive cell type of bone (osteoclast), which may con-tain 15 to 20 nuclei. Sometimes a multinucleate mass is formed by cell fusion rather than nuclear proliferation. This arrangement is called a syncytium. An example is vertebrate skeletal muscle, which is composed of multinucleate fi bers formed by fusion of numerous embryonic cells. Figure 3.22 Structure of a metaphase chromosome. Sister chromatids are still attached at their centromere. Each chromatid has a kinetochore, to which the kinetochore microtubules or fi bers are attached. Kinetochore microtubules from each chromatid run to one of the centrosomes, which are located at opposite poles.
Sister chromatids Kinetochore Kinetochore microtubules Centromere www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 53 Mitosis is artifi cially divided into four successive stages or phases, although one stage merges into the next without sharp lines of transition. These phases are prophase, metaphase, ana-phase, and telophase ( Figures 3.23 and 3.24 ). When cells are not actively dividing, they are in interphase, a major part of the cell cycle described in some detail on page 54. Prophase At the beginning of prophase, centrosomes (along with their cen-trioles) replicate, the nuclear envelope disintegrates, and the two centrosomes migrate to opposite poles of the cell ( Figure 3.23 ). At the same time, microtubules are manufactured between the two centrosomes to form a football-shaped spindle, so named because of its resemblance to nineteenth-century wooden spin-dles, used to twist thread together in spinning. Other microtu-bules radiate outward from each centrosome to form asters. The asters will develop into the microtubular portion of the cytoskel-eton in each new daughter cell formed during cell division. At this time the diffuse nuclear chromatin condenses to form visible chromosomes. These actually consist of two iden-tical sister chromatids ( Figure 3.22 ) formed by DNA replica-tion (see Chapter 5, p. 93) during interphase and joined at their centromere. Dynamic spindle microtubules repeatedly extend and retract from each centrosome. When a microtubule encounters a kinetochore, it binds to the kinetochore, ceases extending and retracting, and is now called a kinetochore microtubule or fi ber. Thus, centrosomes send out “feelers” to fi nd chromosomes. Metaphase Each centromere has two kinetochores, and each of the kineto-chores is attached to one of the centrosomes by a kinetochore fi ber. By a kind of tug-of-war during metaphase, the condensed sister chromatids are moved to the middle of the nuclear region, called the metaphase plate ( Figures 3.23 and 3.24 ). The centro-meres line up precisely in this region with the arms of the sister chromatids trailing off randomly in various directions. Anaphase The single centromere that has held two sister chromatids together now splits so that the two sister chromatids separate to become two independent chromosomes, each with its own centromere. The chromosomes move toward their respective poles, pulled by their kinetochore fibers. The arms of each chromosome trail behind as the microtubules shorten to drag a complete set of chromosomes toward each pole of the cell ( Figures 3.23 and 3.24 ). Present evidence indicates that the force moving the chromosomes is disassembly of the tubulin subunits at the kinetochore end of the each microtubule. As chromosomes approach their respective centrosomes, the centrosomes move farther apart as the microtubules are gradually disassembled. Telophase When daughter chromosomes reach their respective poles, telophase has begun ( Figures 3.23 and 3.24 ). The daughter chromosomes are crowded together and stain intensely with histological stains. Spindle fi bers disappear and the chromo-somes lose their identity, reverting to a diffuse chromatin network characteristic of an interphase nucleus. Finally, nuclear membranes reappear around the two daughter nuclei. Cytokinesis: Cytoplasmic Division During the fi nal stages of nuclear division a cleavage furrow appears on the surface of a dividing cell and encircles it at the midline of the spindle ( Figures 3.23 and 3.24 ). The cleavage furrow deepens and pinches the plasma membrane as though it were being tightened by an invisible Figure 3.23 Stages of mitosis, showing division of a cell with two pairs of chromosomes. One chromosome of each pair is shown in red.
Chromosomes Aster Microtubules Centrosome with pair of centrioles Kinetochore fiber Spindle Metaphase Anaphase Telophase Interphase Prophase Early metaphase Astral fiber Metaphase plate Daughter nuclei Nucleus 54 PART ONE Introduction to Living Animals rubber band. Actin microfi laments are present just beneath the surface in the furrow between the cells. Interaction with myo-sin and other actin-binding proteins, similar to muscle contrac-tion mechanisms (p. 658), draws the furrow inward. Finally, the infolding edges of the plasma membrane meet and fuse, com-pleting cell division. Cell Cycle Cycles are conspicuous attributes of life. The descent of a spe-cies through time is in a very real sense a sequence of life cycles. Similarly, cells undergo cycles of growth and replication as they repeatedly divide. A cell cycle is the interval between one cell division and the next ( Figure 3.25 ). Actual nuclear division, or mitosis, occupies only about 5% to 10% of the cell cycle; the rest of the cell’s time is spent in inter-phase, the stage between nuclear divisions. For many years it was thought that interphase was a period of rest, because nuclei appeared inactive when observed by ordinary light microscopy. In the early 1950s new techniques for revealing DNA replica-tion in nuclei were introduced at the same time that biologists came to appreciate fully the signifi cance of DNA as the genetic material. It was then discovered that DNA replication occurred during the interphase stage. Further studies revealed that many other protein and nucleic acid components essential to normal cell function, growth, and division were synthesized during the seemingly quiescent interphase period. Replication of DNA occurs during a phase called the S phase (period of synthesis). In mammalian cells in tissue culture, the S phase lasts about six of the 18 to 24 hours required to complete one cell cycle. In this phase both strands of a DNA molecule must replicate; new complementary partners are synthesized for each strand so that two identical DNA molecules are produced from the original strand (see Chapter 5, p. 93). These comple-mentary partners are the sister chromatids that are separated dur-ing the next mitosis. The S phase is preceded and succeeded by G 1 and G 2 phases, respectively (G stands for “gap”), during which no DNA synthesis is occurring. For most cells, G 1 is an important preparatory stage for the replication of DNA that follows. Dur-ing G 1 , transfer RNA, ribosomes, messenger RNA, and many enzymes are synthesized. During G 2 , spindle and aster proteins Figure 3.24 Stages of mitosis in whitefi sh.
Prophase Metaphase Telophase Interphase Anaphase Metaphase plate Chromosomes Aster Nucleus www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 55 are synthesized in preparation for chromosome separation dur-ing mitosis. G 1 is typically of longer duration than G 2 , although there is much variation in different cell types. Embryonic cells divide very rapidly because there is no cell growth between divisions, only subdivision of mass. DNA synthesis may pro-ceed a hundred times more rapidly in embryonic cells than in adult cells, and the G 1 phase is very shortened. As an organism develops, the cycle of most of its cells lengthens, and many cells may be arrested for long periods of time in G 1 and enter a nonproliferative or quiescent phase called G 0 . Most neurons or nerve cells, for example, divide no further and are essentially in a permanent G 0 . The events in cell cycles are exquisitely regulated. Transi-tions during cell cycles are mediated by enzymes called cyclin-dependent kinases (cdk’s) and regulatory protein subunits that activate them, called cyclins. In general, kinases are enzymes that add phosphate groups to other proteins to activate or inacti-vate them, and kinases themselves may require activation. Cdk’s become active only when they are bound to the appropriate cyclin, and cyclins are synthesized and degraded during each cell cycle ( Figure 3.26 ). It seems likely that phosphorylation and dephosphorylation of specifi c cdk’s and their interaction with phase-specifi c cyclins regulates the passage from one cell cycle phase to the next. Current research focuses on the checkpoints that regulate this phase to phase passage since disregulation of these mechanisms has been implicated in cancer. Flux of Cells Cell division is important for growth, for replacement of cells lost to natural attrition and for wound healing. Cell division is espe-cially rapid during early development of an organism. At birth a human infant has about 2 trillion cells from repeated division of a single fertilized egg or zygote. This immense number could be attained by just 42 cell divisions, with each generation dividing once every six to seven days. With only fi ve more cell divisions, the cell number would increase to approximately 60 trillion, the number of cells in a human adult weighing 75 kg. Of course no organism develops in this machinelike manner. Cell division is rapid during early embryonic development, then slows with age. Furthermore, different cell populations divide at widely differ-ent rates. In some the average period between divisions is mea-sured in hours, whereas in others it is measured in days, months, or even years. Some cells in the central nervous system stop dividing after the early months of fetal development and gener-ally persist without further division for the life of the individual. Muscle cells also stop dividing during the third month of fetal development, and most future growth depends on enlargement of fi bers already present. In other tissues that are subject to abrasion, lost cells must be constantly replaced. It is estimated that in humans about 1% to 2% of all body cells—a total of 100 billion—are shed daily. Mechanical rubbing wears away the outer cells of the skin, and food in the alimentary canal removes lining epithelial cells. In addition, the restricted life cycle of blood cells involves enor-mous numbers of replacements. Such lost cells are replaced by mitosis. Normal development, however, does entail cell death in which cells are not replaced. As cells age, they accumulate damage from destructive oxidizing agents and eventually die. Other cells undergo a programmed cell death, or apoptosis (a-puh-TOE-sis) (Gr. apo-, from, away from; ptosis, a falling), which is in many cases necessary for continued health and development of an organism. For example, during embryonic development of vertebrates, fi ngers and toes develop as tissues between them die, excess immune cells that would attack the body’s own tissues “commit suicide,” and nerve cells die to create cerebral convolu-tions. Apoptosis consists of a well-coordinated and predictable series of events: the cells shrink, disintegrate, and their compo-nents are absorbed up by surrounding cells. Figure 3.25 Cell cycle, showing relative duration of recognized phases. S, G1, and G2 are phases within interphase; S, synthesis of DNA; G1, presynthetic phase; G2, postsynthetic phase. After mitosis and cytokinesis the cell may go into an arrested, quiescent stage known as G0. Actual duration of the cycle and the different phases varies considerably in different cell types.
R N A a n d f u n c ti o n al p r o t e i n s y n t h e si s Cytokinesis Mitosis G1 G2 S S tr u ct u r a l p r o te in s y n t h e si s D N A r e p li c a ti o n CELL CYCLE Figure 3.26 Variations in the level of cyclin in dividing cells of early sea urchin embryos. Cyclin binds with its cyclin-dependent kinase to activate the enzyme.
Time INTERPHASE Amount of protein INTERPHASE INTERPHASE MITOSIS MITOSIS Cyclin Other proteins 56 PART ONE Introduction to Living Animals S U M M A R Y Cells are the basic structural and functional units of all living organ-isms. Eukaryotic cells differ from the prokaryotic cells of bacteria and Archaebacteria in several respects, the most distinctive of which is presence of a membrane-bound nucleus containing hereditary material composed of DNA, bound to proteins to form chromatin. Chromatin consists of fl exible, linear chromosomes, which become condensed and visible only during cell division. Cells are surrounded by a plasma membrane that regulates the fl ow of molecules between the cell and its surroundings. The nucleus, enclosed by a double membrane, contains chromatin, associated proteins, and one or more nucleoli. Outside the nuclear envelope is cell cytoplasm, subdivided by a membranous network, the endoplasmic reticulum. Among the organelles within cells are the Golgi complex, mitochondria, lysosomes, and other membrane-bound vesicles. The cytoskeleton is composed of microfi laments (actin), microtubules (tubulin), and intermediate fi laments (several types). Cilia and fl agella are hairlike, motile appendages that con-tain microtubules. Ameboid movement by pseudopodia operates by means of the assembly and disassembly of actin microfi laments. Tight junctions, adhesion junctions, desmosomes, and gap junctions are structurally and functionally distinct connections between cells. Membranes in a cell are composed of a phospholipid bilayer and other materials including cholesterol and transmembrane proteins. Hydrophilic ends of the phospholipid molecules are on the outer and inner surfaces of membranes, and the fatty acid portions are directed inward, toward each other, to form a hydrophobic core. Substances can enter cells by diffusion, mediated transport, and endocytosis. Osmosis is diffusion of water through channels in a selectively permeable membrane as a result of osmotic pressure. Solutes to which the membrane is impermeable require channels or a transporter molecule to traverse the membrane. Water and ions move through open channels by diffusion (in the direction of a concentration gradient). Transport-mediated systems include facilitated diffusion and active transport (against a concentration gradient, which requires energy). Endocytosis includes bringing droplets (pinocytosis) or particles (phagocytosis) into a cell. In exocytosis the process of endocytosis is reversed. The cell cycle in eukaryotes includes mitosis, or division of the nuclear chromosomes, and cytokinesis, the division of the cyto-plasm, and interphase. During interphase, G 1 , S, and G 2 phases are recognized, and the S phase is the time when DNA is synthesized (the chromosomes are replicated). Cell division is necessary for the production of new cells from preexisting cells and is the basis for growth in multicellular organ-isms. During this process, replicated nuclear chromosomes divide by mitosis followed by cytoplasmic division or cytokinesis. The four stages of mitosis are prophase, metaphase, anaphase, and telophase. In prophase, replicated chromosomes composed of sister chromatids condense into recognizable bodies. A spindle forms between the centrosomes as they separate to opposite poles of the cell. At the end of prophase the nuclear envelope disintegrates, and the kinetochores of each chromosome become attached to the cen-trosomes by microtubules (kinetochore fi bers). At metaphase the sister chromatids are moved to the center of the cell, held there by the kinetochore fi bers. At anaphase the centromeres divide, and the sister chromatids are pulled apart by the attached kinetochore fi bers of the mitotic spindle. At telophase the sister chromatids, now called chromosomes, gather in the position of the nucleus in each cell and revert to a diffuse chromatin network. A nuclear membrane reappears, and cytokinesis occurs. At the end of mitosis and cyto-kinesis, two cells genetically identical to the parent cell have been produced. Cells divide rapidly during embryonic development, then more slowly with age. Some cells continue to divide throughout the life of an animal to replace cells lost by attrition and wear, whereas others, such as nerve and muscle cells, complete their division during early development and many do not divide again. Some cells undergo a programmed cell death, or apoptosis. R E V I E W Q U E S T I O N S Explain the difference (in principle) between a light microscope and a transmission electron microscope. Briefl y describe the structure and function of each of the following: plasma membrane, chromatin, nucleus, nucleolus, rough endoplasmic reticulum (rough ER), Golgi complex, lysosomes, mitochondria, microfi laments, microtubules, intermediate fi laments, centrioles, basal body (kinetosome), tight junction, gap junction, desmosome, glycoprotein, microvilli. Name two functions each for actin and for tubulin. Distinguish among cilia, fl agella, and pseudopodia. What are the functions of each of the main constituents of the plasma membrane? Our current concept of the plasma membrane is known as the fl uid-mosaic model. Why? You place some red blood cells in a solution and observe that they swell and burst. You place some cells in another solution, and they shrink and become wrinkled. Explain what has happened in each case. 1.
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7. Explain why a beaker containing a salt solution, placed on a table in your classroom, can have a high osmotic pressure, yet be subjected to a hydrostatic pressure of only one atmosphere. The plasma membrane is an effective barrier to molecular movement across it, yet many substances do enter and leave the cell. Explain the mechanisms through which this is accomplished and comment on the energy requirements of these mechanisms. Distinguish among phagocytosis, pinocytosis, receptor-mediated endocytosis, and exocytosis. Defi ne the following: chromosome, centromere, centrosome, kinetochore, mitosis, cytokinesis, syncytium. Explain phases of the cell cycle, and comment on important cellular processes that characterize each phase. What is G 0 ? Name the stages of mitosis in order, and describe the behavior and structure of the chromosomes at each stage. Briefl y describe ways that cells may die during the normal life of a multicellular organism. 8.
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www . m hhe . com /hi ckmani pz14e CHAPTER 3 Cells as Units of Life 57 S E L E C T E D R E F E R E N C E S Alberts, B., D. Bray, K. Hopkin, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. 2003. Essential cell biology, ed. 2. New York, Garland Science Publishing. A well-written text describing more detailed cellular mechanisms. Kaksonen, M., C. P. Toret, and D. G. Drubin. 2006. Harnessing action dynamics for clathrin-coated endocytosis. Nature Reviews, Molecular Cell Biology 7 :404–414. While receptor-mediated endocytosis is the focus of this article, the role of actin in phagocytosis and pinocytosis are discussed. Roth, R. 2006. Clathrin-mediated endocytosis before fl uorescent proteins. Nature Reviews, Molecular Cell Biology 7 :63–68. An excellent discussion of the history of the scientifi c timeline underlying the dynamic process of receptor-mediated endocytosis. Wolfe, S. L. 1995. Introduction to cell and molecular biology, ed. 1. Belmont, CA, Thomson Brooks/Core Publishers. Good coverage of cell structure and function for those interested in getting more detail. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! C H A P T E R 4 Cellular Metabolism Deferring the Second Law Living systems appear to contradict the second law of thermodynam-ics, which states that energy in the universe has direction and that it has been, and always will be, running down. In effect all forms of energy inevitably will be degraded to heat. This increase in disorder, or randomness, in any closed system is termed entropy. Living sys-tems, however, decrease their entropy by increasing the molecular orderliness of their structure. An organism becomes vastly more com-plex during its development from fertilized egg to adult. The second law of thermodynamics, however, applies to closed systems, and living organisms are not closed systems. Animals grow and maintain themselves by borrowing free energy from the environment. When a deer feasts on the acorns and beechnuts of summer, it transfers potential energy, stored as chemical bond energy in the nuts’ tissues, to its own body. Then, in step-by-step sequences called biochemical pathways, this energy is gradually released to fuel the deer’s many activities. In effect, the deer decreases its own internal entropy by increasing the entropy of its food. The orderly structure of the deer is not permanent, however, but will be dissipated when it dies. The ultimate source of this energy for the deer—and for almost all life on earth—is the sun ( Figure 4.1 ). Sunlight is captured by green plants, which fortunately accumulate enough chemical bond energy to sustain both themselves and animals that feed on them. Thus the second law is not violated; it is simply held at bay by life on earth, which uses the continuous fl ow of solar energy to main-tain a biosphere of high internal order, at least for the period of time that life exists. White-tailed deer ( Odocoileus virginianus ) foraging for acorns. www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 59 A B C D Figure 4.2 Diffusion of a solute through a solution, an example of entropy. When the solute (sugar molecules) is fi rst introduced into a solution, the system is ordered and unstable (B). Without energy to maintain this order, the solute particles become distributed into solution, reaching a state of disorder (equilibrium) (D). Entropy has increased from left diagram to right diagram. A ll cells must obtain energy, synthesize their own inter-nal structure, control much of their own activity, and guard their boundaries. Cellular metabolism refers to the collective chemical processes that occur within living cells to accomplish these activities. Although the enormous number of reactions in their aggregate are extremely complex, the central metabolic routes through which matter and energy are channeled appear to be conserved by the majority of living organisms. ENERGY AND THE LAWS OF THERMODYNAMICS The concept of energy is fundamental to all life processes. We usually express energy as the capacity to do work, to bring about change. Yet energy is a somewhat abstract quantity that is dif-fi cult to defi ne and elusive to measure. Energy cannot be seen; it can be identifi ed only by how it affects matter. Energy can exist in either of two states: kinetic or poten-tial. Kinetic energy is the energy of motion. Potential energy is stored energy, energy that is not doing work but has the capacity to do so. Energy can be transformed from one state to another. Especially important for living organisms is chemical energy, a form of potential energy stored in the chemical bonds of molecules. Chemical energy can be tapped when bonds are rearranged to release kinetic energy. Much of the work done by living organisms involves conversion of potential energy to kinetic energy. Conversion of one form of energy to another is governed by the two laws of thermodynamics. The fi rst law of thermo-dynamics states that energy cannot be created or destroyed. It can change from one form to another, but the total amount of energy remains the same. In short, energy is conserved. If we burn gasoline in an engine, we do not create new energy but merely convert the chemical energy in gasoline to another form, in this example, mechanical energy and heat. The second law of thermodynamics, introduced in the prologue to this chap-ter, concerns the transformation of energy. This fundamental law states that a closed system moves toward increasing disorder, or entropy, as energy is dissipated from the system ( Figure 4.2 ). Living systems, however, are open systems that not only main-tain their organization but also increase it, as during the develop-ment of an animal from egg to adult. Figure 4.1 Solar energy sustains virtually all life on earth. With each energy transfer, however, about 90% of the energy is lost as heat.
Sun’s energy harvested by photosynthesis Net productivity in plants Heat loss Heat loss Heat loss Transfer of chemical bond energy in food molecules Herbivores Transfer of chemical bond energy in food molecules Carnivores All bond energy eventually dissipated 60 PART ONE Introduction to Living Animals Free Energy To describe the energy changes of chemical reactions, bio-chemists use the concept of free energy. Free energy is simply the energy in a system available for doing work. In a mol-ecule, free energy equals the energy present in chemical bonds minus the energy that cannot be used. The majority of reac-tions in cells release free energy and are called exergonic (Gr. ex, out, ergon, work). Such reactions are spontaneous and always proceed “downhill” since free energy is lost from the system. Thus: Loss of some energy Products, containing less free energy than reactant Reactant AB A B Exergonic reaction However, many important reactions in cells require the addi-tion of free energy and are said to be endergonic (Gr. endon, within, ergon, work). Such reactions have to be “pushed uphill” because the products store more energy than the reactants. Energy input Products, containing more free energy than reactants Reactants C D CD Endergonic reaction Described on page 62, ATP is a ubiquitous, energy-rich inter-mediate used by organisms to power important uphill reactions such as those required for active transport of molecules across membranes (see Chapter 3, p. 49) and cellular synthesis. THE ROLE OF ENZYMES Enzymes and Activation Energy For any reaction to occur, even exergonic ones that tend to pro-ceed spontaneously, chemical bonds fi rst must be destabilized. Some energy, termed the activation energy, must be supplied before the bond is stressed enough to break. Only then will an overall loss of free energy and formation of reaction products occur. This requirement can be likened to the energy needed to push a ball over the crest of a hill before it will roll sponta-neously down the other side, the ball liberating its energy as it descends ( Figure 4.3, top panel). One way to activate chemical reactants is to raise the tem-perature. This increases the rate of molecular collisions and pushes chemical bonds apart. Thus, heat can impart the nec-essary activation energy to make a reaction proceed. However metabolic reactions must occur at biologically tolerable tempera-tures, which are usually too low to allow reactions to proceed at a rate capable of sustaining life. Instead, living systems have evolved a different strategy: they employ catalysts. Catalysts are chemical substances that accelerate reaction rates without affecting the products of the reaction and without being altered or destroyed by the reaction. A catalyst cannot make an energetically impossible reaction happen; it simply accelerates a reaction that would proceed at a very slow rate otherwise. Enzymes are catalysts of the living world. The special cat-alytic talent of an enzyme is its power to reduce the amount of activation energy required for a reaction. In effect, an enzyme steers the reaction through one or more intermediate steps, each of which requires much less activation energy than that required for a single-step reaction ( Figure 4.3 ). Note that enzymes do not supply the activation energy. Instead they lower the activa-tion energy barrier, making a reaction more likely to proceed. Enzymes affect only the reaction rate. They do not in any way alter the free energy change of a reaction, nor do they change the proportions of reactants and products in a reaction. Figure 4.3 Energy changes during enzyme catalysis of a substrate. The overall reaction proceeds with a net release of energy (exergonic). In the absence of an enzyme, substrate is stable because of the large amount of activation energy needed to disrupt strong chemical bonds. The enzyme reduces the energy barrier by forming a chemical intermediate with a much lower internal energy state.
Substrate Products Enzyme-substrate complex Reaction process Internal energy Activation energy Energy released With enzyme AB AB A B Without enzyme Activation energy www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 61 Nature of Enzymes Enzymes are complex molecules that vary in size from small, simple proteins with a molecular weight of 10,000 to highly complex molecules with molecular weights up to 1 million. Many enzymes are pure proteins—highly folded and interlinked chains of amino acids. Other enzymes require participation of small nonprotein groups called cofactors to perform their enzy-matic function. In some cases these cofactors are metallic ions (such as ions of iron, copper, zinc, magnesium, potassium, and calcium) that form a functional part of the enzyme. Examples are carbonic anhydrase (see Chapter 31, p. 705), which contains zinc; the cytochromes (some enzymes of the electron transport chain, p. 67), which contain iron; and troponin (a muscle con-traction enzyme, see Chapter 29, p. 658), which contains cal-cium. Another class of cofactors, called coenzymes, is organic. Coenzymes contain groups derived from vitamins, most of which must be supplied in the diet. All B-complex vitamins are coenzymatic compounds. Since animals have lost the ability to synthesize the vitamin components of coenzymes, it is obvious that a vitamin defi ciency can be serious. However, unlike dietary fuels and nutrients that must be replaced after they are burned or assembled into structural materials, vitamin components of coenzymes are recovered in their original form and are used repeatedly. Examples of coenzymes that contain vitamins are nicotinamide adenine dinucleotide (NAD), which contains the vitamin nicotinic acid (niacin); coenzyme A, which contains the vitamin pantothenic acid; and fl avin adenine dinucleotide (FAD), which contains ribofl avin (vitamin B 2 ). Action of Enzymes An enzyme functions by associating in a highly specifi c way with its substrate, the molecule whose reaction it catalyzes. Enzymes bear an active site located within a cleft or pocket and that contains a unique molecular confi guration. The active site has a fl exible surface that enfolds and conforms to the substrate ( Figure 4.4 ). The binding of enzyme to substrate forms an enzyme-substrate complex (ES complex), in which the sub-strate is secured by covalent bonds to one or more points in the active site of the enzyme. The ES complex is not strong and will quickly dissociate, but during this fl eeting moment the enzyme provides a unique chemical environment that stresses certain chemical bonds in the substrate so that much less energy is required to complete the reaction. If the formation of an enzyme-substrate complex is so rapidly fol-lowed by dissociation, how can biochemists be certain that an ES complex exists? The original evidence offered by Leonor Michaelis in 1913 is that, when the substrate concentration is increased while the enzyme concentration is held constant, the reaction rate reaches a maximum velocity. This saturation effect is interpreted to mean that all catalytic sites become fi lled at high substrate concentration. A saturation effect is not seen in uncatalyzed reactions. Other evi-dence includes the observation that the ES complex displays unique spectroscopic characteristics not displayed by either the enzyme or the substrate alone. Furthermore, some ES complexes can be isolated in pure form, and at least one kind (nucleic acids and their polymerase enzymes) has been directly visualized by electron microscopy. Enzymes that engage in crucial energy-providing reactions of cells often proceed constantly and usually operate in sets rather than in isolation. For example, conversion of glucose to carbon dioxide and water proceeds through 19 reactions, each requiring a specific enzyme. Such crucial enzymes occur in A B C Figure 4.4 How an enzyme works. The ribbon model (A) and the space-fi lling model (B) show that the enzyme lysozyme bears a pocket containing the active site. When a chain of sugars (substrate) enters the pocket (C), the protein enzyme changes shape slightly so that the pocket enfolds the substrate and conforms to its shape. This positions the active site (amino acids in the protein) next to a bond between adjacent sugars in the chain, causing the sugar chain to break.
62 PART ONE Introduction to Living Animals relatively high concentrations in cells, and they may implement quite complex and highly integrated enzymatic sequences. One enzyme performs the fi rst step, then another enzyme binds to the product and catalyzes the next step. This process continues until the end of the enzymatic pathway is reached. The reactions are said to be coupled. Coupled reactions are explained in a section on chemical energy transfer by ATP (see this page). Specifi city of Enzymes One of the most distinctive attributes of enzymes is their high specifi city. Specifi city is a consequence of the exact molecular fi t required between enzyme and substrate. Furthermore, an enzyme catalyzes only one reaction. Unlike reactions performed in an organic chemist’s laboratory, no side reactions or by-products result. Specifi city of both substrate and reaction is obvi-ously essential to prevent a cell from being swamped with use-less by-products. However, there is some variation in degree of specifi city. Some enzymes catalyze the oxidation (dehydrogenation) of only one substrate. For example, succinic dehydrogenase catalyzes the oxidation of succinic acid only (see the Krebs cycle, p. 66). Oth-ers, such as proteases (for example, pepsin and trypsin, released into the digestive tract during digestion, pp. 716 and 719), act on almost any protein, although each protease has its particular point of attack in the protein ( Figure 4.5 ). Usually an enzyme binds one substrate molecule at a time, catalyzes its chemical change, releases the product, and then repeats the process with another substrate molecule. An enzyme may repeat this process billions of times until it is fi nally worn out (after a few hours to several years) and is degraded by scavenger enzymes in the cell. Some enzymes undergo successive catalytic cycles at speeds of up to a million cycles per minute, but most operate at slower rates. Many enzymes are repeatedly activated and inactivated; several mecha-nisms for regulating enzyme activity are well known (p. 72). the proteolytic enzyme pepsin degrades proteins into amino acids (a catabolic reaction), but it does not accelerate the rebuilding of amino acids into any signifi cant amount of protein (an anabolic reaction). The same is true of most enzymes that catalyze the cleavage of large molecules such as nucleic acids, polysaccharides, lipids, and proteins. There is usually one set of reactions and enzymes that degrade them (catabolism; Gr. kata, down, bole, throw), but they must be resynthesized by a different set of reactions catalyzed by different enzymes (anabo-lism; Gr. ana, up, bole, throw). The net direction of any chemical reaction depends on the relative energy contents of the substances involved. If there is little change in chemical bond energy of substrate and products, the reaction is more easily reversible. However, if large quantities of energy are released as the reaction proceeds in one direction, more energy must be provided in some way to drive the reac-tion in the reverse direction. For this reason many if not most enzyme-catalyzed reactions are in practice irreversible unless the reaction is coupled to another one that makes energy available. In cells both reversible and irreversible reactions are combined in complex ways to make possible both synthesis and degradation. Hydrolysis literally means “breaking with water.” In hydrolysis reac-tions, a molecule is cleaved by the addition of water at the cleavage site. A hydrogen is attached to one subunit and a hydroxyl (OH) unit is attached to another. This breaks the covalent bond between subunits. Hydrolysis is the opposite of condensation (water-losing) reactions in which subunits of molecules are linked together by removal of water. Macromolecules are built by condensation reactions. CHEMICAL ENERGY TRANSFER BY ATP We have seen that endergonic reactions are those that do not proceed spontaneously because their products require an input of free energy. However, an endergonic reaction may be driven by coupling the energy-requiring reaction with an energy-yielding reaction. ATP is one of the most common intermediates in coupled reactions, and because it can drive such energeti-cally unfavorable reactions, it is of central importance in meta-bolic processes. An ATP molecule consists of adenosine (the purine ade-nine and the 5-carbon sugar ribose) and a triphosphate group ( Figures 4.6 and 4.7 ). Most free energy in ATP resides in the tri-phosphate group, especially in two phosphoanhydride bonds between the three phosphate groups called “high-energy bonds.” Usually, only the most exposed high-energy bond is hydrolyzed to release free energy when ATP is converted to ade-nosine diphosphate (ADP) and inorganic phosphate. ATP H O ADP P 2 i → where P i represents inorganic phosphate (i inorganic). The high-energy groups in ATP are often designated by the Figure 4.5 High specifi city of trypsin. It splits only peptide bonds adjacent to lysine or arginine.
C C N C C N H R2 O H O H Hydrolysis site Lysine or arginine Enzyme-Catalyzed Reactions Enzyme-catalyzed reactions are reversible, which is signifi ed by double arrows between substrate and products. For example: Fumaric acid H O Malic acid 2 ⇋ However, for various reasons reactions catalyzed by most enzymes tend to go predominantly in one direction. For example, www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 63 “tilde” symbol ~ ( Figure 4.6 ). A high-energy phosphate bond is shown as ~P and a low-energy bond (such as the bond linking the triphosphate group to adenosine) as P. Thus, ATP may be symbolized as AP~P~P and ADP as AP~P. The way in which ATP can drive a coupled reaction is shown in Figure 4.8. A coupled reaction is really a system involving two reactions linked by an energy shuttle (ATP). The conversion of substrate A to product A is endergonic because the product con-tains more free energy than the substrate. Therefore energy must be supplied by coupling the reaction to an exergonic one, the conversion of substrate B to product B. Substrate B in this reac-tion is commonly called a fuel (for example, glucose or a lipid). Bond energy released in reaction B is transferred to ADP, which in turn is converted to ATP. ATP now contributes its phosphate-bond energy to reaction A, and ADP and P i are produced again. The high-energy bonds of ATP are actually rather weak, unstable bonds. Because they are unstable, the energy of ATP is readily released when ATP is hydrolyzed in cellular reactions. Note that ATP is an energy-coupling agent and not a fuel. It is not a storehouse of energy set aside for some future need. Rather it is produced by one set of reactions and is almost imme-diately consumed by another set. ATP is formed as it is needed, primarily by oxidative processes in mitochondria. Oxygen is not consumed unless ADP and phosphate molecules are available, and these do not become available until ATP is hydrolyzed by some energy-consuming process. Metabolism is therefore mostly self-regulating. CELLULAR RESPIRATION How Electron Transport Is Used to Trap Chemical Bond Energy Having seen that ATP is the one common energy denominator by which most cellular machines are powered, we must ask how this energy is captured from fuel substrates. This question directs Figure 4.6 A, Structure of ATP. B, ATP formation from ADP and AMP. ATP: adenosine triphosphate; ADP: adenosine diphosphate; AMP: adenosine monophosphate.
O O O O P –O P CH2 O O P O– O– O– Adenine Triphosphate Ribose C H C C H C N NH2 C N N N H H O OH OH H H ("high-energy" bonds) Adenine Ribose Phosphate Phosphate Phosphate AMP ATP ADP Phosphoanhydride A B Figure 4.7 Space-fi lling model of ATP. In this model, carbon is shown in black; nitrogen in blue; oxygen in red; and phosphorus in orange.
Phosphate Phosphate Phosphate Phosphoanhydride (high-energy) bonds Adenine Ribose Figure 4.8 A coupled reaction. The endergonic conversion of substrate A to product A will not occur spontaneously but requires an input of energy from another reaction involving a large release of energy. ATP is the intermediate through which the energy is shuttled.
Endergonic reaction Exergonic reaction Enzyme B Enzyme A Product A ADP P P Substrate A ATP P P P Product B Substrate B Pi Pi 64 PART ONE Introduction to Living Animals us to an important generalization: all cells obtain their chemical energy requirements from oxidation-reduction reactions. This means that in the degradation of fuel molecules, hydrogen atoms (electrons and protons) are passed from electron donors to elec-tron acceptors with a release of energy. A portion of this energy can be trapped and used to form the high-energy bonds of mol-ecules such as ATP. An oxidation-reduction (“redox”) reaction involves a trans-fer of electrons from an electron donor (the reducing agent) to an electron acceptor (the oxidizing agent). As soon as the electron donor loses its electrons, it becomes oxidized. As soon as the electron acceptor accepts electrons, it becomes reduced ( Figure 4.9 ). In other words, a reducing agent becomes oxidized when it reduces another compound, and an oxidizing agent becomes reduced when it oxidizes another compound. Thus for every oxidation there must be a corresponding reduction. In an oxidation-reduction reaction the electron donor and electron acceptor form a redox pair: Electron donor e Electron acceptor Energy ( ⇋ reducing agent; (oxidizing agent; becomes oxi idized) becomes reduced) When electrons are accepted by the oxidizing agent, energy is liberated because the electrons move to a more stable position. ATP may be produced in a cell when electrons fl ow through a series of carriers. Each carrier is reduced as it accepts electrons and then is reoxidized when it passes electrons to the next carrier in the series. By transferring electrons stepwise in this manner, energy is gradually released, and ATP is produced. Ultimately, the electrons are transferred to a fi nal electron acceptor. The nature of this fi nal acceptor is the key that determines the overall effi ciency of cellular metabolism. Aerobic Versus Anaerobic Metabolism Heterotrophs (organisms that cannot synthesize their own food but must obtain nutrients from the environment, including ani-mals, fungi, and many single-celled organisms) are divided into two groups based on their overall effi ciency of energy produc-tion during cellular metabolism: aerobes, those that use molecu-lar oxygen as the fi nal electron acceptor, and anaerobes, those that employ another molecule as the fi nal electron acceptor. As discussed in Chapter 2, life originated in the absence of oxygen, and an abundance of atmospheric oxygen was produced only after photosynthetic organisms (autotrophs) evolved. Some strictly anaerobic organisms still exist and indeed play impor-tant roles in specialized habitats. However, evolution has favored aerobic metabolism, not only because oxygen became available, but also because aerobic metabolism is vastly more effi cient in energy production than anaerobic metabolism. In the absence of oxygen, only a very small fraction of the bond energy pres-ent in nutrients can be released. For example, when an anaero-bic microorganism degrades glucose, the fi nal electron acceptor (such as pyruvic acid) still contains most of the energy of the orig-inal glucose molecule. An aerobic organism on the other hand, using oxygen as the fi nal electron acceptor, completely oxidizes glucose to carbon dioxide and water. Almost 20 times as much energy is released when glucose is completely oxidized as when it is degraded only to the stage of pyruvic acid. Thus, an obvious advantage of aerobic metabolism is that a much smaller quantity of food is required to maintain a given rate of metabolism. Overview of Respiration Aerobic metabolism is more familiarly called cellular respi-ration, defi ned as the oxidation of fuel molecules to produce energy with molecular oxygen as the fi nal electron acceptor. We emphasize that oxidation of fuel molecules describes the removal of electrons and not the direct combination of molecular oxygen with fuel molecules. Let us look at this process in general before considering it in more detail. Hans Krebs, the British biochemist who contributed so much to our understanding of respiration, described three stages in the complete oxidation of fuel molecules to carbon dioxide and water ( Figure 4.10 ). In stage I, food passing through the intestinal tract are digested into small molecules that can be absorbed into the circulation. There is no useful energy yield during digestion, which is discussed in Chapter 32. In stage II, also called glycoly-sis, most of the digested food is converted into two 3-carbon units (pyruvic acid) in the cell cytoplasm. The pyruvic acid molecules then enter mitochondria, where in another reaction they join with a coenzyme (coenzyme A or CoA) to form acetyl coenzyme A or acetyl-CoA. Some ATP is generated in stage II, but the yield is small compared with that obtained in stage III of respiration. In stage III the fi nal oxidation of fuel molecules occurs, with a large yield of ATP. This stage takes place entirely in mitochondria. Ace-tyl-CoA is channeled into the Krebs cycle where the acetyl group is completely oxidized to carbon dioxide. Electrons released from acetyl groups are transferred to special carriers that pass them to electron acceptor compounds in the electron transport chain. At the end of the chain the electrons (and the protons accompany-ing them) are accepted by molecular oxygen to form water. Glycolysis We begin our journey through the stages of respiration with gly-colysis, a nearly universal pathway in living organisms that con-verts glucose into pyruvic acid. In a series of reactions occurring Figure 4.9 A redox pair. The molecule at left is oxidized by the loss of an electron. The molecule at right is reduced by gaining an electron.
+ + Oxidation Reduction www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 65 in the cell cytosol, glucose and other 6-carbon monosaccharides are split into 3-carbon molecules of pyruvic acid ( Figure 4.11 ). A single oxidation occurs during glycolysis, and each molecule of glucose yields two molecules of ATP. In this pathway the carbohydrate molecule is phosphorylated twice by ATP, fi rst to glucose-6-phosphate (not shown in Figure 4.11 ) and then to fructose-1,6-bisphosphate. The fuel has now been “primed” with phosphate groups in this uphill portion of glycolysis and is suf-fi ciently reactive to enable subsequent reactions to proceed. This is a kind of defi cit fi nancing required for an ultimate energy return many times greater than the original energy investment. In the downhill portion of glycolysis, fructose-1,6-bisphos-phate is cleaved into two 3-carbon sugars, which undergo an oxidation (electrons are removed), with the electrons and one of the hydrogen ions being accepted by nicotinamide adenine dinucleotide ( NAD , a derivative of the vitamin niacin) to pro-duce a reduced form called NADH. NADH serves as a carrier molecule to convey high-energy electrons to the fi nal electron transport chain, where ATP is produced. The two 3-carbon sugars next undergo a series of reactions, ending with the formation of two molecules of pyruvic acid ( Figure 4.11 ). In two of these steps, a molecule of ATP is pro-duced. In other words, each 3-carbon sugar yields two ATP molecules, and since there are two 3-carbon sugars, four ATP molecules are generated. Recalling that two ATP molecules were used to prime the glucose initially, the net yield at this point is two ATP molecules. The 10 enzymatically catalyzed reactions in glycolysis can be summarized as: Glucose 2 ADP 2 P NAD pyruvic acid 2 NA i 2 2 → DH 2 ATP Acetyl-CoA: Strategic Intermediate in Respiration In aerobic metabolism the two molecules of pyruvic acid formed during glycolysis enter a mitochondrion. There, each molecule is oxidized, and one of the carbons is released as carbon dioxide ( Figure 4.12 ). The 2-carbon residue condenses with coenzyme A (CoA) to form acetyl coenzyme A, or acetyl-CoA, and an NADH molecule is also produced. Figure 4.10 Overview of cellular respiration, showing the three stages in the complete oxidation of food molecules to carbon dioxide and water.
Glycerol Fatty acids Glucose and other sugars Acetyl-CoA Pyruvic acid Krebs cycle Stage I: Digestion of food molecules to small units Stage II: Small molecules converted to acetyl-CoA Stage III: Final common pathway for oxidation of fuel molecules CARBOHYDRATES FATS PROTEINS Amino acids G L Y C O L Y S I S Krebs cycle ATP ATP CO2 ATP Electron transport chain H2O e-Figure 4.11 Glycolysis. Glucose is phosphorylated in two steps and raised to a higher energy level. High-energy fructose-1,6-bisphosphate is split into triose phosphates that are oxidized exergonically to pyruvic acid, yielding ATP and NADH.
2 ADP Fructose-1,6-bisphosphate ADP ADP NAD+ NAD+ Glucose 3-Phospho-glyceraldehyde 1,3-Diphospho-glyceric acid 3-Phospho-glyceric acid 2-Phospho-glyceric acid Phosphoenol pyruvic acid Pyruvic acid ADP ADP 2 ATP NADH NADH ATP ATP ATP ATP G L Y C O L Y S I S 66 PART ONE Introduction to Living Animals Pyruvic acid is the undissociated form of the acid: O • CH3—C—COOH Under physiological conditions pyruvic acid typically dissociates into pyruvate O • (CH3—C—COO–) and H . It is correct to use either term in describing this and other organic acids (such as lactic acid, or lactate) in metabolism. Acetyl-CoA is a critically important compound. Its oxida-tion in the Krebs cycle provides energized electrons to gener-ate ATP, and it is also a crucial intermediate in lipid metabolism (p. 70). Krebs Cycle: Oxidation of Acetyl-CoA Degradation (oxidation) of the 2-carbon acetyl group of acetyl-CoA occurs within the mitochondrial matrix in a cyclic sequence called the Krebs cycle (also called citric acid cycle and tricarbox-ylic acid cycle [TCA cycle]) ( Figure 4.13 ). Acetyl-CoA condenses with a 4-carbon acid (oxaloacetic acid), releasing CoA to react again with more pyruvic acid. Through a cyclic series of reac-tions in the Krebs cycle, the two carbons from the acetyl group Figure 4.12 Formation of acetyl-CoA from pyruvic acid.
NAD+ NADH C CH3 COOH Pyruvic acid Acetyl-CoA CoA CoA O CO2 C CH3 S O Figure 4.13 The Krebs cycle in outline form, showing production of three molecules of reduced NAD, one molecule of reduced FAD, one molecule of ATP, and two molecules of carbon dioxide. The molecules of NADH and FADH2 will yield 11 molecules of ATP when oxidized in the electron transport system.
CO2 ATP GTP GDP ADP 1. The cycle begins with 2C acetyl-CoA condensing with 4C oxaloacetic acid to form 6C citric acid. CoA is released to react again with more pyruvic acid.
2. An isomer of citric acid is oxidized by NAD+, yielding 5C ketoglutaric acid, NADH, and a molecule of CO2.
3. Oxidation by NAD+ occurs again, yielding 4C succinic acid, NADH, and CO2.
4. One molecule of ATP is formed directly with each cycle.
5. Another oxidation by FAD yields FADH2.
6. A final oxidation by NAD+ yields NADH and returns the cycle to its start point.
Succinic acid (4C) Oxaloacetic acid (4C) Citric acid (6C) NADH NADH NAD+ NAD+ NADH NAD+ FAD FADH2 H2O H2O CO2 CoA Acetyl-CoA CO2 KREBS (Citric Acid) CYCLE www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 67 are released as carbon dioxide, and oxaloacetic acid is regener-ated. Hydrogen ions and electrons in the oxidations transfer to NAD and to FAD (fl avine adenine dinucleotide, another elec-tron acceptor), and a pyrophosphate bond is generated in the form of guanosine triphosphate (GTP). This high-energy phos-phate readily transfers to ADP to form ATP. Thus, the overall products of the Krebs cycle are CO 2 , ATP, NADH, and FADH 2 : Acetyl unit 3 NAD FAD ADP P CO NADH F i 2 → 2 3 ADH ATP 2 The molecules of NADH and FADH 2 formed yield 11 molecules of ATP when oxidized in the electron transport chain. The other molecules in the cycle behave as intermediate reactants and products which are continuously regenerated as the cycle turns. Aerobic cellular respiration uses oxygen as the fi nal electron accep-tor and releases carbon dioxide and water from the complete oxida-tion of fuels. The carbon dioxide that we, and other aerobic organ-isms, produce is removed from our bodies into the atmosphere during external respiration (see Chapter 31, p. 698). Fortunately for us and other aerobes, oxygen is continuously produced by cyano-bacteria (blue-green algae), eukaryotic algae, and plants by the pro-cess of photosynthesis. In this process, hydrogen atoms obtained from water react with carbon dioxide from the atmosphere to gen-erate sugars and molecular oxygen. Thus, a balance between oxy-gen used and produced, and carbon dioxide produced and used, is obtained across our planet. Unfortunately, excessive production of carbon dioxide due to human industrialization, and decreased production of oxygen due to our continuous removal of the world’s forests, are threatening this delicate balance. Carbon dioxide levels continue to rise, leading to global atmospheric warming caused by the “greenhouse effect” (see Chapter 37, p. 807). Electron Transport Chain Transfer of hydrogen ions and electrons from NADH and FADH 2 to the fi nal electron acceptor, molecular oxygen, is accomplished in an elaborate electron transport chain embedded in the inner membrane of mitochondria ( Figure 4.14 , see also p. 44). Each carrier molecule in the chain (labeled I to IV in Figure 4.14 ) is a large transmembrane protein-based complex that accepts and releases electrons at lower energy levels than the carrier preced-ing it in the chain. As electrons pass from one carrier molecule to the next, free energy is released. Some of this energy is used to transport H ions across the inner mitochondrial membrane and in this way creates a H gradient across the membrane. The H gradient produced drives the synthesis of ATP. This process is called chemiosmotic coupling ( Figure 4.14 ). According to this model, as electrons contributed by NADH and FADH 2 are carried down the electron transport chain, they activate proton transport-ing molecules which move protons (hydrogen ions) outward and into the space between the two mitochondrial membranes. This causes the proton concentration in this intermembrane space to rise, producing a diffusion gradient that is used to drive the pro-tons back into the mitochondrial matrix through special proton channels. These channels are ATP-forming transmembrane pro-tein complexes (ATP synthase) that use the inward movement of protons to induce the formation of ATP. By this means, oxida-tion of one NADH yields three ATP molecules. FADH 2 from the Krebs cycle enters the electron transport chain at a lower level than NADH and so yields two ATP molecules. This method of energy capture is called oxidative phosphorylation because the formation of high-energy phosphate is coupled to oxygen consumption, and these reactions depend on demand for ATP by other metabolic activities within the cell. Effi ciency of Oxidative Phosphorylation We can now calculate the ATP yield from the complete oxida-tion of glucose ( Figure 4.15 ). The overall reaction is: Glucose 2 ATP 36 ADP 36 P 6 O CO ADP 2 2 → 6 2 36 ATP 6 H O 2 ATP has been generated at several points along the way ( Table 4.1 ). The cytoplasmic NADH generated in glycolysis requires a molecule of ATP to fuel transport of each molecule of NADH into a mitochondrion; therefore, each NADH from glycolysis yields only two ATP (total of four), compared with the three ATP per NADH (total of six) formed within mito-chondria. Accounting for the two ATP used in the priming reactions in glycolysis, the net yield may be as high as 36 mol-ecules of ATP per molecule of glucose. The yield of 36 ATP is a theoretical maximum because some of the H gradient produced by electron transport may be used for other func-tions, such as transporting substances in and out of the mito-chondrion. Overall effi ciency of aerobic oxidation of glucose is about 38%, comparing very favorably with human-designed energy conversion systems, which seldom exceed 5% to 10% effi ciency. The capacity for oxidative phosphorylation is also increased by the elaborate folding of the inner mitochondrial membrane (the cristae shown in Figure 4.14 and labeled in Figure 3.11, p. 44), providing a much greater surface area for more electron transport chain and ATP synthase proteins. Anaerobic Glycolysis: Generating ATP Without Oxygen Under anaerobic conditions, glucose and other 6-carbon sugars are fi rst converted stepwise to a pair of 3-carbon pyruvic acid molecules during glycolysis, described on page 64 (see also Figure 4.11 ). This series of reactions yields two molecules of ATP and two molecules of NADH. In the absence of molecular oxygen, further oxidation of pyruvic acid cannot occur because without oxygen as the fi nal electron acceptor in the electron transport chain, the Krebs cycle and electron transport chain cannot operate and cannot, therefore, reoxidize the NADH pro-duced in glycolysis. The problem is neatly solved in most animal 68 PART ONE Introduction to Living Animals cells by reducing pyruvic acid to lactic acid ( Figure 4.16 ). Pyru-vic acid becomes the fi nal electron acceptor and lactic acid the end product of anaerobic glycolysis. This step converts NADH to NAD , effectively freeing it to recycle and pick up more H and electrons. In alcoholic fermentation (as in yeast, for example) the steps are identical to glycolysis down to pyruvic acid. One of its carbons is then released as carbon dioxide, and the resulting 2-carbon compound is reduced to ethanol, thus regenerating the NAD . Anaerobic glycolysis is only one-eighteenth as efficient as complete oxidation of glucose to carbon dioxide and water, but its key virtue is that it provides some high-energy phosphate in situations in which oxygen is absent or in short supply. Many microorganisms live in places where oxygen is severely depleted, such as waterlogged soil, in mud of a lake or sea bottom, or within a decaying carcass. Vertebrate skeletal muscle may rely heavily on glycolysis during short Figure 4.14 Oxidative phosphorylation. Most of the ATP in living organisms is produced in the electron transport chain. Electrons removed from fuel molecules in cellular oxidations (glycolysis and the Krebs cycle) fl ow through the electron transport chain, the major components of which are four transmembrane protein complexes (I, II, III, and IV). Electron energy is tapped by the major complexes and used to push H outward across the inner mitochondrial membrane. The H gradient created drives H inward through proton channels (ATP synthase) that couple H movement to ATP synthesis.
Downhill movement of H+ through proton channel is coupled to ATP synthesis Outer membrane of mitochondrion Inner membrane of mitochondrion H+ 2 H+ + H+ H+ H+ I II III IV ADP + Pi H2O CO2 O2 CO2 Electron transport coupled to H+ transport across membrane GLYCOLYSIS Glucose Pyruvic acid Acetyl-CoA CoA NADH NADH FADH2 e-2 e- + O Krebs cycle ATP ATP MATRIX T A B L E 4 . 1 Calculation of Total ATP Molecules Generated in Respiration ATP Generated Source 4 Directly in glycolysis 2 As GTP (→ATP) in Krebs cycle 4 From NADH in glycolysis 6 From NADH produced in pyruvic acid to acetyl-CoA reaction 4 From reduced FAD in Krebs cycle 18 From NADH produced in Krebs cycle 38 Total −2 Used in priming reactions in glycolysis 36 Net www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 69 Figure 4.15 Pathway for oxidation of glucose and other carbohydrates. Glucose is degraded to pyruvic acid by cytoplasmic enzymes (glycolytic pathway). Acetyl-CoA is formed from pyruvic acid and is fed into the Krebs cycle. An acetyl-CoA molecule (two carbons) is oxidized to two molecules of carbon dioxide with each turn of the cycle. Pairs of electrons are removed from the carbon skeleton of the substrate at several points in the pathway and are carried by oxidizing agents NADH or FADH2 to the electron transport chain where 32 molecules of ATP are generated. Four molecules of ATP are also generated by substrate phosphorylation in the glycolytic pathway, and two molecules of ATP (initially GTP) are formed in the Krebs cycle. This yields a total of 38 molecules of ATP (36 molecules net) per glucose molecule. Molecular oxygen is involved only at the very end of the pathway as the fi nal electron acceptor at the end of the electron transport chain to yield water.
ADP ATP ADP ATP Krebs cycle 2 FADH2 Fructose, other hexoses Glucose Glucose-6-phosphate 2 ATP 2 ATP 2 ATP 2 ADP 2 ADP Fructose-1,6-bisphosphate Triose phosphate Triose phosphate Pyruvic acid Pyruvic acid Acetyl-CoA Acetyl-CoA Glycogen Plasma membrane Mitochondrial membranes 6 O2 32 ATP Electron transport chain 6 H2O 2 CO2 2 CO2 CO2 CO2 Krebs cycle (2 turns) NADH NADH NADH NADH Electron transport chain Krebs cycle Mobilization of acetyl-CoA Glycolysis CYTOSOL MATRIX 2 NADH 2 NADH 2 NADH 70 PART ONE Introduction to Living Animals bursts of activity when contraction is so rapid and powerful that oxygen delivery to tissues cannot supply energy demands by oxidative phosphorylation alone. At such times an animal has no choice but to supplement oxidative phosphorylation with anaerobic glycolysis. One kind of muscle fi ber (white muscle) has few mitochondria and primarily uses anaerobic glycolysis for ATP production (see Chapter 29, p. 661). In all muscle types, intense or strenuous activity is followed by a period of increased oxygen consumption as lactic acid, the end product of anaerobic glycolysis, diffuses from muscle to the liver where it is metabolized. Because oxygen consumption increases following heavy activity, the animal is said to have acquired an oxygen debt during such activity, which is repaid when activity ceases, and accumulated lactic acid is metabolized. Some animals rely heavily on anaerobic glycolysis during normal activities. For example, diving birds and mammals use glycolysis almost entirely to give them the energy needed to sustain long dives without breathing (that is, without requiring oxygen). Salmon would never reach their spawning grounds were it not for anaerobic glycolysis providing almost all of the ATP used in the powerful muscular bursts needed to carry them up rapids and falls. Many parasitic animals have dis-pensed with oxidative phosphorylation entirely at some stages of their life cycles. They secrete relatively reduced end prod-ucts of their energy metabolism, such as succinic acid, acetic acid, and propionic acid. These compounds are produced in mitochondrial reactions that derive several more molecules of ATP than does the path from glycolysis to lactic acid, although such sequences are still far less effi cient than the aerobic elec-tron transport chain. METABOLISM OF LIPIDS The fi rst step in the breakdown of a triglyceride is its hydrolysis to glycerol and three fatty acid molecules ( Figure 4.17 ). Glyc-erol is phosphorylated and enters the glycolytic pathway (see Figure 4.10 ). The remainder of the triglyceride molecule consists of fatty acids. For example, an abundant naturally occurring fatty acid is stearic acid. H3C CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 C OH O Stearic acid The long hydrocarbon chain of a fatty acid is broken down by oxidation, two carbons at a time; these are released from the end of the molecule as acetyl-CoA. Although two high-energy phosphate bonds are required to prime each 2-carbon fragment, energy is derived both from the reduction of NAD and FAD to NADH and FADH 2 , respectively, and from the acetyl group as it is degraded in the Krebs cycle. The complete oxidation of one molecule of 18-carbon stearic acid nets 146 ATP molecules. By comparison, three molecules of glucose (also totaling 18 carbons) yield 108 ATP molecules. Since there are three fatty acids in each triglyceride molecule, a total of 440 ATP molecules are formed. An additional 22 molecules of ATP are generated in the break-down of glycerol, giving a grand total of 462 molecules of ATP— little wonder that fat is considered the king of animal fuels! Fats are more concentrated fuels than carbohydrates, because fats are almost pure hydrocarbons; they contain more hydrogen per car-bon atom than sugars do, and it is the energized electrons of hydrogen that generate high-energy bonds when they are car-ried through the mitochondrial electron transport chain. Fat stores are derived principally from surplus fats and carbo-hydrates in the diet. Acetyl-CoA is the source of carbon atoms used to build fatty acids. Because all major classes of organic molecules (carbohydrates, fats, and proteins) can be degraded to acetyl-CoA, all can be converted into stored fat. The biosynthetic pathway for fatty acids resembles a reversal of the catabolic pathway already described, but it requires an entirely different set of enzymes. From acetyl-CoA, the fatty acid chain is assembled two carbons at a time. Because fatty acids release energy when they are oxidized, Figure 4.16 Anaerobic glycolysis, a process that proceeds in the absence of oxygen. Glucose is broken down to two molecules of pyruvic acid, with a net production of two molecules of ATP. Pyruvic acid, the fi nal electron acceptor for the hydrogen ions and electrons released during pyruvic acid formation, is converted to lactic acid. Hydrogen and electrons are recycled through the carrier, NAD+.
C O O– CH3 2 Lactate C H OH Glucose 2 Pyruvic acid G L Y C O L Y S I S 2 ATP 2 ADP C O O– C O CH3 2 NAD+ 2 NADH Figure 4.17 Hydrolysis of a triglyceride (neutral fat) by intracellular lipase. The R groups of each fatty acid represent a hydrocarbon chain.
3 H2O O C R1 OH OH OH O CH CH2 O Lipases C R3 O O C R2 O C R1 O C R3 O C R2 O CH2 OH CH CH2 OH OH CH2 + Fatty acids Glycerol Triglyceride www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 71 they obviously require an input of energy for their synthesis. This energy is provided principally by electron energy from glucose degradation. Thus the total ATP derived from oxidation of a mol-ecule of triglyceride is not as great as calculated, because varying amounts of energy are required for synthesis and storage. Stored fats are the greatest reserve fuel in the body. Most usable fat resides in white adipose tissue composed of specialized cells packed with globules of triglycerides. White adipose tissue is widely distributed in the abdominal cavity, in muscles, around deep blood vessels and large organs (for example, heart and kidneys), and especially under the skin. Women average about 30% more fat than men, which is largely responsible for differ-ences in shape between males and females. Humans can only too easily deposit large quantities of fat, generating hazards to health. Physiological and psychological aspects of obesity are now being investigated by many researchers. There is increasing evidence that food intake, and therefore the amount of fat deposition, is regu-lated by feeding centers located in the brain (lateral and ventral hypothalamus and brain stem). The set point of these regions deter-mines normal food intake and body weight for an individual, which may be maintained above or below what is considered normal for humans. Although evidence is accumulating for a genetic compo-nent to obesity, the epidemic proportions of obesity in the United States are more easily explained by lifestyle and feeding habits. Other developed countries show a similar, but less pronounced, trend toward development of an obesity problem. Research also reveals that lipid metabolism in obese individuals appears to be abnormal compared to lean individuals. This research has resulted in the development of drugs that act at various stages of lipid metabolism, such as decreasing lipid digestion and absorp-tion from the digestive tract, or increasing metabolism of lipids once they have been absorbed into the body. METABOLISM OF PROTEINS Since proteins are composed of amino acids, of which 20 kinds commonly occur (p. 26), the central topic of our consideration is amino acid metabolism. Amino acid metabolism is complex. Each of the 20 amino acids requires a separate pathway for bio-synthesis and degradation. Amino acids are precursors to tissue proteins, enzymes, nucleic acids, and other nitrogenous constitu-ents that form the fabric of cells. The central purpose of carbo-hydrate and fat oxidation is to provide energy, much of which is needed to construct and maintain these vital macromolecules. Let us begin with the amino acid pool in blood and extra-cellular fl uid from which the tissues draw their requirements. When animals eat proteins, most are digested in the digestive tract, releasing their constituent amino acids, which are then absorbed ( Figure 4.18 ). Tissue proteins also are hydrolyzed dur-ing normal growth, repair, and tissue restructuring; their amino acids join those derived from protein found in food to enter the amino acid pool. A portion of the amino acid pool is used to rebuild tissue proteins, but most animals ingest a surplus of pro-tein. Since amino acids are not excreted as such in any signifi cant amounts, they must be disposed of in some other way. In fact, amino acids can be and are metabolized through oxidative path-ways to yield high-energy phosphate. In short, excess proteins serve as fuel as do carbohydrates and fats. Their importance as fuel obviously depends on the nature of the diet. In carnivores that ingest a diet of almost pure protein and fat, nearly half of their high-energy phosphate comes from amino acid oxidation. Before an amino acid molecule may enter the fuel depot, nitrogen must be removed by deamination (the amino group splits to form ammonia and a keto acid) or by transamination (the amino group is transferred to a keto acid to yield a new amino acid). Thus amino acid degradation yields two main products, carbon skeletons and ammonia, which are handled in different ways. Once nitrogen atoms are removed, the carbon skeletons O C NH2 NH2 Dietary protein Amino acid pool Undigested Digested and absorbed Consumed by intestinal microflora Rebuild tissue proteins Deaminate and use for energy ATP Ammonia NH3 Gills Urine Urine, feces Insects, reptiles, birds Amphibians, mammals Aquatic animals Urea Uric acid HN N H O O H N N H O Figure 4.18 Fate of dietary protein.
72 PART ONE Introduction to Living Animals of amino acids can be completely oxidized, usually by way of pyruvic acid or acetic acid. These residues then enter routes used by carbohydrate and fat metabolism (see Figure 4.10 ). The other product of amino acid degradation is ammonia. Ammonia is highly toxic because it inhibits respiration by react-ing with α-ketoglutaric acid to form glutamic acid (an amino acid), and effectively removes α-ketoglutarate from the Krebs cycle (see Figure 4.13 ). Disposal of ammonia offers little prob-lem to aquatic animals because it is soluble and readily diffuses into the surrounding medium, often through respiratory surfaces. Terrestrial animals cannot get rid of ammonia so conveniently and must detoxify it by converting it to a relatively nontoxic compound. The two principal compounds formed are urea and uric acid, although a variety of other detoxifi ed forms of ammonia are excreted by different animals. Among vertebrates, amphibians and especially mammals produce urea. Reptiles and birds, as well as many terrestrial invertebrates, produce uric acid (the excretion of uric acid by insects and birds is described on pp. 451 and 597, respectively). The key feature that determines choice of nitrogenous waste is availability of water in the environment. When water is abun-dant, the chief nitrogenous waste is ammonia. When water is restricted, it is urea. Animals living in truly arid habitats use uric acid. Uric acid is highly insoluble and easily precipitates from solution, allowing its removal in solid form. Embryos of birds and reptiles benefi t greatly from excretion of nitrogenous waste as uric acid, because waste cannot be eliminated through their eggshells. During embryonic development, harmless, solid uric acid is retained in one of the extraembryonic membranes. When a hatchling emerges into its new world, accumulated uric acid, along with the shell and membranes that supported develop-ment, is discarded. MANAGEMENT OF METABOLISM The complex pattern of enzymatic reactions that constitutes metabolism cannot be explained entirely in terms of physico-chemical laws or chance happenings. Although some enzymes appear to function automatically, the activity of others is rigidly controlled. In the former case, suppose that the function of an enzyme is to convert A to B. If B is removed by conversion into another compound, the enzyme tends to restore the origi-nal ratio of B to A. Since many enzymes act reversibly, either synthesis or degradation may result. For example, an excess of an intermediate in the Krebs cycle would contribute to glycogen synthesis; a depletion of such a metabolite would lead to glyco-gen breakdown. This automatic compensation (equilibration) is not, however, suffi cient to explain regulation of metabolism. Mechanisms exist for critically regulating enzymes in both quantity and activity. In bacteria, genes leading to synthesis of an enzyme are switched on or off, depending on the presence or absence of a substrate molecule. In this way the quantity of an enzyme is controlled. It is a relatively imprecise process. Mechanisms that alter activity of enzymes can quickly and fi nely adjust metabolic pathways to changing conditions in a cell. The presence or increase in concentration of some molecules can alter the shape (conformation) of particular enzymes, thus activat-ing or inhibiting the enzyme ( Figure 4.19 ). For example, phos-phofructokinase, which catalyzes phosphorylation of glucose-6-phosphate to fructose-1,6-bisphosphate (see Figure 4.15 ), is inhibited by high concentrations of ATP or citric acid. Their presence means that a sufficient amount of precursors has reached the Krebs cycle and additional glucose is not needed. In some cases, the fi nal end product of a particular metabolic pathway inhibits the fi rst enzyme in the pathway. This method is termed feedback inhibition. As well as being subject to alteration in physical shape, many enzymes exist in both an active and an inactive form. These forms may be chemically different. For example, one com-mon way to activate or inactivate an enzyme is to add a phos-phate group to the molecule, thus changing its conformational shape and either exposing or blocking the enzyme’s active site. Enzymes that degrade glycogen (phosphorylase) and synthesize it (synthase) are both found in active and inactive forms. Condi-tions that activate phosphorylase tend to inactivate synthase and vice versa. Figure 4.19 Enzyme regulation. A, The active site of an enzyme may only loosely fi t its substrate in the absence of an activator. B, With the regulatory site of the enzyme occupied by an activator, the enzyme binds the substrate, and the site becomes catalytically active.
Substrate Activator A B Regulatory site www . m hhe . com /hi ckmani pz14e CHAPTER 4 Cellular Metabolism 73 Living systems are subject to the same laws of thermodynamics that govern nonliving systems. The fi rst law states that energy cannot be destroyed, although it may change form. The second law states that the structure of systems proceeds toward total randomness, or increasing entropy, as energy is dissipated. Solar energy trapped by photosynthesis as chemical bond energy is passed through the food chain where it is used for biosynthesis, active transport, and motion, before fi nally being dissipated as heat. Living organisms are able to decrease their entropy and to maintain high internal order because the biosphere is an open system from which energy can be cap-tured and used. Energy available for use in biochemical reactions is termed “free energy.” Enzymes are usually proteins, often associated with nonprotein cofactors, that vastly accelerate rates of chemical reactions in living systems. An enzyme acts by temporarily binding its reactant (sub-strate) onto an active site in a highly specifi c fi t. In this confi guration, internal activation energy barriers are lowered enough to modify the substrate, and the enzyme is restored to its original form. Cells use the energy stored in chemical bonds of organic fuels by degrading fuels through a series of enzymatically controlled steps. This bond energy is transferred to ATP and packaged in the form of “high-energy” phosphate bonds. ATP is produced as it is required in cells to power various synthetic, secretory, and mechani-cal processes. Glucose is an important source of energy for cells. In aero-bic metabolism (respiration), the 6-carbon glucose is split into two 3-carbon molecules of pyruvic acid. Pyruvic acid is decarboxylated to form 2-carbon acetyl-CoA, a strategic intermediate that enters the Krebs cycle. Acetyl-CoA can also be derived from breakdown of fat. In the Krebs cycle, acetyl-CoA is oxidized in a series of reactions to carbon dioxide, yielding, in the course of the reactions, energized electrons that are passed to electron acceptor molecules (NAD and FAD). In the fi nal stage, the energized electrons are passed along an electron transport chain consisting of a series of electron carriers located in the inner membranes of mitochondria. A hydrogen gradi-ent is produced as electrons are passed from carrier to carrier and fi nally to oxygen, and ATP is generated as the hydrogen ions fl ow down their electrochemical gradient through ATP synthase mole-cules located in the inner mitochondrial membrane. A net total of 36 molecules of ATP may be generated from one molecule of glucose. In the absence of oxygen (anaerobic glycolysis), glucose is degraded to two 3-carbon molecules of lactic acid, yielding two mol-ecules of ATP. Although anaerobic glycolysis is vastly less effi cient than aerobic metabolism, it provides essential energy for muscle contraction when heavy energy expenditure outstrips the oxygen-delivery system of an animal; it also is the only source of energy generation for microorganisms living in oxygen-free environments. Triglycerides (neutral fats) are especially rich depots of meta-bolic energy because the fatty acids of which they are composed are highly reduced and free of water. Fatty acids are degraded by sequential removal of 2-carbon units, which enter the Krebs cycle through acetyl-CoA. Amino acids in excess of requirements for synthesis of proteins and other biomolecules are used as fuel. They are degraded by deam-ination or transamination to yield ammonia and carbon skeletons. The latter enter the Krebs cycle to be oxidized. Ammonia is a highly toxic waste product that aquatic animals quickly expel, often through respiratory surfaces. Terrestrial animals, however, convert ammonia into much less toxic compounds, urea or uric acid, for disposal. Integration of metabolic pathways is fi nely regulated by mecha-nisms that control both amount and activity of enzymes. The quan-tity of some enzymes is regulated by certain molecules that switch on or off enzyme synthesis. Enzyme activity may be altered by the presence or absence of metabolites that cause conformational changes in enzymes and thus improve or diminish their effective-ness as catalysts. S U M M A R Y R E V I E W Q U E S T I O N S State the fi rst and second laws of thermodynamics. Living systems may appear to violate the second law of thermodynamics because living things maintain a high degree of organization despite a universal trend toward increasing disorganization. What is the explanation for this apparent paradox? Explain what is meant by “free energy” in a system. Will a reaction that proceeds spontaneously have a positive or negative change in free energy? Many biochemical reactions proceed slowly unless the energy barrier to the reaction is lowered. How is this accomplished in living systems? What happens in the formation of an enzyme-substrate complex that favors the disruption of substrate bonds? What is meant by a “high-energy bond,” and why might the production of molecules with such bonds be useful to living organisms? Although ATP supplies energy to an endergonic reaction, why is it not considered a fuel? 1.
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6. What is an oxidation-reduction reaction and why are such reactions considered so important in cellular metabolism? Give an example of a fi nal electron acceptor found in aerobic and anaerobic organisms. Why is aerobic metabolism more effi cient than anaerobic metabolism? Why must glucose be “primed” with a high-energy phosphate bond before it can be degraded in the glycolytic pathway? What happens to the electrons removed during the oxidation of triose phosphates during glycolysis? Why is acetyl-CoA considered a “strategic intermediate” in respiration? Why are oxygen atoms important in oxidative phosphorylation? What are the consequences if they are absent for a short period of time in tissues that routinely use oxidative phosphorylation to produce useful energy? Explain how animals can generate ATP without oxygen. Given that anaerobic glycolysis is much less effi cient than oxidative phosphorylation, why has anaerobic glycolysis not been discarded during animal evolution? 7.
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74 PART ONE Introduction to Living Animals Why are animal fats sometimes called “the king of fuels”? What is the signifi cance of acetyl-CoA to lipid metabolism? The breakdown of amino acids yields two products: ammonia and carbon skeletons. What happens to these products? 14.
15. Explain the relationship between the amount of water in an animal’s environment and the kind of nitrogenous waste it produces. Explain three ways that enzymes may be regulated in cells. 16.
17. S E L E C T E D R E F E R E N C E S Alberts, B., D. Bray, K. Hopkin, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. 2003. Essential cell biology, ed. 2. New York, Garland Science Publishing. Provides a more in-depth and well-written description of cellular metabolism. Berg, J., J. Tymoczko, and L. Stryer. 2002. Biochemistry, ed. 5. San Francisco, W. H. Freeman & Company. One of the best undergraduate biochemistry texts. Lodish, H., A. Berk, S. L. Zipursky, P. Matsudaira, D. Baltimore, and J. Darnell. 2000. Molecular cell biology, ed. 4. San Francisco, W. H. Freeman & Company. Chapter 16 is a comprehensive, well-illustrated treatment of energy metabolism. Wolfe, S. L. 1995. Introduction to cell and molecular biology. Belmont, CA. Thomson Brooks/Cole Publishers. Covers the same topics as Wolfe’s big book, but in less detail. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! P A R T T W O 2 Genetics: A Review Organic Evolution The Reproductive Process Principles of Development 5 6 7 8 Continuity and Evolution of Animal Life A female Cardinalis cardinalis (left) and a female Cardinalis sinuatus (right). of the coded instructions in genes. The genetic material (deoxyribo-nucleic acid, DNA) is composed of nitrogenous bases arranged on a chemical chain of sugar-phosphate units. The genetic code lies in the linear order or sequence of bases in the DNA strand. Because the DNA molecules replicate and pass from generation to generation, genetic variations can persist and spread in a popula-tion. Such molecular alterations, called mutations, are the ultimate source of biological variation and the raw material of evolution. C H A P T E R 5 Genetics: A Review A Code for All Life The principle of hereditary transmission is a central tenet of life on earth: all organisms inherit a structural and functional organization from their progenitors. What is inherited by an offspring is not an exact copy of the parent but a set of coded instructions that a devel-oping organism uses to construct a body resembling its parents. These instructions are in the form of genes, the fundamental units of inheritance. One of the great triumphs of modern biology was the discovery in 1953 by James Watson and Francis Crick of the nature The site of Gregor Mendel’s experimental garden, Brno, Czech Republic. www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 77 A basic principle of modern evolutionary theory is that organ-isms attain their diversity through hereditary modifi cations of populations. All known lineages of plants and animals are related by descent from common ancestral populations. Heredity establishes the continuity of living forms. Although off-spring and parents in a particular generation may look different, there is nonetheless a genetic continuity that runs from generation to genera-tion for any species of plant or animal. An offspring inherits from its parents a set of coded information (genes), which a fertilized egg uses, together with environmental factors, to guide its development into an adult bearing unique physical characteristics. Each generation passes to the next the instructions required for maintaining continuity of life. The gene is the unit entity of inheritance, the germinal basis for every characteristic that appears in an organism. The study of what genes are, how they are transmitted, and how they work is the sci-ence of genetics. It is a science that reveals the underlying causes of resemblance, as seen in the remarkable fi delity of reproduction, and of variation, the working material for organic evolution. All liv-ing forms use the same information storage, transfer, and translation system, which explains the stability of all life and reveals its descent from a common ancestral form. This is one of the most important unifying concepts of biology. MENDEL’S INVESTIGATIONS The fi rst person to formulate the principles of heredity was Gregor Johann Mendel (1822 to 1884) ( Figure 5.1 and p. 18), an Augustin-ian monk living in Brünn (Brno), Moravia. Brünn was then part of Austria but now lies in the eastern part of the Czech Repub-lic. While conducting breeding experiments in a small monastery garden from 1856 to 1864, Mendel examined with great care the progeny of many thousands of plants. He presented in elegant sim-plicity the laws governing transmission of characters from parents to offspring. His discoveries, published in 1866, were of great sig-nifi cance, coming just after Darwin’s publication of On the Origin of Species by Means of Natural Selection. Yet Mendel’s discoveries remained unappreciated and forgotten until 1900—35 years after the completion of the work and 16 years after Mendel’s death. Mendel chose garden peas for his classic experiments because they had pure strains differing from each other by discrete characters. For example, some varieties were defi nitely dwarf and others tall; some strains produced smooth seeds and others wrinkled seeds ( Figure 5.1 ). Mendel studied single characters that displayed sharply contrasting traits. He carefully avoided mere quantitative, continuously varying characteristics. A second reason for selecting peas was that they were self-fertilizing but subject to experimental cross-fertilization. A giant advance in chromosomal genetics was made when the American geneticist Thomas Hunt Morgan and his colleagues selected a species of fruit fl y, Drosophila melanogaster, for their studies (1910–1920). Flies were cheaply and easily reared in bottles in the laboratory, fed on a simple medium of bananas and yeast. Most importantly, they produced a new generation every 10 days, enabling Morgan to collect data at least 25 times more rapidly than with organisms that take longer to mature, such as garden peas. Morgan’s work led to the mapping of genes on chromosomes and founded the discipline of cytogenetics. Mendel crossed varieties having contrasting traits, making crosses for each of the seven characters shown in Figure 5.1 . He removed the stamens (male part, containing the pollen) from a fl ower to prevent self-fertilization and then placed on the stigma (female part of fl ower) pollen from the fl ower of a plant true-breeding for the contrasting trait. Pollination from other sources such as wind and insects was rare and did not affect his results. Offspring from these crosses are called hybrids, meaning that they contain genetic information from two different parental strains. He collected seeds from the cross-fertilized flowers, planted these hybrid seeds, and examined the resulting plants for the contrasting traits being studied. These hybrid plants then produced offspring by self-pollination. Mendel knew nothing of the cytological basis of hered-ity, since chromosomes and genes were not yet discovered. Although we can admire Mendel’s power of intellect in his dis-covery of the principles of inheritance without knowledge of chromosomes, these principles are easier to understand if we fi rst review chromosomal behavior, especially in meiosis. CHROMOSOMAL BASIS OF INHERITANCE In sexually reproducing organisms, special sex cells, or gam-etes (ova and sperm), transmit genetic information from par-ents to offspring. A scientifi c explanation of genetic principles required a study of germ cells and their behavior, and correla-tions between their transmission and certain visible results of inheritance. Nuclei of sex cells, especially the chromosomes, were early suspected of furnishing the real answer to the hereditary mechanism. Chromosomes are apparently the only entities transmitted in equal quantities from both parents to offspring. When Mendel’s laws were rediscovered in 1900, their paral-lelism with the cytological behavior of chromosomes was obvi-ous. Later experiments showed that chromosomes carried hered-itary material. Meiosis: Reduction Division of Gametes Although animal species differ greatly in the characteristic num-bers, sizes, and shapes of chromosomes present in their body cells, a common feature is that chromosomes occur in pairs. The two members of a chromosomal pair contain similar genes encoding the same set of characteristics and usually, but not always, have the same size and shape. The members of such a pair are called homologous chromosomes; each individual member of a pair is called a homolog. One homolog comes from the mother and the other from the father. Meiosis is a special pair of cell divisions in which the genetic material replicates once followed by two rounds of cell division ( Figure 5.2 ). The result is a set of four daughter cells, each of which has only one member of each homologous chromosome pair. The chromosomes present in a meiotic daughter cell or gamete are collectively called a single set of chromosomes. The number of chromosomes in a single set, which varies among species, is called the haploid ( n ) number of chromosomes. When 78 PART TWO Continuity and Evolution of Animal Life a pair of gametes unites in fertilization, each gamete contributes its set of chromosomes to the newly formed cell, called a zygote, which has two complete sets of chromosomes. The number of chromosomes in two complete sets is called the diploid (2 n ) number. In humans the zygotes and all body cells normally have the diploid number (2 n ), or 46 chromosomes; the gametes have the haploid number ( n ), or 23, and meiosis reduces the number of chromosomes per cell from diploid to haploid. Thus each cell normally has two copies of each gene coding for a given trait, one on each of the homologous chromosomes. Alternative forms of genes for the same trait are allelic forms, or alleles. Sometimes only one of the alleles has a visible effect on the organism, although both are present in each cell, and either may be passed to progeny as a result of meiosis and subsequent fertilization. Alleles are alternative forms of the same gene that have arisen by mutation of the DNA sequence. Like a baseball team with several pitchers, only one of whom can occupy the pitcher’s mound at a time, only one allele can occupy a chromosomal locus (position). Alternative alleles for the locus may be on homologous chromo-somes of a single individual, making that individual heterozygous for the gene in question. Numerous allelic forms of a gene may be found among different individuals in a population, a condition called “multiple alleles” (p. 85). Round vs. wrinkled seeds F1 = all round F2 = 5474 round 1850 wrinkled Ratio: 2.96:1 Purple vs. white flowers F1 = all purple F2 = 705 purple 224 white Ratio: 3.15:1 Green vs. yellow pods F1 = all green F2 = 428 green 152 yellow Ratio: 2.82:1 Long vs. short stems F1 = all long F2 = 787 long 277 short Ratio: 2.84:1 Terminal vs. axial flowers F1 = all axial F2 = 651 axial 207 terminal Ratio: 3.14:1 Inflated vs. constricted pods F1 = all inflated F2 = 882 inflated 299 constricted Ratio: 2.95:1 Yellow vs. green seeds F1 = all yellow F2 = 6022 yellow 2001 green Ratio: 3.01:1 Figure 5.1 Seven experiments on which Gregor Mendel based his postulates. These are the results of monohybrid crosses for fi rst and second generations.
www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 79 Prophase I Late prophase I Metaphase I Anaphase I Prophase II Metaphase II Anaphase II MEIOSIS I MEIOSIS II Telophase II Homolog Homolog Sister chromatids Centromere Region of close association, where crossing over occurs SYNAPSIS B A Figure 5.2 A, Meiosis in a sex cell with two pairs of chromosomes. Prophase I, homologous chromosomes come to lie with side-to-side contact, or synapsis, forming bivalents. A bivalent comprises a pair of homologous chromosomes, with each of the chromosomes containing a pair of identical chromatids joined by a centromere. Metaphase I, bivalents align at the spindle equator. Anaphase I, chromosomes of former bivalents are pulled toward opposite poles. Prophase II, daughter cells contain one of each homologous chromosome (haploid) but each chromosome is in replicated form (two chromatids attached at a centromere). Metaphase II, chromosomes align at the spindle equator. Anaphase II, chromatids of each chromosome separate. Telophase II, four haploid cells (gametes) formed, each with unreplicated chromosomes (one chromatid per chromosome). B, Synapsis occurs in prophase I, in which homologous chromosomes can break and exchange corresponding portions. The labelled sister chromatids and region of close association extend the full length of the bivalent.
80 PART TWO Continuity and Evolution of Animal Life During an individual’s growth, all dividing cells contain the double set of chromosomes (mitosis is described on p. 52). In the reproductive organs, gametes (germ cells) are formed after meiosis, which separates the chromosomes of each homologous pair. Without this reductional division, the union of ovum (egg) and sperm would produce an individual with twice as many chromosomes as the parents. Continuation of this process in just a few generations could yield astronomical numbers of chromo-somes per cell. Most unique features of meiosis occur during prophase of the fi rst meiotic division ( Figure 5.2 ). Prior to meiosis, each chro-mosome has already replicated to form two chromatids joined at one point, the centromere. The two members of each pair of homologous chromosomes make side-by-side contact (synap-sis) to form a bivalent, which permits genetic recombination between the paired homologous chromosomes (p. 89). Each bivalent is composed of two pairs of chromatids (each pair is a dyad, sister chromatids held together at their centromere), or four future chromosomes, and is thus called a tetrad. The position or location of any gene on a chromosome is the gene locus (pl., loci ), and in synapsis all gene loci on a chromatid normally lie exactly opposite the corresponding loci on the sis-ter chromatid and both chromatids of the homologous chromo-some. Toward the end of prophase, the chromosomes shorten and thicken and then enter the fi rst meiotic division. In contrast to mitosis, the centromeres holding the chro-matids together do not divide at anaphase. As a result, each of the dyads is pulled toward one of the opposite poles of the cell by microtubules of the division spindle. At telophase of the fi rst meiotic division, each pole of the cell has one dyad from each tetrad formed at prophase. Therefore at the end of the fi rst meiotic division, the daughter cells contain one chromosome of each homologous pair from the parent cell, so that the total chromosome number is reduced to haploid. However, because each chromosome contains two chromatids joined at a centro-mere, each cell contains twice the amount of DNA present in a gamete. The second meiotic division more closely resembles events in mitosis. The dyads are split at the beginning of anaphase by division of their centromeres, and single-stranded chromosomes move toward each pole. Thus by the end of the second meiotic division, the cells have the haploid number of chromosomes, and each chromatid of the original tetrad exists in a separate nucleus. Four products are formed, each containing one com-plete haploid set of chromosomes and only one copy of each gene. Only one of the four products in female gametogenesis becomes a functional gamete (p. 146). Sex Determination Before the importance of chromosomes in heredity was real-ized in the early 1900s, genetic control of gender was totally unknown. The fi rst scientifi c clue to chromosomal determina-tion of sex came in 1902 when C. McClung observed that bugs (Hemiptera) produced two kinds of sperm in approximately equal numbers. One kind contained among its regular set of chromosomes a so-called accessory chromosome lacking in the other kind of sperm. Since all eggs of these species had the same number of haploid chromosomes, half the sperm would have the same number of chromosomes as the eggs, and half of them would have one chromosome less. When an egg was fertilized by a spermatozoon carrying the accessory (sex) chromosome, the resulting offspring was a female; when fertilized by a sper-matozoon without an accessory chromosome, the offspring was a male. Therefore a distinction was made between sex chromo-somes, which determine sex (and sex-linked traits); and auto-somes, the remaining chromosomes, which do not infl uence sex. The particular type of sex determination just described is often called the XX-XO type, which indicates that females have two X chromosomes and males only one X chromosome (the O indicates absence of the chromosome). The XX-XO method of sex determination is depicted in Figure 5.3 . Later, other types of sex determination were discovered. In humans and many other animals each sex contains the same number of chromosomes; however, the sex chromosomes (XX) are alike in females but unlike (XY) in males. Hence a human egg contains 22 autosomes 1 X chromosome. Sperm are of two kinds; half carry 22 autosomes 1 X and half bear 22 auto-somes 1 Y. The Y chromosome is much smaller than the X and carries very little genetic information. At fertilization, when the 2 X chromosomes come together, offspring are female; when X and Y come together, offspring are male. The XX-XY kind of sex determination is shown in Figure 5.4 . A third type of sex determination is found in birds, moths, butterfl ies, and some fi sh, in which the male has 2 X (or some-times called ZZ) chromosomes and the female an X and Y (or ZW). Finally, there are both invertebrates (p. 380) and verte-brates (p. 571) in which sex is determined by environmental or behavioral conditions rather than by sex chromosomes, or by genetic loci whose variation is not associated with visible differ-ence in chromosomal structure. Figure 5.3 XX-XO sex determination. Only the sex chromosomes are shown.
O O Eggs Female Female Male Male Zygotes (chromosome absent) Sperm X X X X X X X X www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 81 In the case of X and Y chromosomes, homologous chromo-somes are unlike in size and shape. Therefore, they do not both carry the same genes. Genes of the X chromosome often do not have allelic counterparts on the diminutive Y chromosome. This fact is very important in sex-linked inheritance (p. 87). MENDELIAN LAWS OF INHERITANCE Mendel’s First Law Mendel’s law of segregation states that in the formation of gametes, paired factors that may specify alternative phenotypes (visible traits) separate so that each gamete receives only one member of the pair. In one of Mendel’s original experiments, he pollinated pure-line tall plants with the pollen of pure-line dwarf plants. Thus the visible characteristics, or phenotypes, of the parents were tall and dwarf. Mendel found that all progeny in the fi rst generation (F 1 ) were tall, just as tall as the tall parents of the cross. The reciprocal cross—dwarf plants pollinated with tall plants—gave the same result. The tall phenotype appeared in all progeny no matter which way the cross was made. Obviously, this kind of inheritance was not a blending of two traits, because none of the progeny was intermediate in size. Next Mendel self-fertilized (“selfed”) the tall F 1 plants and raised several hundred progeny, the second (F 2 ) generation. This time, both tall and dwarf plants appeared. Again, there was no blending (no plants of intermediate size), but the appearance of dwarf plants from all tall parental plants was surprising. The dwarf trait, seen in half of the grandparents but not in the parents, had reappeared. When he counted the actual number of tall and dwarf plants in the F 2 generation, he discovered that there were almost exactly three times more tall plants than dwarf ones. Mendel then repeated this experiment for the six other contrasting traits that he had chosen, and in every case he obtained ratios very close to 3:1 (see Figure 5.1 ). At this point it must have been clear to Mendel that he was dealing with hereditary determinants for the contrasting traits that did not blend when brought together. Even though the dwarf trait dis-appeared in the F 1 generation, it reappeared fully expressed in the F 2 generation. He realized that the F 1 generation plants carried determinants (which he called “factors”) of both tall and dwarf parents, even though only the tall trait was visible in the F 1 generation. Mendel called the tall factor dominant and the short reces-sive. Similarly, the other pairs of traits that he studied showed dominance and recessiveness. Whenever a dominant factor is present, the recessive one is not visible. The recessive trait appears only when both factors are recessive, or in other words, in a pure condition. In representing his crosses, Mendel used letters as symbols; a capital letter denotes a dominant trait, and the corresponding lowercase letter denotes its recessive alternative. Modern geneti-cists still often follow this custom. Thus the factors for pure tall plants might be represented by T/T, the pure recessive by t/t, and the mix, or hybrid, of the two plants by T/t. The slash mark indicates that the alleles are on homologous chromosomes. The zygote bears the complete genetic constitution of the organism. All gametes produced by T/T must necessarily be T, whereas those produced by t/t must be t. Therefore a zygote produced by union of the two must be T/t, or a heterozygote. On the other hand, the pure tall plants ( T/T ) and pure dwarf plants ( t/t ) are homozygotes, meaning that the paired factors are alike on the homologous chromosomes and represent copies of the same allele. A cross involving variation at only a single locus is called a monohybrid cross. In the cross between tall and dwarf plants there were two phenotypes: tall and dwarf. On the basis of genetic formulas there are three hereditary types: T/T, T/t, and t/t. These are called genotypes. A genotype is an allelic combination present in a diploid organism ( T/T, T/t, or t/t ), and the phenotype is the cor-responding appearance of the organism (tall or dwarf). One of Mendel’s original crosses (tall plant and dwarf plant) could be represented as follows: Parents (P) Gametes F1 T/T (tall) all T t /t (dwarf) all t T/t (tall) x T/t Hybrids Gametes F2 genotypes F2 phenotypes T/t T/t T/t x T t T t T/T t /t Tall Tall Tall Dwarf All possible combinations of F 1 gametes in the F 2 zygotes yield a 3:1 phenotypic ratio and a 1:2:1 genotypic ratio. It is convenient in such crosses to use the checkerboard method devised by Punnett (Punnett square) for representing the various combina-tions resulting from a cross. In the F 2 cross this scheme would apply: Eggs Female Male Zygotes X X X Y X X Y X X X Y Figure 5.4 XX-XY sex determination. Only the sex chromosomes are shown.
82 PART TWO Continuity and Evolution of Animal Life 1/2 T 1/2 t 1/4 T/t (hybrid tall) 1/4 T/T (homozygous tall) 1/2 T Ratio: 3 tall to 1 dwarf 1/4 t /t (homozygous dwarf) 1/4 T/t (hybrid tall) 1/2 t Ova Pollen The next step was an important one because it enabled Mendel to test his hypothesis that every plant contained non-blending factors from both parents. He self-fertilized the plants in the F 2 generation; the pollen of a fl ower fertilized the stigma of the same fl ower. The results showed that self-pollinated F 2 dwarf plants produced only dwarf plants, whereas one-third of the F 2 tall plants produced tall and the other two-thirds produced both tall and dwarf in the ratio of 3:1, just as the F 1 plants had done. Genotypes and phenotypes were as follows: 1/4 T/T 1/2 T/t 1/4 t /t all T/T (homozygous tall) 1 T/T: 2 T/t: 1 t/t (3 tall: 1 dwarf) all t/t (homozygous dwarf) Tall F2 plants: Dwarf Selfed Selfed Selfed This experiment showed that the dwarf plants were pure because they at all times gave rise to short plants when self-pollinated; the tall plants contained both pure tall and hybrid tall. It also demonstrated that, although the dwarf trait disappeared in the F 1 plants, which were all tall, dwarfness appeared in the F 2 plants. Mendel reasoned that the factors for tallness and dwarfness were units that did not blend when they were together in a hybrid individual. The F 1 generation contained both of these units or fac-tors, but when these plants formed their germ cells, the factors sep-arated so that each germ cell had only one factor. In a pure-breed-ing plant both factors were alike; in a hybrid they were different. He concluded that individual germ cells were always pure with respect to a pair of contrasting factors, even when the germ cells were formed from hybrid individuals possessing both contrasting factors. This idea formed the basis for Mendel’s law of segregation, which states that whenever two factors are brought together in a hybrid, they segregate into separate gametes produced by the hybrid. The paired factors of the parent pass with equal frequency to the gametes. We now understand that the factors segregate because they occur on different chromosomes of a homologous pair, but the gametes receive only one chromosome of each pair in meiosis. Thus in current usage the law of segregation refers to the parting of homologous chromosomes during meiosis. Mendel’s great contribution was his quantitative approach to inheritance. His approach marks the birth of genetics, because before Mendel, people assumed that traits were blended like mixing together two colors of paint, a notion that unfortunately still lingers in the minds of many and was a problem for Darwin’s theory of natural selection when he fi rst proposed it (p. 17). If traits blended, variability would be lost in hybridization. With particulate inheritance, different alleles remain intact through the hereditary process and can be resorted like particles. In not reporting confl icting fi ndings, which must surely have arisen as they do in any original research, Mendel has been accused of “cooking” his results. The chances are, however, that he carefully avoided ambiguous material to strengthen his central message. Mendel’s results have withstood repeated testing by other research-ers, which confi rms their scientifi c integrity. Testcross When an allele is dominant, heterozygous individuals containing that allele are identical in phenotype to individuals homozygous for it. Therefore one cannot determine the genotypes of these individuals just by observing their phenotypes. For instance, in Mendel’s experiment of tall and dwarf traits, it is impossible to determine the genetic constitution of the tall plants of the F 2 generation by mere inspection of the tall plants. Three-fourths of this generation are tall, but which ones are heterozygotes? As Mendel reasoned, the test is to cross the questionable individuals with pure recessives. If the tall plant is homozygous, all offspring in such a testcross are tall, thus: t t T/t (hybrid tall) T/t (hybrid tall) T Parents T/T (tall) x t/t (dwarf) T/t (hybrid tall) T/t (hybrid tall) T Ova Pollen All of the offspring are T/t (hybrid tall). If the tall plant is hetero-zygous, half of the offspring are tall and half dwarf, thus: t t T/t (hybrid tall) T/t (hybrid tall) T Parents T/t (hybrid tall) x t /t (dwarf) t /t (homozygous dwarf) t /t (homozygous dwarf) t Ova Pollen The testcross is often used in modern genetics to assess the genetic constitution of offspring and to make desirable homozy-gous stocks of animals and plants. Intermediate Inheritance In some cases neither allele is completely dominant over the other, and the heterozygous phenotype is distinct from those of the parents, often intermediate between them. This is called www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 83 intermediate inheritance, or incomplete dominance. In the four-o’clock fl ower ( Mirabilis ), two allelic variants determine red versus pink or white fl owers; homozygotes are red or white fl ow-ered, but heterozygotes have pink fl owers. In a certain strain of chickens, a cross between those with black and splashed white feathers produces offspring that are not gray but a distinctive color called Andalusian blue ( Figure 5.5 ). In each case, if the F 1 s are crossed, the F 2 s have a ratio of 1:2:1 in colors, or 1 red: 2 pink: 1 white in four-o’clock fl owers and 1 black: 2 blue: 1 white for Andalusian chickens. This phenomenon can be illustrated for the chickens as follows: Parents Gametes F1 (black feathers) (white feathers) B/B' (all blue) B/B' B,B' B/B all B B'/B' all B' B/B' B,B' Crossing hybrids Gametes F2 genotypes F2 phenotypes B/B' B/B' B/B B'/B' Black Blue Blue White When neither of the alleles is recessive, it is customary to represent both by capital letters and to distinguish them by the addition of a “prime” sign ( B ') or by superscript letters, for example, B b (equals black feathers) and B w (equals white feathers). In this kind of cross, the heterozygous phenotype is indeed a blending of both parental types. It is easy to see how such observations would encourage the notion of blending inheri-tance. However, in the cross of black and white chickens or red and white fl owers, only the hybrid phenotype is a blend; its hereditary factors do not blend and homozygous offspring breed true to the parental phenotypes. Mendel’s Second Law Mendel’s second law pertains to studies of two pairs of heredi-tary factors at the same time. For example, does the inheritance of factors for yellow versus green seeds infl uence the inheri-tance of factors for tall versus short plants when the strains being crossed differ for both seed color and plant height? Mendel per-formed crossing experiments between pea strains that differ by two or more phenotypic characters controlled by variation at different genes located on different chromosomes. According to Mendel’s law of independent assortment, genes located on different pairs of homologous chromosomes assort independently during meiosis. Mendel had already established that tall plants were domi-nant to dwarf. He also noted that crosses between plants bearing yellow seeds and plants bearing green seeds produced plants with yellow seeds in the F 1 generation; therefore yellow was dominant to green. The next step was to make a cross between plants differing in these two characteristics. When a tall plant with yellow seeds ( T/T Y/Y ) was crossed with a dwarf plant with green seeds ( t/t y/y ), the F 1 plants were tall and yellow as expected ( T/t Y/y ). The F 1 hybrids were then self-fertilized giving the F 2 results shown in Figure 5.6 . Parents Gametes F1 T/T Y/Y (tall, yellow) all TY T/t Y/y (tall, yellow) t/t y/y (dwarf, green) all ty Mendel already knew that a cross between two plants bear-ing a single pair of alleles of the genotype T/t would yield a 3:1 ratio. Similarly, a cross between two plants with the genotypes Y/y would yield the same 3:1 ratio. If we examine only the tall and dwarf phenotypes expected in the outcome of the dihybrid experiment, they produce a ratio of 12 tall to 4 dwarf, which reduces to a ratio of 3:1. Likewise, a total of 12 plants have yel-low seeds for every 4 plants that have green—again a 3:1 ratio. Thus the monohybrid ratio prevails for both traits when they are considered independently. The 9:3:3:1 ratio is nothing more than a combination of the two 3:1 ratios. 3 1 3 1 9 3 3 1 : : : : : Figure 5.5 Cross between chickens with black and splashed white feathers. Black and white are homozygous; Andalusian blue is heterozygous.
B/B (black) B/B (white splashed) B/B(all blue) B/B B/B 1 Black (B/B) 2 Blue (B/B) 1 White splashed (B/B) 84 PART TWO Continuity and Evolution of Animal Life Parents generation generation T/T Y/Y t/t y/y T/T Y/Y TY T/T Y/y Ty T/t Y/Y tY T/t Y/y T/T Y/y T/T y/y T/t Y/y T/t y/y T/t Y/Y T/t Y/y t/t Y/Y t/t Y/y T/t Y/y T/t y/y t/t Y/y t/t y/y ty TY Ty tY ty F1 F2 Tall, yellow Tall, yellow All tall yellow Ratio: 9 tall yellow : 3 tall green : 3 dwarf yellow : 1 dwarf green T/t Y/y T/t Y/y Tall, yellow Dwarf, green Figure 5.6 Punnett square method for determining ratios of genotypes and phenotypes expected in a dihybrid cross for independently assorting genes.
www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 85 When one of the alleles is unknown, it can be designated by a dash ( T/— ). This designation is used also when it is immaterial whether the genotype is heterozygous or homozygous, as when we count all of a genetically dominant phenotype. The dash could be either T or t. The F 2 genotypes and phenotypes are as follows: 1 T/T Y/Y 2 T/t Y/Y 2 T/T Y/y 4 T/t Y/y 1 T/T y/y 2 T/t y/y 1 t/t Y/Y 2 t/t Y/y 1 t/t y/y 9 T/—Y/— 9 Tall yellow 3 T/—y/y 3 Tall green 3 t/t—Y/— 3 Dwarf yellow 1 t/t y/y 1 Dwarf green The results of this experiment show that segregation of alleles for plant height is entirely independent of segregation of alleles for seed color. Thus another way to state Mendel’s law of independent assortment is that paired copies of two dif-ferent genes located on different ( nonhomologous) chromo-somes segregate independently of one another. The reason is that during meiosis the member of any pair of homologous chromo-somes transmitted to a gamete is independent of which mem-ber of any other pair of chromosomes it receives. Of course, if the genes were close together on the same chromosome, they would assort together (be linked) unless crossing over occurred. Genes located very far apart on the same chromosome show independent assortment because crossing over occurs between them in nearly every meiosis. Linked genes and crossing over are discussed on p. 88. One way to estimate proportions of progeny expected to have a given genotype or phenotype is to construct a Punnett square. With a monohybrid cross, this is easy; with a dihybrid cross, a Punnett square is laborious; and with a trihybrid cross, it is very tedious. We can make such estimates more easily by taking advantage of simple probability calculations. The basic assump-tion is that the genotypes of gametes of one sex have a chance of uniting with the genotypes of gametes of the other sex in propor-tion to the numbers of each present. This is generally true when the sample size is large enough, and the actual numbers observed come close to those predicted by the laws of probability. We defi ne probability, which is the expected frequency of an event, as follows: Probability p Number of times an event ha ( ) ppens Total number of trials or possibilities for the event to happen For example, the probability (p) of a coin falling heads when tossed is 1/2, because the coin has two sides. The probability of rolling a three on a die is 1/6, because the die has six sides. The probability of independent events occurring together (ordered events) involves the product rule, which is simply the product of their individual probabilities. When two coins are tossed together, the probability of getting two heads is 1/2 1/2 1/4, or 1 chance in 4. The probability of rolling two threes simultaneously with two dice is as follows: Probability of two threes 1 6 1 6 1 36 / / / We can use the product rule to predict the ratios of inheri-tance in monohybrid or dihybrid (or larger) crosses if the genes sort independently in the gametes (as they did in all of Mendel’s experiments) ( Table 5.1 ). Note, however, that a small sample size may give a result quite different from that predicted. Thus if we tossed the coin three times and it fell heads each time, we would not be sur-prised. If we tossed the coin 1000 times and the number of heads diverged greatly from 500, we would strongly suspect something wrong with the coin. However, probability has no “memory.” The probability of a coin toss yielding heads remains 1/2, no matter how many times the coin was tossed previously or results of the tosses. Multiple Alleles On page 78 we defi ned alleles as alternate forms of a gene. Whereas an individual can have no more than two alleles at a given locus (one each on each chromosome of the homologous pair, p. 78), many more dissimilar alleles can exist in a popula-tion. An example is the set of multiple alleles that affects coat color in rabbits. The different alleles are C (normal color), c ch (chinchilla color), c h (Himalayan color), and c (albino). The four alleles form a dominance series with C dominant over every-thing. The dominant allele is always written to the left and the recessive to the right: C c c c c h ch h h / / / Normal color Chinchilla color c c c c Himalayan color albino / Multiple alleles arise through mutations at the same gene locus at different times. Any gene can mutate (p. 100) if given time and thus can show many different alleles at the same locus. Gene Interaction The types of crosses previously described are simple in that the character variation results from the action of a single gene with one phenotypic effect. However, many genes have more than a single effect on organismal phenotypes, a phenomenon called pleiotropy. A gene whose variation infl uences eye color, for instance, could at the same time infl uence the development of other characters. An allele at one locus can mask or prevent the expression of an allele at another locus acting on the same trait, a phenomenon called epistasis. Another case of gene interaction 86 PART TWO Continuity and Evolution of Animal Life is one in which several sets of alleles produce a cumulative effect on the same character. Many cases are known in which the variation of a charac-ter results from two or more genes. Mendel probably did not appreciate the real signifi cance of the genotype, as contrasted with the visible character—the phenotype. We now know that many different genes can affect a single phenotype (polygenic inheritance). Several characters in humans are polygenic. In such cases the characters, instead of having discrete alternative phenotypes, show continuous variation between two extremes. Each of sev-eral genes has an allele that adds () and one that fails to add () an incremental dose to the value of the phenotype. This dosage-dependent inheritance is sometimes called quantitative inheri-tance. In this kind of inheritance the children are often phe-notypically intermediate between the two parents. Variation at multiple genes infl uences phenotypic variation, but the different allelic forms of each gene remain unaltered as discrete hereditary factors as they are sorted into various genotypes. As the number of variable genes affecting a quantitative phenotype increases, intermediate conditions between the opposite extreme values of the phenotype become more continuous. One illustration of such a type is the degree of pigmen-tation in matings between people having dark and light skin tones. The cumulative genes in such matings have a quantitative expression. Three or four genes are probably involved in skin pigmentation, but we simplify our explanation by referring to only two pairs of independently assorting genes. Thus a per-son with very dark pigment has two genes for pigmentation on separate chromosomes ( A/A B/B ). Each dominant allele contrib-utes one unit of pigment. A person with very light pigment has alleles ( a/a b/b ) that contribute no color. (Freckles that com-monly appear in the skin of very light people represent pigment contributed by entirely separate genes.) The offspring of very dark and very light parents would have an intermediate skin color ( A/a B/b ). Children of parents having intermediate skin color show a range of skin color, depending on the number of genes for pig-mentation that they inherit. Their skin color ranges from very dark ( A/A B/B ), to dark ( A/A B/b or A/a B/B ), intermediate ( A/A b/b or A/a B/b or a/a B/B ), light ( A/a b/b or a/a B/b ), to very light ( a/a b/b ). It is thus possible for parents heterozygous for skin color to produce children with darker or lighter colors than themselves. T A B L E 5 . 1 Use of Product Rule for Determining Genotypic and Phenotypic Ratios in a Dihybrid Cross for Independently Assorting Genes Parents’ genotypes T/t Y/y T/t Y/y Equivalent monohybrid crosses T/t T/t and Y/y Y/y Genotype ratios in F1s of monohybrid crosses 1/4 T/T 1/4 Y/Y 2/4 T/t 2/4 Y/y 1/4 t/t 1/4 y/y Combine two monohybrid ratios to determine dihybrid genotype ratios 1/4 T/T 1/4 Y/Y 1/16 T/T Y/Y 2/4 Y/y 2/16 T/T Y/y 1/4 y/y 1/16 T/T y/y 2/4 T/t 1/4 Y/Y 2/16 T/t Y/Y 2/4 Y/y 4/16 T/t Y/y 1/4 y/y 2/16 T/t y/y 1/4 t/t 1/4 Y/Y 1/16 t/t Y/Y 2/4 Y/y 2/16 t/t Y/y 1/4 y/y 1/16 t/t y/y Phenotype ratios in F1s of monohybrid crosses 3/4 T/—(tall), 1/4 t/t (dwarf) 3/4 Y/—(yellow), 1/4 y/y (green) Combine two monohybrid ratios to determine phenotype ratios 3/4 T/— 3/4 Y/— 9/16 T/—Y/—(tall, yellow) 1/4 y/y 3/16 T/—y/y (tall, green) 1/4 t/t 3/4 Y/— 3/16 t/t Y/—(dwarf, yellow) 1/4 y/y 1/16 t/t y/y (dwarf, green) Therefore phenotype ratios = 9 tall, yellow: 3 tall, green: 3 dwarf, yellow: 1 dwarf, green www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 87 Inheritance of eye color in humans is another example of gene interaction. One allele ( B ) determines whether pigment is present in the front layer of the iris. This allele is dominant over the allele for the absence of pigment ( b ). The genotypes B/B and B/b pig-ment generally produce brown eyes, and b/b produces blue eyes. However, these phenotypes are greatly affected by many modifi er genes infl uencing, for example, the amount of pigment present, the tone of the pigment, and its distribution. Thus a person with B/b may even have blue eyes if modifi er genes determine a lack of pig-ment, thus explaining the rare instances of a brown-eyed child of blue-eyed parents. Sex-Linked Inheritance It is known that inheritance of some characters depends on the sex of the parent carrying the gene and the sex of the offspring. One of the best-known sex-linked traits of humans is hemophilia (see Chapter 31, p. 690). Another example is red-green color blindness in which red and green colors are indistinguishable to varying degrees. Color-blind men greatly outnumber color-blind women. When color blindness does appear in women, their fathers are color blind. Furthermore, if a woman with normal vision who is a carrier of color blindness (a carrier is heterozygous for the gene and is phenotypically normal) bears sons, half of them are likely to be color blind, regardless of whether the father had normal or affected vision. How are these observations explained? Color blindness and hemophilia defects are recessive traits carried on the X chromosome. They are phenotypically expressed either when both genes are defective in the female or when only one defective gene is present in the male. The inheritance pat-tern of these defects is shown for color blindness in Figure 5.7 . When the mother is a carrier and the father is normal, half of the sons but none of the daughters are color blind. However, if the father is color blind and the mother is a carrier, half of the sons and half of the daughters are color blind (on the average and in a large sample). It is easy to understand then why such defects are much more prevalent in males: a single sex-linked recessive gene in the male has a visible effect because he has only one X chromosome. What would be the outcome of a mating between a homozygous normal woman and a color-blind man? Another example of a sex-linked character was discovered by Thomas Hunt Morgan (1910) in Drosophila. Normal eye Figure 5.7 Sex-linked inheritance of red-green color blindness in humans. A, Carrier mother and normal father produce color blindness in one-half of their sons but in none of their daughters. B, Half of both sons and daughters of carrier mother and color-blind father are color blind.
Carrier woman (XX) A B Normal man (XY) X X X Y Carrier woman (XX) Color-blind man (XY) X X X Y F1 Generation Normal (homozygous) Carrier (heterozygous) Normal (hemizygous) Color-blind (hemizygous) Carrier (heterozygous) Color-blind (homozygous) Normal (hemizygous) Color-blind (hemizygous) 88 PART TWO Continuity and Evolution of Animal Life color of this fl y is red, but mutations for white eyes do occur ( Figure 5.8 ). A gene for eye color is carried on the X chromo-some. If true-breeding white-eyed males and red-eyed females are crossed, all F 1 offspring have red eyes because this trait is dominant ( Figure 5.8 ). If these F 1 offspring are interbred, all F 2 females have red eyes; half of the males have red eyes and the other half have white eyes. No white-eyed females are found in this generation; only males have the recessive character (white eyes). The allele for white eyes is recessive and should affect eye color only in a homozygous condition. However, since the male has only one X chromosome (the Y does not carry a gene for eye color), white eyes appear whenever the X chromosome carries the allele for this trait. Males are said to be hemizygous (only one copy of a genetic locus is present) for traits carried on the X chromosome. If the reciprocal cross is made in which females are white eyed and males red eyed, all F 1 females are red eyed and all males are white eyed ( Figure 5.9 ). If these F 1 offspring are inter-bred, the F 2 generation shows equal numbers of red-eyed and white-eyed males and females. Autosomal Linkage and Crossing Over Linkage Since Mendel’s laws were rediscovered in 1900, it became clear that, contrary to Mendel’s second law, not all factors segregate independently. Indeed, many traits are inherited together. Since the number of chromosomes in any organism is relatively small compared with the number of traits, each chromosome must contain many genes. All genes present on a chromosome are said to be linked. Linkage simply means that the genes are on the same chromosome, and all genes present on homologous chromosomes belong to the same linkage groups. Therefore there should be as many linkage groups as there are chromo-some pairs. Geneticists commonly use the word “linkage” in two somewhat different meanings. Sex linkage refers to inheritance of a trait on the sex chromosomes, and thus its phenotypic expression depends on the sex of the organism and the factors already discussed. Autosomal linkage, or simply, linkage, refers to inheritance of genes on a given autosomal chromosome. Letters used to represent such genes are normally written without a slash mark between them, indicating that they are on the same chromosome. For example, AB/ab shows that genes A and B are on the same chromosome. Interestingly, Mendel studied seven characteristics of garden peas that assorted independently because they are on seven different chro-mosomes. If he had studied eight characteristics, he would not have found independent assortment in two of the traits because garden peas have only seven pairs of homologous chromosomes. In Drosophila, in which this principle has been studied most extensively, there are four linkage groups that correspond to the four pairs of chromosomes found in these fruit fl ies. Usually, Figure 5.8 Sex-linked inheritance of eye color in fruit fl y Drosophila melanogaster. A, White and red eyes of D. melanogaster. B, Genes for eye color are carried on X chromosome; Y carries no genes for eye color. Normal red is dominant to white. Homozygous red-eyed female mated with white-eyed male gives all red-eyed in F1. F2 ratios from F1 cross are one homozygous red-eyed female and one heterozygous red-eyed female to one red-eyed male and one white-eyed male.
F2 F1 P A White and red eyes of D. melanogaster B Figure 5.9 Reciprocal cross of Figure 5.8 (homozygous white-eyed female with red-eyed male) gives white-eyed males and red-eyed females in F1. F2 shows equal numbers of red-eyed and white-eyed females and red-eyed and white-eyed males.
F2 F1 P www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 89 small chromosomes have small linkage groups, and large chro-mosomes have large groups. Crossing Over Linkage, however, is usually not complete. If we perform an experiment in which animals such as Drosophila are crossed, we fi nd that linked traits separate in some percentage of the offspring. Separation of alleles located on the same chromosome occurs because of crossing over. During the protracted prophase of the fi rst meiotic division, paired homologous chromosomes break and exchange equiva-lent portions; genes “cross over” from one chromosome to its homolog, and vice versa ( Figure 5.10 ). Each chromosome con-sists of two sister chromatids held together by means of a pro-teinaceous structure called a synaptonemal complex. Breaks and exchanges occur at corresponding points on nonsister chro-matids. (Breaks and exchanges also occur between sister chro-matids but usually have no genetic signifi cance because sister chromatids are identical.) Crossing over is a means for exchang-ing genes between homologous chromosomes and thus greatly increases the amount of genetic recombination. The frequency of crossing over varies depending on the species, but usually at least one and often several crossovers occur each time chromo-somes pair. Because the frequency of recombination is proportional to the distance between loci, the relative linear position of each locus can be determined. Genes located far apart on very large chromosomes may assort independently because the probabil-ity of a crossover occurring between them in each meiosis is close to 100%. Such genes are found to be carried on the same chromosome only because each one is genetically linked to additional genes located physically between them on the chro-mosome. Laborious genetic experiments over many years have produced gene maps that indicate the positions of more than 500 genes distributed on the four chromosomes of Drosophila melanogaster. Chromosomal Aberrations Structural and numerical deviations from the norm that affect many genes at once are called chromosomal aberrations. They are sometimes called chromosomal mutations, but most cyto-geneticists prefer to use the term “mutation” to refer to quali-tative changes within a gene; gene mutations are discussed on page 100. Despite the incredible precision of meiosis, chromosomal aberrations do occur, and they are more common than one might think. They are responsible for great economic benefi t in agriculture. Unfortunately, they are responsible also for many human genetic malformations. It is estimated that fi ve out of every 1000 humans are born with serious genetic defects attrib-utable to chromosomal anomalies. An even greater number of embryos with chromosomal defects abort spontaneously, far more than ever reach birth. Changes in chromosome numbers are called euploidy when there is the addition or deletion of whole sets of chromosomes and aneuploidy when a single chromosome is added to or sub-tracted from a set. A “set” of chromosomes contains one member of each homologous pair as would be present in the nucleus of a gamete. The most common kind of euploidy is polyploidy, the carrying of three or more sets of chromosomes by an organism. An organism with three or more complete sets of chromosomes is called a polyploid. Such aberrations are much more common in plants than in animals. Animals are much less tolerant of chro-mosomal aberrations, especially those in which sex determination requires a delicate balance between the numbers of sex chromo-somes and autosomes. Many domestic plant species are polyploid (cotton, wheat, apples, oats, tobacco, and others), and perhaps 40% of flowering plants may have originated in this manner. Horticulturists favor polyploids because they often have more intensely colored fl owers and more vigorous vegetative growth. Aneuploidy is usually caused by failure of chromosomes to separate during meiosis (nondisjunction). If a pair of chromo-somes fails to separate during the fi rst or second meiotic divi-sions, both members go to one pole and none to the other. This condition results in at least one gamete or polar body having n 1 chromosomes and another having n 1 chromosomes. If an n 1 gamete is fertilized by a normal n gamete, the result is a monosomic animal. Survival is rare because the lack of one chromosome gives an uneven balance of genetic instruc-tions. Trisomy, the result of the fusion of a normal n gamete and an n 1 gamete, is much more common, and several kinds of trisomic conditions are known in humans. Perhaps the most familiar is trisomy 21, or Down syndrome. As the name indi-cates, it involves an extra chromosome 21 combined with the chromosome pair 21, and it is caused by nondisjunction of that pair during meiosis. It occurs spontaneously, and there is seldom any family history of the abnormality. However, the risk of its appearance rises dramatically with increasing age of the mother; Figure 5.10 Crossing over during meiosis. Nonsister chromatids exchange portions, so that none of the resulting gametes is genetically the same as any other. Gene X is farther from gene Y than Y is from Z; therefore, gene X is more frequently separated from Y in crossing over than Y is from Z.
x y z X Y Z X Y Z x y z X y z X y Z x Y z x Y Z 90 PART TWO Continuity and Evolution of Animal Life it occurs 40 times more often in women over 40 years old than among women between the ages of 20 and 30. In cases where maternal age is not a factor, 20% to 25% of trisomy 21 results from nondisjunction during spermatogenesis; it is paternal in ori-gin and is apparently independent of the father’s age. A syndrome is a group of symptoms associated with a particular disease or abnormality, although every symptom is not necessarily shown by every patient with the condition. An English physician, John Langdon Down, described in 1866 the syndrome that we now know is caused by trisomy 21. Because of Down’s belief that the facial features of affected individuals were mongoloid in appear-ance, the condition has been called mongolism. The resemblances are superfi cial, however, and currently accepted names are trisomy 21 and Down syndrome. Among the numerous characteristics of the condition, the most disabling is severe mental retardation. This, as well as other conditions caused by chromosomal aberrations and several other birth defects, can be diagnosed prenatally by a proce-dure involving amniocentesis. The physician inserts a hypodermic needle through the abdominal wall of the mother and into fl uids surrounding the fetus ( not into the fetus) and withdraws some of the fl uid, which contains some fetal cells. The cells are grown in culture, their chromosomes are examined, and other tests done. If a severe birth defect is found, the mother has the option of having an abortion performed. As an extra “bonus,” the sex of the fetus is learned after amniocentesis. How? Alternatively, determination of concentrations of certain substances in the maternal serum, which is less invasive than amniocentesis, can detect about 60% of Down syndrome fetuses. Ultrasound scanning may be more than 80% accurate. In all diploid species, normal development requires exactly two of each kind of autosome (not sex chromosomes). Nondis-junction can cause trisomies of other chromosomes, but because these lead to imbalance of many gene products, they almost always cause death before or soon after birth. However, each cell requires only one functional X chromosome (the other is inactivated in females). Nondisjunction of sex chromosomes is better tolerated but usually causes sterility and abnormalities of sex organs. For example, a human with XXY (Klinefelter syn-drome) is a phenotypic male, usually infertile and with some female sexual characteristics. Presence of only one X (and no Y) is usually lethal in embryos, but the occasional live birth produces a phenotypic female with a variety of developmental abnormalities (Turner syndrome). Structural aberrations involve whole sets of genes within a chromosome. A portion of a chromosome may be reversed, plac-ing the linear arrangement of genes in reverse order (inversion); nonhomologous chromosomes may exchange sections (translo-cation); entire blocks of genes may be lost (deletion), usually causing serious developmental defects; or an extra section of chro-mosome may attach to a normal chromosome (duplication). These structural changes often produce phenotypic changes. Duplications, although rare, are important for evolution because they supply additional genetic information that may enable new functions. GENE THEORY Gene Concept The term “gene” (Gr. genos, descent) was coined by W. Johannsen in 1909 for the hereditary factors of Mendel. Initially, genes were thought to be indivisible subunits of the chromosomes on which they occurred. Later studies with multiple mutant alleles demon-strated that alleles are in fact divisible by recombination; portions of a gene are separable. Furthermore, parts of many genes in eukaryotes are separated by sections of DNA that do not specify a part of the fi nished product (introns). As the chief unit of genetic information, genes encode prod-ucts essential for specifying the basic architecture of every cell, details of protein synthesis, cell division, and, directly or indi-rectly, the entire metabolic function of the cell. Because of their ability to mutate, to be assorted and shuffl ed in different com-binations, genes are important units of heredity and variation in evolution. Genes maintain their identities for many generations despite mutational changes in some parts of their structure. One Gene–One Enzyme Hypothesis Since genes act to produce different phenotypes, we may infer that their action follows the scheme: gene → gene product → phenotypic expression. Furthermore, we may suspect that the gene product is usually a protein, because proteins act as enzymes, antibodies, hormones, and structural elements through-out the body. The fi rst clear, well-documented study to link genes and enzymes was performed on the common bread mold Neurospora by Beadle and Tatum in the early 1940s. This organism was ide-ally suited to a study of gene function for several reasons: these molds are much simpler to handle than fruit fl ies, they grow readily in well-defi ned chemical media, and they are haploid organisms unencumbered with dominance relationships among alleles. Furthermore, mutations were readily induced by irra-diation with ultraviolet light. Ultraviolet-light-induced mutants, grown and tested in specifi c nutrient media, had single-gene mutations that were inherited. Each mutant strain was defective in one enzyme, which prevented that strain from synthesizing one or more complex molecules. The ability to synthesize a par-ticular molecule was controlled by a single gene. From these experiments Beadle and Tatum made an impor-tant and exciting formulation: one gene produces one enzyme. For this work they received the Nobel Prize for Physiology or Medicine in 1958. The new hypothesis was soon validated by research on many biosynthetic pathways. Hundreds of inher-ited disorders, including dozens of human hereditary diseases, are caused by single mutant genes causing loss of a specifi c enzyme. We now know that a particular protein may contain several chains of amino acids (polypeptides), each one speci-fi ed by a different gene, and not all proteins specifi ed by genes are enzymes (for example, structural proteins, antibodies, trans-port proteins, and hormones). Furthermore, genes directing the synthesis of various kinds of RNA were not included in Beadle www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 91 and Tatum’s formulation. Therefore a gene now may be defi ned more inclusively as a nucleic acid sequence (usually DNA) that encodes a functional polypeptide or RNA sequence. STORAGE AND TRANSFER OF GENETIC INFORMATION Nucleic Acids: Molecular Basis of Inheritance Cells contain two kinds of nucleic acids: deoxyribonucleic acid (DNA), which is the genetic material, and ribonucleic acid (RNA), which functions in protein synthesis. Both DNA and RNA are polymers built of repeated units called nucleotides. Each nucle-otide contains three parts: a sugar, a nitrogenous base, and a phosphate group. The sugar is a pentose (5-carbon) sugar; in DNA it is deoxyribose and in RNA it is ribose ( Figure 5.11 ). Nitrogenous bases of nucleotides are also of two types: pyrim-idines, whose characteristic structure is a single, 6-membered ring, and purines, which contain two fused rings. Purines and pyrimidines contain nitrogen as well as carbon in their rings, which is why they are called “nitrogenous” bases. The purines in both RNA and DNA are adenine and guanine ( Table 5.2 ). The pyrimidines in DNA are thymine and cytosine, and in RNA they are uracil and cytosine. Carbon atoms in the bases are numbered (for identifi cation) according to standard biochemical notation ( Figure 5.12 ). Carbons in ribose and deoxyribose are also num-bered, but to distinguish them from the carbons in the bases, numbers for carbons in the sugars are given prime signs (see Figure 5.11 ). The sugar, phosphate group, and nitrogenous base are linked as shown in the generalized scheme for a nucleotide: Nitrogenous base Sugar Phosphate H H O H OH OH H P O HO C H O H H 1' 2' 3' 4' 5' Base Figure 5.11 Ribose and deoxyribose, the pentose sugars of nucleic acids. A carbon atom lies in each of the four corners of the pentagon (labeled 1′ to 4′). Ribose has a hydroxyl group (—OH) and a hydrogen on the number 2′ carbon; deoxyribose has two hydrogens at this position.
HOCH2 OH Ribose H H O OH OH H H 1' 2' 3' 4' 5' HOCH2 OH Deoxyribose H H O H OH H H 1' 2' 3' 4' 5' T A B L E 5. 2 Chemical Components of DNA and RNA DNA RNA Purines Adenine Adenine Guanine Guanine Pyrimidines Cytosine Cytosine Thymine Uracil Sugar 2-Deoxyribose Ribose Phosphate Phosphoric acid Phosphoric acid PYRIMIDINES Thymine H H H O O CH3 1 2 4 3 6 5 N N Cytosine H NH2 N N H O H Uracil H N N H H O O PURINES H H H NH2 Adenine 1 2 3 4 5 8 7 9 6 N N N N H H H NH2 O Guanine N N N N Figure 5.12 Purines and pyrimidines of DNA and RNA. In DNA the “backbone” of the molecule is built of phos-phoric acid and deoxyribose; to this backbone are attached the nitrogenous bases ( Figure 5.13 ). The 5′ end of the backbone has a free phosphate group on the 5′ carbon of the ribose, and the 3′ end has a free hydroxyl group on the 3′ carbon. However, one of the most interesting and important discoveries about the nucleic acids is that DNA is not a single polynucleotide chain; it has two complementary chains that are precisely cross-linked by specifi c hydrogen bonding between purine and pyrimidine bases. The number of adenines equals the number of thymines, and the number of guanines equals the number of cytosines. This fact suggested a pairing of bases: adenine with thymine (AT) and guanine with cytosine (GC) (see Figures 1.6 and 5.14 ). The result is a ladder structure ( Figure 5.15 ). The upright portions are the sugar-phosphate backbones, and the connect-ing rungs are the paired nitrogenous bases, AT or GC. However, the ladder is twisted into a double helix with approximately 10 base pairs for each complete turn of the helix ( Figure 5.16 ). The two DNA strands run in opposite directions (antiparallel), and the 5′ end of one strand is opposite the 3′ end of the other ( Figure 5.16 ). The two strands are also complementary —the 92 PART TWO Continuity and Evolution of Animal Life sequence of bases along one strand specifi es the sequence of bases along the other strand. The structure of DNA is widely considered the single most important biological discovery of the twentieth century. It was H H H H H H O O CH3 N N • • • • • H H N N N N N N H H H H O N N N N N H H O N N To deoxyribose of chain To deoxyribose of chain To deoxyribose of chain To deoxyribose of chain Hydrogen bond Hydrogen bond Adenine Thymine Guanine Cytosine Figure 5.14 Positions of hydrogen bonds between thymine and adenine and between cytosine and guanine in DNA.
Figure 5.15 DNA, showing how the complementary pairing of bases between the sugar-phosphate “backbones” keeps the double helix at a constant diameter for the entire length of the molecule. Dots represent the three hydrogen bonds between each cytosine and guanine and the two hydrogen bonds between each adenine and thymine.
• • • • • • • • • • • • • • • P S P S P S P S P S P S P S P S P S P S P S P S C C C G G G T T A A T A 3 3 5 5 Figure 5.13 Section of a strand of DNA. Polynucleotide chain is built of a “backbone” of phosphoric acid and deoxyribose sugar molecules. Each sugar holds a nitrogenous base. Shown from top to bottom are adenine, guanine, thymine, and cytosine.
3 end 5 end O O O O O P O HO C O O P O HO C O O P O HO C O O O P O HO C O H H NH2 H H NH2 O N N N N H H O O CH3 N N NH2 H H O N N N N N N Adenine Guanine Thymine Cytosine Phosphate group Deoxyribose sugar based on X-ray diffraction studies of Maurice H. F. Wilkins and Rosalind Franklin and on ingenious proposals of Francis H. C. Crick and James D. Watson published in 1953. Watson, Crick, and Wilkins were later awarded the Nobel Prize for Physiology or Medicine for their momentous work. Rosalind Franklin was not included because she died prior to the award. RNA is similar to DNA in structure except that it consists of a single polynucleotide chain (except in some viruses), has ribose instead of deoxyribose, and has uracil instead of thymine. www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 93 Ribosomal, transfer, and messenger RNAs are the most abundant and well-known types (function described on pp. 95–96), but many structural and regulatory RNAs, such as micro RNAs, are known. Every time a cell divides, the structure of DNA must be pre-cisely copied in the daughter cells. This is called replication ( Figure 5.17 ). During replication, the two strands of the double helix unwind, and each separated strand serves as a template against which a complementary strand is synthesized. An enzyme (DNA polymerase) catalyzes assembly of a new strand of poly-nucleotides with a thymine group going opposite the adenine group in the template strand, a guanine group opposite the cytosine group, and the two reverse conditions. DNA polymerase synthesizes new strands only in the direc-tion of 5′ to 3′. Because the parent DNA strands are antiparallel, one of which runs 5′ to 3′ and the other running 3′ to 5′, synthesis along one of the strands is continuous, and the other must be formed in a series of fragments, each of which begins with a 5′ end run-ning toward a 3′ end ( Figure 5.17 ). DNA Coding by Base Sequence Because DNA is the genetic material and contains a lin-ear sequence of base pairs, an obvious extension of the Watson-Crick model is that the sequence of base pairs in DNA codes for, and is colinear with, the sequence of amino acids in a protein. The coding hypothesis must explain how a string of four different bases—a four-letter alphabet—could specify the sequence of 20 different amino acids. In the coding procedure, obviously there cannot be a 1:1 correspondence between four bases and 20 amino acids. If a coding unit (often called a word, or codon ) were two bases, only 16 words (4 2 ) could be formed, which could not specify 20 amino acids. Therefore the codon must contain at least three bases or three letters, because 64 possible words (4 3 ) could be formed by four bases when taken as triplets. A trip-let code permits considerable redundancy of triplets (codons), because DNA encodes just 20 amino acids. Later work confi rmed that nearly all amino acids are specifi ed by more than one triplet code ( Table 5.3 ). DNA shows surprising stability, both in prokary-otes and in eukaryotes. Interestingly, it is susceptible to damage by harmful chemicals in the environment and by radiation. Such damage is usually not permanent, because cells have an effi cient repair system. Various types of damage and repair are known, one of which is excision repair. Ultraviolet irradiation often dam-ages DNA by linking adjacent pyrimidines by cova-lent bonds (dimerize), preventing transcription and replication. A series of several enzymes “recognizes” the damaged strand and excises the pair of dimerized pyrimidines and several bases following them. DNA polymerase then synthesizes the missing strand using the remaining one as a template, according to the base-pairing rules, and the enzyme DNA ligase joins the end of the new strand to the old one. Transcription and the Role of Messenger RNA Information is coded in DNA, but DNA does not participate directly in protein synthesis. The intermediary molecule between DNA and protein is another nucleic acid called messenger RNA (mRNA). The triplet codes in DNA are transcribed into mRNA, with uracil substituting for thymine ( Table 5.3 ). Figure 5.16 DNA molecule.
OH end end Phosphodiester bond Hydrogen bonds between nitrogenous bases Sugar-phosphate "backbone" P P P P P O O O O O O A T G C T A C G G O O O O P P P P C P O 3 5 ••• • • • •• ••• • •• ••••• • • ••• ••••• • • • • • • •••• • • • •• ••• • •••• •• ••••• •• 94 PART TWO Continuity and Evolution of Animal Life Ribosomal, transfer, and messenger RNAs are transcribed directly from DNA, each encoded by different sets of genes. RNA is formed as a complementary copy of one strand of the appropriate gene using an enzyme called RNA polymerase. (In eukaryotes each type of RNA [ribosomal, transfer, and messen-ger] is transcribed by a different type of RNA polymerase.) The RNA contains a sequence of bases that complements the bases in one of the two DNA strands, just as the DNA strands comple-ment each other. Thus A in the template DNA strand is replaced by U in RNA; C is replaced by G; G is replaced by C; and T is replaced by A. Only one of the two chains is used as the tem-plate for RNA synthesis ( Figure 5.18 ). A codon is referenced as the sequence of bases present in a mRNA molecule ( Table 5.3 ), which is complementary and antiparallel to the template DNA strand (often called the “sense” strand) from which it is made. The DNA strand not used as a template during transcription of a gene is called the “antisense” strand. A bacterial gene is encoded on a continuous stretch of DNA, transcribed into mRNA, and then translated (see the next section). The hypothesis that eukaryotic genes had a similar structure was rejected by the surprising discovery that some stretches of DNA are transcribed in the nucleus but are not found in the correspond-ing mRNA in the cytoplasm. In other words, pieces of the nuclear mRNA were removed in the nucleus before the fi nished mRNA was transported to the cytoplasm ( Figure 5.19 ). Thus many genes are split, interrupted by sequences of bases that do not code for the fi nal product, and mRNA transcribed from them must be edited or “matured” before translation in the cytoplasm. The intervening segments of DNA are called introns, and those that encode part of Figure 5.17 Replication of DNA. Parent strands of DNA part, and DNA polymerase synthesizes daughter strands using the base sequence of parent strands as a template. Because synthesis always proceeds in a 5′ to 3′ direction, synthesis of one strand is continuous, and the other strand must be synthesized as a series of fragments.
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3 3 5 3 5 3 5 3 5 3 5 5 3 5 3 5 3 5 Parent DNA Replication Lagging strand Leading strand Daughter strands T A B L E 5 . 3 The Genetic Code: Amino Acids Specifi ed by Codons of Messenger RNA Second Letter U C A G First Letter U Phenylalanine Serine Tyrosine Cysteine U C A G Third Letter Leucine End chain UGA End chain UGG Tryptophane C Leucine Proline Histidine Arginine U C A G Glutamine A Isoleucine Threonine Asparagine Serine U C A G Lysine Arginine AUG Methionine G Valine Alanine Aspartic acid Glycine U C A G Glutamic acid Also, begin polypeptide chain.
UUU UUC UCU UCC UCA UCG UAU UAC UGU UGC UUA UUG UAA UAG CUU CUC CUA CUG CCU CCC CCA CCG CAA CAG CAU CAC CGU CGC CGA CGG AUU AUC AUA ACU ACC ACA ACG AAU AAC AAA AAG AGU AGC AGA AGG GUU GUC GUA GUG GCU GCC GCA GCG GAA GAG GGU GGC GGA GGG GAU GAC www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 95 the mature RNA and are translated into protein are called exons. Before mRNA leaves the nucleus, a methylated guanine “cap” is added at the 5′ end, and a tail of adenine nucleotides (poly- A ) is often added at the 3′ end ( Figure 5.19 ). The cap and the poly- A tail distinguish mRNA from other kinds of RNA molecules. In mammals, genes coding for histones and for interferons are on continuous stretches of DNA. However, we now know that genes coding for many proteins are split. In lymphocyte differentiation the parts of the split genes coding for immuno-globulins are actually rearranged during development, so that different proteins result from subsequent transcription and trans-lation. This rearrangement partly explains the enormous diversity of antibodies manufactured by descendants of the lymphocytes (p. 775). Base sequences in some introns are complementary to other base sequences in the intron, suggesting that the intron could fold so that complementary sequences would pair. This folding may be necessary to control proper alignment of intron boundar-ies before splicing. Most surprising of all is the discovery that, in some cases, RNA can “self-catalyze” the excision of introns. The ends of the intron join; the intron thus becomes a small circle of RNA, and the exons are spliced together. This process does not fi t the classical defi nition of an enzyme or other catalyst because the molecule itself is changed by the reaction. Translation: Final Stage in Information Transfer The translation process occurs on ribosomes, granular struc-tures composed of protein and ribosomal RNA (rRNA). Ribo-somal RNA contains a large and a small subunit, and the small subunit comes to lie in a depression of a large subunit to form a functional ribosome ( Figure 5.20 ). Messenger RNA molecules attach themselves to ribosomes to form a messenger RNA-ribosome complex. Because only a short section of mRNA makes contact with a single ribosome, the mRNA usually has several ribosomes attached along its length, each one at a different stage of synthesizing the encoded polypeptide. The entire complex, called a polyribosome or polysome, allows several molecules of the same kind of polypeptide to be synthesized concurrently, one on each ribosome of the polysome ( Figure 5.20 ). Assembly of polypeptides on the mRNA-ribosome complex requires another kind of RNA called transfer RNA (tRNA). Transfer RNAs have a complex secondary structure of folded stems and loops, often illustrated in the form of a cloverleaf ( Figure 5.21 ), although the three-dimensional shape is somewhat different. Molecules of tRNA collect free amino acids from the cytoplasm and deliver them to the polysome, where they are assembled into a polypeptide. There are special tRNA molecules mRNA RNA polymerase DNA 5 5 5 3 3 3 Figure 5.18 Transcription of mRNA from a DNA template. Transcription is similar for mRNA, rRNA, and tRNA except that each kind of RNA uses a different form of the enzyme, RNA polymerase. This diagram shows transcription midway to completion. Transcription began by unwinding the DNA helix, annealing of a RNA primer to the template strand of DNA, and extension of the primer at its 3′ end by adding nucleotides (not shown) complementary to the sequence of bases in the template DNA strand. The primer is at the 5′ end of the mRNA, which continues to grow in length by adding nucleotides at its annealed 3′ end. When transcription is fi nished, the mRNA will detach completely from the DNA template.
DNA Primary RNA transcript Mature mRNA transcript Exon (coding region) Intron (noncoding region) 1 2 3 4 5 6 7 3 poly-A tail 5 cap Introns are removed and coding regions are spliced together Transcription Figure 5.19 Expression of ovalbumin gene of chicken. The entire gene of 7700 base pairs is transcribed to form the primary mRNA, then the 5′ cap of methyl guanine and the 3′ polyadenylate tail are added. After the introns are spliced out, the mature mRNA is transferred to the cytoplasm.
96 PART TWO Continuity and Evolution of Animal Life for every amino acid. Furthermore, each tRNA is accompanied by a specifi c tRNA synthetase. Transfer RNA synthetases are enzymes that attach the correct amino acid to the terminal adenine on the 3′ end of each tRNA by a process called charging. On the cloverleaf-shaped molecule of tRNA, a special sequence of three bases (the anticodon ) is exposed in just the right way to form base pairs with complementary bases (the codon) in the mRNA. The codons are read and polypeptides assembled along the mRNA in a 5′ to 3′ direction. The anticodon of each tRNA is the key to the correct ordering of amino acids in the polypeptide being assembled. For example, alanine is assembled into a polypeptide when it is signaled by the codon GCG in an mRNA. The translation is accomplished by alanine tRNA in which the anticodon is CGC. An alanine tRNA is fi rst charged with alanine by its tRNA synthetase. The alanine-tRNA complex enters the ribosome where it fi ts precisely into the right place on the mRNA strand. Then the next charged tRNA specified by the mRNA code (glycine tRNA, for example) enters the ribosome and attaches itself beside the alanine tRNA. The two amino acids are united by a peptide bond, and the alanine tRNA then detaches from the ribosome. The process continues stepwise as the polypeptide chain is built ( Figure 5.22 ). A polypeptide of 500 amino acids can be assembled in less than 30 seconds. Figure 5.20 How the polypeptide chain is formed. As ribosomes move along messenger RNA in a 5′ to 3′ direction, amino acids are added stepwise to form the polypeptide chain.
Ribosome Messenger RNA Attachment of ribosome to mRNA Growing peptide chains Complete peptide 3 5 Figure 5.21 Diagram of a tRNA molecule. The anticodon loop bears bases complementary to those in the mRNA codon. The other two loops function in binding to the ribosome in polypeptide synthesis. The amino acid is added to the free single-stranded 3′ end by tRNA synthetase.
•• •• •• •• •• •• •• •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• G A A U U C OH Amino acid attaches here Anticodon loop Anticodon Codon mRNA 3 5 3 5 Figure 5.22 Formation of polypeptide chain on messenger RNA. As a ribosome moves down the messenger RNA molecule, transfer RNA molecules with attached amino acids enter the ribosome (top). Amino acids are joined together into a polypeptide chain, and transfer RNA molecules leave the ribosome (bottom).
A C C Ala Phe Trp Codon Transfer RNA A U U U U C C C G G G G Leu A C C Ala Phe Trp A U U U U C C C G G G G Leu A C G A C C Ala Phe Trp Leu A C G Anticodon Ribosome Messenger RNA New peptide bond A U U U U C C C G G G G 3 5 3 5 3 5 3 5 A C G www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 97 Regulation of Gene Expression In Chapter 8 we show how the orderly differentiation of an organism from fertilized ovum to adult requires expression of genetic material at every stage of development. Developmental biologists have provided convincing evidence that every cell in a developing embryo is genetically equivalent. Thus as tissues dif-ferentiate (change developmentally), each one uses only a part of the genetic instruction present in every cell. Genes express themselves only at certain times and not at others. Indeed, most of the genes are inactive at any given moment in a particular cell or tissue. The problem in development is to explain how, if every cell has a full gene complement, certain genes are “turned on” to produce proteins required for a particular developmental stage while other genes remain silent. Although developmental changes bring the question of gene activation clearly into focus, gene regulation is neces-sary throughout an organism’s existence. The cellular enzyme systems that control all functional processes obviously require genetic regulation because enzymes have powerful effects even in minute amounts. Enzyme synthesis must respond to the infl u-ences of supply and demand. Gene Regulation in Eukaryotes Several metabolic stages in eukaryotic cells may serve as con-trol points for gene expression. Transcriptional and translational control are the primary stages for control of gene expression in animals, with gene rearrangement also used in some cases. Transcriptional Control Control of transcription is prob-ably the most important mechanism for regulating gene expres-sion. Transcription factors are molecules that can have a posi-tive or a negative effect on transcription of RNA from the DNA of target genes. The factors in some cases act within cells that produce them and in other cases are transported to different parts of the body prior to action. Examples of transcription fac-tors are steroid receptors when bound to a steroid hormone. Steroid hormones produced by endocrine glands elsewhere in the body enter a target cell and bind with a receptor protein in the nucleus. The steroid-receptor complex then binds with DNA near the target gene (p. 755). Progesterone, for example, binds with a nuclear receptor in cells of the chicken oviduct; the hormone-receptor complex then activates transcription of genes encoding egg albumin and other substances. An important mechanism for silencing genes is methylation of cytosine bases; a methyl group (CH 3 ) binds the carbon in the 5 position in the cytosine ring ( Figure 5.23A ). This usually happens when the cytosine is next to a guanine base; thus, the bases in the complementary DNA strand would also be a cyto-sine and a guanine ( Figure 5.23B ). When the DNA is replicated, an enzyme recognizes the CG sequence and quickly methylates the daughter strand, keeping the gene inactive. Translational Control Genes can be transcribed and the mRNA sequestered so that translation is delayed. Develop-ment of eggs of many animals commonly uses this mechanism. Oocytes accumulate large quantities of messenger RNA during their development; then, fertilization activates metabolism and initiates translation of maternal mRNA. Gene Rearrangement Vertebrates contain cells called lym-phocytes that bear genes encoding proteins called antibodies (p. 775). Each type of antibody binds only a particular foreign substance (antigen). Because the number of different antigens is enormous, diversity of antibody genes must be equally great. One source of this diversity is rearrangement of DNA sequences coding for antibodies during development of lymphocytes. Molecular Genetics Progress in our understanding of genetic mechanisms on the molecular level, as discussed in the last few pages, has been almost breathtaking in the last few years. We expect many more discoveries in the near future. This progress results from many biochemical techniques now used in molecular biology. We describe briefl y the most important techniques. Recombinant DNA An important tool in this technology is a series of enzymes called restriction endonucleases. Each of these enzymes, derived from bacteria, cleaves double-stranded DNA at particular sites deter-mined by their base sequences. Many of these endonucleases cut the DNA strands so that one has several bases projecting farther Figure 5.23 Some genes in eukaryotes are turned off by methylation of some cytosine residues in the chain. A, Structure of 5-methyl cytosine. B, Cytosine residues next to guanine are those that are methylated in a strand, thus allowing both strands to be symmetrically methylated.
• • • • • • • • • • P S S S S S S P P P P P S P P P P S C C G G T T A A 3 3 5 5 NH2 CH3 H C H O N C N C C me me 4 1 3 5 2 6 A B 98 PART TWO Continuity and Evolution of Animal Life than the other strand ( Figure 5.24 ), leaving what are called “sticky ends.” When these DNA fragments are mixed with others that have been cleaved by the same endonuclease, their sticky ends tend to anneal (join) by the rules of complementary base pairing. The ends are sealed into their new position by the enzyme DNA ligase in a process called ligation. Besides their chromosomes, most prokaryotic and at least some eukaryotic cells have small circles of double-stranded DNA called plasmids. Although constituting only 1% to 3% of the bacterial genome, they may carry important genetic information, for exam-ple, resistance to an antibiotic. Plastids in plant cells (for example, chloroplasts) and mitochondria, found in most eukaryotic cells, are self-replicating and have their own complement of DNA in the form of small circles reminiscent of plasmids. The DNA of mitochondria codes for some mitochondrial proteins, whereas other mitochon-drial proteins are specifi ed by nuclear genes. If DNA from a foreign source (such as a mammal) is ligated into a plasmid (see preceding note), the product is recombi-nant DNA. To produce the recombinant DNA in large quantities, the modifi ed plasmid must be cloned in bacteria. The bacteria are treated with dilute calcium chloride to make them more sus-ceptible to entry by the recombinant DNA, but plasmids do not enter most bacterial cells. Bacterial cells that have acquired the recombinant DNA can be identifi ed if the plasmid has a marker, for example, resistance to an antibiotic. Then, only bacteria that can grow in the presence of the antibiotic are ones that have absorbed the recombinant DNA. Some bacteriophages (bacterial viruses) also are used as carriers for recombinant DNA. Plasmids and bacteriophages that carry recombinant DNA are called vec-tors. The vectors retain the ability to replicate in the bacterial cells; therefore the recombinant insert is produced in large quan-tities, a process called amplifi cation. A clone is a collection of individuals or cells all derived by asexual reproduction from a single individual. When we speak of cloning a gene or plasmid in bacteria, we mean that we isolate a colony or group of bacteria derived from a single ancestor into which the gene or plasmid was inserted. Cloning is used to obtain large quan-tities of a gene that has been ligated into a bacterial plasmid. Polymerase Chain Reaction Recent advances permit a specifi c gene to be cloned enzymati-cally from any organism as long as part of the sequence of that gene is known. The technique is called the polymerase chain reaction (PCR). Two short chains of nucleotides called primers are synthesized; primers are complementary to different DNA strands in the known sequence at opposite ends of the gene to be cloned. A large excess of each primer is added to a sample of DNA from the organism, and the mixture is heated to separate the double helix into single strands. When the mixture is cooled, there is a much greater probability that each strand of the gene of interest will anneal to a primer than to the other strand of the gene—because the primer is present in a much higher con-centration. A heat-stable DNA polymerase and the four deoxy-ribonucleotide triphosphates are added to the reaction mixture. DNA synthesis proceeds from the 3′ end of each primer, extend-ing the primer in the 5′ to 3′ direction. Primers are designed so that the free 3′ end of each faces toward the gene whose sequence is to be cloned. Entire new complementary strands are synthesized, and the number of copies of the gene has doubled ( Figure 5.25 ). The reaction mixture is then reheated and cooled again to allow more primers to bind original and new copies of each strand. With each cycle of DNA synthesis, the number of copies of the gene doubles. Since each cycle can take less than fi ve minutes, the number of copies of a gene can increase from one to over one million in less than two hours! The PCR allows cloning a known gene from an individual patient, identifi cation of a drop of dried blood at a crime scene, or cloning DNA of a 40,000-year-old woolly mammoth. Recombinant DNA technology and PCR are currently being used to engineer crop plants, including soybeans, cotton, rice, corn, and tomato. Transgenic mice are commonly used in research, and gene therapy for human genetic diseases is being developed. Genomics and Proteomics The scientific field of mapping, sequencing, and analyzing genomes is now called genomics. Some researchers divide Figure 5.24 Action of restriction endonuclease, EcoRI. Such enzymes recognize specifi c base sequences that are palindromic (a palindrome is a word spelled the same backward and forward). EcoRI leaves “sticky ends,” which anneal to other DNA fragments cleaved by the same enzyme. The strands are joined by DNA ligase.
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • C C C G G G C C G G T T A A T T A A T A T A T A C C C G G G C C G G T T A A T T A A T A T A T A EcoRI EcoRI www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 99 genomic analysis into “structural genomics” (mapping and sequencing) and “functional genomics” (development of genome-wide or system-wide experimental approaches to understand gene function). In the 1970s Allan Maxam and Walter Gilbert in the United States and Frederick Sanger in England reported practical tech-niques for identifying the sequence of bases in DNA. By 1984 and 1985 scientists proposed to sequence and to map the entire human genome, an effort called the Human Genome Project. It was a most ambitious undertaking: the genome was estimated at 50,000 to 100,000 genes and regulatory subunits encoded in a linear sequence of about 3 to 6 billion pairs of bases. Using techniques available in 1988, it would have taken until 2700 to sequence the genome completely, but biologists then expected technical improvements to make it possible by the twenty-second century. In fact, development and improvement of auto-mated sequencers, as well as competition between the publicly supported Human Genome Sequencing Consortium and a large group of privately supported scientists (Celera Genomics and collaborators) led to publication of draft sequences in 2001! Whether determination of the draft sequence was “the great-est scientifi c discovery of our time,” as claimed by Davies’s book (in Selected References), is debatable. Nevertheless, it was very exciting and yielded many surprises. For example, the human genome has fewer genes than thought previously, with 21,724 genes currently known. Only 5% of the 28% of the genome that is actually transcribed into RNA encodes protein. More than half the DNA present is repeated sequences of several types, includ-ing 45% in parasitic DNA elements. Parasitic DNA (also called “selfi sh” and “junk” DNA) is DNA that seems to serve no cellular or organismal function except its own propagation, but it may have utility in ways not yet clear. Animal species vary by several orders of magnitude in the total amount of DNA present in their nuclear genomes (from slightly less than 10 8 to 10 11 base pairs in a haploid gamete nucleus). At the low end are sponges (p. 248), some single-celled forms (see Chapter 11) and some arthropods (see Chapter 19), although the latter two groups include a wide range of genome sizes exceeding 10 9 in single-celled forms and reaching 10 10 base pairs in arthro-pods. Most vertebrates have genomes of approximately 10 9 base pairs, but salamanders (p. 548), caecilians (p. 548), and lungfi shes (p. 529) have genome sizes exceeding 10 10 base pairs, with some salamanders reaching 10 11 base pairs. Large genomes should not be considered advantageous, however, because most of the dif-ference in genome size is contributed by accumulation of large amounts of the “selfi sh” or “junk” DNA in the larger genomes rather than DNA sequences useful to cellular metabolism and organismal function. The metabolic demands of replicating large amounts of DNA and the physical demands of housing it within the cell nucleus produce selection against accumulation of too much para-sitic DNA in the genome. Animal groups with the largest genomes are likely those most able to tolerate accumulation of large amounts of parasitic DNA in their nuclear genomes without harming cellular and organismal functions. Because the emphasis of our book is on organismal biology, our review of genetics concentrates on genes that have clear roles in cellular and organismal functions, although these genes are a small minority of the DNA sequences present in animal nuclear genomes. Some DNA sequences considered useless to the organism exhibit variation that is useful in studies of popula-tion genetics (p. 126) and evolutionary relationships among species. A thousand human diseases, such as cystic fibrosis and Huntington’s chorea, result from defects in single genes. Almost 300 disease-associated genes are known. Information developed from knowledge of gene sequences can permit new diagnostic tests, treatments, possible preventive strategies, and advances in molecular understanding of genetic diseases. However, to realize Figure 5.25 Steps in the polymerase chain reaction (PCR). Note that two different primers are required, one for each end of the target sequence.
Target sequence Primers DNA polymerase Free nucleotides 2 copies 4 copies 8 copies Cycle 3 Cycle 2 Cycle 1 P1 P2 P P P P P P P1 P1 P2 P2 3 Primer extension Annealing of primers P1 P1 P1 P1 P1 P2 P1 P1 P1 P1 P2 P2 P2 P2 P1 P2 P2 P2 P2 P2 Heat Heat Heat Cool Cool Cool Denaturation 3 3 5 5 100 PART TWO Continuity and Evolution of Animal Life such benefi ts it is not suffi cient simply to know the sequence of amino acids encoded by a nucleotide sequence in a gene. The human genome is responsible for hundreds of thousands of different proteins (proteome) . The polypeptide encoded by a gene may be cleaved into separate functional parts or associ-ated with polypeptides coded by other genes to produce diverse protein functions. Many scientists are now engaged in the dif-fi cult fi eld of proteomics: to identify all the proteins in a cell, tissue, or organism; to determine how the proteins interact to accomplish their functions; and to outline the folding structures of the proteins. GENETIC SOURCES OF PHENOTYPIC VARIATION The creative force of evolution is natural selection acting on biological variation. Without variability among individuals, there could be no continued adaptation to a changing environment and no evolution (see Chapter 6). Although natural selection acts on varying organismal phenotypes, phenotypic variation within a population in a particular environment is often caused by variation in genotype. Preservation of favored phenotypes by natural selection therefore increases the abundance in a pop-ulation of alleles associated with favored phenotypes, leading to adaptive evolution of the population. Through this process, a population evolves organismal phenotypes molded for effective use of environmental resources; such phenotypes are termed adaptations. There are several genetic sources of phenotypic variation, all of which involve mutation at individual genes and the com-bining of the resulting alleles at variable genes into gametes and zygotes. Independent assortment of chromosomes during meio-sis is a random process that creates new chromosomal combina-tions in gametes. In addition, chromosomal crossing over during meiosis allows recombination of linked genes between homolo-gous chromosomes, further increasing variability. Random fusion of gametes from both parents also produces variation. Thus sexual reproduction multiplies variation and provides the diversity and plasticity necessary for a species to survive envi-ronmental change. Sexual reproduction with its sequence of gene segregation and recombination across generations is what geneti-cist T. Dobzhansky called the “master adaptation” that makes all other evolutionary adaptations more accessible. Although sexual reproduction reshuffl es and amplifi es what-ever genetic diversity exists in a population, new genetic varia-tion happens through gene mutations, chromosomal aberrations, and possibly by participation of parasitic DNA. Gene Mutations Gene mutations are chemicophysical changes that alter the sequence of bases in DNA. These mutations are studied directly by determining the DNA sequence and indirectly through their effects on organismal phenotype, if such effects occur. Some mutations produce a codon substitution as in the human condition called sickle cell anemia. Homozygotes for the sickle cell allele often die before the age of 30 because the ability of their red blood cells to carry oxygen is greatly impaired by substitution of only a single amino acid in their hemoglobin. Other mutations involve deletion of one or more bases or insertion of additional bases into a DNA chain. Trans-lation of mRNA is thus shifted, producing codons that specify incorrect amino acids and usually a nonfunctional or dysfunc-tional protein product. Once a gene is mutated, it faithfully reproduces its new form. Many mutations are harmful; many are neither helpful nor harmful, and sometimes mutations are advantageous. Helpful mutations are of great signifi cance to evolution because they furnish new possibilities with which natural selection can build adaptations. Natural selection determines which new alleles merit survival; the environment imposes a screening process that accumulates benefi cial and eliminates harmful alleles. When an allele of a gene is mutated to a new allele, the new form tends to be recessive, and its effects are normally masked by its partner allele. Only in the homozygous condition can such mutant alleles infl uence phenotype. Thus a population carries a reservoir of mutant recessive alleles, some of which are homozygous lethals but which are rarely present in the homozygous condition. Inbreeding encourages formation of homozygotes and increases the probability of recessive mutants being expressed in the phenotype. Most mutations are destined for a brief existence. There are cases, however, in which mutations harmful or neutral under one set of environmental conditions become helpful under a different set. The earth’s changing environment has provided numerous opportunities for favoring new gene mutations, as evi-denced by the great diversity of animal life. Frequency of Mutations Although mutation occurs randomly with respect to an organ-ism’s needs, different mutation rates prevail at different loci. Some kinds of mutations are more likely to occur than others, and individual genes differ considerably in length. A long gene (more base pairs) is more likely to have a mutation than a short gene. Nevertheless, it is possible to estimate average spontane-ous rates of mutation for different organisms and traits. Genes are extremely stable. In the well-studied fruit fl y, Drosophila melanogaster, there is approximately one detectable mutation per 10,000 loci (rate of 0.01% per locus per generation). The rate for humans is one per 10,000 to one per 100,000 loci per generation. If we accept the latter, more conservative fi g-ure, then a single normal allele is expected to undergo 100,000 generations before it is mutated. However, since human chro-mosomes contain approximately 21,724 loci, about every third person carries a new mutation. Similarly, each ovum or sperma-tozoon contains, on average, one mutant allele. Since most mutations are deleterious, these statistics are anything but cheerful. Fortunately, most mutant genes are recessive and are not expressed in heterozygotes. Only a few by chance will increase enough in frequency for homozygotes to be produced. www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 101 MOLECULAR GENETICS OF CANCER The crucial defect in cancer cells is that they proliferate in an unrestrained manner (neoplastic growth). The mechanism that controls the rate of division of normal cells has somehow been lost, and cancer cells multiply much more rapidly, invading other tissues in the body. Cancer cells originate from normal cells that lose their constraint on division and become dedifferentiated (less specialized) to some degree. Thus there are many kinds of cancer, depending on the original founder cells of the tumor. The change in many cancerous cells, perhaps all, has a genetic basis, and investigation of the genetic damage that causes cancer is now a major thrust of cancer research. Oncogenes and Tumor-Suppressor Genes We now recognize that specifi c genetic changes occurring in a particular clone of cells produce cancer. These genetic changes include alterations in numerous genes of two types, oncogenes and tumor-suppressor genes . Oncogenes (Gr. onkos, bulk, mass, genos, descent) occur normally in cells, and in their normal form they are called proto-oncogenes. One of these encodes a protein called Ras. Ras protein is a guanosine triphosphatase (GTPase) located just beneath the cell membrane. When a receptor on the cell surface binds a growth factor, Ras is activated and initiates a cascade of reactions causing cell division. The oncogene form encodes a protein that initiates the cell-division cascade even when the growth factor is absent from the surface receptor. Of the many ways that cellular DNA can sustain damage, the three most important are ionizing radiation, ultraviolet radiation, and chemical mutagens. The high energy of ionizing radiation (X rays and gamma rays) causes electrons to be ejected from the atoms it encounters, producing ionized atoms with unpaired electrons (free radicals). The free radicals (principally from water) are highly reac-tive chemically, and they react with molecules in the cell, including DNA. Some damaged DNA is repaired, but if the repair is inac-curate, a mutation results. Ultraviolet radiation is of much lower energy than ionizing radiation and does not produce free radicals; it is absorbed by pyrimidines in DNA and causes formation of a double covalent bond between the adjacent pyrimidines. UV repair mechanisms can also be inaccurate. Chemical mutagens react with the DNA bases and cause mispairing during replication. Gene products of tumor-suppressor genes act as a constraint on cell proliferation. One such product is called p53 (for “53-kilodalton protein,” a reference to its molecular weight). Muta-tions in the gene encoding p53 occur in about half of the 6.5 million cases of human cancer diagnosed each year. Normal p53 has several crucial functions, depending on the circumstances of the cell. It can trigger apoptosis (p. 55), act as a transcription activator or repressor (turning genes on or off), control progres-sion from G 1 to S phase in the cell cycle, and promote repair of damaged DNA. Many of the mutations known in p53 interfere with its binding to DNA and thus its function. S U M M A R Y In sexual animals genetic material is distributed to offspring via gametes (ova and sperm), produced by meiosis. Each somatic cell in an organism has two chromosomes of each kind (homologous chromosomes) and is thus diploid. Meiosis separates homologous chromosomes, so that each gamete has half the somatic chromosome number (haploid). In the fi rst meiotic division, centromeres do not divide, and each daughter cell receives one of each pair of replicated homologous chromo-somes with sister chromatids still attached to the centromere. At the beginning of the fi rst meiotic division, replicated homologous chromosomes come to lie alongside each other (synapsis), form-ing a bivalent. The gene loci on one set of chromatids lie opposite the corresponding loci on the homologous chromatids. Portions of adjacent chromatids can exchange with the nonsister chromatids (crossing over) to produce new genetic combinations. At the second meiotic division, the centromeres divide, completing the reduction in chromosome number and amount of DNA. The diploid number is restored when male and female gametes fuse to form a zygote. Gender is determined in many animals by the sex chromo-somes; in humans, fruit fl ies, and many other animals, females have two X chromosomes, and males have an X and a Y. Genes are the unit entities that infl uence all characteristics of an organism and are inherited by offspring from their parents. Allelic variants of genes might be dominant, recessive, or intermediate; a recessive allele in the heterozygous genotype will not be expressed in the phenotype but requires the homozygous condition for overt expression. In a monohybrid cross involving a dominant allele and a recessive alternative allele (both parents homozygous), the F 1 gen-eration will be all heterozygous, whereas F 2 genotypes will occur in a 1:2:1 ratio, and phenotypes in a 3:1 ratio. This result demonstrates Mendel’s law of segregation. Heterozygotes in intermediate inheri-tance show phenotypes distinct from homozygous phenotypes, sometimes intermediate forms, with corresponding alterations in phenotypic ratios. Dihybrid crosses (in which genes for two different character-istics are carried on separate pairs of homologous chromosomes) demonstrate Mendel’s law of independent assortment, and phe-notypic ratios are 9:3:3:1 with dominant and recessive characters. Expected ratios in crosses of two or more characters are calculated from laws of probability. Genes can have more than two alleles in a population, and different combinations of alleles can produce different phenotypic effects. Alleles of different genes can interact in producing a pheno-type, as in polygenic inheritance and epistasis, in which one gene affects the expression of another gene. A gene on the X chromosome shows sex-linked inheritance and produces an effect in males, even if a recessive allele is present, because the Y chromosome does not carry a corresponding allele. 102 PART TWO Continuity and Evolution of Animal Life complementary to the respective codons of the mRNA. In eukary-otic nuclear DNA the sequences of bases in DNA coding for amino acids in a protein (exons) are interrupted by intervening sequences (introns). The introns are removed from the primary mRNA before it leaves the nucleus, and the protein is synthesized in the cytoplasm. Genes, and the synthesis of the products for which they are responsible, must be regulated: turned on or off in response to vary-ing environmental conditions or cell differentiation. Gene regulation in eukaryotes occurs at several levels, with control of transcription being a particularly important point of regulation. Molecular genetic methods have made spectacular advances possible. Restriction endonucleases cleave DNA at specifi c base sequences, and DNA from different sources can be rejoined to form recombinant DNA. Combining mammalian with plasmid or viral DNA, a mammalian gene can be introduced into bacterial cells, which then multiply and produce many copies of the mammalian gene. The polymerase chain reaction (PCR) is used to clone specifi c genes if the base sequence of short pieces of DNA surrounding the gene are known. Draft sequences of the human genome were published in 2001. Among many exciting results was a revision of the number of genes to 21,724, down from previous estimates of 100,000. These genes are responsible for hundreds of thousands of proteins in a typical cell. A mutation is a physicochemical alteration in the bases of DNA that may change the phenotypic effect of a gene. Although rare and usually detrimental to survival and reproduction of an organism, mutations are occasionally benefi cial and may be accumulated in populations by natural selection. Cancer (neoplastic growth) results from genetic changes in a clone of cells allowing unrestrained proliferation of those cells. Oncogenes (such as the gene coding for Ras protein) and inactiva-tion of tumor-suppressor genes (such as that coding for p53 protein) are implicated in many cancers. All genes on a given autosomal chromosome are linked, and their variants do not assort independently unless they are very far apart on the chromosome, in which case crossing over occurs between them in nearly every meiosis. Crossing over increases the amount of genetic recombination in a population. Occasionally, a pair of homologous chromosomes fails to sep-arate in meiosis causing the gametes to get one chromosome too many or too few. Resulting zygotes usually do not survive; humans with 2 n 1 chromosomes sometimes live, but they have serious abnormalities, such as Down syndrome. Nucleic acids in the cell are DNA and RNA, which are large polymers of nucleotides composed of a nitrogenous base, pentose sugar, and phosphate group. The nitrogenous bases in DNA are adenine (A), guanine (G), thymine (T), and cytosine (C), and those in RNA are the same except that uracil (U) is substituted for thymine. DNA is a double-stranded, helical molecule in which the bases extend toward each other from the sugar-phosphate backbone: A always pairs with T and G with C. The strands are antiparallel and complementary, being held in place by hydrogen bonds between the paired bases. In DNA replication the strands part, and the enzyme DNA polymerase synthesizes a new strand along each parental strand, using the parental strand as a template. A gene can encode either a ribosomal RNA (rRNA), a transfer RNA (tRNA), or a messenger RNA (mRNA); a gene of the latter kind specifi es the sequence of amino acids in a polypeptide (one gene– one polypeptide hypothesis). In mRNA, each triplet of three bases specifi es a particular amino acid. Proteins are synthesized by transcription of DNA into the base sequence of a molecule of messenger RNA (mRNA), which functions in concert with ribosomes (containing ribosomal RNA [rRNA] and protein) and transfer RNA (tRNA). Ribosomes attach to the strand of mRNA and move along it, assembling the amino acid sequence of the protein. Each amino acid is brought into position for assembly by a molecule of tRNA, which itself bears a base sequence (anticodon) R E V I E W Q U E S T I O N S What is the relationship between homologous chromosomes, copies of a gene, and alleles? Describe or diagram the sequence of events in meiosis (both divisions). What are the designations of the sex chromosomes in males of bugs, humans, and butterfl ies? How do the chromosomal mechanisms determining sex differ in the three taxa in question 3? Diagram by Punnett square a cross between individuals with the following genotypes: A / a A / a; A / a B / b A / a B / b. Concisely state Mendel’s law of segregation and his law of independent assortment. Assuming brown eyes ( B ) are dominant over blue eyes ( b ), determine the genotypes of all the following individuals. The blue-eyed son of two brown-eyed parents marries a brown-eyed woman whose mother was brown eyed and whose father was blue eyed. Their child is blue eyed. Recall that red color ( R ) in four-o’clock fl owers is incompletely dominant over white ( R ′). In the following crosses, give the genotypes of the gametes produced by each parent and the fl ower color of the offspring: R / R ′ R / R ′; R ′/ R ′ R / R ′; R / R R / R ′; R / R R ′/ R ′. 1.
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8. A brown male mouse is mated with two female black mice. In several litters of young, the fi rst female has had 48 black and the second female has had 14 black and 11 brown young. Can you deduce the pattern of inheritance of coat color and the genotypes of the parents? Rough coat ( R ) is dominant over smooth coat ( r ) in guinea pigs, and black coat ( B ) is dominant over white ( b ). If a homozygous rough black is mated with a homozygous smooth white, give the appearance of each of the following: F 1 ; F 2 ; offspring of F 1 mated with smooth, white parent; offspring of F 1 mated with rough, black parent. Assume right-handedness ( R ) is genetically dominant over left-handedness ( r ) in humans, and that brown eyes ( B ) are genetically dominant over blue ( b ). A right-handed, blue-eyed man marries a right-handed, brown-eyed woman. Their two children are (1) right handed, blue eyed and (2) left handed, brown eyed. The man marries again, and this time the woman is right handed and brown eyed. They have 10 children, all right handed and brown eyed. What are the probable genotypes of the man and his two wives? In Drosophila melanogaster, red eyes are dominant to white and the variation for this characteristic is on the X 9.
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www . m hhe . com /hi ckmani pz14e CHAPTER 5 Genetics: A Review 103 Why could a codon not consist of only two bases? Explain the transcription and processing of mRNA in the nucleus. Explain the role of mRNA, tRNA, and rRNA in polypeptide synthesis. What are four ways that genes can be regulated in eukaryotes? In modern molecular genetics, what is recombinant DNA, and how is it prepared? Name three sources of genetic recombination that contribute to phenotypic variation. Distinguish between proto-oncogene and oncogene. Describe two mechanisms by which genetic change causes cancer? What are Ras protein and p53? How can mutations in the genes for these proteins contribute to cancer? Outline the essential steps in the polymerase chain reaction. Draft sequences of the human genome have been published. What are some interesting observations based on the results? What are some potential benefi ts? What is the proteome? 17.
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chromosome. Vestigial wings ( v ) are recessive to normal ( V ) for an autosomal gene. What will be the appearance of offspring of the following crosses: X W / X w V / v X w / Y v / v, X w / X w V / v X W / Y V / v. Assume that color blindness is a recessive character on the X chromosome. A man and woman with normal vision have the following offspring: daughter with normal vision who has one color-blind son and one normal son; daughter with normal vision who has six normal sons; and a color-blind son who has a daughter with normal vision. What are the probable genotypes of all individuals? Distinguish the following: euploidy, aneuploidy, and polyploidy; monosomy and trisomy. Name the purines and pyrimidines in DNA and tell which pairs occur in the double helix. What are the purines and pyrimidines in RNA and to what are they complementary in DNA? Explain how DNA is replicated. 13.
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16. S E L E C T E D R E F E R E N C E S Burt, A., and R. Trivers. 2006. Genes in confl ict: the biology of selfi sh genetic elements. Cambridge, Massachusetts, Belknap Press of Harvard Univ. Press. A thorough discussion of the means by which some genes acquire the ability to promote their own transmission or expression at the expense of others. Conery, J. S., and M. Lynch. 2000. The evolutionary fate and consequences of duplicate genes. Science 290: 1151–1155. Gene duplications are an important source of genetic variation. Davies, K. 2001. Cracking the genome: inside the race to unlock human DNA. Craig Venter, Francis Collins, James Watson, and the story of the greatest scientifi c discovery of our time. New York, The Free Press. Fascinating story of competition between the Human Genome Sequencing Project and Craig Venter’s Celera Genomics. Of course, the genome is not “cracked” until its meaning is deciphered, and we are a long way from that. Dhand, R. 2000. Functional genomics. Nature 405: 819. Introduction to a number of articles on functional genomics. Ezzell, C. 2002. Proteins rule. Sci. Am. 286: 40–47 (April). An excellent explanation of the current status and problems of proteomics. Futreal, P. A., A. Kasprzyk, E. Birney, J. C. Mullikin, R. Wooster, and M. R. Stratton. 2001. Cancer and genomics. Nature 409: 850–852. Good list of cancer genes. Griffi ths, A. J., S. R. Wessler, R. C. Lewontin, W. M. Gelbart, D. T. Suzuki and J. H. Miller. 2005. Introduction to genetic analysis, ed. 8. New York, W. H. Freeman & Company. A good and recent general genetics text. Hartwell, L. H., L. Hood, M. L. Goldberg, A. E. Reynolds, L. M. Silver, and R. C. Veres. 2004. Genetics: from genes to genomes, ed. 2. Boston, Massachusetts, McGraw-Hill Higher Education. A good and recent text of genetics and genomics. Jimenez-Sanchez, G., B. Childs, and D. Valle. 2001. Human disease genes. Nature 409: 853–855. They found “striking correlation between the function of the gene product and features of disease. . . .” Lewin, B. 2004. Genes VIII. Upper Saddle River, New Jersey, Pearson/Prentice Hall. Thorough, up-to-date coverage on molecular biology of genes. Mange, E. J., and A. P. Mange. 1999. Basic human genetics, ed. 2. Sunderland, Massachusetts, Sinauer Associates. A readable, introductory text concentrating on the genetics of the animal species of greatest concern to most of us. Mullis, K. B. 1990. The unusual origin of the polymerase chain reaction. Sci. Am. 262: 56–65 (April). How the author had the idea for the simple production of unlimited copies of DNA while driving through the mountains of California. Pennisi, E. 2000. Genomics comes of age. Science 290: 2220–2221. Determination of genomes of a number of organisms is “Breakthrough of the Year.” Roberts, L. 2001. A history of the Human Genome Project. Science 291: chart. Each major discovery from double helix to full genome (1953–2001). Has glossary. The International Human Genome Mapping Consortium. 2001. A physical map of the human genome. Nature 409: 934–941. The draft sequence published by the publicly supported consortium. Venter, J. C., M. D. Adams, E. W. Myers, P. W. Li, R. J. Mural, and 265 others. 2001. The sequence of the human genome. Science 291: 1304– 1351. The draft sequence published by Celera Genomics and its collaborators. Watson, J. D., and A. Berry. 2003. DNA: the secret of life. New York, Alfred A. Knopf. An exciting account of the history and applications of genetics. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! A trilobite fossilized in Paleozoic rock. C H A P T E R 6 Organic Evolution A Legacy of Change Life’s history is a legacy of perpetual change. Despite the apparent permanence of the natural world, change characterizes all things on earth and in the universe. Earth’s rock strata record the irreversible, historical change that we call organic evolution. Countless kinds of animals and plants have fl ourished and disappeared, leaving behind a sparse fossil record of their existence. Many, but not all, have left living descendants that bear some resemblance to them. Life’s changes are observed and measured in many ways. On a short evolutionary timescale, we see changes in the frequencies of different genetic traits within populations. Evolutionary changes in the relative frequencies of light- and dark-colored moths were observed within a single human lifetime in the polluted towns of industrial England. The formation of new species and dramatic changes in organismal form, as illustrated by the evolutionary diversifi cation of Hawaiian birds, requires longer timescales cov-ering 100,000 to 1 million years. Major evolutionary trends and episodic mass extinctions occur on even larger timescales, cover-ing tens of millions of years. The fossil record of horses through the past 50 million years shows a series of different species replacing older ones through time and ending with the horses alive today. The fossil record of marine invertebrates shows us a series of mass extinctions separated by intervals of approximately 26 million years. Because every feature of life as we know it today is a product of evolution, biologists consider organic evolution the keystone of all biological knowledge. www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 105 I n Chapter 1, we introduced Darwinian evolutionary theory as the dominant paradigm of biology. Charles Robert Darwin and Alfred Russel Wallace ( Figure 6.1 ) fi rst established evolu-tion as a powerful scientifi c theory. Today the reality of organic evolution can be denied only by abandoning reason. As the noted English biologist Sir Julian Huxley wrote, “Charles Darwin effected the greatest of all revolutions in human thought, greater than Einstein’s or Freud’s or even Newton’s, by simultaneously establishing the fact and discovering the mechanism of organic evolution.” Darwinian theory helps us to understand both the genetics of populations and long-term trends in the fossil record. Darwin and Wallace did not originate the basic idea of organic evolution, which has an ancient history. We review the history of evolutionary thinking as it led to Darwin’s theory, evidence supporting it, and changes to the theory that have produced our modern synthetic theory of evolution. ORIGINS OF DARWINIAN EVOLUTIONARY THEORY Pre-Darwinian Evolutionary Ideas Before the eighteenth century, speculation on origins of species rested on mythology and superstition, not on anything resem-bling a testable scientifi c theory. Creation myths often described the world remaining constant after a short period of creation. Nevertheless, some people approached the idea that nature has a long history of perpetual and irreversible change. Early Greek philosophers, notably Xenophanes, Emped-ocles, and Aristotle, developed an early idea of evolutionary change. They recognized fossils as evidence for former life that they believed had been destroyed by natural catastrophe. Despite their intellectual inquiry, the Greeks failed to establish an evo-lutionary concept, and the issue declined well before the rise of Christianity. The opportunity for evolutionary thinking became even more restricted as a biblical account of the earth’s creation became accepted as a tenet of faith. The year 4004 B.C. was fi xed by Archbishop James Ussher (mid-seventeenth century) as the date of life’s creation. Evolutionary views were considered rebel-lious and heretical, but they refused to die. The French natural-ist Georges Louis Buffon (1707 to 1788) stressed the infl uence of environment on the modifi cations of animal form. He also extended the age of the earth to 70,000 years. Lamarckism: The First Scientifi c Explanation of Evolution French biologist Jean Baptiste de Lamarck (1744 to 1829; Figure 6.2 ) authored the fi rst complete explanation of evolu-tion in 1809, the year of Darwin’s birth. He made a convincing case that fossils were remains of extinct animals. Lamarck’s proposed evolutionary mechanism, inheritance of acquired characteristics, was engagingly simple: organisms, by striv-ing to meet the demands of their environments, acquire adap-tations and pass them by heredity to their offspring. According to Lamarck, the giraffe evolved its long neck because its ances-tors lengthened their necks by stretching to obtain food and then passed the lengthened neck to their offspring. Over many generations, these changes accumulated to produce the long necks of modern giraffes. We call Lamarck’s concept of evolution transfor-mational, because it claims that as individual organisms transform their characteristics through the use and disuse of parts, heredity makes corresponding adjustments to produce evolution. We now reject transformational the-ories because genetic studies show that traits acquired by an organism during its lifetime, such as strength-ened muscles, are not inherited by offspring. Darwin’s evolutionary theory differs from Lamarck’s in being a variational theory, based on the distribution of genetic variation in populations. Evolutionary change is caused by differential survival and reproduction among organ-isms that differ in hereditary traits, not by inheritance of acquired characteristics. Charles Lyell and Uniformitarianism The geologist Sir Charles Lyell (1797 to 1875; Figure 6.3 ) established in his Principles of Geology (1830 to 1833) the principle of uniformitarianism. Uniformitarianism encom-passes two important principles that guide scientifi c study of the history of nature: (1) that the laws of physics and chemistry have not changed throughout the history of the earth, and (2) that past geological events occurred by natural processes similar to those observed today. Lyell showed that natural forces, acting over long periods A B Figure 6.1 Founders of the theory of evolution by natural selection. A, Charles Robert Darwin (1809 to 1882), as he appeared in 1881, the year before his death. B, Alfred Russel Wallace (1823 to 1913) in 1895. Darwin and Wallace independently developed the same theory. A letter and essay from Wallace written to Darwin in 1858 spurred Darwin into writing On The Origin of Species, published in 1859.
106 PART TWO Continuity and Evolution of Animal Life of time, could explain the formation of fossil-bearing rocks. Lyell’s geological studies led him to conclude that the earth’s age must be measured in millions of years. These principles were important for discrediting miraculous and supernatural explana-tions of the history of nature and replacing them with scientifi c explanations. Lyell also stressed the gradual nature of geological changes that occur through time, and he argued further that such changes have no inherent tendency to occur in any particular direction. Both of these claims left important marks on Darwin’s evolutionary theory. Darwin’s Great Voyage of Discovery “After having been twice driven back by heavy southwestern gales, Her Majesty’s ship Beagle, a ten-gun brig, under the com-mand of Captain Robert FitzRoy, R.N., sailed from Devonport on the 27th of December, 1831.” Thus began Charles Darwin’s account of the historic fi ve-year voyage of the Beagle around the world ( Figure 6.4 ). Darwin, not quite 23 years old, had been asked to accompany Captain FitzRoy on the Beagle, a small ves-sel only 90 feet in length, which was about to depart on an Figure 6.2 Jean Baptiste de Lamarck (1744 to 1829), French naturalist who offered the fi rst scientifi c explanation of evolution. Lamarck’s hypothesis that evolution proceeds by inheritance of acquired characteristics has been rejected and replaced by neo-Darwinian theories.
Figure 6.4 Five-year voyage of H.M.S. Beagle.
Figure 6.3 Sir Charles Lyell (1797 to 1875), English geologist and friend of Darwin. His book Principles of Geology greatly infl uenced Darwin during Darwin’s formative period. This photograph was made about 1856.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 107 extensive surveying voyage to South America and the Pacifi c ( Figure 6.5 ). It was the beginning of the most important scientifi c voyage of the nineteenth century. During the voyage (1831 to 1836), Darwin endured sea-sickness and the erratic companionship of Captain FitzRoy, but Darwin’s youthful physical strength and early training as a naturalist equipped him for his work. The Beagle made many stops along the coasts of South America and adjacent islands. Darwin made extensive collections and observations on the fauna and fl ora of these regions. He unearthed numerous fos-sils of animals long extinct and noted the resemblance between fossils of the South American pampas and the known fossils of North America. In the Andes he encountered seashells embed-ded in rocks at 13,000 feet. He experienced a severe earth-quake and watched mountain torrents that relentlessly wore away the earth. These observations, and his reading of Lyell’s Principles of Geology during the voyage, strengthened his con-viction that natural forces could explain the geological features of the earth. In mid-September of 1835, the Beagle arrived at the Galá-pagos Islands, a volcanic archipelago straddling the equator 600 miles west of Ecuador ( Figure 6.6 ). The fame of the islands stems from their oceanic isolation and rugged volcanic terrain. Circled by capricious currents, surrounded by shores of twisted lava bearing skeletal brushwood baked by the equatorial sun, inhabited by strange reptiles and by convicts stranded by the Ecuadorian government, the islands had few admirers among mariners. By the middle of the seventeenth century, the islands were known to Spaniards as “Las Islas Galápagos”—the tortoise islands. The giant tortoises, used for food fi rst by buccaneers and later by American and British whalers, sealers, and ships of war, were the islands’ principal attraction. At the time of Darwin’s visit, the tortoises already were heavily exploited. During the Beagle ’s fi ve-week visit to the Galápagos, Darwin documented the unique character of the Galápagos plants and animals, including the giant tortoises, marine iguanas, mocking-birds, and ground fi nches. Darwin later described these studies as the “origin of all my views.” Darwin was struck by the fact that, although the Galápa-gos Islands and the Cape Verde Islands (visited earlier in this voyage of the Beagle ) were similar in climate and topography, Galápagos plants and animals were related to those of the South American mainland and were entirely different from the African-derived forms of the Cape Verde Islands. Each Galápagos Island often contained a unique species related to forms on other Galá-pagos Islands. In short, Galápagos life must have originated in continental South America and then undergone modifi cation in the various environmental conditions of the different islands. He Figure 6.5 Charles Darwin and H.M.S. Beagle. A, Darwin in 1840, four years after the Beagle returned to England, and a year after his marriage to his cousin, Emma Wedgwood. B, The H.M.S. Beagle sails in Beagle Channel, Tierra del Fuego, on the southern tip of South America in 1833. The watercolor was painted by Conrad Martens, one of two offi cial artists on the voyage of the Beagle.
A B Figure 6.6 The Galápagos Islands viewed from the rim of a volcano.
108 PART TWO Continuity and Evolution of Animal Life concluded that living forms were neither divinely created nor immutable; they were, in fact, products of evolution. On October 2, 1836, the Beagle returned to England, where Darwin conducted most of his scientifi c work ( Figure 6.7 ). Most of Darwin’s extensive collections had preceded him there, as had notebooks and diaries kept during the cruise. Darwin’s journal, published three years after the Beagle ’s return to England, was an instant success and required two additional printings within the fi rst year. Darwin later revised his journal as The Voyage of the Beagle , one of the most lasting and popular travel books. The main product of Darwin’s voyage, his theory of evolu-tion, would continue to develop for more than 20 years after the Beagle ’s return. In 1838, he “happened to read for amusement” an essay on populations by T. R. Malthus (1766 to 1834), who stated that animal and plant populations, including human popu-lations, tend to increase beyond the capacity of the environment to support them. Darwin already had been gathering information on artifi cial selection of animals under domestication. Darwin was especially fascinated by artifi cial breeds of pigeons. Many pigeon breeds differed so much in appearance and behavior that they would be considered different species if found in nature, yet all had clearly been derived from a single wild species, the rock pigeon ( Columba livia ). After reading Malthus’s article, Dar-win realized that a process of selection in nature, a “struggle for existence” because of overpopulation, could be a powerful force for evolution of wild species. Darwin allowed the idea to develop in his own mind until it was presented in 1844 in a still-unpublished essay. Finally in 1856, he began to assemble his voluminous data into a work on the origin of species. He expected to write four volumes, a very big book, “as perfect as I can make it.” However, his plans took an unexpected turn. In 1858, he received a manuscript from Alfred Russel Wallace (1823 to 1913), an English naturalist in Malaya with whom he corre-sponded. Darwin was stunned to fi nd that in a few pages, Wallace summarized the main points of the natural selection theory on which Darwin had been working for two decades. Rather than to withhold his own work in favor of Wallace as he was inclined to do, Darwin was persuaded by two close friends, the geologist Lyell and the botanist Hooker, to publish his views in a brief statement that would appear together with Wallace’s paper in the Journal of the Linnean Society. Portions of both papers were read to an unimpressed audience on July 1, 1858. For the next year, Darwin worked urgently to prepare an “abstract” of the planned four-volume work. This book was pub-lished in November 1859, with the title On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. The 1250 copies of the fi rst printing sold the fi rst day! The book instantly generated a storm that has never abated. Darwin’s views were to have extraordinary conse-quences on scientifi c and religious beliefs and remain among the greatest intellectual achievements of all time. “Whenever I have found that I have blundered, or that my work has been imperfect, and when I have been contemptuously criti-cized, and even when I have been overpraised, so that I have felt mortifi ed, it has been my greatest comfort to say hundreds of times to myself that ‘I have worked as hard and as well as I could, and no man can do more than this.’ ” Charles Darwin, in his autobiography, 1876. Once Darwin’s caution had been swept away by the pub-lication of On the Origin of Species, he entered an incredibly productive period of evolutionary thinking for the next 23 years, producing book after book. He died on April 19, 1882, and was buried in Westminster Abbey. The little Beagle had already dis-appeared, having been retired in 1870 and sold for scrap. DARWINIAN EVOLUTIONARY THEORY: THE EVIDENCE Perpetual Change The main premise underlying Darwinian evolution is that the liv-ing world is neither constant nor perpetually cycling, but always changing. Perpetual change in the form and diversity of animal life throughout its 600- to 700-million-year history is seen most directly in the fossil record. A fossil is a remnant of past life uncovered from the crust of the earth ( Figure 6.8 ). Some fossils constitute complete remains (insects in amber and mammoths), actual hard parts (teeth and bones), and petrifi ed skeletal parts infi ltrated with silica or other minerals (ostracoderms and mol-luscs). Other fossils include molds, casts, impressions, and fossil excrement (coprolites). In addition to documenting organismal evolution, fossils reveal profound changes in the earth’s environ-ment, including major changes in the distributions of lands and seas. Because many organisms left no fossils, a complete record of the past is always beyond our reach; nonetheless, discovery of new fossils and reinterpretation of familiar ones expand our knowledge of how the form and diversity of animals changed through geological time. Figure 6.7 Darwin’s study at Down House in Kent, England, is preserved today much as it was when Darwin wrote On The Origin of Species.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 109 Fossil remains may on rare occasions include soft tissues pre-served so well that recognizable cellular organelles are revealed by electron microscopy! Insects are frequently found entombed in amber, the fossilized resin of trees. One study of a fl y entombed in 40-million-year-old amber revealed structures corresponding to muscle fi bers, nuclei, ribosomes, lipid droplets, endoplasmic reticulum, and mitochondria ( Figure 6.8D ). This extreme case of mummifi cation probably occurred because chemicals in the plant sap diffused into the embalmed insect’s tissues. Interpreting the Fossil Record The fossil record is biased because preservation is selective. Ver-tebrate skeletal parts and invertebrates with shells and other hard structures left the best records ( Figure 6.8 ). Soft-bodied animals, including jellyfi shes and most worms, are fossilized only under very unusual circumstances such as those that formed the Bur-gess Shale of British Columbia ( Figure 6.9 ). Exceptionally favor-able conditions for fossilization produced the Precambrian fossil bed of South Australia, the tar pits of Rancho La Brea (Hancock Park, Los Angeles), the great dinosaur beds (Alberta, Canada, and Jensen, Utah; Figure 6.10 ), and the Yunnan and Lianoning provinces of China. Fossils are deposited in stratifi ed layers with new depos-its forming on top of older ones. If left undisturbed, which is rare, a sequence is preserved with the ages of fossils being directly proportional to their depth in the stratifi ed layers. Char-acteristic fossils often serve to identify particular layers. Certain widespread marine invertebrate fossils, including various fora-miniferans (p. 242) and echinoderms (p. 472), are such good indicators of specifi c geological periods that they are called “index,” or “guide,” fossils. Unfortunately, the layers are usually tilted or show faults (cracks). Old deposits exposed by erosion may be covered with new deposits in a different plane. When exposed to tremendous pressures or heat, stratifi ed sedimentary rock metamorphoses into crystalline quartzite, slate, or marble, which destroys fossils. Stratigraphy for two major groups of African antelopes and its evolutionary interpretation are shown on Figure 6.11 . Species in this group are identifi ed by characteristic sizes and shapes of horns, which form much of the fossil record of this group. Solid vertical lines in Figure 6.11 denote the temporal distributions of species determined by presence of their characteristic horns in rock strata of various ages. Red lines denote the fossil records of living species, and gray lines denote the fossil records of extinct species. The dotted gray lines show the inferred relationships among living and fossil species based on their sharing of homol-ogous structural features. Geological Time Long before the earth’s age was known, geologists divided its history into a table of succeeding events based on the ordered layers of sedimentary rock. The “law of stratigraphy” produced a relative dating with the oldest layers at the bottom and the youngest at the top of the sequence. Time was divided into eons, eras, periods, and epochs as shown on the endpaper inside the back cover of this book. Time during the last eon (Phanerozoic) is expressed in eras (for example, Cenozoic), periods (for exam-ple, Tertiary), epochs (for example, Paleocene), and sometimes smaller divisions of an epoch. In the late 1940s, radiometric dating methods were devel-oped for determining the absolute age in years of rock forma-tions. Several independent methods are now used, all based on Figure 6.8 Four examples of fossil material. A, Fish fossil from rocks of the Green River Formation, Wyoming. Such fi sh swam here during the Eocene epoch of the Tertiary period, approximately 55 million years ago. B, Stalked crinoids (class Crinoidea, p. 487) from 85-million-year-old Cretaceous rocks. The fossil record of these echinoderms shows that they reached their peak millions of years earlier and began a slow decline to the present. C, An insect fossil that got stuck in the resin of a tree 40 million years ago and that has since hardened into amber. D, Electron micrograph of tissue from a fl y fossilized as shown in C; the nucleus of a cell is marked in red.
A B C D 110 PART TWO Continuity and Evolution of Animal Life Amiskwia, from an extinct phylum Odontogriphus, from an extinct phylum Anomalocaris canadensis, from an extinct phylum Eldonia, a possible echinoderm Pikaia, an early chordate Canadia, a polychaete Marrella splendens, a unique arthropod Opabinia, from an extinct phylum Aysheaia, an onychophoran Ottoia, a priapulid Wiwaxia, from an extinct phylum Hallucigenia, from an extinct phylum Yohoia, a unique arthropod Sidneyia, a unique arthropod Dinomischus, from an extinct phylum Xianguangia, an anenome-like animal Halichondrites, a sponge Figure 6.9 A, Fossil trilobites visible at the Burgess Shale Quarry, British Columbia. B, Animals of the Cambrian period, approximately 580 million years ago, as reconstructed from fossils preserved in the Burgess Shale of British Columbia, Canada. The main new body plans that appeared rather abruptly at this time established the body plans of animals familiar to us today.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 111 the radioactive decay of naturally occurring elements into other elements. These “radioactive clocks” are independent of pres-sure and temperature changes and therefore are not affected by often violent earth-building activities. One method, potassium-argon dating, uses the decay of potassium-40 ( 40 K) to argon-40 ( 40 Ar) (12%) and calcium-40 ( 40 Ca) (88%). The half-life of potassium-40 is 1.3 billion years; half of the original atoms will decay in 1.3 billion years, and half of the remaining atoms will be gone at the end of the next 1.3 billion years. This decay continues until all radioactive potassium-40 atoms are gone. To measure the age of the rock, one calcu-lates the ratio of remaining potassium-40 atoms to the amount of potassium-40 originally there (the remaining potassium-40 atoms plus the argon-40 and calcium-40 into which other potassium-40 atoms have decayed). Several such isotopes exist for dating purposes, some for dating the age of the earth itself. One of the most useful radioactive clocks depends on the decay of uranium into lead. With this method, rocks over 2 billion years old can be dated with a probable error of less than 1%. Figure 6.10 A dinosaur skeleton partially excavated from rock at Dinosaur Provincial Park, Alberta.
Blesbok — Hartebeest — Wildebeest Group Millions of years before present 0 1 2 3 4 5 6 ?
Impalas Figure 6.11 Stratigraphic record and inferred evolutionary relationships among alcelaphine (blesboks, hartebeests, wildebeests) and aepycerotine (impalas) antelopes in Africa. Species in this group are identifi ed by characteristic sizes and shapes of horns found in rock strata of various ages. Solid vertical lines show the temporal distribution of species in rock strata whose ages are shown on the scale at the left side of the fi gure. Red lines show the temporal distributions of living species, and gray lines show the temporal distributions of extinct species in rock strata. Dotted gray lines show the inferred relationships among species based on their sharing of homologous structural features. The relative constancy of horn structure within species through geological time is consistent with the theory of punctuated equilibrium (p. 123). This fossil record shows that rates of speciation and extinction are higher for alcelaphine antelopes than for impalas.
112 PART TWO Continuity and Evolution of Animal Life The fossil record of macroscopic organisms begins near the start of the Cambrian period of the Paleozoic era, approximately 600 million years BP . Geological time before the Cambrian is called the Precambrian era or Proterozoic eon. Although the Precambrian era occupies 85% of all geological time, it has received much less attention than later eras, partly because oil, which provides the com-mercial incentive for much geological work, seldom exists in Precambrian formations. The Precambrian era contains well-preserved fossils of bacteria and algae, and casts of jellyfi shes, sponge spicules, soft corals, segmented fl atworms, and worm trails. Most, but not all, are microscopic fossils. Evolutionary Trends The fossil record allows us to view evolutionary change across the broadest scale of time. Species arise and go extinct repeatedly throughout the geological history recorded by the fossil record. Animal species typically survive approximately 1 to 10 million years, although their durations are highly variable. When we study patterns of species or taxon replacement through time, we observe trends. Trends are directional changes in the char-acteristic features or patterns of diversity in a group of organisms. Fossil trends clearly demonstrate Darwin’s principle of perpetual change. Our use of the phrase “evolutionary trend” does not imply that more recent forms are superior to older ones or that the changes represent progress in adap-tation or organismal complexity. Although Darwin predicted that such trends would show progressive adaptation, many contemporary paleontologists con-sider progressive adaptation rare among evolutionary trends. Observed trends in the evolution of horses do not imply that contemporary horses are superior in any general sense to their Eocene ancestors. A well-studied fossil trend is the evolution of horses from the Eocene epoch to the present. Looking back at the Eocene epoch, we see many different genera and species of horses replaced by others through time ( Figure 6.12 ). George Gaylord Simpson (p. 208) showed that this trend is compat-ible with Darwinian evolutionary theory. The three characteristics that show the clearest trends in horse evolution are body size, foot structure, and tooth structure. Compared to modern horses, those of extinct genera were small; their teeth had a rela-tively small grinding surface, and their feet had a Figure 6.12 A reconstruction of genera of horses from Eocene to present. Evolutionary trends toward increased size, elaboration of molars, and loss of toes are shown together with a hypothetical phylogeny of extant and fossil genera.
Eohippus Hyracotherium Orohippus Epihippus Mesohippus Miohippus Parahippus Merychippus Callipus Pliohippus Nannipus Equus Styohipparion Hippidium and other genera Archaeohippus Hypohippus Anchitherium Megahippus Hypohippus Hipparion Neohipparion Pachynolophus Propalaeotherium Paleotherium RECENT PLEISTOCENE PLIOCENE MIOCENE OLIGOCENE EOCENE www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 113 relatively large number of toes (four). Throughout the subse-quent Oligocene, Miocene, Pliocene, and Pleistocene epochs are continuing patterns of new genera arising and old ones going extinct. In each case, a net increase in body size, expansion of the grinding surface of the teeth, and reduction in the number of toes occurred. As the number of toes was reduced, the central digit became increasingly more prominent in the foot, and even-tually only this central digit remained. The fossil record shows a net change not only in the charac-teristics of horses but also variation in the numbers of different horse genera (and numbers of species) through time. The many horse genera of past epochs have been lost to extinction, leav-ing only a single survivor, Equus. Evolutionary trends in diver-sity are observed in fossils of many different groups of animals ( Figure 6.13 ). Trends in fossil diversity through time are produced by dif-ferent rates of species formation versus extinction through time. Why do some lineages generate large numbers of new species whereas others generate relatively few? Why do different lin-eages undergo higher or lower rates of extinction (of species, genera, or taxonomic families) throughout evolutionary time? To answer these questions, we must turn to Darwin’s other four theories of evolution. Regardless of how we answer these ques-tions, however, the observed trends in animal diversity clearly illustrate Darwin’s principle of perpetual change. Because the remaining four theories of Darwinism rely on the theory of per-petual change, evidence supporting these theories strengthens Darwin’s theory of perpetual change. Common Descent Darwin proposed that all plants and animals have descended from an ancestral form into which life was fi rst breathed. Life’s history is depicted as a branching tree, called a phylogeny. Pre-Darwinian evolutionists, including Lamarck, advocated multiple independent origins of life, each of which gave rise to lineages that changed through time without extensive branching. Like all good scientifi c theories, common descent makes several important predictions that can be tested and potentially used to reject it. According to this theory, we should be able to trace the genealogies of all mod-ern species backward until they converge on ancestral lineages shared with other species, both living and extinct. We should be able to continue this process, moving farther backward through evolutionary time, until we reach the primor-dial ancestor of all life on earth. All forms of life, including many extinct forms that represent dead branches, will connect to this tree somewhere. Although reconstructing the history of life in this manner may seem almost impossible, phylogenetic research has been extraordinarily successful. How has this diffi cult task been accomplished? Figure 6.13 Diversity profi les of taxonomic families from different animal groups in the fossil record. The scale marks the Precambrian (PC), Paleozoic (P), Mesozoic (M), and Cenozoic (C) eras. The relative number of families is indicated from the width of the profi le.
Polychaeta Pycnogonida Trilobita Ostracoda Copepoda Cirripedia Malacostraca Echinoidea Stelleroidea Holothuroidea Crinoidea Agnatha Placodermi Chondrichthyes Osteichthyes Reptilia Mammalia Hexactinellida Demospongiae Calcarea Sclerospongiae Hydrozoa Scyphozoa Anthozoa Nemertea Polyplacophora Monoplacophora Gastropoda Scaphopoda Bivalvia Cephalopoda Sipuncula Echiura C M P PC C M P PC 10 Families Scale: Time Time 114 PART TWO Continuity and Evolution of Animal Life Darwin’s theory of common descent illustrates the scientifi c impor-tance of general theories that give unifi ed explanations to diverse kinds of data. Darwin proposed his theory of descent with modifi ca-tion of all living forms because it explained the patterns of similar-ity and dissimilarity among organisms in anatomical structures and cellular organization.
Anatomical similarities between humans and apes led Darwin to propose that humans and apes share more recent common ances-try with each other than they do with any other species. Darwin was unaware that his theory, a century later, would provide the pri-mary explanation for similarities and dissimilarities among species in the structures of their chromosomes, sequences of amino acids in homologous proteins, and sequences of bases in homologous genomic DNA.
The accompanying figure shows photographs of a complete haploid set of chromosomes from each of four ape species: human (Homo sapiens), bonobo (the pygmy chimpanzee, Pan paniscus), gorilla (Gorilla gorilla), and orangutan (Pongo pygmaeus). Each chromosome in the human genome has a corresponding chromo-some with similar structure and gene content in the genomes of other ape species. The most obvious difference between human and ape chromosomes is that the large second chromosome in the human nuclear genome was formed evolutionarily by a fusion of two smaller chromosomes characteristic of the ape genomes. Detailed study of the human and other ape chromosomes shows remarkable correspondence between them in genic content and organization. Ape chromosomes are more similar to each other than they are to chromosomes of any other animals.
Comparison of DNA and protein sequences among apes likewise confi rms their close genetic relationships, with humans and the two chimpanzee species being closer to each other than any of these species are to other apes. DNA sequences from the nuclear and mitochondrial genomes independently support the close relation-ships among ape species and especially the grouping of humans and chimpanzees as close relatives. Homologous DNA sequences of humans and chimpanzees are approximately 99% similar in base sequence.
Studies of variation in chromosomal structure, mitochondrial DNA sequences, and nuclear DNA sequences produced multiple independent data sets, each one potentially capable of reject-ing Darwin’s theory of common descent. Darwin’s theory would be rejected, for example, if the chromosomal structures and DNA sequences of apes were no more similar to each other than to those of other animals. The data in this case support rather than reject predictions of Darwin’s theory. The ability of Darwin’s theory of common descent to make precise predictions of genetic similarities among these and other species, and to have those predictions con-fi rmed by numerous empirical studies, illustrates its great strength. As new kinds of biological data have become available, the scope and strength of Darwin’s theory of common descent have increased enormously. Indeed, nothing in biology makes sense in the absence of this powerful explanatory theory.
The Power of a Theory The human haploid genome contains 22 autosomes (I–XXII) and a sex chromosome (X or Y). The human chromosome is shown fi rst in each group of four, followed by the corresponding chromosomes of bonobo, gorilla, and orangutan, in that order. Note that the chromatin of human chromosome II corresponds to that of two smaller chromosomes (marked p and q) in other apes.
Comparative Karyotype of Great Apes www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 115 Figure 6.15 This 1873 advertisement for Merchant’s Gargling Oil ridicules Darwin’s theory of the common descent of humans and apes, which received only limited acceptance by the general public during Darwin’s lifetime.
Figure 6.14 Forelimbs of fi ve vertebrates show skeletal homologies: brown, humerus; orange, radius and ulna; purple, “hand” (carpals, metacarpals, and phalanges). Clear homologies of bones and patterns of connection are evident despite evolutionary modifi cation for different functions.
Human Bat Porpoise Horse Frog Homology and Phylogenetic Reconstruction Darwin recognized the major source of evidence for common descent in the concept of homology. Darwin’s contemporary, Richard Owen (1804 to 1892), used this term to denote “the same organ in different organisms under every variety of form and function.” A classic example of homology is the limb skel-eton of vertebrates. Bones of vertebrate limbs maintain char-acteristic structures and patterns of connection despite diverse modifi cations for different functions ( Figure 6.14 ). According to Darwin’s theory of common descent, the structures that we call homologies represent characteristics inherited with some modifi cation from a corresponding feature in a common ancestor. Darwin devoted an entire book, The Descent of Man and Selection in Relation to Sex, largely to the idea that humans share common descent with apes and other animals. This idea was repugnant to many Victorians, who responded with predictable outrage ( Figure 6.15 ). Darwin built his case mostly on anatomi-cal comparisons revealing homology between humans and apes. To Darwin, the close resemblances between apes and humans could be explained only by common descent. Throughout the history of all forms of life, evolutionary processes generate new characteristics that are then inherited by subsequent generations. Every time a new feature arises on an evolving lineage, we see the origin of a new homology. That homology gets transmitted to all descendant lineages unless it is subsequently lost. The pattern formed by the sharing of homol-ogies among species provides evidence for common descent and allows us to reconstruct the branching evolutionary history of life. We can illustrate such evidence using a phylogenetic tree for a group of large, ground-dwelling birds ( Figure 6.16 ). A new skeletal homology arises on each of the lineages shown (descriptions of specifi c homologies are not included because they are highly technical). The different groups of species located at the tips of the branches contain different combina-tions of these homologies, which refl ect ancestry. For example, ostriches show homologies 1 through 5 and 8, whereas kiwis show homologies 1, 2, 13, and 15. Branches of the tree com-bine these species into a nested hierarchy of groups within groups (see Chapter 10). Smaller groups (species grouped near terminal branches) are contained within larger ones (species grouped by basal branches, including the trunk of the tree). If we erase the tree structure but retain patterns of homology observed in the living species, we are able to reconstruct the branching structure of the entire tree. Evolutionists test the theory of common descent by observing patterns of homology present within all groups of organisms. The pattern formed by all homologies taken together should specify a single branch-ing tree that represents the evolutionary genealogy of all living organisms. The nested hierarchical structure of homology is so per-vasive in the living world that it forms the basis for our sys-tematic classifi cation of all forms of life (genera grouped into families, families grouped into orders, and other categories). Hierarchical classification even preceded Darwin’s theory 116 PART TWO Continuity and Evolution of Animal Life because this pattern is so evident, but it was not explained sci-entifi cally before Darwin. Once the idea of common descent was accepted, biologists began investigating the structural, molecular, and chromosomal homologies of animal groups. Taken together, the nested hierarchical patterns uncovered by these studies have permitted us to reconstruct evolutionary trees of many groups and to continue investigating others. Use of Darwin’s theory of common descent to reconstruct the evo-lutionary history of life and to classify animals is the subject of Chapter 10. Note that the earlier evolutionary hypothesis that life arose many times, forming unbranched lineages, predicts linear sequences of evolutionary change with no nested hierarchy of homologies among species. Because we do observe nested hier-archies of homologies, that hypothesis is rejected. Note also that because the creationist argument is not a scientifi c hypothesis, it can make no testable predictions about any pattern of homology and therefore fails to meet the criteria of a scientifi c theory of animal diversity. Characters of different organisms that perform similar functions are not necessarily homologous. The wings of bats and birds, although homologous as vertebrate forelimbs, are not homologous as wings. The most recent common ancestor of bats and birds had forelimbs, but the forelimbs were not in the form of wings. Wings of bats and birds evolved independently and have only superfi cial similarity in their fl ight structures. Bat wings are formed by skin stretched over elongated digits, whereas bird wings are formed by feathers attached along the fore-limb. Such functionally similar but nonhomologous structures are often termed analogues. Ontogeny, Phylogeny, and Recapitulation Ontogeny is the history of the development of an organism through its entire life. Early developmental and embryological features contribute greatly to our knowledge of homology and common descent. Comparative studies of ontogeny show how the evolutionary alteration of developmental timing generates new characteristics, thereby producing evolutionary divergence among lineages. The German zoologist Ernst Haeckel, a contemporary of Darwin, proposed that each successive stage in the develop-ment of an individual represented one of the adult forms that appeared in its evolutionary history. The human embryo with gill depressions in the neck corresponded, for example, to the adult appearance of a fi shlike ancestor. On this basis Haeckel gave his generalization: ontogeny (individual development) reca-pitulates (repeats) phylogeny (evolutionary descent). This notion later became known simply as recapitulation or the biogene-tic law. Haeckel based his biogenetic law on the fl awed prem-ise that evolutionary change occurs by successively adding new features onto the end of an unaltered ancestral ontogeny while condensing the ancestral ontogeny into earlier developmen-tal stages. This notion was based on Lamarck’s concept of the inheritance of acquired characteristics (p. 105). The nineteenth-century embryologist, K. E. von Baer, gave a more satisfactory explanation of the relationship between ontog-eny and phylogeny. He argued that early developmental features were simply more widely shared among different animal groups than later ones. Figure 6.17 shows, for example, the early embry-ological similarities of organisms whose adult forms are very different (see Figure 8.21 , p. 176). The adults of animals with relatively short and simple ontogenies often resemble pre-adult stages of other animals whose ontogeny is more elaborate, but embryos of descendants do not necessarily resemble the adults of their ancestors. Even early development undergoes evolution-ary divergence among lineages, however, and it is not as stable as von Baer proposed. We now know many parallels between ontogeny and phy-logeny, but features of an ancestral ontogeny can be shifted either to earlier or later stages in descendant ontogenies. Evolution-ary change in timing of development is called heterochrony, Figure 6.16 The phylogenetic pattern specifi ed by 15 homologous structures in the skeletons of a group of fl ightless birds. Homologous features are numbered 1 through 15 and are marked both on the branches of the tree on which they arose and on the birds that have them. If you were to erase the tree structure, you would be able to reconstruct it without error from the distributions of homologous features shown for the birds at the terminal branches.
1 2 3 4 13 12 15 14 11 10 9 6 5 8 7 Rhea (1, 2, 3, 4, 5, 7) Ostrich (1, 2, 3, 4, 5, 8) Emu (1, 2, 3, 4, 6, 9) Cassowary (1, 2, 3, 4, 6, 10) Elephant bird (1, 2, 3, 11) Moa (1, 2, 13, 14) Kiwi (1, 2, 13, 15) Tinamou (1, 12) www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 117 a term initially used by Haeckel to denote exceptions to reca-pitulation. If a descendant’s ontogeny extends beyond its ances-tral one, new characteristics can be added late in development, beyond the point at which development would have ended in an evolutionary ancestor. Features observed in the ancestor often are moved to earlier stages of development in this process, and ontogeny therefore does recapitulate phylogeny to some degree. Ontogeny also can be shortened during evolution, however. Ter-minal stages of the ancestor’s ontogeny can be deleted, causing adults of descendants to resemble pre-adult stages of their ances-tors ( Figure 6.18 ). This outcome reverses the parallel between ontogeny and phylogeny (reverse recapitulation) producing pae-domorphosis (the retention of ancestral juvenile characters by descendant adults). Because lengthening or shortening of ontog-eny can change different parts of the body independently, we often see a mosaic of different kinds of developmental evolution-ary change occurring concurrently. Therefore, cases in which an entire ontogeny recapitulates phylogeny are rare. Developmental Modularity and Evolvability Evolutionary innovations occur not only by simple changes in rates of developmental processes but by changes in the physical location in the body where a process is activated. Heterotopy is the term traditionally used to describe a change in the physical location of a developmental process in an organism’s body. For such a change to be successful, the developmental process must be compartmentalized into semiautonomous modules whose expression can be activated in new locations. An interesting example of modularity and heterotopy occurs in some geckos. Geckos as a group typically have toepads, adhesive structures on the ventral side of the toes that permit climbing and clinging to smooth surfaces. Toepads consist of modifi ed scales containing long protrusions, called setae, which can be molded to the surface of a substrate. A module responsible for toepad devel-opment is expressed in an unusual gecko species not only on the toes but also on the ventral side of the tip of the tail. This species has thereby acquired an additional adhesive appendage by ectopic expression of a standard developmental module. Modularity is evident also in the homeotic mutations of the fruit fl y, Drosophila melanogaster . Such mutations can substitute a developmental module for a leg in place of one normally spec-ifying an antenna, thereby producing a fl y with a pair of legs on its head. Another homeotic mutation in fruit fl ies transforms the balancer organs on the thorax into a second pair of wings; the balancer module is replaced by activation of the wing module, which in fl ies is activated normally only in a more anterior por-tion of the thorax. Modularity is important in explaining some major evolution-ary changes, such as evolution of tetrapod limbs (p. 546). The evolutionary transition from fi nlike limbs to the standard tetrapod limbs occurred by activating at the site of limb formation a set of homeobox genes (p. 173) whose expression pattern evolved initially as a module for forming part of the vertebral column. Shared patterns of gene expression between the vertebral col-umn and the forelimbs and hindlimbs of tetrapods revealed the genetic and developmental mechanics of this module. The term evolvability has been introduced recently to denote the great evolutionary opportunities created by having semiautonomous developmental modules whose expression can be moved from one part of the body to another. An evolv-ing lineage that contains a large modular developmental toolkit can “experiment” with the construction of many new structures, some of which will persist and give rise to new homologies. Multiplication of Species Multiplication of species through time is a logical corollary to Darwin’s theory of common descent. A branch point on the evolutionary tree means that an ancestral species has split into two different ones. Darwin’s theory postulates that genetic variation present within a species, especially variation that occurs between geographically separated populations, provides the material from which new species are produced. Because evolution is a branching process, the total number of species Figure 6.17 Comparison of gill arches of different embryos. All are shown separated from the yolk sac. Note the remarkable similarity of the four embryos at this early stage in development.
Human Bird Reptile Fish Figure 6.18 Aquatic and terrestrial forms of axolotls. Axolotls retain the juvenile, aquatic morphology (top) throughout their lives unless forced to metamorphose (bottom) by hormone treatment. Axolotls evolved from metamorphosing ancestors, an example of paedomorphosis.
Transformation to adult form Axolotl (nontransformed) HORMONE TREATMENT 118 PART TWO Continuity and Evolution of Animal Life produced by evolution increases through time, although most of these species eventually become extinct without leaving descendant species. A major challenge for evolutionists is to discover the process by which an ancestral species “branches” to form two or more descendant species. Before we explore multiplication of species, we must decide what we mean by “species.” As explained in Chapter 10, no con-sensus exists regarding defi nition of species. Most biologists agree, however, that important criteria for recognizing species include (1) descent of all members from a common ancestral population forming a lineage of ancestor-descendant populations, (2) repro-ductive compatibility (ability to interbreed) within and reproduc-tive incompatibility between species for sexually reproducing animals, and (3) maintenance within species of genotypic and phenotypic cohesion (lack of abrupt differences among popula-tions in allelic frequencies and organismal characteristics). The criterion of reproductive compatibility has received the greatest attention in studies of species formation, also called speciation. Biological features that prevent different species from inter-breeding are called reproductive barriers. The primary prob-lem of speciation is to discover how two initially compatible populations evolve reproductive barriers that cause them to become distinct, separately evolving lineages. How do popula-tions diverge from each other in their reproductive properties while maintaining complete reproductive compatibility within each population? Reproductive barriers between populations usually evolve gradually. Evolution of reproductive barriers requires that diverg-ing populations must be kept physically separate for long peri-ods of time. If diverging populations reunite before reproductive barriers have evolved, interbreeding occurs between the popula-tions and they merge. Speciation by gradual divergence in ani-mals may require extraordinarily long periods of time, perhaps 10,000 to 100,000 years or more. Geographical isolation followed by gradual divergence is the most effective way for reproductive barriers to evolve, and many evolutionists consider geographical separation a prerequisite for branching speciation. Geographical barriers between populations are not the same thing as reproductive barriers. Geographical barriers refer to spatial sepa-ration of two populations. They prevent gene exchange and are usually a precondition for speciation. Reproductive barriers result from evolution and refer to various behavioral, physical, physi-ological, and ecological factors that prevent interbreeding between different species. Behavioral barriers often evolve faster than other kinds of reproductive barriers. Geographical barriers do not guar-antee that reproductive barriers will evolve. Reproductive barriers are most likely to evolve under conditions that include small popu-lation size, a favorable combination of selective factors, and long periods of geographical isolation. One or both of a pair of geo-graphically isolated populations may become extinct prior to evolu-tion of reproductive barriers between them. Over the vast span of geological time, however, conditions suffi cient for speciation have occurred millions of times. Allopatric Speciation Allopatric (“in another land”) populations of a species are those that occupy separate geographical areas. Because of their geographical separation, they cannot interbreed, but would be expected to do so if the geographic barriers between them were removed. If populations are allopatric immediately pre-ceding and during evolution of reproductive barriers between them, the resulting speciation is called allopatric speciation or geographic speciation. The separated populations evolve independently and adapt to their respective environments, generating reproductive barriers between them as a result of their separate evolutionary paths. Because their genetic varia-tion arises and changes independently, physically separated populations will diverge genetically even if their environments remain very similar. Environmental change between popula-tions also can promote genetic divergence between them by favoring different phenotypes in the separated populations. Ernst Mayr ( Figure 6.19 ) has contributed greatly to our knowl-edge of allopatric speciation through his studies of speciation in birds. Allopatric speciation begins when a species splits into two or more geographically separated populations. This splitting can happen in either of two ways: by vicariant speciation or by a founder event. Vicariant speciation is initiated when climatic or geological changes fragment a species’s habitat, producing impenetrable barriers that separate different populations geo-graphically. For example, a mammalian species inhabiting a low-land forest could be divided by uplifting of a mountain barrier, sinking and fl ooding of a geological fault, or climatic changes that cause prairie or desert conditions to encroach on the forest. Formation of the isthmus of Panama separated populations of the sea urchin genus Eucidaris , leading to formation of the pair of species shown in Figure 1.5D. Figure 6.19 Ernst Mayr (1904 to 2005), a major contributor to our knowledge of speciation and of evolution in general.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 119 Vicariant speciation has two important consequences. Although the ancestral population is fragmented, individual frag-ments are usually left fairly intact. The vicariant process itself does not induce genetic change by reducing populations to a small size or by transporting them to unfamiliar environments. Another important consequence is that the same vicariant events may fragment several different species simultaneously. For example, fragmentation of a lowland forest most likely would disrupt numerous and diverse species, including salamanders, frogs, snails, and many other forest dwellers. Indeed, the same geographic patterns are observed among closely related species in different groups of organisms whose habitats are similar. Such patterns provide strong evidence for vicariant speciation. An alternative means of initiating allopatric speciation is for a small number of individuals to disperse to a distant place where no other members of their species occur. The dispersing individu-als may establish a new population in what is called a founder event. Allopatric speciation caused by founder events has been observed, for example, in the native fruit fl ies of Hawaii. Hawaii contains numerous patches of forest separated by volcanic lava fl ows. On rare occasions, strong winds can transport a few fl ies from one forest to another, geographically isolated forest where the fl ies are able to start a new population. Sometimes, a single fertilized female may found a new population. Unlike what hap-pens in vicariant speciation, the new population initially has a very small size, which can cause its genetic structure to change quickly and dramatically from that of its ancestral population (see p. 129). When this event happens, phenotypic characteristics that were sta-ble in the ancestral population often reveal unprecedented varia-tion in the new population. As the newly expressed variation is sorted by natural selection, large changes in phenotype and repro-ductive properties occur, hastening evolution of reproductive bar-riers between the ancestral and newly founded populations. The term founder event in its most general usage denotes a disper-sal of organisms from an ancestral population across a geographic barrier to start a new, allopatric population. A founder event does not always directly cause important changes in the genetic constitu-tion of the new population relative to the old one, although such changes are expected when the number of founding individuals is very small (less than 5 to 10 individuals, for example) and the ancestral population has a large amount of genetic variation. A change in the genetic constitution of a newly founded population because of a small number of founders is termed a founder effect , which includes a population bottleneck (p. 129). If a founder effect is so profound that selection acts in new ways on reproductively important characters, the founder event can induce speciation. Founder-induced speciation describes the subset of founder events in which a founder effect hastens species-level divergence of the newly founded population. Speciation in Hawaiian Drosophila as described in the text illustrates founder-induced speciation. Excluded from founder-induced speciation are founder events whose role in speciation is strictly to establish a new allopatric population capable of independent evolutionary change. Surprisingly, we often learn most about the genetics of allo-patric speciation from cases in which formerly separated popula-tions regain geographic contact following evolution of incipient reproductive barriers that are not absolute. The occurrence of mating between divergent populations is called hybridization and offspring of these matings are called hybrids ( Figure 6.20 ). By studying the genetics of hybrid populations, we can identify the genetic bases of reproductive barriers. Biologists often distinguish between reproductive barriers that impair fertilization (premating barriers) and those that impair growth and development, survival, or reproduction of hybrid indi-viduals (postmating barriers). Premating barriers cause members of divergent populations either not to recognize each other as potential mates or not to complete the mating ritual successfully. Details of the horn structures of African antelopes ( Figure 6.11 ) are important for recognizing members of the same species as poten-tial mates. In some other cases, female and male genitalia of the different populations may be incompatible or the gametes may be incapable of fusing to form a zygote. In others, premating barriers may be strictly behavioral, with members of different species being otherwise nearly identical in phenotype. Different species that are indistinguishable in organismal appearance are called sibling species. Sibling species arise when allopatric populations diverge in the seasonal timing of reproduction or in auditory, behavioral, or chemical signals required for mating. Evolutionary divergence in these features can produce effective premating barriers without obvious changes in organismal appearance. Sibling species occur in groups as diverse as ciliates, fl ies, and salamanders. Nonallopatric Speciation Can speciation ever occur without prior geographic separa-tion of populations? Allopatric speciation may seem an unlikely Figure 6.20 Pure and hybrid salamanders. Hybrids are intermediate in appearance between parental populations. A, Pure white-spotted Plethodon teyahalee; B, a hybrid between white-spotted P. teyahalee and red-legged P. shermani, intermediate in appearance for both spotting and leg color; C, pure red-legged P. shermani.
C B A 120 PART TWO Continuity and Evolution of Animal Life explanation for situations where many closely related species occur together in restricted areas that have no traces of physi-cal barriers to animal dispersal. For example, several large lakes around the world contain very large numbers of closely related species of fi sh. The great lakes of Africa (Lake Malawi, Lake Tanganyika, and Lake Victoria) each contain many species of cichlid fi shes that are found nowhere else. Likewise, Lake Baikal in Siberia contains many different species of sculpins that occur nowhere else in the world ( Figure 6.21 ). It is diffi cult to con-clude that these species arose anywhere other than in the lakes they inhabit, and yet those lakes are young on an evolution-ary timescale and have no obvious environmental barriers that would fragment fi sh populations. To explain speciation of fi sh in freshwater lakes and other examples like these, sympatric (“same land”) speciation has been hypothesized. According to this hypothesis, differ-ent individuals within a species become specialized for occu-pying different components of the environment. By seeking and using very specifi c habitats in a single geographic area, different populations achieve suffi cient physical and adaptive separation to evolve reproductive barriers. For example, cich-lid species of African lakes are very different from each other in their feeding specializations. In many parasitic organisms, particularly parasitic insects, different populations may use dif-ferent host species, thereby providing the physical separation necessary for reproductive barriers to evolve. Supposed cases of sympatric speciation have been criticized, however, because the reproductive distinctness of the different populations often is not well demonstrated, so that we may not be observing formation of distinct evolutionary lineages that will become different species. The occurrence of sudden sympatric speciation is perhaps most likely among higher plants. Between one-third and one-half of fl owering plant species may have evolved by polyploidy (doubling of chromosome numbers), without prior geographic isolation of populations. In animals, however, speciation through polyploidy is an exceptional event. Another possible mode of speciation, termed parapatric speciation , is geographically intermediate between allopatric and sympatric speciation. Two species are called parapatric with respect to each other if their geographic ranges are primarily allo-patric but make contact along a borderline that neither species successfully crosses. In parapatric speciation, a geographically continuous ancestral species evolves within its range a border-line across which populations evolve species-level differences while maintaining geographic contact along the border. The simplest model of parapatric speciation is one in which a change in environmental conditions splits the geographic range of a species into two environmentally distinct but geographi-cally adjoining parts. An increase in temperature on a Caribbean island, for example, might cause part of a wet tropical forest to become a dry, sandy one. A lizard species occupying the formerly wet forest might become divided into geographically adjacent wet and dry forest populations. Unlike vicariant allopat-ric speciation, however, the populations in the different habitat types are not isolated by a physical barrier but maintain genetic Figure 6.21 The sculpins of Lake Baikal, products of speciation that occurred within a single lake.
Batrachocottus nikolskii Cottocomephorus inermis Asprocottus herzensteini Abyssocottus godlewskii Procottus jeittelesi minor Comephorus baikalensis Cottus knere Cottinella boulengeri www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 121 interactions along the geographic borderline between the dif-ferent habitat types. The disparity in environmental conditions across the border, nonetheless, causes the populations to evolve as separate lineages adapted to different environments despite some gene exchange between them. A parapatric distribution of species does not necessarily imply that speciation occurred parapatrically. Most cases of para-patrically distributed species show evidence of past allopatry, with subsequent removal of a geographic barrier permitting the two species to make geographic contact but with each species excluding the other one from its geographic territory. The prevalence of parapatric speciation is controversial. This model of speciation predicts that parapatrically distributed populations differ mainly in adaptive features associated with the observed environmental differences but show relative homo-geneity for other genetic variation. Comparisons of parapatrically distributed populations, including those of lizards occupying dif-ferent forest types on Caribbean islands, often show extensive divergence for molecular variation unrelated to adaptive differ-entiation of the populations; such results are better explained by vicariant allopatric speciation than by parapatric speciation. In some cases, geological evidence shows that what is now a single island was physically fragmented into separate islands during warm periods when the sea level was higher than it is now; such evidence likewise favors the interpretation of allopatric specia-tion for parapatrically distributed species whose geographic con-tact occurs in formerly inundated areas. Adaptive Radiation The production of several ecologically diverse species from a common ancestral species is called adaptive radiation , espe-cially when many disparate species arise within a short inter-val of geological time (a few million years). Some of our best examples of adaptive radiation are associated with lakes and young islands, which provide new evolutionary opportunities for aquatic and terrestrial organisms, respectively. Oceanic islands formed by volcanoes are initially devoid of life. They are gradu-ally colonized by plants and animals from a continent or from other islands in separate founder events. The founders encounter ideal situations for evolutionary diversifi cation, because environ-mental resources that were heavily exploited by other species on the mainland are free for colonization on the sparsely populated island. Archipelagoes, such as the Galápagos Islands, greatly increase opportunities for both founder events and ecological diversifi cation. The entire archipelago is isolated from the conti-nent and each island is geographically isolated from the others by sea; moreover, each island is different from every other one in its physical, climatic, and biotic characteristics. Galápagos fi nches illustrate adaptive radiation on an oce-anic archipelago ( Figures 6.22 and 6.23 ). Galápagos finches (the name “Darwin’s fi nches” was popularized in the 1940s by the British ornithologist David Lack) are closely related to each other, but each species differs from others in size and shape of the beak and in feeding habits. If the fi nches were specially created, it would require the strangest kind of coincidence for 13 similar kinds of finches to be created on the Galápagos Islands and nowhere else. Darwin’s fi nches descended from a single ancestral population that arrived from the mainland and subsequently colonized the different islands of the Galápagos archipelago. The fi nches underwent adaptive radiation, occupy-ing habitats that on the mainland were denied to them by the presence of other species better able to exploit those habitats. Galápagos finches thus assumed characteristics of mainland birds as diverse and unfi nchlike as warblers and woodpeckers. A fourteenth Darwin’s fi nch, found on isolated Cocos Island far north of the Galápagos archipelago, is similar in appearance to the Galápagos fi nches and almost certainly descended from the same ancestral founder. Gradualism Darwin’s theory of gradualism opposes arguments for the sud-den origin of species. Small differences, resembling those that we observe among organisms within populations today, are the raw material from which the different major forms of life evolved. This theory shares with Lyell’s uniformitarianism the notion that we must not explain past changes by invoking unusual catastrophic events that are not observed today. If new species originated in single, catastrophic events, we should be able to see such events happening today and we do not. Instead, what we usually observe in natural populations are small, con-tinuous changes in phenotypes. Such continuous changes can produce major differences among species only by accumulating over many thousands to millions of years. A simple statement of Darwin’s theory of gradualism is that accumulation of quantita-tive changes leads to qualitative change. Mayr (see Figure 6.19 ) makes an important distinction between populational gradualism and phenotypic gradualism. Populational gradualism states that new traits become estab-lished in a population by increasing their frequency initially from a small fraction of the population to a majority of the population. Populational gradualism is well established and is not controver-sial. Phenotypic gradualism states that new traits, even those that are strikingly different from ancestral ones, are produced in a series of small, incremental steps. Phenotypic Gradualism Phenotypic gradualism was controversial when Darwin fi rst pro-posed it, and it is still controversial. Not all phenotypic changes are small, incremental ones. Some mutations that appear during artifi cial breeding change the phenotype substantially in a single mutational step. Such mutations traditionally are called “sports.” Sports that produce dwarfi ng are observed in many species, including humans, dogs, and sheep, and have been used by ani-mal breeders to achieve desired results; for example, a sport that deforms the limbs was used to produce Ancon sheep, which cannot jump hedges and are therefore easily contained ( Fig-ure 6.24 ). Many colleagues of Darwin who accepted his other theories considered phenotypic gradualism too extreme. If sporting mutations can be used in animal breeding, why must 122 PART TWO Continuity and Evolution of Animal Life Figure 6.22 Model for evolution of the 13 Darwin’s fi nches on the Galápagos Islands. The model postulates three steps: (1) Immigrant fi nches from the South American mainland reach the Galápagos and colonize an island; (2) once the population becomes established, fi nches disperse to other islands where they adapt to new conditions and change genetically; and (3) after a period of isolation, secondary contact is established between different populations. The two populations are then recognized as separate species if they cannot interbreed successfully.
Española Baltra Santa Fe Santa Cruz Floreana San Cristóbal 3 2 2 2 1 Española Figure 6.23 A, Adaptive radiation in 10 species of Darwin’s fi nches from Santa Cruz, one of the Galápagos Islands. Differences in bills and feeding habits are shown. All apparently descend from a single common ancestral fi nch from the South American continent. B, Woodpecker fi nch, one of the 13 species of Galápagos Islands fi nches, using a slender twig as a tool for feeding. This fi nch worked for about 15 minutes before spearing and removing a wood roach from a break in the tree.
A B www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 123 states that phenotypic evolution is concentrated in relatively brief events of branching speciation, followed by much longer intervals of morphological evolutionary stasis ( Figure 6.26 ). Spe-ciation is an episodic event, having a duration of approximately 10,000 to 100,000 years. Because species may survive for 5 to 10 million years, the speciation event is a “geological instant,” representing 1% or less of a species’s life span. Ten thousand years is plenty of time, however, for Darwinian gradual evo-lution to accomplish dramatic changes. A small fraction of the evolutionary history of a group therefore contributes most of the morphological evolutionary change that we observe. Punctuated equilibrium contrasts with the views of paleontologist George Simpson, who attributed only moderate rates of morphological Figure 6.25 The phyletic gradualist model of evolutionary change in morphology, viewed as proceeding more or less steadily through geological time (millions of years). Bifurcations followed by gradual divergence led to speciation. Note that most morphological change accumulates incrementally within species lineages between branch points, which are not accompanied by unusually large amounts of morphological change.
Time Morphological divergence Figure 6.26 The punctuated equilibrium model sees morphological evolutionary change being concentrated in relatively rapid bursts of branching speciation (lateral lines) followed by prolonged periods of no change throughout geological time (millions of years).
Time Morphological divergence Figure 6.24 The Ancon breed of sheep arose from a “sporting mutation” that caused dwarfi ng of legs. Many of his contemporaries criticized Darwin for his claim that such mutations are not important for evolution by natural selection.
we exclude them from our evolutionary theory? In favor of grad-ualism, Darwin and others have replied that sporting mutations always have negative side-effects that would cause selection to eliminate them from natural populations. Indeed, it is ques-tionable whether Ancon sheep, despite their attractiveness to farmers, would propagate successfully in the presence of their long-legged relatives without human intervention. A mutation of large effect appears responsible, however, for an adaptive bill size polymorphism in an African fi nch species ( Pyrenestes ostrinus ) in which large-billed forms eat hard seeds and small-billed forms eat softer ones. Recent work in evolutionary devel-opmental genetics (p. 174) illustrates the continuing controversy surrounding phenotypic gradualism. Punctuated Equilibrium When we view Darwinian gradualism on a geological time-scale, we may expect to fi nd in the fossil record a long series of intermediate forms connecting the phenotypes of ancestral and descendant populations ( Figure 6.25 ). This predicted pattern is called phyletic gradualism. Darwin recognized that phyletic gradualism is not often revealed by the fossil record. Studies conducted since Darwin’s time generally have not revealed the continuous series of fossils predicted by phyletic gradualism. Is the theory of gradualism therefore refuted by the fossil record? Darwin and others claim that it is not, because the fossil record is too imperfect to preserve transitional series. Although evolu-tion is a slow process by our standards, it is rapid relative to the rate at which good fossil deposits accumulate. Others have argued, however, that abrupt origins and extinctions of species in the fossil record force us to conclude that phyletic gradualism is rare. Niles Eldredge and Stephen Jay Gould proposed punctuated equilibrium to explain the discontinuous evolutionary changes observed throughout geological time. Punctuated equilibrium 124 PART TWO Continuity and Evolution of Animal Life evolution to branching speciation and expected most morpho-logical change to accumulate gradually in the “phyletic” mode between events of branching speciation. Founder-induced allopatric speciation provides a possible explanation for punctuated equilibria. Remember that founder-induced speciation requires the breaking of genetic equilibrium in a small, geographically isolated population. Such small popu-lations have very little chance of being preserved in the fos-sil record. After a new genetic equilibrium forms and stabilizes, the new population may increase in size, thereby increasing the chance that some of its members will be preserved as fossils. Founder-induced speciation cannot be the exclusive cause of punctuated equilibrium, however, because punctuated equilib-rium may be observed in groups where speciation by founder events is unlikely. Evolutionists who lamented the imperfect state of the fossil record were treated in 1981 to the opening of an uncensored page of fossil history in Africa. Peter Williamson, a British pale-ontologist working in fossil beds 400 m deep near Lake Turkana, documented a remarkably clear record of speciation in freshwa-ter snails. The geology of the Lake Turkana basin reveals a his-tory of instability. Earthquakes, volcanic eruptions, and climatic changes caused the waters episodically to rise and to fall, some-times by hundreds of feet. Thirteen lineages of snails show long periods of stability interrupted by relatively brief periods of rapid change in shell shape when snail populations were fragmented by receding waters. These populations diverged to produce new species that then remained unchanged through thick deposits before becoming extinct and being replaced by descendant spe-cies. The transitions occurred within 5000 to 50,000 years. In the few meters of sediment where speciation occurred, transi-tional forms were visible. Williamson’s study conforms well to the punctuated equilibrium model of Eldredge and Gould. Natural Selection Natural selection is the major process by which evolution occurs in Darwin’s theory of evolution. It gives us a natural explana-tion for the origins of adaptation, including all developmental, behavioral, anatomical, and physiological attributes that enhance an organism’s ability to use environmental resources to survive and to reproduce. Evolution of color patterns that conceal moths from predators (Figure 1.11, p. 12), and of bills adapted to dif-ferent modes of feeding in fi nches ( Figure 6.23 ), illustrate natural selection leading to adaptation. Darwin developed his theory of natural selection as a series of fi ve observations and three infer-ences drawn from them: Observation 1—Organisms have great potential fertility. All populations produce large numbers of gametes and potentially large numbers of offspring each generation. Population size would increase exponentially at an enormous rate if all individuals that were produced each generation sur-vived and reproduced. Darwin calculated that, even in slow-breeding animals such as elephants, a single pair breeding from age 30 to 90 and having only six young could produce 19 million descendants in 750 years. Observation 2—Natural populations normally remain constant in size, except for minor fl uctuations. Natural populations fl uctuate in size across generations and some-times go extinct, but no natural populations show the con-tinued exponential growth that their reproductive biology theoretically could sustain. Observation 3—Natural resources are limited. Exponen-tial growth of a natural population would require unlimited natural resources to provide food and habitat for the expand-ing population, but natural resources are fi nite. Inference 1—A continuing struggle for existence exists among members of a population. Survivors represent only a part, usually a very small part, of the individuals pro-duced each generation. Darwin wrote in On The Origin of Species that “it is the doctrine of Malthus applied with mani-fold force to the whole animal and vegetable kingdoms.” The struggle for food, shelter, and space becomes increasingly severe as overpopulation develops. Observation 4—Populations show variation among organisms. No two individuals are exactly alike. They differ in size, color, physiology, behavior, and many other ways. Observation 5—Some variation is heritable. Darwin noted that offspring tend to resemble their parents, although he did not understand how. The hereditary mechanism dis-covered by Gregor Mendel would be applied to Darwin’s theory many years later. Inference 2—Varying organisms show differential sur-vival and reproduction favoring advantageous traits ( natural selection). Survival in the struggle for existence is not random with respect to hereditary variation present in the population. Some traits give their possessors an advantage in using the environment for effective survival and repro-duction. Survivors transmit their favored traits to offspring, thereby causing those traits to accumulate in the population. Inference 3—Over many generations, natural selection generates new adaptations and new species. The differ-ential reproduction of varying organisms gradually transforms species and causes long-term “improvement” of populations. Darwin knew that people often use hereditary variation to produce useful new breeds of livestock and plants. Natural selection acting over millions of years should be even more effective in producing new types than the artifi cial selection imposed during a human lifetime. Natural selection acting independently on geographically separated populations would cause them to diverge from each other, thereby generating reproductive barriers that lead to speciation. www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 125 The popular phrase “survival of the fi ttest” was not originated by Darwin but was coined a few years earlier by the British philoso-pher Herbert Spencer, who anticipated some of Darwin’s principles of evolution. Unfortunately the phrase later came to be coupled with unbridled aggression and violence in a bloody, competitive world. In fact, natural selection operates through many other characteristics of living organisms. The fi ttest animal may be one that enhances the living conditions of its population. Fighting prowess is only one of several means toward survival and reproductive advantage. Natural selection may be considered a two-step process with a random component and a nonrandom component. Production of variation among organisms is the random component. The mutational process does not preferentially generate traits that are favorable to the organism; new variants are probably more likely to be unfavorable. The nonrandom component is the survival of different traits. This differential survival is determined by the effectiveness of different traits in permitting their possessors to use environmental resources to survive and to reproduce. The phenomenon of differential survival and reproduction among varying organisms is now called sorting and should not be equated with natural selection. We now know that even random processes (genetic drift, p. 127) can produce sorting among vary-ing organisms. When selection operates, sorting occurs because certain traits give their possessors advantages in survival and reproduction relative to others that lack those traits. Selection is therefore a specifi c cause of sorting. Darwin’s theory of natural selection has been challenged repeatedly. One challenge claims that directed (nonrandom) variation governs evolutionary change. In the decades around 1900, diverse evolutionary hypotheses collectively called ortho-genesis proposed that variation has momentum that forces a lineage to evolve in a particular direction that is not always adaptive. The extinct Irish elk was a popular example of ortho-genesis. Newly produced variation was considered biased toward enlarging their antlers, thereby generating an evolution-ary momentum for producing larger antlers. Natural selection was considered ineffective at stopping the antlers eventually from becoming so large and cumbersome that they forced the Irish elk into extinction ( Figure 6.27 ). Orthogenesis explained apparently nonadaptive evolutionary trends that supposedly forced species into decline. Because extinction is the expected evolutionary fate of most species, disappearance of the Irish elk is not extraordinary and probably not related to large antlers. Subsequent genetic research on the nature of variation clearly has rejected the genetic predictions of orthogenesis. Another recurring criticism of natural selection is that it can-not generate new structures or species but can only modify old ones. Most structures in their early evolutionary stages could not have performed the biological roles that the fully formed structures perform, and it is therefore unclear how natural selec-tion could have favored them. What use is half a wing or the rudiment of a feather for a fl ying bird? To answer this criticism, we propose that many structures evolved initially for purposes different from the ones they have today. Rudimentary feathers would have been useful in thermoregulation, for example. The Observation 1 Organisms have great potential fertility, which permits exponential growth of populations. (Source: Thomas Malthus) Observation 2 Natural populations normally do not increase exponentially but remain fairly constant in size.
(Source: Charles Darwin and many others) Inference 2 Varying organisms show differential survival and reproduction, favoring advantageous traits (= natural selection).
(Source: Charles Darwin) Inference 3 Natural selection, acting over many generations, gradually produces new adaptations and new species.
(Source: Charles Darwin) Observation 3 Natural resources are limited.
(Source: Thomas Malthus) Inference 1 A struggle for existence occurs among organisms in a population.
(Source: Thomas Malthus) Observation 4 Variation occurs among organisms within populations.
(Source: Animal breeding and systematics) Observation 5 Variation is heritable.
(Source: Animal breeding) Darwin's Explanatory Model of Evolution by Natural Selection 126 PART TWO Continuity and Evolution of Animal Life feathers later became useful for fl ying after they incidentally acquired aerodynamic properties. Natural selection then could act to improve the usefulness of feathers for fl ying. Exaptation denotes the utility of a structure for a biological role that was not part of the structure’s evolutionary origin. Exaptation contrasts with adaptation, which implies that a structure arose by natural selection for a particular biological role. Bird feathers are there-fore adaptations for thermoregulation but exaptations for fl ight. Because structural changes that separate members of different species are similar in kind to variation that we observe within species, it is reasonable to propose that selection can produce new species. REVISIONS OF DARWIN’S THEORY Neo-Darwinism The most serious weakness in Darwin’s theory was his failure to identify correctly the mechanism of inheritance. Darwin saw heredity as a blending phenomenon in which the hereditary fac-tors of parents melded together in their offspring. Darwin also invoked the Lamarckian hypothesis that an organism could alter its heredity through use and disuse of body parts and through the direct influence of the environment. August Weismann rejected Lamarckian inheritance by showing experimentally that modifi cations of an organism during its lifetime do not change its heredity (see Chapter 5), and he revised Darwin’s theory accord-ingly. We now use the term neo-Darwinism to denote Darwin’s theory as revised by Weismann. Mendelian genetics eventually clarifi ed the particulate inher-itance that Darwin’s theory of natural selection required (p. 18). Ironically, when Mendel’s work was rediscovered in 1900, it was considered antagonistic to Darwin’s theory of natural selec-tion. When mutations were discovered in the early 1900s, most geneticists thought that they produced new species in single large steps. These geneticists relegated natural selection to the role of executioner, a negative force that merely eliminated the obviously unfi t. Emergence of Modern Darwinism: The Synthetic Theory In the 1930s a new generation of geneticists began to reevaluate Darwin’s theory from a mathematical perspective. These were population geneticists, scientists who studied variation in natural populations using statistics and mathematical models. Gradually, a new comprehensive theory emerged that brought together population genetics, paleontology, biogeography, embryology, systematics, and animal behavior in a Darwinian framework. Population geneticists study evolution as a change in the genetic composition of populations. With the establishment of population genetics, evolutionary biology became divided into two different subfi elds. Microevolution pertains to evolution-ary changes in frequencies of different allelic forms of genes (p. 78) within populations. Macroevolution refers to evolution on a grand scale, encompassing the origins of new organismal structures and designs, evolutionary trends, adaptive radiation, phylogenetic relationships of species, and mass extinction. Mac-roevolutionary research is based in systematics and the com-parative method (p. 206). Following the evolutionary synthesis, both macroevolution and microevolution have operated fi rmly within the tradition of neo-Darwinism, and both have expanded Darwinian theory in important ways. MICROEVOLUTION: GENETIC VARIATION AND CHANGE WITHIN SPECIES Microevolution is the study of genetic change occurring within natural populations. Occurrence of different allelic forms of a gene in a population is called polymorphism. All alleles of all genes possessed by members of a population collectively form the gene pool of that population. The amount of polymorphism present in large populations is potentially enormous, because at observed mutation rates, many different alleles are expected for all genes. Population geneticists study polymorphism by identifying the different allelic forms of a gene present in a population and then measuring the relative frequencies of the different alleles in the population. The relative frequency of a particular allelic form of a gene in a population is called its allelic frequency. For exam-ple, in the human population, there are three different allelic forms of the gene encoding the ABO blood types (p. 782). Using the symbol I to denote the gene encoding the ABO blood types, I A and I B denote genetically codominant alleles encoding blood types A and B, respectively. Allele i is a recessive allele encoding blood group O. Therefore genotypes I A I A and I A i produce type A blood, genotypes I B I B and I B i produce type B blood, genotype I A I B produces type AB blood, and genotype ii produces type O Figure 6.27 Irish elk, a fossil species that once was used to support the orthogenetic idea that momentum in variation caused the antlers to become so large that the species was forced into extinction.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 127 blood. Because each individual contains two copies of this gene, the total number of copies present in the population is twice the number of individuals. What fraction of this total is represented by each of the three different allelic forms? In France, we fi nd the following allelic frequencies: I A .46, I B .14, and i .40. In Russia, the corresponding allelic frequencies differ ( I A .38, I B .28, and i .34), demonstrating microevolutionary diver-gence between these populations ( Figure 6.28 ). Although alleles I A and I B are dominant to i, i is nearly as frequent as I A and exceeds the frequency of I B in both populations. Dominance describes the phenotypic effect of an allele in heterozygous indi-viduals, not its relative abundance in a population of individuals. We will demonstrate that Mendelian inheritance and dominance do not alter allelic frequencies directly or produce evolutionary change in a population. Genetic Equilibrium In many human populations, genetically recessive traits, including the O blood type, blond hair, and blue eyes, are very common. Why have not the genetically dominant alternatives gradually supplanted these recessive traits? It is a common misconception that a characteristic associated with a dominant allele increases in frequency because of its genetic dominance. This misconception is refuted by a principle called Hardy-Weinberg equilibrium (see box next page), which forms the foundation for popula-tion genetics. According to this theorem, the hereditary process alone does not produce evolutionary change. In large biparental populations, allelic frequencies and genotypic ratios attain an equilibrium in one generation and remain constant thereafter unless disturbed by recurring mutations, natural selection, migra-tion, nonrandom mating, or genetic drift (random sorting). Such disturbances are the sources of microevolutionary change. A rare allele, according to this principle, does not disap-pear from a large population merely because it is rare. Certain rare traits, such as albinism and cystic fi brosis, persist for many generations. For example, albinism in humans is caused by a rare recessive allele a. Only one person in 20,000 is an albino, and this individual must be homozygous ( a/a ) for the recessive allele. Obviously the population contains many carriers, people with normal pigmentation who are heterozygous ( A/a ) for albi-nism. What is their frequency? A convenient way to calculate the frequencies of genotypes in a population is with the binomial expansion of ( p q ) 2 (see box next page). We let p represent the allelic frequency of A and q the allelic frequency of a. Assuming that mating is random, the distribution of geno-typic frequencies is p 2 A / A , 2 pq A / a , and q 2 a / a . Only the frequency of genotype a/a is known with certainty, 1/20,000; therefore: q q p q 2 1 2 1 20 000 1 20 000 1 141 1 140 1 / , ( / , ) / / / 41 The frequency of carriers is: A a pq / / / / 2 2 140 141 1 141 1 70 One person in every 70 is a carrier! Although the reces-sive trait is rare, the recessive allele is surprisingly common in the population. There is a message here for anyone proposing to eliminate a “bad” recessive allele from a population by con-trolling reproduction. It is practically impossible. Because only the homozygous recessive individuals reveal the phenotype against which artifi cial selection could act (by sterilization, for example), the allele would persist through heterozygous carri-ers. For a recessive allele present in 2 of every 100 persons (but homozygous in only 1 in 10,000 persons), it would require 50 generations of complete selection against the homozygotes just to reduce its frequency to one in 100 persons. How Genetic Equilibrium Is Upset Genetic equilibrium is disturbed in natural populations by (1) random genetic drift, (2) nonrandom mating, (3) recurring muta-tion, (4) migration, (5) natural selection, and interactions among these factors. Recurring mutation is the ultimate source of vari-ability in all populations, but it usually requires interaction with one or more of the other factors to upset genetic equilibrium. We consider these other factors individually. Genetic Drift Some species, such as cheetahs ( Figure 6.29 ), contain very lit-tle genetic variation, probably because their ancestral lineages Figure 6.28 Frequencies of the blood-type B allele among humans in Europe. The allele is more common in the east and rarer in the west. The allele may have arisen in the east and gradually diffused westward through the genetic continuity of human populations. This allele has no known selective advantage; its changing frequency probably represents the effects of random genetic drift.
25-30% 20-25% 15-20% 15-20% 10-15% 10-15% 10-15% 5-10% 5-10% 0-5% EUROPE 128 PART TWO Continuity and Evolution of Animal Life The Hardy-Weinberg law is a logical consequence of Mendel’s fi rst law of segregation and expresses the tendency toward equilibrium inherent in Mendelian heredity.
Let us select for our example a population having a single locus bearing just two alleles T and t. The phenotypic expression of this gene might be, for example, the ability to taste a chemical compound called phenylthiocarbamide. Individuals in the population will be of three genotypes for this locus, T/T, T/t (both tasters), and t/t (non-tasters). In a sample of 100 individuals, let us suppose that we have 20 of T/T genotype, 40 of T/t genotype, and 40 of t/t genotype. We could then make a table showing the allelic frequencies (remember that every individual has two copies of the gene): Genotype Number of Individuals Copies of the T Allele Copies of the t Allele T/T 20 40 T/t 40 40 40 t/t 40 80 Total 100 80 120 Of the 200 copies, the proportion of the T allele is 80/200 0.4 (40%), and the proportion of the t allele is 120/200 0.6 (60%). It is customary to use “p” and “q” to represent the two allelic frequen-cies. The genetically dominant allele is represented by p, and the genetically recessive by q. Thus: p T q t frequency of frequency of The 0 4 0 6 .
.
refore p q 1 Having calculated allelic frequencies in the sample, let us deter-mine whether these frequencies will change spontaneously in a new generation of the population. Assuming that mating is random (gametes are sampled independently in pairs), each individual will contribute an equal number of gametes to the “common pool” from which the next generation is formed. Frequencies of gametes in the “pool” then will equal the allelic frequencies in the sample: 40% of the gametes will be T, and 60% will be t (ratio of 0.4:0.6). Both ova and sperm will, of course, show the same frequencies. The next generation is formed: Collecting genotypes, we have: frequency of / frequency of / f T T T t 0 16 0 48 .
.
requency of / t t 0 36 .
Next, we determine the values of p and q from the randomly mated populations. From the table above, we see that the frequency of T will be the sum of genotypes T/T, which is 0.16, and one-half of the genotype T/t, which is 0.24: T p ( ) .
. ( .
) .
0 16 5 0 48 0 4 Similarly, the frequency of t will be the sum of genotypes t/t, which is 0.36, and one-half the genotype T/t, which is 0.24: t p ( ) .
. ( .
) .
0 36 5 0 48 0 6 The new generation bears exactly the same allelic frequencies as the parent population! Note that there has been no increase in the frequency of the genetically dominant allele T. Thus, in a freely interbreeding, sexually reproducing population, the frequency of each allele would remain constant generation after generation in the absence of natural selection, migration, recurring mutation, and genetic drift (see text). A mathematically minded reader will rec-ognize that the genotype frequencies T/T, T/t, and t/t are actually a binomial expansion of (p q)2: ( ) p q p pq q 2 2 2 2 1 A statistically minded reader will note that the equilibrium calcu-lations give expected frequencies, which are unlikely to be realized exactly in a population of fi nite size. For this reason, fi nite population size is a cause of evolutionary change.
Hardy-Weinberg Equilibrium: Why the Hereditary Process Does Not Change Allelic Frequencies Ova Sperm T 0.4 t 0.6 T 0.4 T/T 0.16 T/t 0.24 t 0.6 T/t 0.24 t/t 0.36 passed through periods when the total number of individuals in the population was very small. A small population clearly can-not contain large amounts of genetic variation. Each individual organism has at most two different allelic forms of each gene, and a single breeding pair contains at most four different allelic forms of each gene. Suppose that we have such a breeding pair. We know from Mendelian genetics (see Chapter 5) that chance decides which of the different allelic forms of a gene gets passed to offspring. It is therefore possible by chance alone that one or two of the parental alleles in this example will not be passed to any offspring. It is highly unlikely that the different alleles pres-ent in a small ancestral population are all passed to descendants without any change of allelic frequency. This chance fl uctuation in allelic frequency from one generation to the next, including loss of alleles from the population, is called genetic drift. Genetic drift occurs to some degree in all populations of fi nite size. Perfect constancy of allelic frequencies, as predicted by Hardy-Weinberg equilibrium, occurs only in infi nitely large popu-lations, and such populations occur only in mathematical mod-els. All populations of animals are fi nite and therefore experience www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 129 some effect of genetic drift, which becomes greater, on average, as population size declines. Genetic drift erodes genetic variabil-ity of a population. If population size remains small for many generations in a row, genetic variation can be greatly depleted. This loss is harmful to a species’s evolutionary success because it restricts potential genetic responses to environmental change. Indeed, biologists are concerned that cheetah populations may have insuffi cient variation for continued survival. A large reduction in the size of a population that increases evolutionary change by genetic drift is commonly called a bot-tleneck. A bottleneck associated with the founding of a new geographic population is called a founder effect and may be associated with formation of a new species (p. 119). Nonrandom Mating If mating is nonrandom, genotypic frequencies will deviate from Hardy-Weinberg expectations. For example, if two different alleles of a gene are equally frequent ( p q .5), we expect half of the genotypes to be heterozygous (2 pq 2[.5] [.5] .5) and one-quarter to be homozygous for each of the respective alleles ( p 2 q 2 [.5] 2 .25). If we have positive assortative mating, individuals mate preferentially with others of the same genotype, such as albinos mating with other albinos. Matings among individuals homozygous for the same allele generate offspring that are homozygous like themselves. Matings among individuals heterozygous for the same pair of alleles produce on average 50% heterozygous offspring and 50% homozygous offspring (25% of each alternative type) each generation. Positive assortative mating increases the frequency of homozygous geno-types and decreases the frequency of heterozygous genotypes in a population but does not change allelic frequencies. Preferential mating among close relatives also increases homozygosity and is called inbreeding. Whereas positive assortative mating usually affects one or a few traits, inbreed-ing simultaneously affects all variable traits. Strong inbreeding greatly increases chances that rare recessive alleles will become homozygous and be expressed. Because inbreeding and genetic drift are both promoted by small population size, they are often confused with each other. Their effects are very different, however. Inbreeding alone can-not change allelic frequencies in the population, only the ways that alleles are combined into genotypes. Genetic drift changes allelic frequencies and consequently also changes genotypic frequencies. Even very large populations have the potential for being highly inbred if there is a behavioral preference for mat-ing with close relatives, although this situation rarely occurs in animals. Genetic drift, however, will be relatively weak in very large populations. Inbreeding has surfaced as a serious problem in zoos holding small populations of rare mammals. Matings of close relatives tend to bring together genes from a common ancestor and increase the probability that two copies of a deleterious gene will come together in the same organism. The result is “inbreeding depression.” Our management solution is to enlarge genetic diversity by bringing together captive animals from different zoos or by introducing new stock from wild populations if possible. Paradoxically, where zoo populations are extremely small and no wild stock can be obtained, deliberate inbreeding is recommended. This procedure selects for genes that tolerate inbreeding; deleterious genes disappear if they kill animals homozygous for them. Migration Migration prevents different populations of a species from diverging. If a large species is divided into many small popula-tions, genetic drift and selection acting separately in the different populations can produce evolutionary divergence among them. A small amount of migration in each generation keeps the dif-ferent populations from becoming too distinct genetically. For example, the French and Russian populations whose ABO allele frequencies were discussed previously show some genetic diver-gence, but their genetic connection through intervening popu-lations by continuing migration prevents them from becoming completely distinct. Natural Selection Natural selection can change both allelic frequencies and geno-typic frequencies in a population. Although the effects of selec-tion are often reported for particular polymorphic genes, we must stress that natural selection acts on the whole animal, not on isolated traits. An organism that possesses a superior combination of traits will be favored. An animal may have traits that confer no advantage or even a disadvantage, but it is successful overall if its combination of traits is favorable. When we claim that a genotype at a particular gene has a higher relative fi tness than others, Figure 6.29 Cheetahs, a species whose genetic variability has been depleted to very low levels because of small population size in the past.
130 PART TWO Continuity and Evolution of Animal Life we state that on average that genotype confers an advantage in survival and reproduction in the population. If alternative geno-types have unequal probabilities of survival and reproduction, Hardy-Weinberg equilibrium is upset. Using the genetic theory of natural selection, one can mea-sure relative fi tness values associated with different genotypes in a population. Geneticists often use W to denote the expected average fi tness of a genotype in a population, with the genotype of highest fi tness given a value of one and fi tnesses of other genotypes indicated as fractions. We illustrate measurement of fi tness using genetic variation associated with the disease sickle-cell anemia in human popula-tions. Considering only the alleles for normal hemoglobin ( A ) and sickle-cell hemoglobin ( S ) for the beta-hemoglobin gene in human populations (p. 100), the possible genotypes are AA , AS , and SS . Measurements of viability of individuals of these three genotypes in nonmalarial environments give a fi tness value of 1 to genotypes AA and AS and a fi tness of 0.2 to genotype SS . People having the SS genotype, who are susceptible to severe anemia, are expected to contribute only 20% as many offspring to the next generation on average as are individuals having the AA or AS genotypes. In malarial environments, genotype AS has the highest fi tness (1); genotype AA has a slightly decreased fi tness (0.9) because these individuals have a greater incidence of malaria than AS individuals, and SS has a low fi tness (0.2) because of anemia. From these measured fitness values and knowledge of the frequencies of alleles in a population and its system of mating, one can calculate the average effect that an allele has on the phenotype of relative fi tness in that population. In the example of sickle-cell anemia, the average effect of allele S on fi tness in a malarial environment is a balance between the strongly negative effect it has when homozygous and the positive effect that it has when heterozygous with allele A . In Chapter 36, we discuss the related concept of inclusive fi tness . The average effect of an allele on fi tness is expressed not only by its direct contribution to the fi tness of its possessors but by aid that its possessors give to close relatives, who are likely also to contain copies of the allele. The term “inclusive fi tness” pertains to cases where the average effect of an allele would be calculated incorrectly if only its direct effects on fi tness were measured. Some traits and combinations of traits are advantageous for certain aspects of an organism’s survival or reproduction and dis-advantageous for others. Darwin used the term sexual selection to denote the selection of traits that are advantageous for obtaining mates but not for survival. Bright colors and elaborate feathers can enhance a male bird’s competitive ability in obtaining mates while simultaneously increasing his visibility to predators ( Figure 6.30 ). Environmental changes, such as extinction of a predator popula-tion, can alter the selective value of different traits. The action of selection on character variation is therefore very complex. Interactions of Selection, Drift, and Migration Subdivision of a species into small populations that exchange migrants is an optimal situation for promoting rapid adaptive evolution of a species. Interaction of genetic drift and selection in different populations permits many different genetic combi-nations of many polymorphic genes to be tested against natu-ral selection. Migration among populations permits particularly favorable new genetic combinations to spread throughout the species as a whole. Interaction of selection, genetic drift, and migration in this example produces evolutionary change qualita-tively different from what would result if any of these three fac-tors acted alone. Geneticist Sewall Wright called this interaction shifting balance because it permits a population to explore dif-ferent adaptive combinations of variable traits. Natural selection, genetic drift, mutation, nonrandom mating, and migration inter-act in natural populations to create an enormous opportunity for evolutionary change; perpetual stability, as predicted by Hardy-Weinberg equilibrium, almost never occurs across any signifi cant amount of evolutionary time. Measuring Genetic Variation Within Populations How do we measure the genetic variation that occurs in natural populations? Genetic dominance, interactions between alleles of different genes, and environmental effects on a phenotype make it diffi cult to quantify genetic variation indirectly by observing organismal phenotypes. Variability can be quantifi ed, however, at the molecular level. Protein Polymorphism Different allelic forms of genes encode proteins that often differ slightly in their amino acid sequence. This phenomenon is called Figure 6.30 A pair of wood ducks. Brightly colored feathers of male birds probably confer no survival advantage and might even be harmful by alerting predators. Such colors nonetheless confer advantage in attracting mates, which overcomes, on average, the negative consequences of these colors for survival. Darwin used the term “sexual selection” to denote evolution of traits that give an individual an advantage in reproduction, even if the traits are neutral or harmful for survival.
www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 131 protein polymorphism. If these differences affect the protein’s net electric charge, the different allelic forms can be separated using protein electrophoresis ( Figure 6.31 ). We can identify the genotypes of particular individuals for protein-coding genes and measure allelic frequencies in a population. Over the last 40 years, geneticists using this approach have discovered far more variation than was previously expected. Despite the high levels of polymorphism discovered using pro-tein electrophoresis ( Table 6.1 ), these studies underestimate both protein polymorphism and the total genetic variation present in a population. For example, protein polymorphism that does not involve charge differences is not detected. Furthermore, because the genetic code is degenerate (more than one codon for most amino acids, p. 94), protein polymorphism does not reveal all of the genetic variation present in protein-coding genes. Genetic changes that do not alter protein structure sometimes alter pat-terns of protein synthesis during development and can be very important to an organism. When all kinds of variation are consid-ered, it is evident that most species have an enormous potential for further evolutionary change. Quantitative Variation Quantitative traits are those that show continuous variation with no obvious pattern of Mendelian segregation in their inheri-tance. The values of the trait in offspring often are intermediate between the values in the parents. Such traits are infl uenced by variation at many genes, each of which follows Mendelian inheritance and contributes a small, incremental amount to the total phenotype. Examples of traits that show quantitative varia-tion include tail length in mice, length of a leg segment in grass-hoppers, number of gill rakers in sunfi shes, number of peas in pods, and height of adult males of the human species. When trait Figure 6.31 Study of genetic variation in proteins using gel electrophoresis. A, An electrophoretic apparatus separates allelic variants of proteins that differ in charge because of differences in their sequence of amino acids. B, Genetic variation in the protein leucine aminopeptidase for nine brown snails, Helix aspersa. Two different sets of allelic variants are revealed. The top set contains two alleles [denoted fast (F) and slow (S) according to their relative movement in the electric fi eld]. Individuals homozygous for the fast allele show only a single fast band on the gel (FF), those homozygous for the slow allele show only a single slow band (SS), and heterozygous individuals have both bands (FS). The lower set contains three different alleles denoted fast (F), medium (M), and slow (S). Note that no individuals shown are homozygous for the medium (M) allele.
Buffer Positive pole Positions of proteins after migration Initial position of proteins Negative pole Buffer Power supply A Gel Lower set of allelic variants B Top set of allelic variants FF FS SS FM MS FM FS FS FS FS FF FS SS SS FS SS FF FS T A B LE 6. 1 Values of Polymorphism (P) and Heterozygosity (H) for Various Animals and Plants as Measured Using Protein Electrophoresis (a) Species Number of Proteins P H Humans 71 0.28 0.067 Northern elephant seal 24 0.0 0.0 Horseshoe crab 25 0.25 0.057 Elephant 32 0.29 0.089 Drosophila pseudoobscura 24 0.42 0.12 Barley 28 0.30 0.003 Tree frog 27 0.41 0.074 (b) Taxa Number of Species P H Plants — 0.31 0.10 Insects (excluding Drosophila) 23 0.33 0.074 Drosophila 43 0.43 0.14 Amphibians 13 0.27 0.079 Reptiles 17 0.22 0.047 Birds 7 0.15 0.047 Mammals 46 0.15 0.036 Average 0.27 0.078 Source: Data from P.W. Hedrick, Population biology. Jones and Bartlett, Boston, 1984.
P, the average number of alleles per gene per species; H, the proportion of heterozygous genes per individual.
132 PART TWO Continuity and Evolution of Animal Life values are graphed with respect to frequency distribution, they often approximate a normal, or bell-shaped, probability curve ( Figure 6.32A ). Most individuals fall near the average; fewer fall somewhat above or below the average, and extremes form the “tails” of the frequency curve with increasing rarity. Usually, the larger the population sample, the more closely the frequency distribution resembles a normal curve. Selection can act on quantitative traits to produce three different kinds of evolutionary response ( Figure 6.32B , C, and D). One outcome is to favor average values of the trait and to disfavor extreme ones; this outcome is called stabilizing selection ( Figure 6.32B ). Directional selection favors a phe-notypic value either above or below the average and causes the population average to shift toward the favored value over time ( Figure 6.32C ). When we think about natural selection produc-ing evolutionary change, it is usually directional selection that we have in mind, although we must remember that this is not the only possibility. A third alternative is disruptive selection in which two different extreme phenotypes are simultane-ously favored, but their average is disfavored ( Figure 6.32D ). The population then becomes bimodal, meaning that two very different phenotypic values predominate. MACROEVOLUTION: MAJOR EVOLUTIONARY EVENTS Macroevolution describes large-scale events in organic evolu-tion. Speciation links macroevolution and microevolution. Major trends in the fossil record (see Figures 6.11 , 6.12, and 6.13 ) are clearly within the realm of macroevolution. Patterns and pro-cesses of macroevolutionary change emerge from those of microevolution, but they acquire some degree of autonomy in doing so. The emergence of new adaptations and species, and the varying rates of speciation and extinction observed in the fossil record go beyond the fl uctuations of allelic frequencies within populations. Stephen Jay Gould recognized three different “tiers” of time at which we observe distinct evolutionary processes. The fi rst tier constitutes the timescale of population genetic pro-cesses, from tens to thousands of years. The second tier cov-ers millions of years, the scale on which rates of speciation and extinction are measured and compared among different groups of organisms. Punctuated equilibrium is a theory of the second tier, explaining the occurrence of speciation and mor-phological change and their association over millions of years. The third tier covers tens to hundreds of millions of years, and is marked by occurrence of episodic mass extinctions. In the fossil record of marine organisms, mass extinctions recur at intervals of approximately 26 million years. Five of these mass extinctions have been particularly disastrous ( Figure 6.33 ). The study of long-term changes in animal diversity focuses on the third-tier timescale (see Figures 6.13 and 6.33 ). Speciation and Extinction Through Geological Time Evolutionary change at the second tier provides a new per-spective on Darwin’s theory of natural selection. Although a species may persist for many millions of years, it ultimately has two possible evolutionary fates: it may give rise to new species or become extinct without leaving descendants. Rates of spe-ciation and extinction vary among lineages, and lineages that have the highest speciation rates and lowest extinction rates produce the greatest number of living species. The character-istics of a species may make it more or less likely than oth-ers to undergo speciation or extinction events. Because many characteristics are passed from ancestral to descendant species (analogous to heredity at the organismal level), lineages whose characteristics increase the probability of speciation and confer resistance to extinction should dominate the living world. This species-level process that produces differential rates of specia-tion and extinction among lineages is analogous in many ways Figure 6.32 Responses to selection on a continuous (polygenic) character, coloration in a snail. A, The frequency distribution of coloration before selection. B, Stabilizing selection culls extreme variants from the population, in this case eliminating individuals that are unusually light or dark, thereby stabilizing the mean. C, Directional selection shifts the population mean, in this case by favoring darkly colored variants. D, Disruptive selection favors both extremes but not the mean; the mean is unchanged but the population no longer has a bell-shaped distribution of phenotypes.
Disruptive selection Directional selection Stabilizing selection Light A B C D Dark Coloration Number of individuals www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 133 to natural selection. It represents an expansion of Darwin’s the-ory of natural selection. This expansion is particularly important for macroevolution if one accepts the theory of punctuated equi-librium, which states that the evolutionarily important variation occurs primarily among rather than within species. Species selection encompasses the differential survival and multiplication of species through geological time based on variation among lineages, especially in emergent, species-level properties. These species-level properties include mating rituals, social structuring, migration patterns, geographic distribution, and all other properties that emerge at the species level (see p. 6). Descendant species usually resemble their ancestors in these properties. For example, a “harem” system of mating in which a single male and several females compose a breeding unit char-acterizes some mammalian lineages but not others. We expect speciation rates to be enhanced by social systems that promote founding of new populations by small numbers of individuals. Certain social systems may increase the likelihood that a spe-cies will survive environmental challenges through cooperative action. Such properties would be favored by species selection over geological time. Differential speciation and extinction among lineages also can be caused by variation in organismal-level properties (such as specialized versus generalized feeding) rather than species-level properties (see p. 6). Organisms that specialize in eating a restricted range of foods, for example, may be subjected more readily than generalized feeders to geographic isolation among populations, because areas where their preferred food is scarce or absent will function as geographic barriers to dispersal. Such geographic isolation could generate more frequent opportunities for speciation to occur throughout geological time. The fossil records of two major groups of African antelopes suggest this result ( Figure 6.11 ). A group of spe-cialized grazers that contains blesboks, hartebeests, and wildebeests shows high speciation and extinc-tion rates. Since the late Miocene, 33 extinct and 7 living species are known, representing at least 18 events of branching speciation and 12 termi-nal extinctions. In contrast, a group of generalist grazers and browsers that contains impalas shows neither branching speciation nor terminal extinc-tion during this same interval of time. Interestingly, although these two lineages differ greatly in specia-tion rates, extinction rates, and species diversity, they do not differ signifi cantly in total number of individual animals alive today. Paleontologist Elisabeth Vrba, whose research produced the results in Figure 6.11 , uses the term effect macroevolution to describe differ-ential speciation and extinction rates among lin-eages caused by organismal-level properties. She reserves the term species selection for cases where species-level emergent properties are of primary importance. Some other evolutionary paleon-tologists consider effect macroevolution a subset of species selection because fi tness differences occur among different species lineages rather than among varying organisms within species. Mass Extinctions When we study evolutionary change on an even larger time scale, we observe episodic events in which large numbers of taxa go extinct simultaneously. These events are called mass extinc-tions (see Figure 6.33 ). The most cataclysmic of these extinction episodes happened about 225 million years ago, when at least half of the families of shallow-water marine invertebrates, and fully 90% of marine invertebrate species disappeared within a few million years. This event was the Permian extinction. The Cretaceous extinction, which occurred about 65 million years ago, marked the end of dinosaurs, as well as numerous marine invertebrates and many small reptilian taxa. Causes of mass extinctions and evolutionary timing of mass extinctions at intervals of approximately 26 million years are dif-fi cult to explain. Some people have proposed biological expla-nations for these episodic mass extinctions and others consider many mass extinctions artifacts of our statistical and taxonomic analyses. Walter Alvarez proposed that the earth was periodically bombarded by asteroids, causing these mass extinctions ( Fig-ure 6.34 ). The drastic effects of such bombardment of a planet were observed in July 1994 when fragments of Comet Shoemaker-Levy 9 bombarded Jupiter. The fi rst fragment to hit Jupiter was estimated to have the force of 10 million hydrogen bombs. Twenty additional fragments hit Jupiter within the following week, one of which was 25 times more powerful than the fi rst fragment. This bombardment was the most violent event in the recorded history of the solar system. A similar bombardment on earth would send Figure 6.33 Changes in numbers of families of marine animals through time from the Cambrian period to the present. Sharp drops represent fi ve major extinctions of skeletonized marine animals. Note that despite the extinctions, the overall number of marine families has increased to the present.
Ordovician extinction Devonian extinction Cambrian "explosion" Permian extinction Triassiac a extinction n Cretaceous extinction 0 200 400 60 00 600 900 300 Ge eo ol logic time (106 ef fore pre esen nt) ye ea ars s be Number of families of marine animals PA ALE EO OZ ZO OIC ER RA A ME ESO OZ ZOI IC C ERA A ERA E C C I O ZO Z O N E C 134 PART TWO Continuity and Evolution of Animal Life debris into the atmosphere, blocking sunlight and causing drastic changes of climate. Temperature changes would challenge eco-logical tolerances of many species. Alvarez’s hypothesis is being tested in several ways, including a search for impact craters left by asteroids and for altered mineral content of rock strata where mass extinctions occurred. Atypical concentrations of the rare-earth ele-ment iridium in strata at the Cretaceous-Tertiary boundary imply that this element entered the earth’s atmosphere through asteroid bombardment. Sometimes, lineages favored by species selection are unusu-ally susceptible to mass extinction. Climatic changes produced by the hypothesized asteroid bombardments could produce selec-tive challenges very different from those encountered at other times in the earth’s history. Selective discrimination of particu-lar biological traits by events of mass extinction is termed cata-strophic species selection. For example, mammals survived the end-Cretaceous mass extinction that destroyed the dinosaurs and other prominent vertebrate and invertebrate groups. Fol-lowing this event, mammals were able to use environmental resources that previously had been denied them, leading to their adaptive radiation. Natural selection, species selection, and catastrophic species selection interact to produce the macroevolutionary trends seen in the fossil record. Studies of these interacting causal processes have made modern evolutionary paleontology an active and exciting fi eld. Figure 6.34 Twin craters of Clearwater Lakes in Canada show that multiple impacts on the earth are not as unlikely as they might seem. Evidence suggests that at least two impacts within a short time were responsible for the Cretaceous mass extinction. S U M M A R Y Organic evolution explains the diversity of living organisms as the historical outcome of gradual change from previously existing forms. Evolutionary theory is strongly identifi ed with Charles Robert Darwin, who presented the fi rst credible explanation for evolution-ary change. Darwin derived much of the material used to construct his theory from his experiences on a fi ve-year voyage around the world aboard the H.M.S. Beagle. Darwin’s evolutionary theory has fi ve major components. Its most basic proposition is perpetual change, the theory that the world is neither constant nor perpetually cycling but is steadily undergoing irreversible change. The fossil record amply demon-strates perpetual change in the continuing fl uctuation of animal form and diversity following the Cambrian explosion 600 million years ago. Darwin’s theory of common descent states that all organ-isms descend from a common ancestor through a branching of genealogical lineages. This theory explains morphological homol-ogies among organisms as characteristics inherited with modifi ca-tion from a corresponding feature in their common evolutionary ancestor. Patterns of homology formed by common descent with modifi cation permit us to classify organisms according to their evolutionary relationships. Changes in the timing of developmental processes, termed het-erochrony, and changes in their physical location within the body plan, termed heterotopy, explain the evolution of new morphologi-cal homologies. A developmental evolutionary module is a set of developmental processes and associated genes that can be expressed as a unit at different parts of the body to produce different structures with some shared developmental properties. Evolution of limbs in terrestrial vertebrates occurred by expressing at the limb bud a set of developmental processes that evolved initially to construct part of the vertebral column. Evolvability denotes the potential of a lineage to evolve new morphological features by using a set of developmental modules as an evolutionary toolkit. A corollary of common descent is the multiplication of species through evolutionary time. Allopatric speciation denotes the evo-lution of reproductive barriers between geographically separated populations to generate new species. In some animals, especially parasitic insects that specialize on different host species, speciation may occur without geographical isolation, which is called sympatric speciation. Intermediate between allopatric speciation and sympatric speciation is a third mode, parapatric speciation, in which an envi-ronmental change splits a species into two environmentally distinct parts that maintain contact along a geographic borderline as they diverge to become separate species. Adaptive radiation is the proliferation of many adaptively diverse species from a single ancestral lineage within a relatively short period of evolutionary time, such as a few million years. Oce-anic archipelagoes, such as the Galápagos Islands, are particularly conducive to adaptive radiation of terrestrial organisms. Darwin’s theory of gradualism states that large phenotypic dif-ferences between species are produced by accumulation through evolutionary time of many individually small changes. Gradualism www . m hhe . com /hi ckmani pz14e CHAPTER 6 Organic Evolution 135 is still controversial. Mutations that have large effects on an organ-ism have been useful in animal breeding, leading some to dispute Darwin’s claim that such mutations are not important in evolution. On a macroevolutionary perspective, punctuated equilibrium states that most evolutionary change occurs in relatively brief events of branching speciation, separated by long intervals in which little phenotypic change accumulates. Darwin’s fi fth major statement is that natural selection is the guiding force of evolution. This principle is founded on observa-tions that all species overproduce their kind, causing a struggle for the limited resources that support existence. Because no two organisms are exactly alike, and because variable traits are at least partially heritable, those organisms whose hereditary endowment enhances their use of resources for survival and reproduction con-tribute disproportionately to the next generation. Over many gen-erations, the sorting of variation by selection produces new species and new adaptations. Mutations are the ultimate source of all new variation on which selection acts. Darwin’s theory emphasizes that variation is produced at random with respect to an organism’s needs and that differential survival and reproduction provide the direction for evolutionary change. Darwin’s theory of natural selection was modifi ed around 1900 and in subsequent decades by correction of his genetic errors. This modifi ed theory is called neo-Darwinism. Population geneticists discovered the principles by which genetic properties of populations change through time. A particu-larly important discovery, known as Hardy-Weinberg equilibrium, showed that the hereditary process itself does not change the genetic composition of populations. Important sources of evolu-tionary change include mutation, genetic drift, nonrandom mating, migration, natural selection, and their interactions. Neo-Darwinism, as elaborated by population genetics, formed the basis for the Synthetic Theory of the 1930s and 1940s. Genet-ics, natural history, paleobiology, and systematics were unifi ed by the common goal of expanding our knowledge of Darwinian evo-lution. Microevolution comprises studies of genetic change within contemporary populations. These studies show that most natural populations contain enormous amounts of variation. Macroevolution comprises studies of evolutionary change on a geological timescale. Macroevolutionary studies measure rates of speciation, extinction, and changes of diversity through time. These studies have expanded Darwinian evolutionary theory to include higher-level processes that regulate rates of speciation and extinction among lineages, including species selection and catastrophic species selection. R E V I E W Q U E S T I O N S Briefl y summarize Lamarck’s concept of the evolutionary process. What is wrong with this concept? What is “uniformitarianism”? How did it infl uence Darwin’s evolutionary theory? Why was the Beagle ’s journey so important to Darwin’s thinking? What was the key idea contained in Malthus’s essay on populations that was to help Darwin formulate his theory of natural selection? Explain how each of the following contributes to Darwin’s evolutionary theory: fossils; geographic distributions of closely related animals; homology; animal classifi cation. How do modern evolutionists view the relationship between ontogeny and phylogeny? Explain how the observation of paedomorphosis confl icts with Haeckel’s “biogenetic law.” What are the important differences between the vicariant and founder-event modes of allopatric speciation? What are reproductive barriers? How do premating and postmating barriers differ? Under what conditions is sympatric speciation proposed? What is the main evolutionary lesson provided by Darwin’s fi nches on the Galápagos Islands? How is the observation of “sporting mutations” in animal breeding used to challenge Darwin’s theory of gradualism? Why did Darwin reject such mutations as having little evolutionary importance? What does the theory of punctuated equilibrium state about the occurrence of speciation throughout geological time? What observation led to this theory? Describe the observations and inferences that compose Darwin’s theory of natural selection. Identify the random and nonrandom components of Darwin’s theory of natural selection. 1.
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14. Describe some recurring criticisms of Darwin’s theory of natural selection. How can these criticisms be refuted? It is a common but mistaken belief that because some alleles are dominant and others are recessive, the dominants will eventually replace all the recessives in a population. How does the Hardy-Weinberg equilibrium refute this notion? Assume that you are sampling a trait in animal populations; the trait is controlled by a single allelic pair A and a, and you can distinguish all three phenotypes AA, Aa, and aa (intermediate inheritance). Your sample includes: Population AA Aa aa TOTAL I 300 500 200 1000 II 400 400 200 1000 Calculate the distribution of phenotypes in each population as expected under Hardy-Weinberg equilibrium. Is population I in equilibrium? Is population II in equilibrium? If after studying a population for a trait determined by a single pair of alleles you fi nd that the population is not in equilibrium, what possible reasons might explain the lack of equilibrium? Explain why genetic drift is more powerful in small populations. Describe how the effects of genetic drift and natural selection can interact in a subdivided species. Is it easier for selection to remove a deleterious recessive allele from a randomly mating population or a highly inbred population? Why? Distinguish between microevolution and macroevolution, and describe some evolutionary processes evident only at the macroevolutionary level. 15.
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136 PART TWO Continuity and Evolution of Animal Life S E L E C T E D R E F E R E N C E S Avise, J. C. 2004. Molecular markers, natural history, and evolution, ed. 2. Sunderland, Massachusetts, Sinauer Associates. An exciting and readable account of using molecular studies to help us understand evolution. Browne, E. J. 2002. Charles Darwin: a biography. New York, Knopf. A comprehensive two-volume biography of Darwin completed in 2002 . Conner, J. K., and D. L. Hartl. 2004. A primer of ecological genetics. Sunderland, Massachusetts, Sinauer Associates. An introductory text on population genetics . Coyne, J. A., and H. A. Orr. 2004. Speciation. Sunderland, Massachusetts, Sinauer Associates. A detailed coverage of speciation with emphasis on controversies in this fi eld. Darwin, C. 1859. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. London, John Murray. There were fi ve subsequent editions by the author. Freeman, S., and J. C. Herron. 2007. Evolutionary analysis, ed. 4. Upper Saddle River, New Jersey, Pearson/Prentice Hall. An introductory textbook on evolutionary biology designed for undergraduate biology majors. Futuyma, D. J. 2005. Evolution. Sunderland, Massachusetts, Sinauer Associates. A very thorough introductory textbook on evolution. Gould, S. J. 2002. The structure of evolutionary theory. Cambridge, Massachusetts, Belknap Press of Harvard University Press. A provocative discussion of what fossils tell us about the nature of life’s evolutionary history. Graur, D., and W. H. Li. 2000. Fundamentals of molecular evolution. Sunderland, Massachusetts, Sinauer Associates. A current textbook on molecular evolution. Hall, B. K. 1998. Evolutionary developmental biology. New York, Chapman and Hall. An excellent textbook on the emerging fi eld of developmental evolutionary biology. Hartl, D. L., and A. G. Clark. 2007. Principles of population genetics, ed. 4. Sunderland, Massachusetts, Sinauer Associates. A current textbook on population genetics. Jablonski, D. 2005. Mass extinctions and macroevolution. Paleobiology 31 : S192–210. A recent review of extinction theory . Levinton, J. S. 2001. Genetics, paleontology and macroevolution, ed. 2. Cambridge, U.K., Cambridge University Press. A provocative discussion on the Darwinian basis of macroevolutionary theory. Mayr, E. 2001. What evolution is. New York, Basic Books. A general survey of evolution by a leading evolutionary biologist. Mousseau, T. A., B. Sinervo, and J. Endler (eds.). 2000. Adaptive genetic variation in the wild. Oxford, U.K., Oxford University Press. Detailed examples of adaptively important genetic variation in natural populations. Ruse, M. 1998. Philosophy of biology. Amherst, New York, Prometheus Books. A collection of essays on evolutionary biology, including information on the Arkansas Balanced Treatment for Creation-Science and Evolution-Science Act. Stokstad, E. 2001. Exquisite Chinese fossils add new pages to book of life. Science 291: 232–236. Exciting new fossil discoveries help to complete our understanding of life’s evolutionary history. Some related articles immediately follow this one. Templeton, A. R. 2006. Population genetics and microevolutionary theory. Hoboken, New Jersey, Wiley-Liss. An insightful treatment of evolutionary theory at the population level . West-Eberhard, M. J. 2003. Developmental plasticity and evolution. Oxford, U.K., Oxford University Press. A provocative discussion of the role of development and modularity in evolution . O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links, and more! C H A P T E R 7 The Reproductive Process Human egg and sperm at the moment of fertilization. “Omne vivum ex ovo” In 1651, late in a long life, William Harvey, the English physiologist who earlier had founded experimental physiology by explaining the circuit of blood, published a treatise on reproduction. He asserted that all life developed from the egg— omne vivum ex ovo. This was insightful, since Harvey had no means for visualizing eggs of many animals, in particular the microscopic mammalian egg, which is no larger than a speck of dust to the unaided eye. Further, argued Harvey, eggs are launched into their developmental course by some infl uence from semen, a conclusion that was either remark-ably perceptive or a lucky guess, since sperm also were invisible to Harvey. Such ideas differed sharply from existing notions of biogenesis, which saw life springing from many sources of which eggs were but one. Harvey was describing characteristics of sexual reproduction in which two parents, male and female, must produce gametes that fuse to become a new individual. Despite the importance of Harvey’s aphorism that all life arises from eggs, it was not wholly correct. Life springs from reproduction of preexisting life, and reproduction may not be restricted to eggs and sperm. Asexual reproduction, the creation of new, genetically identical individuals by budding or fragmentation or fi ssion from a single parent, is common, indeed characteristic, among some phyla. Nevertheless, most animals have found sex the winning strategy, probably because sexual reproduction promotes diversity, enhancing long-term survival of the lineage in a world of perpetual change. 138 PART TWO Continuity and Evolution of Animal Life R eproduction is one of the ubiquitous properties of life. Evolution is inextricably linked to reproduction, because the ceaseless replacement of aging predecessors with new life gives animal populations the means to adapt to a changing environment. In this chapter we distinguish asexual and sexual reproduction and explore the reasons why, for mul-ticellular animals at least, sexual reproduction appears to offer important advantages over asexual. We then consider, in turn, the origin and maturation of germ cells; plan of reproductive systems; reproductive patterns in animals; and, fi nally, the endo-crine events that orchestrate reproduction. NATURE OF THE REPRODUCTIVE PROCESS Two modes of reproduction are recognized: asexual and sexual. In asexual reproduction ( Figure 7.1A and B) there is only one parent and with no special reproductive organs or cells. Each organism is capable of producing genetically identical copies of itself as soon as it becomes an adult. The production of copies is marvelously simple, direct, and typically rapid. Sexual reproduction ( Figure 7.1C and D) as a rule involves two parents, each of which contributes special germ cells ( gametes or sex cells ) that in union (fertilization) develop into a new individual. The zygote formed from this union receives genetic material from both parents, and the combination of genes (p. 90) produces a genetically unique individual, bearing characteristics of the species but also bearing traits that make it different from its parents. Sexual reproduction, by recombining parental characters, multiplies variations and makes possible a richer and more diversifi ed evolution. Mechanisms for interchange of genes between individuals are more limited in organisms with only asexual reproduction. Of course, in asexual organisms that are haploid (bear only one set of genes, p. 77), mutations are immediately expressed and evolution proceeds quickly. In sexual animals, on the other hand, a gene mutation is often not expressed immediately, since it may be masked by its normal partner on the homologous chromosome. (Homologous chromosomes, discussed on p. 77, are those that pair during meiosis and carry genes encoding the same characteristics.) There is only a remote chance that both members of a gene pair will mutate in the same way at the same moment and thus be expressed immediately. Asexual Reproduction: Reproduction without Gametes Asexual reproduction ( Figure 7.1A and B) is the production of individuals without gametes (eggs or sperm). It includes a num-ber of distinct processes, all without involving sex or a second parent. Offspring produced by asexual reproduction all have the same genotype (unless mutations occur) and are called clones. Asexual reproduction appears in bacteria and unicellular eukaryotes and in many invertebrate phyla, such as cnidarians, bryozoans, annelids, echinoderms, and hemichordates. In Figure 7.1 Examples of asexual and sexual reproduction in animals. A, Binary fi ssion in Paramecium, a single-celled eukaryote, results in two individuals. B, Budding, a simple form of asexual reproduction as shown in a hydra, a radiate animal. The buds, shown growing out of the central, parent hydra, eventually detach themselves and grow into fully formed individuals. C, Earthworms reproduce sexually, but are hermaphroditic, with each individual bearing both male and female organs. Each earthworm passes sperm from genital pores along grooves to seminal receptacles of its mate. D, Frogs, here in mating position (amplexus), represent bisexual reproduction, the most common form of sexual reproduction involving separate male and female individuals. A Binary fission in Paramecium B Hydra budding D Frogs in amplexus C Earthworms mating www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 139 animal phyla in which asexual reproduction occurs, most members also employ sexual reproduction. In these groups, asexual reproduction ensures rapid increase in numbers when development and differentiation of the organism has not advanced to the point of forming gametes. Asexual reproduction is absent among vertebrates (although some forms of parthenogenesis have been interpreted as asexual by some authors; see p. 140). It would be a mistake to conclude that asexual reproduction is in any way a “defective” form of reproduction relegated to the minute forms of life. Given the facts of their abundance, that they have persisted on earth for 3.5 billion years, and that they form the base of the food chain on which all higher forms depend, single-celled asexual organisms are both resoundingly abundant and supremely important. For these forms the advantages of asexual reproduction are its rapidity (many bacteria divide every half hour) and simplic-ity (no germ cells to produce and no time and energy expended in fi nding a mate). The basic forms of asexual reproduction are fi ssion (binary and multiple), budding, gemmulation, and fragmentation. Binary fi ssion is common among bacteria and protozoa ( Figure 7.1A ). In binary fi ssion the body of the unicellular parent divides by mitosis (p. 52) into two approximately equal parts, each of which grows into an individual similar to the parent. Binary fi ssion may be lengthwise, as in fl agellate protozoa, or transverse, as in ciliate protozoa. In multiple fi ssion, or schi-zogony, the nucleus divides repeatedly before division of the cytoplasm, producing many daughter cells simultaneously. Spore formation, called sporogony, is a form of multiple fi ssion common among some parasitic protozoa, for example, malarial parasites. Budding is an unequal division of an organism. A new indi-vidual arises as an outgrowth (bud) from its parent, develops organs like those of the parent, and then detaches itself. Bud-ding occurs in several animal phyla and is especially prominent in cnidarians ( Figure 7.1B ). Gemmulation is the formation of a new individual from an aggregation of cells surrounded by a resistant capsule, called a gemmule. In many freshwater sponges, gemmules develop in the fall and survive the winter in the dried or frozen body of the parent. In spring, the enclosed cells become active, emerge from the capsule, and grow into a new sponge. In fragmentation a multicellular animal breaks into two or more parts, with each fragment capable of becoming a complete individual. Many invertebrates can reproduce asexually by sim-ply breaking into two parts and then regenerating the missing parts of the fragments, for example, most anemones and many hydroids. Many echinoderms can regenerate lost parts, but this is not the same as reproduction by fragmentation. Sexual Reproduction: Reproduction with Gametes Sexual reproduction is the production of individuals from gam-etes. It includes bisexual (or biparental ) reproduction as the most common form, involving two separate individuals. Her-maphroditism and parthenogenesis are less common forms of sexual reproduction. Bisexual Reproduction Bisexual reproduction is the production of offspring formed by the union of gametes from two genetically different parents ( Figures 7.1C and D, and 7.2 ). Offspring will thus have a new genotype different from either parent. Individuals sharing parent-hood are characteristically of different sexes, male and female (there are exceptions among sexually reproducing organisms, such as bacteria and some protozoa in which sexes are lacking). Each has its own reproductive system and produces only one kind of germ cell, spermatozoon or ovum, rarely both. Nearly all vertebrates and many invertebrates have separate sexes, and such a condition is called dioecious (Gr. di, two, oikos, house). Individual animals that have both male and female reproduc-tive organs are called monoecious (Gr. monos, single, oikos, house). These animals are called hermaphrodites (from a com-bination of the names of the Greek god Hermes and goddess Aphrodite); this form of reproduction is described on page 140. Distinctions between male and female are based, not on any differences in parental size or appearance, but on the size and mobility of the gametes they produce. The ovum (egg), pro-duced by the female, is large (because of stored yolk to sustain early development), nonmotile, and produced in relatively small numbers. The spermatozoon (sperm), produced by the male, is small, motile, and produced in enormous numbers. Each sperm is a stripped-down package of highly condensed genetic mate-rial designed for the single purpose of reaching and fertilizing an egg. There is another crucial event that distinguishes sexual from asexual reproduction: meiosis, a distinctive type of gamete-producing nuclear division (described in detail on p. 77). Meiosis differs from ordinary cell division (mitosis) in being a double division. Chromosomes split once, but the cell divides twice, producing four cells, each with half the original number of chro-mosomes (the haploid number). Meiosis is followed by fertil-ization in which two haploid gametes are combined to restore the normal ( diploid ) chromosomal number of the species. The new cell (zygote), which now begins to divide by mito-sis (described on p. 52), has equal numbers of chromosomes from each parent and is a unique individual bearing a recombi-nation of parental characteristics. Genetic recombination is the great strength of sexual reproduction that keeps feeding new genetic combinations into the population. Many unicellular organisms reproduce both sexually and asexually. When sexual reproduction does occur, it may or may not involve male and female gametes. Sometimes two mature sexual parent cells join together to exchange nuclear material or merge cytoplasm ( conjugation, p. 232 in Chapter 11). Distinct sexes do not exist in these cases. The male-female distinction is more clearly evident in most animals. Organs that produce germ cells are called gonads. The gonad that produces sperm is a testis (see Figure 7.12 ) and 140 PART TWO Continuity and Evolution of Animal Life that which forms eggs is an ovary (see Figure 7.13 ). Gonads represent the primary sex organs, the only sex organs found in certain groups of animals. Most metazoa, however, have various accessory sex organs (such as penis, vagina, uterine tubes, and uterus) that transfer and receive germ cells. In the primary sex organs germ cells undergo many complicated changes during their development, the details of which are described on pages 143–146. Hermaphroditism Animals that have both male and female organs in the same individual are called hermaphrodites, and the condition is called hermaphroditism. In contrast to the dioecious state of separate sexes, hermaphrodites are monoecious, meaning that the same organism bears both male and female organs. Many sessile, burrowing, or endoparasitic invertebrate animals are her-maphoditic (for example, most fl atworms, some hydroids and annelids, and all barnacles and pulmonate snails), as well as a few vertebrates (some fi shes). Some hermaphrodites fertilize themselves, but most avoid self-fertilization by exchanging germ cells with another member of the same species ( Figures 7.1C and 7.3 ). An advantage is that with every individual producing eggs, a hermaphroditic species could potentially produce twice as many offspring as could a dioecious species in which half the individuals are nonproductive males. Some fi shes are sequen-tial hermaphrodites, in which a genetically programmed sex change occurs within an individual organism. In many species of reef fi shes, for example, wrasses, an animal begins life as either a female or a male (depending on the species) but later becomes the opposite sex. Parthenogenesis Parthenogenesis (“virgin origin”) is the development of an embryo from an unfertilized egg or one in which the male and female nuclei fail to unite following fertilization. There are many patterns of parthenogenesis. In one type, called ameiotic parthenogenesis, no meiosis occurs, and the egg is formed by mitotic cell division. This “asexual” form of parthenogene-sis occurs in some species of fl atworms, rotifers, crustaceans, insects, and probably others. In these cases, the offspring are clones of the parent because, without meiosis, the parent’s chro-mosomal complement is passed intact to offspring. In meiotic parthenogenesis a haploid ovum is formed by meiosis, and it may or may not be activated by the infl uence of Figure 7.2 A sexual life cycle. The life cycle begins with haploid germ cells, formed by meiosis, combining to form a diploid zygote, which grows by mitosis to an adult. Most of the life cycle is spent as a diploid organism.
Meiosis Meiosis FEMALE (diploid) 2N MALE (diploid) 2N Fertilization Egg (haploid) N Zygote (diploid) 2N Sperm (haploid) N Growth Adult male and female mate www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 141 a male’s sperm. For example, in some species of fi shes, a female may be inseminated by a male of the same or related species, but the sperm serves only to activate the egg; the male’s genetic material is rejected before it can penetrate the egg ( gynogen-esis ). In several species of fl atworms, rotifers, annelids, mites, and insects, the haploid egg begins development spontaneously; no males are required to stimulate activation of an ovum. The diploid condition may be restored by chromosomal duplication or by autogamy (rejoining of haploid nuclei). A variant of this type of parthenogenesis occurs in many bees, wasps, and ants. In honey bees, for example, the queen bee can either fertilize eggs as she lays them or allow them to pass unfertilized. Fer-tilized eggs become diploid females (queens or workers), and unfertilized eggs develop parthenogenetically to become hap-loid males (drones); this type of sex determination is known as haplodiploidy. In some animals meiosis may be so severely modifi ed that offspring are clones of the parent. Certain popula-tions of whiptail lizards of the American southwest are clones consisting solely of females (Cole, 1984). Parthenogenesis is surprisingly widespread in animals. It is an abbreviation of the usual steps of bisexual reproduction. It may have evolved to avoid the problem—which may be great in some animals—of bringing together males and females at the right moment for successful fertilization. The disadvantage of par-thenogenesis is that if the environment should suddenly change parthenogenetic species have limited capacity to shift gene com-binations to adapt to any new conditions. Bisexual species, by recombining parental characteristics, have a better chance of producing variant offspring that can utilize new environments. Occasionally claims arise that spontaneous parthenogenetic devel-opment to term has occurred in humans. A British investigation of about 100 cases in which the mother denied having had intercourse revealed that in nearly every case the child possessed characteristics not present in the mother, and consequently must have had a father. Nevertheless, mammalian eggs very rarely spontaneously start devel-oping into embryos without fertilization. In certain strains of mice, such embryos will develop into fetuses and then die. The most remarkable instance of parthenogenetic development among verte-brates occurs in turkeys in which ova of certain strains, selected for their ability to develop without sperm, grow to reproducing adults. Why Do So Many Animals Reproduce Sexually Rather Than Asexually? Because sexual reproduction is so nearly universal among ani-mals, it might be inferred to be highly advantageous. Yet it is easier to list disadvantages to sex than advantages. Sexual repro-duction is complicated, requires more time, and uses much more energy than asexual reproduction. Males may waste valuable energy in competition for a mate and often possess sexual char-acteristics that can be detrimental to survival—for example, the elongated tail feathers of peacocks. Mating partners must come together, and this can be a disadvantage in sparsely populated areas for some species. Males and females must also coordinate their activities to produce young. Many biologists believe that an even more troublesome problem is the “cost of meiosis.” A female that reproduces asexually passes all of her genes to her offspring, but when she reproduces sexually the genome is divided during meiosis and only half her genes fl ow to the next generation. Another cost is wastage in production of males, many of which fail to reproduce and thus consume resources that could be applied to production of females. Whiptail lizards of the American southwest offer a fascinating example of the potential advantage of parthenogenesis (discussed on p. 140). When unisexual and bisexual species of the same genus are reared under similar conditions in the laboratory, the popula-tion of the unisexual species grows more quickly because all unisexual lizards (all females) deposit eggs, whereas only 50% of the bisexual lizards do so ( Figure 7.4 ). Clearly, the costs of sexual reproduction are substantial. How are they offset? Biologists have disputed this question for years. One hypothesis suggests that sexual reproduction, with its separation and recombination of genetic material, keeps produc-ing novel genotypes that in times of environmental change may be advantageous to survival and thus the organism may live to reproduce, whereas most others die. An often quoted example is the rapidly changing environment of organisms that is produced by parasites continuously evolving new mechanisms of attack and therefore favoring recombination of their hosts. Variability, advocates of this viewpoint argue, is sexual reproduction’s trump card. Another hypothesis suggests that sexual recombination provides a means for the spread of benefi cial mutations without a population being held back by deleterious ones. Experimental support for this hypothesis has recently been provided using the fruit fl y, Drosophila, in which benefi cial mutations increased to a greater degree in sexual populations compared with clonal (asexual) ones. These hypotheses are not mutually exclusive, Figure 7.3 Hermaphroditic earthworms mating. Earthworms are “simultaneous” hermaphrodites; during mating each partner passes sperm from genital pores along grooves to seminal receptacles of its mate. They are held together by mucous secretions during this process.
142 PART TWO Continuity and Evolution of Animal Life however, and both provide possible explanations for the evolu-tion of sexual reproduction. There still remains the question of why sexual reproduction has been maintained in spite of its costs. Considerable evidence suggests that asexual reproduction is most successful in colonizing new environments. When habitats are empty what matters most is rapid reproduction; variability and increased fitness provided by beneficial genetic recombination matter little. As habitats become more crowded, competition between species for resources increases. Selection becomes more intense, and genetic variability—new benefi cial genotypes produced by recombination in sexual reproduction—furnishes the diversity that permits a population to resist extinction. Therefore, on a geological timescale, asexual lineages, because they lack genetic fl exibility, may be more prone to extinction than sexual lineages. Sexual reproduction is therefore favored by species selection (species selection is described on p. 133). There are many invertebrates that use both sexual and asexual reproduction, thus enjoying the advantages each has to offer. THE ORIGIN AND MATURATION OF GERM CELLS Many sexually reproducing organisms are composed of nonre-productive somatic cells, which are differentiated for special-ized functions and die with the individual, and germ cells, which form the gametes: eggs and sperm. Germ cells provide continuity of life between generations, the germ cell line. Germ cells, or their precursors, the primordial germ cells, develop at the beginning of embryonic development (described in Chapter 8), usually in the endoderm, and migrate to the gonads. Here they develop into eggs or sperm. The other cells of the gonads are somatic cells. They cannot form eggs or sperm, but they are necessary for support, protection, and nourishment of the germ cells during their development (gametogenesis). A traceable germ cell line, as present in vertebrates, is also distinguishable in some invertebrates, such as nematodes and arthropods. In many invertebrates, however, germ cells develop directly from somatic cells at some period in the life of an individual. Migration of Germ Cells In vertebrates, the actual tissue from which gonads arise appears in early development as a pair of genital ridges, growing into the coelom from the dorsal coelomic lining on each side of the hindgut near the anterior end of the kidney (mesonephros). Surprisingly perhaps, primordial germ cells do not arise in the developing gonad, but in the yolk-sac endoderm (p. 175). From studies with frogs and toads, it has been possible to trace the germ cell line back to the fertilized egg, in which a localized area of germinal cytoplasm (called germ plasm ) can be identifi ed in the vegetal pole of the uncleaved egg mass. This material can be followed through subsequent cell divisions of the embryo until it becomes situated in primordial germ cells in gut endoderm. From here the cells migrate by ameboid movement to the genital ridges, located on either side of the hindgut. A similar migration of primordial germ cells occurs in mammals ( Figure 7.5 ). Primor-dial germ cells are the future stock of gametes for an animal. Once in the genital ridges and during subsequent gonadal devel-opment, germ cells begin to divide by mitosis, increasing their numbers from a few dozen to several thousand. Sex Determination At fi rst gonads are sexually indifferent. In normal human males, a “male-determining gene” on the Y chromosome called SRY ( sex-determining region Y ) organizes the developing gonad into a testis instead of an ovary. Once formed, the testis secretes the steroid testosterone. This hormone, and its metabolite, dihy-drotestosterone (DHT), masculinizes the fetus, causing the dif-ferentiation of penis, scrotum, and the male ducts and glands. It also destroys the incipient breast primordia, but leaves behind the nipples that are a reminder of the indifferent ground plan from which both sexes develop. Testosterone is also responsible for the masculinization of the brain, but it does so indirectly. Surprisingly, testosterone is enzymatically converted to estrogen in the brain, and it is estrogen that determines the organization of the brain for male-typical behavior. Biologists have often stated that in mammals the indifferent gonad has an inherent tendency to become an ovary. Classic experiments performed in rabbits provide support for the idea that the female is the default sex during development. Removal of the fetal gonads before they have differentiated will invariably produce a female with uterine tubes, uterus, and vagina, even if the rabbit is a genetic male. Localization in 1994 of a region on the X chromosome named DDS (dosage-sensitive sex reversal) or Figure 7.4 Comparison of the growth of a population of unisexual whiptail lizards with a population of bisexual lizards. Because all individuals of the unisexual population are females, all produce eggs, whereas only half the bisexual population are egg-producing females. By the end of the third year the unisexual lizards are more than twice as numerous as the bisexual ones.
Bisexual population Unisexual population 0 1 2 3 Year 1000 800 600 400 200 0 s d r a z i l f o r e b m u N www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 143 SRVX (sex-reversing X), which promotes ovary formation, has challenged this view. In addition, the presence of such a region may help to explain feminization in some XY males. It is clear, however, that absence of testosterone in a genetic female embryo promotes development of female sexual organs: vagina, clitoris, and uterus. The developing female brain does require special protection from the effects of estrogen because, as mentioned earlier, estrogen causes masculinization of the brain. In rats, a blood protein (alpha-fetoprotein) binds to estrogen and keeps the hormone from reaching the developing female brain. This does not appear to be the case in humans, however, and even though circulating fetal estrogen levels can be quite high, the developing female brain does not become masculinized. One possible explanation is that the level of brain estrogen receptors in the developing female brain is low, and therefore, high levels of circulating estrogen would have no effect. The genetics of sex determination are treated in Chapter 5 (p. 80). Sex determination is strictly chromosomal in mammals, birds, amphibians, most reptiles, and probably most fi shes. How-ever, some fi shes and reptiles lack sex chromosomes altogether; in these groups, gender is determined by nongenetic factors such as temperature or behavior. In crocodilians, many turtles, and some lizards the incubation temperature of the nest determines the sex ratio probably by indirectly activating and/or suppressing genes that direct development of the animals’ sex organs. Alliga-tor eggs, for example, incubated at low temperature all become females; those incubated at higher temperature all become males ( Figure 7.6 ). Sex determination of many fi shes is behavior depen-dent. Most of these species are hermaphroditic, possessing both male and female gonads. Sensory stimuli from the animal’s social environment determine whether it will be male or female. Gametogenesis Mature gametes are produced by a process called gametogenesis. Although the same essential processes are involved in maturation of both sperm and eggs in vertebrates, there are some important differences. Gametogenesis in testes is called spermatogenesis, and in ovaries, oogenesis. Spermatogenesis The walls of the seminiferous tubules contain differentiating germ cells arranged in a stratifi ed layer fi ve to eight cells deep ( Figure 7.7 ). Germ cells develop in close contact with large Ser-toli (sustentacular) cells, which extend from the periphery of the seminiferous tubules to the lumen and provide nourishment during germ-cell development and differentiation ( Figure 7.8 ). The outermost layers contain spermatogonia, diploid cells that Figure 7.5 Migration of mammalian primordial germ cells. A, From the yolk sac the primordial germ cells migrate toward the region where the hindgut develops. B, Later-stage embryo in which the hindgut is more developed. Enlarged fi gure shows the germ cells migrating through the hindgut and into the genital ridges. In human embryos, migration is complete by the end of the fi fth week of gestation.
Hindgut Hindgut Coelom Primordial germ cells Genital ridge Primordial germ cells Heart Yolk sac B A Figure 7.6 Temperature-dependent sex determination. In many reptiles that lack sex chromosomes incubation temperature of the nest determines gender. The graph shows that embryos of many turtles develop into males at low temperature, whereas embryos of many lizards and alligators become males at high temperatures. Embryos of crocodiles, leopard geckos, and snapping turtles become males at intermediate temperatures, and become females at higher or lower temperatures.
Source: Data from David Crews, “Animal Sexuality,” Scientifi c American 270(1):108–114, January 1994.
Percent males Incubation temperature Low High 20 0 40 60 80 100 Many lizards, alligators Many turtles Leopard gecko, snapping turtle, crocodiles KEY: 144 PART TWO Continuity and Evolution of Animal Life have increased in number by mitosis. Each spermatogonium increases in size and becomes a primary spermatocyte. Each primary spermatocyte then undergoes the fi rst meiotic division, as described in Chapter 5 (p. 79), to become two secondary spermatocytes ( Figure 7.8 ). For every structure in the reproductive system of males or females, there is a homologous structure in the other. This happens because during early development male and female characteristics begin to differentiate from the embryonic genital ridge, and two duct systems develop, which at fi rst are identical in both sexes. Under the infl uence of sex hormones, the genital ridge develops into the testes of males and the ovaries of females. One duct system (mesonephric or Wolffi an) becomes ducts of the testes in males and regresses in females. The other duct (paramesonephric or Müllerian) develops into the uterine tubes, uterus, and vagina of females and regresses in males. Similarly, the clitoris and labia of females are homologous to the penis and scrotum of males, since they develop from the same embryonic structures. Figure 7.7 Section of a seminiferous tubule containing male germ cells. More than 200 meters long, highly coiled seminiferous tubules are packed in each human testis. This scanning electron micrograph reveals, in the tubule’s central cavity, numerous tails of mature spermatozoa that have differentiated from germ cells in the periphery of the tubule. (525) From R. G. Kessel and R. H. Kardon, Tissues and Organs: A Text-Atlas of Scanning Electron Microscopy, 1979, W. H. Freeman and Co.
Figure 7.8 Spermatogenesis. Section of seminiferous tubule showing spermatogenesis. Germ cells develop within the recesses of large Sertoli (sustentacular) cells, that extend from the periphery of seminiferous tubules to their lumen, and that provide nourishment to the germ cells. Stem germ cells from which sperm differentiate are the spermatogonia, diploid cells located peripherally in the tubule. These divide by mitosis to produce either more spermatogonia or primary spermatocytes. Meiosis begins when primary spermatocytes divide to produce haploid secondary spermatocytes with double-stranded chromosomes. The second meiotic division forms four haploid spermatids with single-stranded chromosomes. As sperm develop, they are gradually pushed toward the lumen of the seminiferous tubule.
Spermatozoa Spermatogenesis Spermatogonium (diploid) MEIOSIS I Primary spermatocyte (diploid) MEIOSIS II Secondary spermatocytes (haploid) Spermatids (haploid) Epididymis Vasa efferentia Testis Coiled seminiferous tubules Vas deferens Cross section of seminiferous tubule Sertoli (sustentacular) cell www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 145 Each secondary spermatocyte enters the second meiotic divi-sion without intervention of a resting period. In the two steps of meiosis each primary spermatocyte gives rise to four spermatids, each containing the haploid number (23 in humans) of chromo-somes. A spermatid usually contains a combination of his parents’ chromosomes but may contain all chromosomes that the male inherited from his mother or from his father. Without further divi-sions the spermatids are transformed into mature spermatozoa or (sperm) ( Figure 7.8 ). Modifi cations include great reduction of cytoplasm, condensation of the nucleus into a head, formation of a middle piece containing mitochondria, and a whiplike, fl agellar tail for locomotion ( Figures 7.8 and 7.9 ). The head consists of a nucleus containing the chromosomes for heredity and an acro-some, a distinctive feature of nearly all metazoa (exceptions are teleost fi shes and certain invertebrates). In many species, both invertebrate and vertebrate, the acrosome contains enzymes that are released to clear an entrance through the layers that surround an egg. In mammals at least, one of the enzymes is hyaluronidase, which allows a sperm to penetrate the follicular cells surrounding an egg. A striking feature of many invertebrate spermatozoa is the acrosome fi lament, an extension of varying length in differ-ent species that projects suddenly from the sperm head when the latter fi rst contacts the surface of an egg. Fusion of the egg and sperm plasma membranes is the initial event of fertilization (see Contact and Recognition between Egg and Sperm, p. 160). The total length of a human sperm is 50 to 70 µm. Some toads have sperm that exceed 2 mm (2000 µm) in length ( Fig-ure 7.9 ) and are easily visible to the unaided eye. Most sperm, however, are microscopic in size (see p. 159 for an early seventeenth-century drawing of mammalian sperm, interpreted by biologists of the time as parasitic worms in the semen). In all sexually reproducing animals the number of sperm in males is far greater than the number of eggs in corresponding females. The number of eggs produced is correlated with the chances of young to hatch and to reach maturity. Oogenesis Early germ cells in the ovary, called oogonia, increase in num-ber by mitosis. Each oogonium contains the diploid number of chromosomes. After the oogonia cease to increase in number, they grow in size and become primary oocytes ( Figure 7.10 ). Before the fi rst meiotic division, the chromosomes in each pri-mary oocyte meet in pairs, paternal and maternal homologues, just as in spermatogenesis. When the fi rst maturation (reduction) division occurs, the cytoplasm is divided unequally. One of the two daughter cells, the secondary oocyte, is large and receives most of the cytoplasm; the other is very small and is called the fi rst polar body ( Figure 7.10 ). Each of these daughter cells, however, has received half of the chromosomes. In the second meiotic division, the secondary oocyte divides into a large ootid and a small polar body. If the fi rst polar body also divides in this division, which sometimes happens, there are three polar bodies and one ootid ( Figure 7.10 ). The ootid devel-ops into a functional ovum. Polar bodies are nonfunctional, and they disintegrate. Formation of nonfunctional polar bodies is necessary to enable an egg to dispose of excess chromosomes, Figure 7.9 Examples of vertebrate and invertebrate sperm. The head and midpiece region of the human sperm is shown in more detail.
Crab Nematode Butterfly Skate Toad Robin Human Head Midpiece Tail Acrosome Nucleus Centrioles Mitochondria and the unequal cytoplasmic division makes possible a large cell with the cytoplasm containing a full set of cytoplasmic compo-nents needed for early development. Thus a mature ovum has N (haploid) number of chromosomes, the same as a sperm. How-ever, each primary oocyte gives rise to only one functional gam-ete instead of four as in spermatogenesis. In most vertebrates and many invertebrates the egg does not actually complete meiotic division before fertilization occurs. The general rule is that development is arrested during prophase I of the fi rst meiotic division (in the primary oocyte phase). Meiosis resumes and is completed either at the time of ovulation (birds 146 PART TWO Continuity and Evolution of Animal Life and most mammals) or shortly after fertilization (many inverte-brates, teleost fi shes, amphibians, and reptiles). In humans, the ova begin the fi rst meiotic division at about the thirteenth week of fetal development. Then their development arrests in prophase I as the primary oocyte until puberty, at which time one of these primary oocytes typically develops into a secondary oocyte each menstrual month. Thus, in humans meiosis II is completed only when the secondary oocyte is penetrated by a spermatozoon. In many animals, the most obvious feature of egg matura-tion is deposition of yolk. Yolk, usually stored as granules or more organized platelets, is not a defi nite chemical substance but may be lipid or protein or both. Yolk may be synthesized within an egg from raw materials supplied by surrounding fol-licle cells, or preformed lipid or protein yolk may be transferred by pinocytosis from follicle cells to the oocyte. Eggs also contain a large amount of mRNA that in not trans-lated (p. 95) into polypeptides/proteins until fertilization triggers activation of these previously quiescent mRNA molecules. At this time the newly formed polypeptides/proteins begin to orches-trate the developmental process (see Chapter 8, p. 160) . Enormous accumulation of yolk granules, other nutrients (glycogen and lipid droplets), and quiescent mRNA cause an egg to grow well beyond the normal limits that force ordinary body (somatic) cells to divide. A young frog oocyte 50 µm in diameter, for example, grows to 1500 µm in diameter when mature after 3 years of growth in the ovary, and its volume is increased by a factor of 27,000. Bird eggs attain even greater absolute size; a hen egg increases 200 times in volume in only the last 6 to 14 days of rapid growth preceding ovulation. Thus eggs are remarkable exceptions to the otherwise uni-versal rule that organisms are composed of relatively minute cellular units. An egg’s large size creates a problematic surface area-to-cell volume ratio, since everything that enters and leaves the ovum (nutrients, respiratory gases, wastes, and so on) must pass through the cell membrane. As the egg becomes larger, the available surface per unit of cytoplasmic volume (mass) becomes smaller. As we would anticipate, the metabolic rate of an egg gradually diminishes until a secondary oocyte or ovum (depend-ing on the species) is in suspended animation until fertilization. REPRODUCTIVE PATTERNS The great majority of invertebrates, as well as many vertebrates, lay their eggs outside the body for development; these animals are called oviparous (“egg-birth”). Fertilization may be either internal (eggs are fertilized inside the body of a female before she lays them) or external (eggs are fertilized by a male after a female lays them). While many oviparous animals simply abandon their eggs rather indiscriminately, others display extreme care in fi nding places that will provide immediate and suitable sources of food for the young when they hatch. Some animals retain eggs in their body (in the oviduct or uterus) while they develop, with embryos deriving all their nourishment from yolk stored within the egg. These animals are Figure 7.10 Oogenesis in humans. Early germ cells (oogonia) increase by mitosis during embryonic development to form diploid primary oocytes. After puberty, each menstrual month a diploid primary oocyte divides in the fi rst meiotic division into a haploid secondary oocyte and a haploid polar body. If the secondary oocyte is fertilized, it enters the second meiotic division. The double-stranded chromosomes separate into a large ootid and small second polar body. The ootid develops into an ovum. Both ovum and second polar body now contain the N amount of DNA. Fusion of the haploid egg nucleus with a haploid sperm nucleus produces a diploid (2N) zygote.
MEIOSIS I MEIOSIS II First polar body Second polar body Ovum (haploid) Secondary oocyte Primary oocyte Oogonium (diploid) Primary oocytes Mature follicle with secondary oocyte Ruptured follicle Corpus luteum Developing follicle Fertilization Fallopian tube Oogenesis www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 147 called ovoviviparous (“egg-live-birth”). Ovoviviparity occurs in several invertebrate groups (for example, various annelids, brachiopods, insects, and gastropod molluscs) and is common among certain fi shes (p. 538) and reptiles (p. 580). In the third pattern, viviparous (“live-birth”), eggs develop in the oviduct or uterus with embryos deriving their nourishment directly from the mother. Usually some kind of intimate anatomical relationship is established between developing embryos and their mother. In both ovoviviparity and viviparity, fertilization must be internal (within the body of the female) and the mother gives birth to young usually in a more advanced stage of development. Vivi-parity is confi ned mostly to lizards, snakes, mammals, and elas-mobranch fi shes, although viviparous invertebrates (scorpions, for example) and amphibians are known. Development of embryos within a mother’s body, whether ovoviviparous or viviparous, obvi-ously affords more protection to the offspring than egg-laying. STRUCTURE OF REPRODUCTIVE SYSTEMS The basic components of reproductive sys-tems are similar in sexual animals, although differences in reproductive habits and meth-ods of fertilization have produced many variations. Sexual systems consist of two components: (1) primary organs, which are the gonads that produce sperm and eggs and sex hormones; and (2) accessory organs, which assist the gonads in forma-tion and delivery of gametes, and may also serve to support the embryo. They are of great variety, and include gonoducts (sperm ducts and oviducts), accessory organs for transferring spermatozoa into the female, storage organs for spermatozoa or yolk, packaging systems for eggs, and nutritional organs such as yolk glands and placenta. Invertebrate Reproductive Systems Invertebrates that transfer sperm from male to female for internal fertilization require organs and plumbing to facilitate this func-tion that may be as complex as those of any vertebrate. In contrast, reproductive systems of invertebrates that simply release their gametes into the water for external fertilization may be little more than centers for gametogenesis. Polychaete annelids, for example, have no permanent reproductive organs. Gametes arise by proliferation of cells lining the body cavity. When mature the gametes are released through coelomic or nephridial ducts or, in some species, may exit through ruptures in the body wall. Insects have separate sexes (dioecious), practice internal fertilization by copulation and insemination, and consequently have complex reproductive systems ( Figure 7.11 ). Sperm from the testes pass through sperm ducts to seminal vesicles (where the sperm are stored) and then through a single ejaculatory duct to a penis. Seminal fl uid from one or more accessory glands is added to the semen in the ejaculatory duct. Females have a pair of ovaries formed from a series of egg tubes (ovarioles). Mature ova pass through oviducts to a common genital chamber and then to a short copulatory bursa (vagina). In most insects, the male transfers sperm by inserting the penis directly into the female’s genital bursa (vagina); from here they migrate, and are stored in a seminal receptacle. Often a single mating provides suffi cient sperm to last the reproductive life of a female. Vertebrate Reproductive Systems In vertebrates the reproductive and excretory systems are together called the urogenital system because of their close anatomical connection, especially in males. This association is Figure 7.11 Reproductive system of crickets. Sperm from the paired testes of males pass through sperm tubes (vas deferens) to an ejaculatory duct housed in the penis. In females, eggs from the ovaries pass through oviducts to the genital bursa. At mating sperm enclosed in a membranous sac (spermatophore) formed by the secretions of the accessory gland are deposited in the genital bursa of the female, then migrate to her seminal receptacle where they are stored. The female controls the release of a few sperm to fertilize her eggs at the moment they are laid, using the needlelike ovipositor to deposit the eggs in the soil.
Ovary with eggs Testis Vas deferens Male accessory gland Penis (Aedeagus) Male genital bulb Epiproct Seminal receptacle Oviduct Bursa (vagina) Ovipositor MALE FEMALE 148 PART TWO Continuity and Evolution of Animal Life This odd arrangement provides an environment of slightly lower temperature, since in most mammals (including humans) viable sperm do not form at temperatures maintained within the body. In marine mammals and all other vertebrates the testes are posi-tioned permanently within the abdomen. The sperm travel from the seminiferous tubules to the vasa efferentia, small tubes passing to a coiled epididymis (one for each testis), where fi nal sperm maturation occurs and then to a vas deferens, the ejaculatory duct ( Figures 7.8 and 7.12 ). In mammals the vas deferens joins the urethra, a duct that carries both sperm and urinary products through the penis, or external intromittent organ. Most aquatic vertebrates have no need for a penis, since sperm and eggs are liberated into the water in close proximity to each other. However, in terrestrial (and some aquatic) vertebrates that bear their young alive or enclose the egg within a shell, sperm must be transferred to the female. Few birds have a true penis (examples of exceptions are the ostrich and the Argentine lake duck) and the mating process simply involves presenting cloaca to cloaca. Most reptiles and mammals have a true penis. In mammals the normally fl accid organ becomes erect when engorged with blood. Some mammals possess a bone in the penis (baculum), which presum-ably helps with rigidity. In most mammals three sets of accessory glands open into the reproductive channels: a pair of seminal vesicles, a single prostate gland, and the pair of bulbourethral glands ( Fig-ure 7.12 ). Fluid secreted by these glands furnishes food to the sperm, lubricates the female reproductive tract for sperm, and counteracts the acidity of the vagina so that the sperm retain their viability longer after being deposited in the female. Female Reproductive System The ovaries of female vertebrates produce both ova and female sex hormones (estrogens and progesterone). In all jawed vertebrates, mature ova from each ovary enter the funnel-like opening of a uterine tube or oviduct, which typically has a fringed margin (fi mbriae) that envelops the ovary at the time of ovu-lation. The terminal end of the uterine tube is unspecialized in most fi shes and amphib-ians, but in cartilaginous fi shes, reptiles, and birds that produce a large, shelled egg, spe-cial regions have developed for production of albumin and shell. In amniotes (reptiles, birds, and mammals; see Amniotes and the Amniotic Egg, p. 175) the terminal portion of the uterine tube is expanded into a mus-cular uterus in which shelled eggs are held before laying or in which embryos complete their development. In placental mammals, the very striking during embryonic development. In male fi shes and amphibians the duct that drains the kidney (opisthonephric duct or Wolffi an duct) also serves as a sperm duct (see p. 674). In male reptiles, birds, and mammals in which the kidney devel-ops its own independent duct (ureter) to carry away waste, the old mesonephric duct becomes exclusively a sperm duct or vas deferens. In all these forms, with the exception of most mammals, the ducts open into a cloaca (derived, appropriately, from the Latin meaning “sewer”), a common chamber into which intestinal, reproductive, and excretory canals empty. Almost all placental mammals have no cloaca; instead the urogenital system has its own opening separate from the anal opening. In females, the uterine duct or oviduct is an independent duct that opens into the cloaca in animals having a cloaca. Male Reproductive System The male reproductive system of vertebrates, such as that of human males ( Figure 7.12 ) includes testes, vasa efferentia, vas deferens, accessory glands, and (in some birds and reptiles, and all mammals) a penis. Paired testes are the sites of sperm production. Each testis is composed of numerous seminiferous tubules, in which the sperm develop ( Figure 7.8 ). The sperm are surrounded by Ser-toli cells (or sustentacular cells ), which nourish the develop-ing sperm. Between the tubules are interstitial cells (or Leydig cells ), which produce the male sex hormone (testosterone). In most mammals the two testes are housed permanently in a saclike scrotum suspended outside the abdominal cavity, or the testes descend into the scrotum during the breeding season. Figure 7.12 Human male reproductive system showing the reproductive structures in sagittal view.
Rectum Ejaculatory duct Vertebral column Seminal vesicle Peritoneum Urinary bladder Ureter Vas deferens Glans penis Prepuce External urinary meatus Prostate gland Anus Bulbourethral glands Testicle Scrotum Penis Urethra Pubic bone Epididymis www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 149 walls of the uterus establish a close vascular association with the embryonic membranes through a placenta (see p. 177). The paired ovaries of the human female ( Figure 7.13 ), slightly smaller than the male testes, contain many thousands of oocytes. Each oocyte develops within a follicle that enlarges and fi nally ruptures to release a secondary oocyte ( Figure 7.10 ). During a woman’s fertile years, except following fertilization, approxi-mately 13 oocytes mature each year, and usually the ovaries alter-nate in releasing oocytes. Because a woman is fertile for only about 30 years, of the approximately 400,000 primary oocytes in her ovaries at birth, only 300 to 400 have a chance to reach matu-rity; the others degenerate and are resorbed. A long-held principle of mammalian reproductive biology has been that in the male, germ cell lines continue to remain functional and form sperm throughout adult life, while females possess a fi nite number of germ cells, and oocyte production ceases at birth. Indeed, we have just described human follicular development as such a case, where the primary oocytes present at birth provide her only source of follicles. Recently, an exciting discovery in mice has challenged this reproductive dogma. Juvenile and adult mouse ovaries have been shown to possess actively dividing germ cells that replenish the oocyte pool. If this fi nding can be extended to other mammalian species, it will have signifi cant implications in the management of endangered species, where assisted reproductive techniques could be used to expand pools of oocytes that might mean the difference between extinction and survival. The uterine tubes, or oviducts, are lined with cilia for propelling the egg away from the ovary from which it was released. The two ducts open into the upper corners of the uterus, or womb, which is specialized for housing the embryo during its intrauterine existence. It consists of thick muscular walls, many blood vessels, and a specialized lining: the endome-trium. The uterus varies among different mam-mals, and in many it is designed to hold more than one developing embryo. Ancestrally it was paired but is fused to form one large chamber in many eutherian mammals. The vagina is a muscular tube adapted to receive the male’s penis and serves as the birth canal during expulsion of a fetus from the uterus. Where vagina and uterus meet, the uterus proj-ects down into the vagina to form a cervix. The external genitalia of human females, or vulva, include folds of skin, the labia majora and labia minora, and a small erectile organ, the clitoris (the female homolog of the glans penis of males). The opening into the vagina is often reduced in size in the virgin state by a membrane, the hymen, although in today’s more physically active females, this membrane may be much reduced in extent. ENDOCRINE EVENTS THAT ORCHESTRATE REPRODUCTION Hormonal Control of Timing of Reproductive Cycles From fi sh to mammals, reproduction in vertebrates is usually a seasonal or cyclic activity. Timing is crucial, because offspring should appear when food is available and other environmen-tal conditions are optimal for survival. The sexual reproductive process is controlled by hormones, which are regulated by environmental cues, such as food intake, and seasonal changes in photoperiod, rainfall, or temperature, and by social cues. A region within the forebrain called the hypothalamus (p. 758) reg-ulates the release of anterior pituitary gland hormones, which in turn stimulate tissues of the gonads (neurosecretion and the pitu-itary gland are described in Chapter 34). This hormonal system controls development of the gonads, accessory sex structures, and secondary sexual characteristics (see next section), as well as timing of reproduction. The cyclic reproductive patterns of female mammals are of two types: estrous cycle, characteristic of most mammals, and menstrual cycle, characteristic only of the anthropoid primates (monkeys, apes, and humans). These two cycles differ in two important ways. First, in estrous cycles, females are receptive to males only during brief periods of estrus, or “heat,” whereas in the menstrual cycle receptivity may occur throughout the cycle. Figure 7.13 Human female reproductive system showing the pelvis in sagittal section.
Labia minora Labia majora Vagina Anus Urinary bladder Uterine tube Ovary Fimbriae Uterus Pubic bone Urethra Clitoris Cervix Rectum 150 PART TWO Continuity and Evolution of Animal Life Second, a menstrual cycle, but not an estrous cycle, ends with breakdown and discharge of the inner portion of the uterus (endometrium). In an estrous cycle, each cycle ends with the endometrium simply reverting to its original state, without the discharge characteristic of the menstrual cycle. Gonadal Steroids and Their Control The ovaries of female vertebrates produce two kinds of steroid sex hormones— estrogens and progesterone ( Figure 7.14 ). There are three kinds of estrogens: estradiol, estrone and estriol, of which estradiol is secreted in the highest amounts during reproductive cycles. Estrogens are responsible for development of female accessory sex structures (oviducts, uterus, and vagina) and for stimulating female reproductive activity. Secondary sex characters, those characteristics that are not primarily involved in formation and delivery of ova (or sperm in males), but that are essential for behavioral and functional success of repro-duction, are also controlled or maintained by estrogens. Sec-ondary sexual characteristics include distinctive skin or feather coloration, bone development, body size and, in mammals, initial development of the mammary glands. In female mam-mals, both estrogen and progesterone are responsible for pre-paring the uterus to receive a developing embryo. These hor-mones are controlled by anterior pituitary gonadotropins: follicle-stimulating hormone (FSH), and luteinizing hor-mone (LH) ( Figure 7.15 ). The release of these two gonado-tropins are in turn governed by gonadotropin-releasing hormone (GnRH) produced by neurosecretory cells in the hypothalamus (see p. 758 and Table 34.1). Through this con-trol system environmental factors such as light, nutrition, and stress may infl uence reproductive cycles. Estrogens and proges-terone feed back to the hypothalamus and anterior pituitary to keep secretion of GnRH, FSH, and LH in check (see Chapter 34, for a discussion of negative feedback of hormones). The male sex steroid, testosterone ( Figure 7.14 ), is manu-factured by the interstitial cells of the testes. Testosterone, and its metabolite, dihydrotestosterone (DHT), are necessary for the growth and development of the male accessory sex struc-tures (penis, sperm ducts, and glands), development of second-ary male sex characters (such as bone and muscle growth, male plumage or pelage coloration, antlers in deer, and, in humans, voice quality), and male sexual behavior. Development of the testes and secretion of testosterone is controlled by FSH and LH, the same anterior pituitary hormones that regulate the female reproductive cycle, and ultimately by GnRH from the hypothala-mus. Like estrogens and progesterone in the female, testosterone and DHT feed back to the hypothalamus and anterior pituitary to regulate secretion of GnRH, FSH, and LH. Recent identifi cation of a peptide in the hypothalamus of birds and mammals that inhibits the secretion of GnRH and LH has led some scientists to believe that a gonadotropin-inhibiting hormone has fi nally been discovered. Further study is necessary, however, before we can be sure that this peptide antagonizes GnRH in all physiological conditions. Both the ovary and testes secrete a peptide hormone, inhibin, which is secreted by the developing follicles in the female and by the Sertoli cells (or sustentacular cells) in the male. This hormone is an additional regulator of FSH secretion from the anterior pituitary in a negative feedback manner. The Menstrual Cycle The human menstrual cycle (L. mensis, month) consists of two distinct phases within the ovary: follicular phase and luteal phase, and three distinct phases within the uterus: menstrual phase, proliferative phase, and secretory phase ( Figure 7.15 ). Menstruation (the “period”) signals the menstrual phase, when part of the lining of the uterus (endometrium) degenerates and sloughs off, producing the menstrual discharge. Meanwhile, the follicular phase within the ovary is occurring, and by day 3 of the cycle blood levels of FSH and LH begin to rise slowly, prompting some of the ovarian follicles to begin growing and to secrete estrogen. As estrogen levels in the blood increase, the uterine endometrium heals and begins to thicken, and uterine glands within the endometrium enlarge (proliferative phase). By day 10 most of the ovarian follicles that began to develop at day 3 now degenerate (become atretic ), leaving only one (sometimes two or three) to continue developing until it appears as a bulge on the surface of the ovary. This is a mature follicle or graafi an follicle. During the latter part of the follicular phase, the graafi an follicle secretes more estrogen, and also inhibin. As the levels of inhibin rise, the levels of FSH fall. At day 13 or 14 in the cycle, the now high levels of estro-gen from the graafi an follicle stimulate a surge of GnRH from the hypothalamus, which induces a surge of LH (and to a lesser extent, FSH) from the anterior pituitary. The LH surge causes the graafi an follicle to rupture (ovulation), releasing an oocyte Figure 7.14 Sex hormones. These three sex hormones show the basic four-ring steroid structure. The main female sex hormone, estradiol (an estrogen) is a C18 (18-carbon) steroid with an aromatic A ring (fi rst ring to left). The main male sex hormone testosterone (an androgen) is a C19 steroid with a carbonyl group (CO) on the A ring. The female sex hormone progesterone is a C21 steroid, also bearing a carbonyl group on the A ring.
O CH3 Progesterone O CH3 CH3 OH Testosterone O CH3 CH3 C Estradiol-17 HO CH3 OH www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 151 from the ovary. The oocyte remains viable for approximately 12 hours, during which time it may be fertilized by a sperm. Dur-ing the ovarian luteal phase, a corpus luteum (“yellow body” for its appearance in cow ovaries) forms from the remains of the ruptured follicle that released the oocyte at ovulation ( Figures 7.10 and 7.15 ). The corpus luteum, responding to con-tinued stimulation by LH, becomes a transitory endocrine gland that secretes progesterone (and estrogen in primates). Pro-gesterone (“before carrying [gestation]”), as its name implies, stimulates the uterus to undergo fi nal maturational changes that prepare it for gestation (secretory phase). The uterus is now fully ready to house and to nourish an embryo. If fertil-ization has not occurred, the corpus luteum degenerates, and its hormones are no longer secreted. Since the uterine lining (endometrium) depends on progesterone and estrogen for its maintenance, their declining levels cause the uterine lining to deteriorate, leading to menstrual discharge of the next cycle. Oral contraceptives (the “Pill”) usually are combined preparations of estrogen and progesterone that act to decrease the output of pituitary gonadotropins FSH and LH. This prevents the ovarian folli-cles from ripening fully and usually prevents ovulation from occur-ring. Oral contraceptives are highly effective, with a failure rate of less than 1% if the treatment procedure is followed properly. More recently, estrogen and progesterone are administered as a once per month injection (Lunelle), as a skin patch (Ortho Evra), or as a vaginal ring (NuvaRing). Progesterone also acts on the reproductive tract as a whole, making it inhospitable for sperm and any fertilized oocyte. This mechanism has been exploited in progesterone-only contraceptives (“mini-pill,” Depo-Provera), which may not block follicular development or ovulation, and in the “morning after pill,” a postcoital emergency contraceptive that is now available over the counter in the United States for women aged 18 years or older. Figure 7.15 Human menstrual cycle, showing changes in blood-hormone levels and uterine endometrium during the 28-day ovarian cycle. FSH promotes maturation of ovarian follicles, which secrete estrogen. Estrogen prepares the uterine endometrium and causes a surge in LH, which in turn causes ovulation and stimulates the corpus luteum to secrete progesterone and estrogen. Progesterone and estrogen production will persist only if an ovum is fertilized; without pregnancy progesterone and estrogen levels decline and menstruation follows.
FSH LH Gonadotropin hormone levels in plasma Ovarian hormone levels in plasma Ovarian cycle Menstrual cycle (uterine cycle) Menstrual phase Secretory phase Proliferative phase 0 7 14 Days 21 28 Ruptured follicle Mature follicle Maturing follicles Primary follicle Corpus luteum Estrogen Progesterone Follicular phase Luteal phase Ovulation 152 PART TWO Continuity and Evolution of Animal Life GnRH from the hypothalamus, and LH and FSH from the anterior pituitary, are controlled by negative feedback of ovar-ian steroids (and inhibin). This negative feedback occurs through-out the menstrual cycle, except for a few days before ovulation. As just mentioned, ovulation is due to the high levels of estrogen causing a surge of GnRH, LH (and FSH). Such positive feedback mechanisms are rare in the body, since they move events away from stable set points. (Feedback mechanisms are described in Chapter 34, p. 756). This event is terminated by ovulation when estrogen levels fall as an oocyte is released from the follicle. Hormones of Human Pregnancy and Birth If fertilization occurs, it normally does so in the fi rst third of the uterine tube (ampulla). The zygote travels from here to the uterus, dividing by mitosis to form a blastocyst (see Chapter 8, p. 177) by the time it reaches the uterus. The developing blasto-cyst adheres to the uterine surface after about 6 days and embeds itself in the endometrium. This process is called implantation. Growth of the embryo continues, producing a spherically shaped trophoblast. This embryonic stage contains three distinct tissue layers, the amnion, chorion, and embryo proper, the inner cell mass (see Figure 8.25, p. 178). The chorion becomes the source of human chorionic gonadotropin (hCG), which appears in the bloodstream soon after implantation. hCG stimulates the cor-pus luteum to continue to synthesize and to release both estro-gen and progesterone ( Figure 7.16 ). The placenta forms the point of attachment between tro-phoblast and uterus (evolution and development of the placenta is described in Chapter 8, p. 177). Besides serving as a medium for the transfer of materials between maternal and fetal blood-streams, the placenta also serves as an endocrine gland. The placenta continues to secrete hCG and also produces estrogen (mainly estriol) and progesterone. After about the third month of pregnancy, the corpus luteum degenerates in some mammals, but by then the placenta itself is the main source of both proges-terone and estrogen ( Figure 7.17 ). Preparation of the mammary glands for secretion of milk requires two additional hormones, prolactin (PRL) and human placental lactogen (hPL) (or human chorionic somatomammotropin ). PRL is produced by the anterior pitu-itary, but in nonpregnant women its secretion is inhibited. Dur-ing pregnancy, elevated levels of progesterone and estrogen depress the inhibitory signal, and PRL begins to appear in the blood. PRL is also produced by the placenta during pregnancy. PRL, in combination with hPL, prepare the mammary glands for secretion. hPL, together with human placental growth hormone (hPGH) and maternal growth hormone, also stimu-late an increase in available nutrients in the mother, so that more are provided to the developing embryo. The placenta also secretes β-endorphin and other endogenous opioids (see Chapter 33, p. 743) that regulate appetite and mood during pregnancy. Opioids may also contribute to a sense of well-being and help to alleviate some of the discomfort associated with the later months of pregnancy. Later the placenta begins to synthesize a peptide hormone called relaxin; this hormone Figure 7.16 The multiple roles of progesterone and estrogen in normal human pregnancy. After implantation of an embryo in the uterus, the trophoblast (the future embryo and placenta) secretes human chorionic gonadotropin (hCG), which maintains the corpus luteum until the placenta, at about the seventh week of pregnancy, begins producing the sex hormones progesterone and estrogen.
Supports secretory endometrium Inhibits contractility of uterine muscle Firms cervix and inhibits dilation ESTROGEN PROGESTERONE www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 153 allows some expansion of the pelvis by increasing the fl ex-ibility of the pubic symphysis, and also dilates the cervix in preparation for delivery. Birth, or parturition, occurs after approximately 9 months in humans and begins with a series of strong, rhythmic contrac-tions of the uterine musculature, called labor. The exact sig-nal that triggers birth is not fully understood in humans, but placental corticotropin-releasing hormone (CRH) appears to initiate the birth process. Just before birth, secretion of estro-gen, which stimulates uterine contractions, rises sharply, while the level of progesterone, which inhibits uterine contractions, declines ( Figure 7.17 ). This removes the “progesterone block” that keeps the uterus quiescent throughout pregnancy. Pros-taglandins, a large group of hormones (long-chain fatty acid derivatives), also increase at this time, making the uterus more “irritable” (see Chapter 34, p. 762, for more on prostaglandins). Finally, stretching of the cervix sets in motion neural refl exes that stimulate secretion of oxytocin from the posterior pitu-itary. Oxytocin also stimulates uterine smooth muscle, leading to stronger and more frequent labor contractions. Secretion of oxy-tocin during childbirth is another example of positive feedback. This time the event is terminated by birth of the baby. Childbirth occurs in three stages. In the fi rst stage the cervix is enlarged by pressure from the baby in its bag of amniotic fl uid, which may be ruptured at this time ( dilation; Figure 7.18B ). In the second stage, the baby is forced out of the uterus and through the vagina to the outside ( expulsion; Figure 7.18C ). In the third stage, the placenta, or afterbirth, is expelled from the mother’s body, usually within 10 minutes after the baby is born ( placental delivery; Figure 7.18D ). Miscarriages during pregnancy, or spontaneous abortions, are quite common and serve as a mechanism to reject prenatal abnormalities such as chromosomal damage and other genetic errors, exposure to drugs or toxins, immune irregularities, or improper hormonal priming of the uterus. Modern hormonal tests show that about 30% of fertile zygotes are spontaneously aborted before or right after implantation; such miscarriages are unknown to the mother or are expressed as a slightly late menstrual period. Another 20% of estab-lished pregnancies end in miscarriage (those known to the mother), giving a spontaneous abortion rate of about 50%. After birth, secretion of milk is triggered when the infant sucks on its mother’s nipple. This leads to a refl ex release of oxytocin from the posterior pituitary; when oxytocin reaches the mammary glands it causes contraction of smooth muscles lining ducts and sinuses of the mammary glands and ejection of milk. Figure 7.17 Hormone levels released from the corpus luteum and placenta during pregnancy. The width of the arrows suggests the relative amounts of hormone released; hCG (human chorionic gonadotropin) is produced solely by the placenta. Synthesis of progesterone and estrogen shifts during pregnancy from the corpus luteum to the placenta.
Full term 120 days 30 days hCG ESTROGEN PROGESTERONE Hormone levels Corpus luteum 9 8 7 6 5 4 3 2 1 0 Conception Delivery Months of pregnancy 154 PART TWO Continuity and Evolution of Animal Life Figure 7.18 Birth, or parturition, in humans.
Human fetus just before birth First stage of labor: dilation Second stage of labor: expulsion Third stage of labor: placental delivery Placenta Intestine Umbilical cord Wall of uterus Vagina Cervix Placenta A B C D Suckling also stimulates release of prolactin from the anterior pituitary, which stimulates continued production of milk by the mammary glands. Multiple Births Many mammals give birth to more than one offspring at a time or to a litter (multiparous), each member of which has come from a separate egg. There are some mammals, however, that have only one offspring at a time (uniparous), although occa-sionally they may have more than one. The armadillo ( Dasypus ) is almost unique among mammals in giving birth to four off-spring at one time—all of the same sex, either male or female, and all derived from the same zygote. Human twins may come from one zygote ( identical, or monozygotic twins; Figure 7.19A ) or two zygotes ( nonidentical, dizygotic, or fraternal twins; Figure 7.19B ). Fraternal twins do not resemble each other any more than other children born separately in the same family, but identical twins are, of course, strikingly alike and always of the same sex. Triplets, quadruplets, and quintuplets may include a pair of identical twins. The other babies in such multiple births usually come from separate zygotes. About 33% of identical twins have separate placentas, indicating that the blastomeres separated at an early, possibly the two-cell, stage ( Figure 7.19A , top ). All other identical twins share a common placenta, indicating that splitting occurred after formation of the inner cell mass (see Figure 8.25 on p. 180). If splitting were to happen after placenta formation, but before the amnion forms, the twins would have individual amniotic sacs ( Figure 7.19A , middle ), as observed in the great majority of identical twins. Finally, a very small percentage of identical twins share one amniotic sac and a single placenta ( Figure 7.19A , bottom ), indicating that separation occurred after day 9 of pregnancy, by which time the amnion has formed. In these cases, the twins are at risk of becoming conjoined, a condition known as Siamese twinning. Embryologically, each member of fraternal twins has its own placenta and amnion ( Figure 7.19B ). The frequency of twin births in comparison to single births is approximately 1 in 86, that of triplets 1 in 86 2 , and that of quadru-plets approximately 1 in 86 3 . Frequency of identical twin births to all births is about the same the world over, whereas frequency of fraternal births varies with race and country. In the United States, three-fourths of all twin births are dizygotic (fraternal), whereas in Japan only about one-fourth are dizygotic. The tendency for frater-nal twinning (but apparently not identical twinning) seems to run in family lines; fraternal twinning (but not identical twinning) also increases in frequency as mothers get older. www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 155 Figure 7.19 Formation of human twins. A, Monozygotic (identical) twin formation. B, Dizygotic (fraternal) twin formation. See text for explanation.
Fertilization of 2 different oocytes DIZYGOTIC (FRATERNAL) TWINS Yolk sac Embryo MONOZYGOTIC (IDENTICAL) TWINS Fertilization of single oocyte Inner cell mass Placenta Splitting at 2-cell stage Complete split of inner cell mass Split of inner cell mass late in development Blastocoel Two amnions B A 156 PART TWO Continuity and Evolution of Animal Life S U M M A R Y Reproduction is the production of new life and provides an opportu-nity for evolution to occur. Asexual reproduction is a rapid and direct process by which a single organism produces genetically identical copies of itself. It may occur by fi ssion, budding, gemmulation, or fragmentation. Sexual reproduction involves production of germ cells (sex cells or gametes), usually by two parents (bisexual reproduc-tion), which combine by fertilization to form a zygote that develops into a new individual. Germ cells are formed by meiosis, reducing the number of chromosomes to haploid, and the diploid chromo-some number is restored at fertilization. Sexual reproduction recom-bines parental characters and thus reshuffl es and amplifi es genetic diversity. Genetic recombination is important for evolution. Two alternatives to typical bisexual reproduction are hermaphroditism, the presence of both male and female organs in the same individual, and parthenogenesis, the development of an unfertilized egg. Sexual reproduction exacts heavy costs in time and energy, requires cooperative investments in mating, and causes a 50% loss of genetic representation of each parent in the offspring. The classi-cal view of why sex is needed is that it maintains variable offspring within the population, which may help the population to survive environmental change. In vertebrates the primordial germ cells arise in the yolk-sac endoderm, then migrate to the gonad. In mammals, a gonad becomes a testis in response to masculinizing signals encoded on the Y chromosome of the male, and the reproductive tract masculin-izes in response to circulating male sex steroids. Female reproduc-tive structures (ovary, uterine tubes, uterus, and vagina) develop in the absence of signals encoded on the Y chromosome, although recent data suggests that a female-determining region on the X chro-mosome has a role in differentiation of female reproductive organs. Germ cells mature in the gonads by a process called game-togenesis (spermatogenesis in males and oogenesis in females), involving both mitosis and meiosis. In spermatogenesis, each pri-mary spermatocyte gives rise by meiosis and growth to four motile sperm, each bearing the haploid number of chromosomes. In oogen-esis, each primary oocyte gives rise to only one mature, nonmotile, haploid ovum. The remaining nuclear material is discarded in polar bodies. During oogenesis an egg accumulates large food reserves within its cytoplasm. Sexual reproductive systems vary enormously in complexity, ranging from some invertebrates, such as polychaete worms that lack any permanent reproductive structures to the complex sys-tems of vertebrates and many invertebrates consisting of permanent gonads and various accessory structures for transferring, packaging, and nourishing gametes and embryos. The male reproductive system of humans includes testes, com-posed of seminiferous tubules in which millions of sperm develop, and a duct system (vasa efferentia and vas deferens) that joins the urethra, glands (seminal vesicles, prostate, bulbourethral), and penis. The human female system includes ovaries, containing thousands of eggs within follicles; oviducts; uterus; and vagina. The seasonal or cyclic nature of reproduction in vertebrates has required evolution of precise hormonal mechanisms that control production of germ cells, signal readiness for mating, and prepare ducts and glands for successful fertilization of eggs. Neurosecretory centers within the hypothalamus of the brain secrete gonadotropin-releasing hormone (GnRH), which stimulates endocrine cells of the anterior pituitary to release follicle-stimulating hormone (FSH) and luteinizing hormone (LH), which in turn stimulate the gonads. Estrogens and progesterone in females, and testosterone and dihydrotestosterone (DHT) in males, control the growth of accessory sex structures and secondary sex characteristics, in addition to feeding back to the hypothalamus and anterior pituitary to regulate GnRH, FSH, and LH secretion. In the human menstrual cycle, estrogen induces the initial pro-liferation of uterine endometrium. A surge in GnRH and LH, induced by rising estrogen levels from the developing follicle(s), midway in the cycle causes ovulation and the corpus luteum to secrete proges-terone (and estrogen in humans), which completes preparation of the uterus for implantation. If an egg is fertilized, pregnancy is main-tained by hormones produced by the placenta and mother. Human chorionic gonadotropin (hCG) maintains secretion of progesterone and estrogen from the corpus luteum, while the placenta grows and eventually secretes estrogen, progesterone, hCG, human placental lactogen (hPL), human placental growth hormone (hPGH), prolactin (PRL), endogenous opioids, placental corticotropin-releasing hormone (CRH), and relaxin. Estrogen, progesterone, PRL, and hPL, as well as maternal prolactin, induce development of the mammary glands in preparation for lactation. hPL, hPGH, and maternal growth hormone also increase nutrient availability for the developing embryo. Birth or parturition (at least in most mammals) appears to be initiated by release of placental CRH. In addition, a decrease in progesterone and an increase in estrogen levels occur so that the uterine muscle begins to contract. Oxytocin (from the posterior pituitary) and uterine prostaglandins continue this process until the fetus (followed by the placenta) is expelled. Placental relaxin makes the birth process easier by enabling expansion of the pelvis and dilation of the cervix. Multiple births in mammals may result from division of one zygote, producing identical, monozygotic twins, or from separate zygotes, producing fraternal, dizygotic twins. Identical twins in humans may have separate placentas, or (most commonly) they may share a common placenta but have individual amniotic sacs. R E V I E W Q U E S T I O N S Defi ne asexual reproduction, and describe four forms of asexual reproduction in invertebrates. Defi ne sexual reproduction and explain why meiosis contributes to one of its great strengths. 1.
2. Explain why genetic mutations in asexual organisms lead to much more rapid evolutionary change than do genetic mutations in sexual forms. Why might harmful mutations be more deleterious to asexual organisms compared with sexual organisms? 3.
www . m hhe . com /hi ckmani pz14e CHAPTER 7 The Reproductive Process 157 S E L E C T E D R E F E R E N C E S Cole, C. J. 1984. Unisexual lizards. Sci. Am. 250: 94–100 (Jan.). Some populations of whiptail lizards from the American southwest consist only of females that reproduce by virgin birth. Crews, D. 1994. Animal sexuality. Sci. Am. 270: 108–114 (Jan.). Sex is determined genetically in mammals and most other vertebrates, but not in many reptiles and fi shes, which lack sex chromosomes. The author describes nongenetic sex determination and suggests a new framework for understanding the origin of sexuality. Crow, J. F. 1994. Advantages of sexual reproduction. Developmental Genetics 15: 205–213. An excellent discussion of the advantages and disadvantages of sexual reproduction with a critique of the various hypotheses presented on this issue. Very readable. Forsyth, A. 1986. A natural history of sex: the ecology and evolution of sexual behavior. New York, Charles Scribner’s Sons. Engagingly written, factually accurate account of the sex lives of animals from unicellular organisms to humans, abounding in imagery and analogy. Highly recommended. Johnson, J., J. Cannling, T. Kaneko, J. P. Pru, and J. L. Tilly. 2004. Germline stem cells and follicular renewal in the postnatal mammalian ovary. Nature 428: 145–150. Exciting new evidence that female mammals possess a renewable germ cell line, refuting an age-old hypothesis of reproductive biology. Johnson, M. H., and B. J. Everitt. 2000. Essential reproduction, ed. 5. Oxford, U.K., Blackwell Sciences Ltd. Excellent coverage of reproductive physiology with emphasis on humans. Jones, R. E. 2006. Human reproductive biology, ed. 3. San Diego, Academic Press. Thorough treatment of human reproductive physiology. Kinsley, C. H., and K. G. Lambert. 2006. The maternal brain. Sci. Am. 294: 72–79. This excellent review discusses how the hormones secreted during pregnancy, and lactation in mammals appears to confer long-lasting benefi ts to the brain that alter skills and behavior associated with better parental care. Kriegsfeld, L. J., D. F. Mei, G. E. Bentley, Y. Ubuka, A. O. Mason, K. Inoue, K. Ukena, K. Tsutsui, and R. Silver. 2006. Identifi cation and characterization of a gonadotropin-inhibitory system in the brains of mammals. Proceedings of the National Academy of Science 103: 2410–2415. An original research paper that presents evidence of a gonadotropin-inhibiting hormone that suppresses the reproductive axis. Lee, D. M., R. R. Yeoman, D. E. Battaglia, R. L. Stouffer, M. B. Zelinski-Wooten, J. W. Fanton, and D. P. Wolf. 2004. Live birth after ovarian tissue transplant. Nature 428: 137–138. New hope in the future for cancer patients that are made prematurely sterile is provided by the recent news of successful ovarian tissue transplants in monkeys. Lombardi, J. 1998. Comparative vertebrate reproduction. Boston, Kluwer Academic Publishers. Comprehensive coverage of vertebrate reproductive physiology. Maxwell, K. 1994. The sex imperative: an evolutionary tale of sexual survival. New York, Plenum Press. Witty survey of sex in the animal kingdom. Michod, R. E. 1995. Eros and evolution: a natural philosophy of sex. Reading, Massachusetts, Addison-Wesley Publishing Company. In this engaging book, the author argues that sex evolved as a way of coping with genetic errors and avoiding homozygosity. Piñón, R. 2002. Biology of human reproduction. Sausalito, University Science Books. An updated examination of human reproductive physiology. Ridley, M. 2001. The advantages of sex. www.pbs.org/wgbh/evolution/sex/ advantage/ An essay adapted from a New Scientist publication (4 Dec, 1993) summarizing hypotheses proposed for the evolution of sex. Defi ne two alternatives to bisexual reproduction— hermaphroditism and parthenogenesis—and offer a specifi c example of each from the animal kingdom. What is the difference between ameiotic and meiotic parthenogenesis? Defi ne the terms dioecious and monoecious. Can either of these terms be used to describe a hermaphrodite? A paradox of sexual reproduction is that despite being widespread in nature, the question of why it exists at all is still unresolved. What are some disadvantages of sex? What are some consequences of sex that make it so important? What is a germ cell line? How do germ cells pass from one generation to the next? Explain how a spermatogonium, containing a diploid number of chromosomes, develops into four functional sperm, each containing a haploid number of chromosomes. In what signifi cant way(s) does oogenesis differ from spermatogenesis? Defi ne, and distinguish among, the terms oviparous, ovoviviparous, and viviparous. Name the general location and give the function of the following reproductive structures: seminiferous tubules, 4.
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vas deferens, urethra, seminal vesicles, prostate gland, bulbourethral glands, mature follicle, oviducts, uterus, vagina, endometrium. How do the two kinds of mammalian reproductive cycles— estrous and menstrual—differ from each other? What are the male sex hormones and what are their functions? Explain how the female hormones GnRH, FSH, LH, and estrogen interact during the menstrual cycle to induce ovulation and, subsequently, formation of the corpus luteum. Explain the function of the corpus luteum in the menstrual cycle. If fertilization of the ovulated egg happens, what endocrine events occur to support pregnancy? Describe the role of pregnancy hormones during human pregnancy. What hormones prepare the mammary glands for lactation and what hormones continue to be important during this process? If identical human twins develop from separate placentas, when must the embryo have separated? When must separation have occurred if the twins share a common placenta but develop within separate amnions? 11.
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16. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links and more! C H A P T E R 8 Principles of Development The Primary Organizer During the fi rst half of the twentieth century, experiments by the German embryologist Hans Spemann (1869 to 1941) and his stu-dent, Hilde Pröscholdt Mangold (1898 to 1924), ushered in the fi rst of two golden ages of embryology. Working with salaman-ders, they found that tissue transplanted from one embryo into another could induce development of a complete organ, such as an eyeball, at the site of the transplant. This phenomenon is called embryonic induction. Mangold later discovered that one particular tissue, the dorsal lip from an embryonic stage called the gastrula, could induce the development of an entirely new salamander joined to the host salamander at the site of the transplant. (This work earned Spemann the Nobel Prize in Physiology or Medicine in 1935, but Hilde Mangold had died in a household accident only a few weeks after her research was published.) Spemann desig-nated this dorsal lip tissue the primary organizer, now often called the Spemann organizer. Recent advances in molecular biology have inaugurated the second golden age of embryology, still in progress. During this current golden age we are beginning to understand that induction is due to secretion of certain mol-ecules that trigger or repress the activity of combinations of genes in nearby cells. For example, cells of the Spemann organizer migrate over the dorsal midline, secreting proteins with names like noggin, chordin, and follistatin. These proteins allow nearby cells to develop into the nervous system and other tissues along the middle of the back, and those tissues in turn release other proteins that induce development of other parts of the body. Such organizer proteins do not occur only in salamanders; remarkably similar proteins function in development of other vertebrates and even invertebrates. Because all animals appear to share similar molecular mechanisms for development, it may now be possible to understand how changes in such developmental controls led to the evolution of the great variety of animals. Research in this area has given rise to the exciting new fi eld called evolutionary developmental biology. Spemann organizer cells (color) migrating from the dorsal lip (arrow) of a gastrula. www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 159 H ow is it possible that a tiny, spherical fertilized human egg, scarcely visible to the naked eye, can develop into a fully formed, unique person, consisting of thousands of billions of cells, each cell performing a predestined functional or structural role? How is this marvelous unfolding controlled? Clearly all information needed must originate from the nucleus and in the surrounding cytoplasm. But knowing where the control system lies is very different from understanding how it guides the conversion of a fertilized egg into a fully differentiated animal. Despite intense scrutiny by thousands of scientists over many decades, it seemed until very recently that developmental biology, almost alone among the biological sciences, lacked a satisfactory explanatory theory. This now has changed. During the last two decades the combination of genetics and evolution with modern techniques of cellular and molecular biology has provided the long-sought explanation of animal development. Causal relationships between development and evolution have also become the focus of research. We do at last appear to have a conceptual framework to account for development. EARLY CONCEPTS: PREFORMATION VERSUS EPIGENESIS Early scientists and laypeople alike speculated at length about the mystery of development long before the process was sub-mitted to modern techniques of biochemistry, molecular biol-ogy, tissue culture, and electron microscopy. An early and per-sistent idea was that young animals were preformed in eggs and that development was simply a matter of unfolding what was already there. Some claimed they could actually see a miniature adult in the egg or sperm ( Figure 8.1 ). Even the more cautious argued that all parts of the embryo were in the egg, but too small and transparent to be seen. The concept of preformation was strongly advocated by most seventeenth- and eighteenth-century naturalist-philosophers. In 1759 German embryologist Kaspar Friedrich Wolff clearly showed that in the earliest developmental stages of the chick, there was no preformed individual, only undifferentiated granular material that became arranged into layers. These lay-ers continued to thicken in some areas, to become thinner in others, to fold, and to segment, until the body of the embryo appeared. Wolff called this process epigenesis (“origin upon or after”), an idea that a fertilized egg contains building material only, somehow assembled by an unknown directing force. Cur-rent ideas of development are essentially epigenetic in concept, although we know far more about what directs growth and differentiation. Development describes the progressive changes in an indi-vidual from its beginning to maturity ( Figure 8.2 ). In sexual mul-ticellular organisms, development usually begins with a fertilized egg that divides mitotically to produce a many-celled embryo. These cells then undergo extensive rearrangements and interact with one another to generate an animal’s body plan and all of the many kinds of specialized cells in its body. This generation of cellular diversity does not occur all at once, but emerges sequentially by a hierarchy of developmental decisions. The many familiar cell types that make up the body do not simply “unfold” at some point, but arise from conditions created in preceding stages. At each stage of development new structures arise from the interaction of less committed rudiments. Each interaction is increasingly restrictive, and the decision made at each stage in the hierarchy further limits developmental fate. Once cells embark on a course of differentiation, they become irrevocably committed to that course. They no longer depend on the stage that preceded them, nor do they have the option Figure 8.1 Preformed human infant in sperm as imagined by seventeenth-century Dutch histologist Niklaas Hartsoeker, one of the fi rst to observe sperm, using a microscope of his own construction. Other remarkable pictures published during this period depicted the fi gure sometimes wearing a nightcap!
Figure 8.2 Key events in animal development.
Egg and sperm fuse Sperm and eggs form, mature Zygote subdivides, determinants partitioned in blastomeres Germ layers form Body organs form, cells interact, differentiate Organs increase in size, adult body form attained Gamete formation Fertilization Cleavage Gastrulation Organogenesis Growth 160 PART TWO Continuity and Evolution of Animal Life of becoming something different. Once a structure becomes committed it is said to be determined. Thus the hierarchy of commitment is progressive and it is usually irreversible. The two basic processes responsible for this progressive subdivision are cytoplasmic localization and induction. We discuss both processes as we proceed through this chapter. FERTILIZATION The initial event in development in sexual reproduction is fer-tilization, the union of male and female gametes to form a zygote. Fertilization accomplishes two things: it provides for recombination of paternal and maternal genes, thus restoring the original diploid number of chromosomes characteristic of a species, and it activates the egg to begin development. How-ever, sperm are not always required for development. Eggs of some species can be artifi cially induced to initiate develop-ment without sperm fertilization (artifi cial parthenogenesis), but in the great majority of cases an embryo will not be able to progress very far down the developmental path before lethal developmental abnormalities arise. However, some species have natural parthenogenesis (p. 140). Of these, some have eggs that develop normally in the absence of sperm. In other species (some fi shes and salamanders), sperm is required for egg activation, but the sperm contributes no genetic material. Thus neither sperm contact nor the paternal genome is always essential for egg activation. Oocyte Maturation During oogenesis, described in the preceding chapter, an egg prepares itself for fertilization and for the beginning of develop-ment. Whereas a sperm eliminates all its cytoplasm and con-denses its nucleus to the smallest possible dimensions, an egg grows in size by accumulating yolk reserves to support future growth. An egg’s cytoplasm also contains vast amounts of messenger RNA, ribosomes, transfer RNA, and other elements required for protein synthesis. In addition, eggs of most spe-cies contain morphogenetic determinants that direct activa-tion and repression of specifi c genes later in postfertilization development. The nucleus also grows rapidly in size during egg maturation, becoming bloated with RNA and so changed in appearance that it is given a special name, the germinal vesicle. Most of this intense preparation occurs during an arrested stage of meiosis. In mammals, for example, it occurs during the prolonged prophase of the fi rst meiotic division. The oocyte is now poised to resume meiotic divisions that are essential to pro-duce a haploid female pronucleus that will join a male hap-loid pronucleus at fertilization. After resumption of meiosis, the egg rids itself of excess chromosomal material in the form of polar bodies (described in Chapter 7, p. 145). A vast amount of synthetic activity has preceded this stage. The oocyte is now a highly structured system, provided with a dowry which, after fertilization, will support nutritional requirements of the embryo and direct its development through cleavage. Fertilization and Activation Our current understanding of fertilization and activation derives in large part from more than a century of research on marine invertebrates, especially sea urchins. Sea urchins produce large numbers of eggs and sperm, which can be combined in the lab-oratory for study. Fertilization also has been studied in many vertebrates and, more recently, in mammals, using sperm and eggs of mice, hamsters, and rabbits. Contact and Recognition Between Egg and Sperm Most marine invertebrates and many marine fi shes simply release their gametes into the ocean. Although an egg is a large target for a sperm, the enormous dispersing effect of the ocean and lim-ited swimming range of a spermatozoon conspire against an egg and a sperm coming together by chance encounter. To improve likelihood of contact, eggs of numerous marine species release a chemotactic factor that attracts sperm to eggs. The chemotactic molecule is species-specifi c, attracting to eggs only sperm of the same species. In sea urchin eggs, sperm fi rst penetrate a jelly layer surround-ing the egg, then contact an egg’s vitelline envelope, a thin mem-brane lying just above the egg plasma membrane ( Figure 8.3 ). At this point, egg-recognition proteins on the acrosomal process of the sperm ( Figure 8.4 ) bind to species-specifi c sperm receptors on the vitelline envelope. This mechanism ensures that an egg recognizes only sperm of the same species. This is important in the marine environment where many closely related species may be spawning at the same time. Similar recognition proteins have been found on sperm of vertebrate species (including mammals) and presumably are a universal property of animals. Prevention of Polyspermy At the point of sperm contact with the egg vitelline envelope a fertilization cone appears into which the sperm head is later drawn ( Figure 8.4 ). This event is followed immediately by Figure 8.3 Structure of sea urchin egg during fertilization.
Vitelline envelope Yolk granule Jelly layer Plasma membrane Sperm Mitochondrion Cortical granule Nucleus www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 161 important changes in the egg surface that block entrance of additional sperm, which, in marine eggs especially, may quickly surround the egg in swarming numbers ( Figure 8.5 ). Entrance of more than one sperm, called polyspermy, must be prevented because union of more than two haploid nuclei would be ruin-ous for normal development. In a sea urchin egg, contact of the fi rst sperm with the egg membrane is instantly followed by an electrical potential change in the egg membrane that prevents additional sperm from fusing with the membrane. This event, called the fast block, is followed immediately by the cortical reaction, in which thousands of enzyme-rich cortical granules, located just beneath the egg membrane, fuse with the membrane and release their contents into the space between the egg mem-brane and the overlying vitelline envelope (see Figure 8.4 ). The cortical reaction creates an osmotic gradient, causing water to rush into this space, elevating the envelope and lifting away all sperm bound to it, except the one sperm that has successfully fused with the egg membrane. One of the cortical granule enzymes causes the vitelline envelope to harden, and it is now called a fertilization membrane. The block to polyspermy is complete. Timing of these early events is summarized in Figure 8.6 . Mam-mals have a similar security system that is erected within seconds after the fi rst sperm fuses with the egg membrane. Fusion of Pronuclei and Egg Activation Once sperm and egg membranes have fused, the sperm loses its fl agellum, which disintegrates. Its nuclear envelope then breaks apart, allowing the sperm chromatin to expand from its extremely condensed state. The enlarged sperm nucleus, now called a pronucleus, migrates inward to contact the female pronucleus. Their fusion forms the diploid zygote nucleus. Nuclear fusion takes only about 12 minutes in sea urchin eggs ( Figure 8.6 ), but requires about 12 hours in mammals. Figure 8.4 Sequence of events during sperm contact and penetration of a sea urchin egg.
Discharging cortical granule Fertilization membrane 4 3 2 1 Fertilization cone Jelly layer Fusion of egg and sperm membranes Egg cortex Cortical granule Other sperm Sperm nucleus Acrosomal process Egg-recognition proteins Vitelline envelope Plasma membrane of egg Figure 8.5 Binding of sperm to the surface of a sea urchin egg. Only one sperm penetrates the egg surface, the others being blocked from entrance by rapid changes in the egg membranes. Unsuccessful sperm are soon lifted away from the egg surface by a newly formed fertilization membrane.
162 PART TWO Continuity and Evolution of Animal Life Fertilization sets in motion several important changes in the cytoplasm of an egg—now called a zygote—that pre-pare it for cleavage. Inhibitors that had blocked metabolism and kept the egg quiescent, in suspended-animation, are removed. Fertilization is immediately followed by a burst of DNA and protein synthesis, the latter utilizing the abundant supply of messenger RNA previously stored in the egg cyto-plasm. Fertilization also initiates an almost complete reorga-nization of the cytoplasm within which are morphogenetic determinants that activate or repress specific genes as devel-opment proceeds. Movement of cytoplasm repositions the determinants into new and correct spatial arrangements that are essential for proper development. The zygote now enters cleavage. In animal eggs, fertilization induces an increase in the amount of free calcium ions inside the egg cytoplasm. This increase in intra-cellular free calcium regulates later developmental events and is essential for normal development to occur in all taxa studied, but the mechanisms controlling calcium levels vary. In some taxa, cal-cium ions are released from intracellular stores, whereas in others calcium enters the egg from outside via voltage-gated calcium chan-nels (see Chapter 3, p. 49). Some organisms combine both mecha-nisms. The calcium signal can occur in a single pulse, as it does in jellyfi sh, starfi sh, and frog zygotes, or in a series of closely spaced pulses identifi ed in ribbon worms, polychaetes, and mammals. Researchers once thought that the calcium signaling pattern might vary as part of the developmental dichotomy between protostomes and deuterostomes, but this is not the case. In even the short list of taxa just given, the two chordate deuterostomes exhibit different calcium release patterns, suggesting that the differing patterns are more likely related to the number and duration of developmental events that require calcium signaling. CLEAVAGE AND EARLY DEVELOPMENT During cleavage the embryo divides repeatedly to convert the large, unwieldy cytoplasmic mass into a large cluster of small, maneuverable cells called blastomeres. No growth occurs dur-ing this period, only subdivision of mass, which continues until normal somatic cell size is attained. At the end of cleavage the zygote has been divided into many hundreds or thousands of cells and the blastula stage is formed. Before cleavage begins, an animal-vegetal axis is visible on the embryo. This axis exists because yolk, nutrition for the developing embryo, occurs only at one end, establishing polarity in the embryo. The yolk-rich end is the vegetal pole and the other end is the animal pole ( Figure 8.7B ); the ani-mal pole contains mostly cytoplasm and very little yolk. The animal-vegetal axis provides a landmark or reference point on the embryo. Cleavage is generally an orderly sequence of cell divisions so that one cell divides to form two cells, these each divide to form four cells, the four make eight cells, and the pro-cess continues. During each division, a distinct cleavage furrow is visible in the cell. This cleavage furrow can be parallel or perpendicular to the animal-vegetal axis. How Amount and Distribution of Yolk Affect Cleavage The amount of yolk at the vegetal pole varies among taxa. Eggs with very little yolk, evenly distributed throughout the egg ( Figure 8.7A , C, and E), are called isolecithal (Gr. isos, equal lekithos, yolk). Mesolecithal (Gr. mesos, middle lekithos, yolk) eggs have a moderate amount of yolk concen-trated at the vegetal pole ( Figure 8.7B ), whereas telolecithal Figure 8.6 Timing of events during fertilization and early development in a sea urchin.
Start of DNA synthesis First cleavage division Fusion of egg and sperm pronuclei Sperm nucleus begins decondensation Sperm begins migration to egg center Fertilization membrane complete Cortical reaction Sperm fusion to egg membrane Fast block to polyspermy begins Gastrulation begins Binding of sperm to egg 0 sec 2 sec 10 sec 30 sec 1 min 1.5 min 2 min 12 min 20 min 90 min 10 hrs www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 163 eggs (Gr. telos, end lekithos, yolk) contain an abundance of yolk densely concentrated at the vegetal pole of the egg ( Figure 8.7D ). Centrolecithal eggs have a large, centrally located, mass of yolk. The presence of yolk disrupts cleavage to varying degrees; when little yolk is present, cleavage furrows extend completely through the egg in holoblastic (Gr. holo, whole blastos, germ) cleavage ( Figure 8.7A , B, C, and E). When much yolk is present, cleavage is meroblastic (Gr. meros, part blastos, germ), with cells sitting atop a mass of undivided yolk ( Figure 8.7D ). Mero-blastic cleavage is incomplete because cleavage furrows cannot cut through the heavy concentration of yolk, but instead stop at the border between the cytoplasm and the yolk below. Holoblastic cleavage occurs in isolecithal eggs and is present in echinoderms, tunicates, cephalochordates, nemerteans, and most molluscs, as well as in marsupial and placental mammals, including humans ( Figure 8.7A , C, and E). Mesolecithal eggs also cleave holoblastically, but cleavage proceeds more slowly in the presence of yolk, leaving the vegetal region with a few large, yolk-fi lled cells, whereas the animal region has many small cells. Amphibian eggs ( Figure 8.7B ) illustrate this process. Meroblastic cleavage occurs in telolecithal and centrolecithal eggs. In telolecithal eggs of birds, reptiles, most fi shes, a few amphibians, cephalopod molluscs, and monotreme mammals, cleavage is restricted to cytoplasm in a narrow disc on top of the yolk (see chick development in Figure 8.7D ). In centrolecithal eggs of insects and many other arthropods, cytoplasmic cleavage is limited to a surface layer of yolk-free cytoplasm, whereas the yolk-rich inner cytoplasm remains uncleaved (see Figure 8.15 ). The function of yolk is to nourish the embryo. When much yolk is present, as in telolecithal eggs, young exhibit direct development, going from an embryo to a miniature adult. When little yolk is present, as in isolecithal or mesolecithal eggs, young develop into various larval stages capable of feeding themselves. In this indirect development, larvae differ from adults and must metamorphose into adult bodies ( Figure 8.8 ). There is another way to compensate for absence of yolk: In most mammals, the mother nourishes the embryos by means of a placenta. What Can We Learn From Development? Biologists study development for different reasons. Some stud-ies focus on understanding how the zygote, a single large cell, can produce the multitude of body parts in an organism. Figure 8.7 Cleavage stages in sea star, frog, nemertean worm, chick, and mouse. Yellow regions represent yolk in each diagram.
RADIAL HOLOBLASTIC CLEAVAGE SPIRAL HOLOBLASTIC CLEAVAGE Sea star: Isolecithal egg Frog: Mesolecithal egg Nemertean worm: Isolecithal egg DISCOIDAL MEROBLASTIC CLEAVAGE ROTATIONAL HOLOBLASTIC Chick: Telolecithal egg Mouse: Isolecithal egg Animal pole Gray crescent Vegetal pole A B C D E Figure 8.8 Indirect development in a sea urchin. After gastrulation, a free-swimming larva develops; it feeds and grows in ocean surface waters. The larva will metamorphose into a tiny bottom-dwelling sea urchin; the urchin feeds and grows, reaching sexual maturity in this body form.
Gastrulation Crawling sea urchin Larval body begins to form Free-swimming feeding larva Larva begins metamorphosis 164 PART TWO Continuity and Evolution of Animal Life Understanding the mechanisms of development requires knowledge of how cleavage partitions cytoplasm, how differ-ent cells interact, and how gene expression proceeds. These topics are covered on pages 170–174. Another reason to study development is to search for com-monalities among organisms. Commonalities in mechanisms of development are discussed on page 174, but there are also commonalities among organisms in the sequence of develop-mental events. All multicellular animals begin as zygotes and all go through cleavage and some subsequent developmental stages. Embryos of sponges, snails, and frogs diverge at some point to produce different adults. When does this divergence occur? All zygotes do not cleave in the same way; do certain types of cleavage characterize particular animal groups? Types of cleavage do characterize particular groups of animals, but cleavage type occurs along with other developmental features to form a suite of characters. Therefore, an overview of a devel-opmental sequence is needed to explain other characters in the suite. Based on these character suites, the 34 multicellular ani-mal phyla fall into several distinct groups. Rather than attempt-ing to grasp the details of 34 phyla, we can understand these phyla as variations on a much smaller number of developmen-tal themes. The character suites are discussed on page 166, and in Chapter 9. AN OVERVIEW OF DEVELOPMENT FOLLOWING CLEAVAGE Blastulation Cleavage subdivides the mass of the zygote until a cluster of cells called a blastula (Gr. blastos, germ, ule, little) is formed ( Figure 8.9 ). In mammals, the cluster of cells is called a blasto-cyst (see Figure 8.13E ). In most animals, the cells are arranged around a central fl uid-fi lled cavity ( Figure 8.9 ) called a blasto-coel (Gr. blastos, germ, koilos, cavity). (A hollow blastula can be called a coeloblastula to distinguish it from a solid stereoblas-tula; the general account here assumes the blastula is hollow.) In the blastula stage, the embryo consists of a few hundred to several thousand cells poised for further development. There has been a great increase in total DNA content because each of the many daughter cell nuclei formed by chromosomal replication at mitosis contains as much DNA as the original zygote nucleus. The whole embryo, however, is no larger than the zygote. Formation of a blastula stage, with its one layer of germ cells, occurs in all multicellular animals. In most animals, devel-opment continues beyond the blastula to form one or two more germ layers in a gastrula stage. Sponges were previously thought to complete embryogenesis with only a single layer of blastula cells, but recent work shows that cell migrations produce external and internal layers in embryos of at least some sponges (see Figure 12.12). Whether such cell layers are homologous to the true germ layers of other organisms is debated. The germ layers ultimately produce all structures of the adult body; the germ layer derivatives for vertebrates are shown in Figure 8.26 . Gastrulation and Formation of Two Germ Layers Gastrulation converts the spherical blastula into a more complex confi guration and forms a second germ layer ( Figure 8.9 ). There is variation in the process (see pp. 167–170). In some cases a second layer forms as cells migrate inward without making an internal cavity, but generally one side of the blastula bends inward in a process called invagination. This inward bending continues until the surface of the bending region extends about one-third of the way into the blastocoel, forming a new internal cavity ( Figure 8.9 ). Picture a sphere being pushed inward on one side—the inward region forms a pouch. The internal pouch is the gut cavity, called an archenteron (Gr. archae, old, enteron, gut) or a gastrocoel (Gr. gaster, stomach, koilos, cavity). It sits inside the now-reduced blastocoel. The opening to the gut, where the inward bending began, is the blastopore (Gr. blastos, germ, poros, hole). The gastrula (Gr. gaster, stomach, ule, little) stage has two layers: an outer layer of cells surrounding the blastocoel, called ectoderm (Gr. ecto, out, deros, skin), and an inner layer of cells lining the gut, called endoderm (Gr. endon, inside, deros, skin). In forming a mental picture of the developmental process, remember that cavities or spaces can only be defi ned by their boundaries. Thus, the gut cavity is a space defi ned by the layer of cells surrounding it ( Figure 8.9 ). Figure 8.9 A generalized developmental sequence showing formation of three germ layers and two body cavities that persist into adulthood.
Blastula Gastrula Complete gut forms Mesoderm lies between endoderm and ectoderm Coelomic cavity forms inside mesoderm Mesoderm Ectoderm Gut tube Blastocoel Endoderm Coelom Gut www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 165 This gut opens only at the blastopore; it is called a blind or incomplete gut. Anything consumed by an animal with a blind gut must either be completely digested, or the undigested parts egested through the mouth. Certain animals, sea anemones and fl atworms, for example, have a blind gut, sometimes called a gastrovascular cavity. However, most animals have a complete gut with a second opening, the anus ( Figure 8.9 ). The blasto-pore becomes the mouth in organisms with one suite of devel-opmental characters, but it becomes the anus in organisms with another such suite of characters (see Figure 8.10 ). Formation of a Complete Gut When a complete gut forms, the inward movement of the arch-enteron continues until the end of the archenteron meets the ectodermal wall of the gastrula. The archenteron cavity extends through the animal, and the ectoderm and endoderm layers join together. This joining produces an endodermal tube, the gut, surrounded by the blastocoel, inside an ectodermal tube, the body wall ( Figure 8.9 ). The endodermal tube now has two openings, the blastopore and a second, unnamed, opening that formed when the archenteron tube merged with the ectoderm ( Figure 8.9 ). Formation of Mesoderm, a Third Germ Layer The vast majority of multicellular animals proceed from a blas-tula to a gastrula, producing two germ layers. In one of many quirks of biological terminology, there is no term for organisms with only a single germ cell layer, but animals with two germ layers are called diploblastic (Gr. diploos, twofold, blastos, germ). Diploblastic animals include sea anemones and comb jel-lies. Most animals have a third germ layer and are triploblastic (L. tres, three, blastos, germ). The third layer, mesoderm (Gr. mesos, middle, deros, skin), eventually lies between the ectoderm and the endoderm ( Figure 8.9 ). Mesoderm can form in two ways: cells arise from a ventral area near the lip of the blasto-pore and proliferate into the space between the archenteron and outer body wall (see Figure 8.13C ), or the central region of the archenteron wall pushes outward into the space between the archenteron and outer body wall (see Figure 8.13A ). Regardless of method, initial cells of mesoderm come from endoderm. (In a few groups, such as amphibians, part of a third layer of cells is made from ectoderm; this is called ectome-soderm (Gr. ecto, out, mesos, middle, deros, skin) to distinguish it from true, endodermally derived, mesoderm). At the end of gastrulation, ectoderm covers the embryo, and mesoderm and endoderm have been brought inside ( Fig-ure 8.9 ). As a result, cells have new posi-tions and new neighbors, so interactions among cells and germ layers then generate more of the body plan. Formation of the Coelom A coelom (Gr. koilos, cavity) is a body cav-ity completely surrounded by mesoderm; the band of mesoderm with its internal coelom lies inside the space previously occupied by the blastocoel ( Figure 8.9 ). How did this happen? During gastrulation, the blastocoel is filled, partially or com-pletely, with mesoderm. The coelomic cav-ity appears inside the mesoderm by one of two methods: schizocoely or enterocoely. These methods are discussed on pages 166 and 169. A coelom made by schizocoely is Figure 8.10 Developmental tendencies of lophotrochozoan protostomes (fl atworms, annelids, molluscs, etc.) and deuterostomes. These tendencies are much modifi ed in some groups, for example vertebrates. Cleavage in mammals is rotational rather than radial; in reptiles, birds, and many fi shes cleavage is discoidal. Vertebrates have also evolved a derived form of coelom formation that is basically schizocoelous.
1 blastomere excised 1 blastomere excised 4-cell embryo 4-cell embryo 2 normal larvae Gut 2 Mosaic embryo 2 Regulative embryo 4 Coelom forms by splitting (schizocoelous) 4 Coelom forms by outpocketing (enterocoelous) Blastopore Mouth 1 Spiral cleavage 1 Radial cleavage 3 Blastopore becomes mouth, anus forms secondarily Blastopore becomes anus, mouth forms secondarily 3 Future intestine Future Anus Future mouth Future intestine Blastopore Blastocoel Development arrested Mesoderm Blastocoel Coelom Coelom Pocket of gut Mesoderm LOPHOTROCHOZOAN PROTOSTOME DEUTEROSTOME Anus 166 PART TWO Continuity and Evolution of Animal Life functionally equivalent to a coelom made by enterocoely. The method by which the coelom forms is an inherited character useful for grouping organisms into the suites of developmental characters mentioned earlier. When coelom formation is complete, the body has three germ layers and two cavities ( Figure 8.9 ). One cavity is the gut cavity and the other is the fl uid-fi lled coelomic cavity. The coe-lom, surrounded by its mesodermal walls, has completely fi lled the blastocoel. Mesoderm around the coelom will eventually produce layers of muscles, among other structures. SUITES OF DEVELOPMENTAL CHARACTERS There are two major groups of triploblastic animals, proto-stomes and deuterostomes. The groups are identifi ed by a suite of four developmental characters: (1) radial or spiral posi-tioning of cells as they cleave, (2) regulative or mosaic cleav-age of cytoplasm, (3) fate of the blastopore to become mouth or anus, and (4) schizocoelous or enterocoelous formation of a coelom. Snails and earthworms, among others, belong to the protostomes. Sea stars, fi shes, and frogs, among others, belong to the deuterostomes. Deuterostome Development Cleavage Patterns Radial cleavage ( Figure 8.10 ) is so named because the embry-onic cells are arranged in radial symmetry around the animal-vegetal axis. In radial cleavage of sea stars, the fi rst cleavage plane passes right through the animal-vegetal axis, yielding two identical daughter cells (blastomeres). For the second cleavage division, furrows form simultaneously in both blastomeres, and these are oriented parallel to the animal-vegetal axis (but per-pendicular to the fi rst cleavage furrow). Cleavage furrows next form simultaneously in the four daughter blastomeres, this time oriented perpendicular to the animal-vegetal axis, yielding two tiers of four cells each. An upper tier of cells sits directly atop the tier of cells below it ( Figure 8.10 ). Subsequent cleavages yield an embryo composed of several tiers of cells. A second characteristic of cleavage concerns the fates of isolated blastomeres and the cytoplasm they contain. This issue did not come to light until biologists undertook developmental experiments with embryos early in cleavage. Imagine a four-cell embryo ( Figure 8.10 ). All cells in an organism are ultimately derived from these four cells, but when are the products of each cell decided? If one cell is removed from the mass, can the other cells continue developing to produce a normal organism? Most deuterostomes have regulative development where the fate of a cell depends on its interactions with neighboring cells, rather than on what piece of cytoplasm it acquired during cleavage. In these embryos, at least early in development, each cell is able to produce an entire embryo if separated from the other cells ( Figure 8.11 ). In other words, an early blastomere originally has the ability to follow more than one path of dif-ferentiation, but its interaction with other cells restricts its fate. If a blastomere is removed from an early embryo, the remain-ing blastomeres can alter their normal fates to compensate for the missing blastomere and produce a complete organism. This adaptability is termed regulative development. Fate of Blastopore A deuterostome (Gr. deuteros, second, stoma, mouth) embryo develops through the blastula and gastrula stages, and forms a complete gut. The blastopore becomes the anus, and a second, unnamed, opening becomes the mouth, as is indicated by root words in the name of this group. Coelom Formation The fi nal deuterostome feature concerns the origin of the coe-lom. In enterocoely (Gr. enteron, gut, koilos, cavity), both mesoderm and coelom are made at the same time. In entero-coely, gastrulation begins with one side of the blastula bending inward to form the archenteron or gut cavity. As the archen-teron continues to elongate inward, the sides of the archenteron push outward, expanding into a pouchlike coelomic compart-ment (see Figure 8.10 ). The coelomic compartment pinches off to form a mesodermally bound space surrounding the gut (see Figure 8.10 ). Fluid collects in this space. Notice that the cells Figure 8.11 Regulative and mosaic cleavage. A, Regulative cleavage. Each of the early blastomeres (such as that of a sea urchin) when separated from the others develops into a small pluteus larva. B, Mosaic cleavage. In a mollusc, when blastomeres are separated, each gives rise to only a part of an embryo. The larger size of one defective larva is the result of the formation of a polar lobe (P) composed of clear cytoplasm of the vegetal pole, which this blastomere alone receives.
Regulative (Sea urchin) Mosaic (Mollusc) Separate blastomeres Separate blastomeres Normal larvae (plutei) Normal larva Defective larvae P P A B www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 167 that form the coelom during enterocoely come from a differ-ent region of the endoderm than do those making the coelom during schizocoely (see Figure 8.10 ). Examples of Deuterostome Development The general outline of deuterostome development just given var-ies in some of its details depending upon the animal being stud-ied. The presence of large amounts of yolk in some embryos further complicates the developmental sequence. A few examples of specifi c developmental sequences illustrate this variation. Variations in Deuterostome Cleavage The typical deu-terostome pattern is radial cleavage, but ascidian chordates (also called tunicates) exhibit bilateral cleavage. In ascidian eggs, the anteroposterior axis is established prior to fertiliza-tion by asymmetrical distribution of several cytoplasmic com-ponents ( Figure 8.12 ). The fi rst cleavage furrow passes through the animal-vegetal axis, dividing the asymmetrically distributed cytoplasm equally between the fi rst two blastomeres. Thus, this fi rst cleavage division separates the embryo into its future right and left sides, establishing its bilateral symmetry (hence the name bilateral holoblastic cleavage). Each successive division orients itself to this plane of symmetry, and the half-embryo formed on one side of the fi rst cleavage is the mirror image of the half embryo on the other side. Most mammals possess isolecithal eggs and a unique cleav-age pattern called rotational cleavage, so called because of the orientation of blastomeres with respect to each other dur-ing the second cleavage division (see mouse development in Figure 8.7E ). Cleavage in mammals is slower than in any other animal group. In humans, the fi rst division is completed about 36 hours after fertilization (compared with about an hour and a half in sea urchins), and the next divisions follow at 12- to 24-hour intervals. As in most other animals, the fi rst cleavage plane runs through the animal-vegetal axis to yield a two-cell embryo. However, during the second cleavage one of these blastomeres divides meridionally (through the animal-vegetal axis) while the other divides equatorially (perpendicular to the animal-vegetal axis). Thus, the cleavage plane in one blasto-mere is rotated 90 degrees with respect to the cleavage plane of the other blastomere (hence the name rotational cleavage). Furthermore, early divisions are asynchronous; not all blasto-meres divide at the same time. Thus, mammalian embryos may not increase regularly from two to four to eight blastomeres, but often contain odd numbers of cells. After the third division, the cells suddenly close into a tightly packed confi guration, which is stabilized by tight junctions that form between outer-most cells of the embryo. These outer cells form the tropho-blast. The trophoblast is not part of the embryo proper but will form the embryonic portion of the placenta when the embryo implants in the uterine wall. Cells that actually give rise to the embryo proper form from the inner cells, called the inner cell mass (see blastula stage in Figure 8.13E ). Telolecithal eggs of reptiles, birds, and most fi sh divide by discoidal cleavage. Because of the great mass of yolk in these eggs, cleavage is confi ned to a small disc of cytoplasm lying atop a mound of yolk (see chick development in Figure 8.7D ). Early cleavage furrows carve this cytoplasmic disc to yield a single layer of cells called the blastoderm. Further cleavages divide the blastoderm into fi ve to six layers of cells ( Figure 8.13D ). Variations in Deuterostome Gastrulation In sea stars, gastrulation begins when the entire vegetal area of the blastula fl attens to form a vegetal plate (a sheet of epithelial tissue). This event is followed by a process called invagination, in which the vegetal plate bends inward and extends about one-third of the way into the blastocoel, forming the archenteron ( Figure 8.13A ). Coelomic formation is typical of enterocoely. As the archenteron continues to elongate toward the animal pole, and its anterior end expands into two pouchlike coelo-mic vesicles, which pinch off to form left and right coelomic compartments ( Figure 8.13A ). The ectoderm gives rise to the epithelium of the body sur-face and to the nervous system. The endoderm gives rise to the epithelial lining of the digestive tube. The outpocketing of the archenteron is the origin of mesoderm. This third germ layer will form the muscular system, reproductive system, peritoneum (lining of the coelomic compartments), and the calcareous plates of the sea star’s endoskeleton. Frogs are deuterostomes with radial cleavage but morpho-genetic movements of gastrulation are greatly infl uenced by the mass of inert yolk in the vegetal half of the embryo. Cleavage divisions are slowed in this half so that the resulting blastula consists of many small cells in the animal half and a few large cells in the vegetal half (see Figures 8.7B and 8.13B ). Gastrula-tion in amphibians begins when cells located at the future dor-sal side of the embryo invaginate to form a slitlike blastopore. Thus, as in sea stars, invagination initiates archen-teron formation, but amphibian gastrulation begins in the marginal zone of the blastula, where ani-mal and vegetal hemispheres come together, and where there is less yolk than in the vegetal region. Gastrulation progresses as sheets of cells in the marginal zone turn inward over the blastopore lip and move inside the gastrula to form mesoderm and endoderm (see opening fi gure of this chapter, p. 158). The three germ layers now formed are the primary structural layers that play crucial roles in further differentiation of the embryo. Figure 8.12 Bilateral cleavage in tunicate embryos. The fi rst cleavage division divides the asymmetrically distributed cytoplasm evenly between the fi rst two blastomeres, establishing the future right and left sides of the adult animal. Bilateral symmetry of the embryo is maintained through subsequent cleavage divisions.
168 PART TWO Continuity and Evolution of Animal Life In bird and reptile embryos (see Figure 8.13D ), gastrulation begins with a thickening of the blastoderm at the caudal end of the embryo, which migrates forward to form a primitive streak ( Figure 8.14 ). The primitive streak becomes the anteroposterior axis of the embryo and the center of early growth. The primi-tive streak is homologous to the blastopore of frog embryos, but in chicks it does not open into the gut cavity because of the obstructing mass of yolk. The blastoderm consists of two layers (epiblast and hypoblast) with a blastocoel between them. Cells of the epiblast move as a sheet toward the primitive streak, then roll over the edge and migrate as individual cells into the blastocoel. These migrating cells separate into two streams. One stream of cells moves deeper (displacing the hypoblast along the midline) and forms endoderm. The other stream moves between the epiblast and hypoblast to form mesoderm. Cells on the sur-face of the embryo compose the ectoderm. The embryo now has three germ layers, at this point arranged as sheetlike layers with ectoderm on top and endoderm at the bottom. This arrange-ment changes, however, when all three germ layers lift from the underlying yolk ( Figure 8.14 ), then fold under to form a three-layered embryo that is pinched off from the yolk except for a stalk attachment to the yolk at midbody (see Figure 8.22 ). Gastrulation in mammals is remarkably similar to gastrula-tion in reptiles and birds (see Figure 8.13E ). Gastrulation move-ments in the inner cell mass produce a primitive streak. Epiblast cells move medially through the primitive streak into the blas-tocoel, and individual cells then migrate laterally through the blastocoel to form mesoderm and endoderm. Endoderm cells Figure 8.13 Blastula and gastrula stages in embryos of sea star, frog, nemertean worm, chick, and mouse.
Blastula Gastrula Blastula Blastocoel Blastocoel Gastrula Blastula Gastrula Blastopore (becomes mouth) Blastopore (becomes anus) Yolk plug (blastopore) Coelomic vesicles Archenteron Archenteron Blastocoel Prospective mesoderm cells Blastocoel Blastocoel Inner cell mass Trophoblast Yolk Blastula Blastula (blastocyst) Gastrula Gastrula Primitive streak Amniotic cavity Amnion Ectoderm Endoderm Migrating cells Yolk sac Migrating cells A Sea star B Frog C Nemertean worm D Chick E Mouse Figure 8.14 Gastrulation in a chick. Transverse sections through the heart-forming region of the chick show development at 18, 25, and 28 hours of incubation.
18 hours 25 hours 28 hours Yolk Yolk Yolk These layers form extra-embryonic membranes Forming heart tube Coelomic (pericardial space) Heart rudiments Superficial ectoderm Blastocoel Hypoblast Migrating cells Foregut Neural tube Notochord Neural groove Primitive streak Subgerminal space Epiblast www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 169 (derived from the hypoblast) form a yolk sac devoid of yolk (since mammalian embryos derive nutrients directly from the mother via the placenta). Amphibians, reptiles, and birds, which have moderate to large amounts of yolk concentrated in the vegetal region of the egg, have evolved derived gastrulation patterns in which the yolk does not participate in gastrulation. Yolk is an impediment to gastrulation and consequently the gastrulation process occurs around (amphibians) or on top (reptiles and birds) of the vegetal yolk. Mammalian eggs are isolecithal, and thus one might expect them to have a gastrulation pattern similar to that of sea stars. Instead they have a pattern more suited to telolecithal eggs. The best explanation for this feature of mammalian egg development is common ancestry with birds and reptiles. Reptiles, birds, and mammals share a common ancestor whose eggs were teloleci-thal. Thus, all three groups inherited their gastrulation patterns from this common ancestor, and mammals subsequently evolved isolecithal eggs but retained the telolecithal gastrulation pattern. A further developmental complication in vertebrates is that coelom formation occurs by a modifi ed form of schizocoely (see Figure 8.10 ), not enterocoely. The nonvertebrate chordates form the coelom by enterocoely, as is typical of deuterostomes. Protostome Development Cleavage Patterns Spiral cleavage (see Figure 8.10 ) occurs in most protostomes. It differs from radial cleavage in two important ways. Rather than dividing parallel or perpendicular to the animal-vegetal axis, blas-tomeres cleave obliquely (approximately 45-degree angle) to this axis and typically produce quartets of cells that come to lie, not on top of but in the furrows between cells of the underlying layer. The upper layer of cells appears offset (shifted in a spiral fashion) from the lower (see Figure 8.10 ). In addition, spirally cleaving blastomeres pack themselves tightly together much like a group of soap bubbles, rather than just lightly contacting each other as do many radially cleaving blastomeres (see Figure 8.10 ). Mosaic development characterizes most protostomes (see Figure 8.10 ). In mosaic development, cell fate is determined by the distribution of certain proteins and messenger RNAs, called morphogenetic determinants, in the egg cytoplasm. As cleav-age occurs, these morphogenetic determinants are partitioned among the cells unequally. When a particular blastomere is iso-lated from the rest of the embryo, it still forms the character-istic structures decided by the morphogenetic determinants it contains (see Figure 8.11 ). In the absence of a particular blasto-mere, the animal lacks those structures normally formed by that blastomere, so it cannot develop normally. This pattern is called mosaic development because the embryo seems to be a mosaic of self-differentiating parts. Fate of Blastopore A protostome (Gr. protos, fi rst, stoma, mouth) is so named because the blastopore becomes the mouth, and the second, unnamed, opening becomes the anus. Coelom Formation In protostomes, a mesodermal band of tissue surrounding the gut forms before a coelom is made. If present, the inner coelomic cavity is made by schizocoely. To form mesoderm, endodermal cells arise ventrally at the lip of the blastopore (see Figure 8.10 ) and move, via ingression, into the space between the walls of the archenteron (endoderm) and outer body wall (ectoderm). These cells divide and deposit new cells, called mesodermal pre-cursors, between the two existing cell layers (see Figure 8.13C ). The proliferating cells become the mesoderm. Meticulous cell-lineage studies by embryologists established that in many organ-isms with spiral cleavage, for example, fl atworms, snails, and related organisms, these mesodermal precursors arise from one large blastomere, called the 4d cell, that is present in a 29- to 64-cell embryo. Some protostomes do not develop a coelom. Flatworms, like Planaria, develop to an early gastrula stage and then form a mesodermal layer as just described. Mesoderm completely fi lls the blastocoel and a coelom never forms (see Figure 9.3). Ani-mals without a coelom are called acoelomate. In other proto-stomes, mesoderm lines only one side of the blastocoel, leaving a fl uid-fi lled blastocoel next to the gut (see Figure 9.3). The fl uid-fi lled cavity surrounding the gut is called a pseudocoe-lom (Gr. pseudés, false, koilos, cavity); it is bordered on the inner edge by the endodermal gut lining, and on the outer edge by a layer of mesoderm next to the ectoderm. Thus, a pseudo-coelom has mesoderm on only one side, whereas a true coe-lom is a fl uid-fi lled cavity completely surrounded by mesoderm (see Figure 9.3). Acoelomate and pseudocoelomate body plans are discussed in more detail in Chapter 9. For coelomate protostomes, like earthworms and snails, the mesodermal layer forms as just described, and a coelom is made by schizocoely (Gr. schizein, to split, koilos, cavity). A coelom arises, as the name suggests, when the mesodermal band around the gut splits open centrally (see Figure 8.10 ). Fluid collects in the coelom. Examples of Protostome Development The protostomes are divided into two clades. One clade, lophotrochozoan protostomes, contains segmented worms, molluscs (snails, slugs, clams, octopus and their kin) and several less familiar taxa. The name of this clade refers to two features present in some members of the group: a horseshoe-shaped whorl of tentacles called a lophophore (see pp. 324–328), and a trochophore larva (see p. 337). Lophotrochozoans have the four protostome features described previously (see Figure 8.10 ). They typically form mesoderm from the embryonic 4d cell. The other clade, ecdysozoan protostomes, includes arthropods (insects, spiders, crabs, and related organisms), roundworms, and other taxa which molt the exoskeleton. The name of this clade refers to shedding of the cuticle, ecdysis (Gr. ekdyo, take off, strip). Variations in Protostome Cleavage Spiral cleavage is typ-ical of protostomes, but one highly specialized class of molluscs, 170 PART TWO Continuity and Evolution of Animal Life cephalopods, has bilateral cleavage like that of ascidian chor-dates (see p. 167 and Figure 8.12 ). Octopus, squid, and cuttlefi sh, among others, are cephalopods. Many ecdysozoans do not exhibit spiral cleavage; in some, cleavage appears radial, and in others, such as insects, cleavage is neither spiral nor radial. Centrolecithal eggs of insects undergo superfi cial cleav-age ( Figure 8.15 ) where the centrally located mass of yolk restricts cleavage to the cytoplasmic rim of the egg. This pattern is highly unusual because cytoplasmic cleavage (cytokinesis) does not occur until after many rounds of nuclear division. After roughly eight rounds of mitosis in the absence of cytoplasmic division (yielding 256 nuclei), the nuclei migrate to the yolk-free periphery of the egg. A few nuclei at the posterior end of the egg become surrounded by cytoplasm to form pole cells, which give rise to germ cells of the adult. Next, the entire egg cell membrane folds inward, partitioning each nucleus into a single cell, and yielding a layer of cells at the periphery sur-rounding the mass of yolk ( Figure 8.15 ). Because yolk is an impediment to cleavage, this pattern avoids cleaving the yolk and instead confi nes cytoplasmic division to small regions of yolk-free cytoplasm. Variations in Protostome Gastrulation In most proto-stomes, mesodermal cells all derive from the 4d cell (see p. 169). However, in some nemertean worms (see Figure 8.13C ), meso-derm derives from an earlier blastomere. Mesodermal origins are diffi cult to determine in many ecdysozoan protostomes due to the modifi ed pattern of cleavage. MECHANISMS OF DEVELOPMENT Nuclear Equivalence How does a developing embryo generate a multitude of cell types of a complete multicellular organism from the starting point of a single diploid nucleus of a zygote? To many nineteenth-century embryologists there seemed only one acceptable answer: as cell division ensued, hereditary material had to be parceled unequally to daughter cells. In this view, the genome gradually became broken into smaller and smaller units until fi nally only the information required to impart the characteristics of a single cell type remained. This became known as the Roux-Weismann hypothesis, after the two German embryologists who developed the concept. However, in 1892 Hans Driesch discovered that if he mechanically shook apart a two-celled sea urchin into separate cells, both half-embryos developed into normal larvae. Driesch concluded that both cells contained all genetic information of the original zygote. Still, this experiment did not settle the argu-ment, because many embryologists thought that even if all cells contained complete genomes, the nuclei might become progres-sively modifi ed in some way to dispense with the information not used in forming differentiated cells. The efforts of Hans Driesch to disrupt egg development are poeti-cally described by Peattie: “Behold Driesch grinding the eggs of Loeb’s favorite sea urchin up between plates of glass, pounding and breaking and deforming them in every way. And when he ceased from thus abusing them, they proceeded with their orderly and normal development. Is any machine conceivable, Driesch asks, which could thus be torn down . . . have its parts all disar-ranged and transposed, and still have them act normally? One cannot imagine it. But of the living egg, fertilized or not, we can say that there lie latent within it all the potentialities presumed by Aristotle, and all of the sculptor’s dream of form, yes, and the very power in the sculptor’s arm.” From Peattie, D. C. 1935. An Almanac for Moderns. New York, G. P. Putnam’s Sons. Around the turn of the century Hans Spemann introduced a new approach to testing the Roux-Weismann hypothesis. Spe-mann placed minute ligatures of human hair around salaman-der zygotes just as they were about to divide, constricting them until they were almost, but not quite, separated into two halves. The nucleus lay in one half of the partially divided zygote; the other side was anucleate, containing only cytoplasm. The zygote then completed its fi rst cleavage division on the side containing Figure 8.15 Superfi cial cleavage in a Drosophila embryo. The zygote nucleus at fi rst divides repeatedly in the yolk-rich endoplasm by mitosis without cytokinesis. After several rounds of mitosis, most nuclei migrate to the surface where they are separated by cytokinesis into separate cells. Some nuclei migrate to the posterior pole to form the primordial germ cells, called pole cells. Several nuclei remain in the endoplasm where they will regulate breakdown of yolk products. The cellular blastoderm stage corresponds to the blastula stage of other embryos.
Cellular blastoderm Pole cells Nuclei www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 171 the nucleus; the anucleate side remained undivided. Eventually, when the nucleated side had divided into about 16 cells, one of the cleavage nuclei would wander across the narrow cytoplas-mic bridge to the anucleate side. Immediately this side began to divide and developed normally. Sometimes, however, Spemann observed that the nucle-ated half of the embryo developed only into an abnormal ball of “belly” tissue. The explanation, Spemann discovered, depended on the presence of the gray crescent, a pigment-free area shown in Figure 8.7B . The gray crescent is required for normal devel-opment because it is the precursor of the Spemann organizer discussed in the opening essay on page 158. Spemann’s experiment demonstrated that every blastomere contains suffi cient genetic information for the development of a complete animal. In 1938 he suggested another experiment that would demonstrate that even somatic cells of an adult contain a complete genome. The experiment, which Spemann charac-terized as being “somewhat fantastical” at that time, would be to remove the nucleus of an egg cell and replace it with the nucleus from a somatic cell from a different individual. If all cells contained the same genetic information as a zygote, then the embryo should develop into an individual that is genetically identical to the animal from which the nucleus was obtained. It took several decades to solve the technical diffi culties, but the experiment was successfully performed on amphibians, and today it is done in a variety of mammals. The procedure is now familiarly called cloning. One of the most famous cloned mam-mals, Dolly the sheep, got the genetic material in her nuclei from the mammary glands of a six-year-old ewe. If all nuclei are equivalent, what causes some cells to develop into neurons while others develop into skeletal muscle? In most animals (excluding insects), there are two major ways by which cells become committed to particular developmental fates: (1) cytoplasmic partitioning of determinative molecules during cleavage and (2) interaction with neighboring cells (inductive interactions). All animals use both of these mecha-nisms to some extent to specify different cell types. However, in some animals cytoplasmic specifi cation is dominant, whereas others rely predominantly on inductive interactions. Cytoplasmic Specifi cation A fertilized egg contains cytoplasmic components that are unequally distributed within the egg. These different cytoplas-mic components are thought to contain morphogenetic determi-nants that control commitment of a cell to a particular cell type. These morphogenetic determinants are partitioned among differ-ent blastomeres as a result of cleavage, and the developmental fate of each cell becomes specifi ed by the type of cytoplasm it acquires during development (see mosaic development, p. 169). This process is especially striking (and easily visualized) in some tunicate species in which the fertilized egg contains as many as fi ve differently colored types of cytoplasm (see Fig-ure 8.12 ). These differently pigmented cytoplasms are segre-gated into different blastomeres, which then proceed to form distinct tissues or organs. For example, yellow cytoplasm gives rise to muscle cells while gray equatorial cytoplasm produces notochord and neural tube. Clear cytoplasm produces larval epidermis and gray vegetal cytoplasm gives rise to the gut. Embryonic Induction Induction, the capacity of some cells to evoke a specifi c devel-opmental response in others, is a widespread phenomenon in development. The classic experiments, cited in the open-ing essay on page 158, were reported by Hans Spemann and Hilde Mangold in 1924. When a piece of dorsal blastopore lip from a salamander gastrula was transplanted into a ventral or lateral position of another salamander gastrula, it invaginated and developed a notochord and somites. It also induced the host ectoderm to form a neural tube. Eventually a whole sys-tem of organs developed where the graft was placed, and then grew into a nearly complete secondary embryo ( Figure 8.16 ). This creature was composed partly of grafted tissue and partly of induced host tissue. It was soon found that only grafts from the dorsal lip of the blastopore were capable of inducing the formation of a com-plete or nearly complete secondary embryo. This area corre-sponds to the presumptive areas of notochord, somites, and prechordal plate (see p. 500). It was also found that only ecto-derm of the host would develop a nervous system in the graft and that the reactive ability was greatest at the early gastrula stage and declined as the recipient embryo got older. Spemann designated the dorsal lip area the primary orga-nizer because it was the only tissue capable of inducing the Figure 8.16 The Spemann-Mangold primary organizer experiment.
Discarded mesoderm opposite dorsal lip Donor mesoderm from dorsal lip Dorsal lip Primary neural fold Secondary neural development Primary notochord and neural development Secondary notochord and neural development 172 PART TWO Continuity and Evolution of Animal Life development of a secondary embryo in the host. It is now often called the Spemann organizer. Spemann also termed this induc-tive event primary induction because he considered it the fi rst inductive event in development. Subsequent studies showed that many other cell types originate by later inductive interactions, a process called secondary induction. Usually cells that have differentiated act as inductors for adjacent undifferentiated cells. Timing is important. Once a pri-mary inductor sets in motion a specifi c developmental pattern in some cells, numerous secondary inductions follow. What emerges is a sequential pattern of development involving not only inductions but cell movement, changes in adhesive proper-ties of cells, and cell proliferation. There is no “hard-wired” mas-ter control panel directing development, but rather a sequence of local patterns in which one step in development is a subunit of another. In showing that each step in the developmental hier-archy is a necessary preliminary for the next, Hans Spemann’s induction experiments were among the most signifi cant events in experimental embryology. GENE EXPRESSION DURING DEVELOPMENT Since every cell with few exceptions receives the same genetic material, cytoplasmic specifi cation and induction must involve the activation of different combinations of genes in different cells. Understanding development is therefore ultimately a prob-lem of understanding the genetics involved. It is not surprising that developmental genetics was fi rst studied in the geneticists’ favorite model organism, the fruit fl y Drosophila. These studies have been repeated in several other model animals, such as the nematode worm Caenorhabditis elegans, zebra fi sh Danio rerio, frog Xenopus laevis, chick Gallus gallus, and mouse Mus mus-culus. This research suggests that epigenesis proceeds in three general stages: pattern formation, determination of position in the body, and induction of limbs and organs appropriate for that position. Each stage is guided by gradients of gene products that function as morphogens. Pattern Formation The fi rst step in organizing development of an embryo is pattern formation: determination of the front-to-rear (anteroposterior), left-to-right, and back-to-front (dorsoventral) axes. As Spemann demonstrated in salamanders, the anteroposterior axis of the embryo is determined by the Spemann organizer, located in the gray crescent of a zygote. In Drosophila the anteroposterior axis is determined even before an egg is fertilized. Christiane Nüsslein-Volhard and her colleagues in Germany found that this determination is due to a gradient of mRNA that is secreted into the egg by nurse cells in the mother. The end of the egg that receives the highest level of this mRNA is fated to become the anterior of the embryo and eventually of the adult. The mRNA is transcribed from a gene called bicoid (pronounced BICK-oyd) in the nurse cells. After an egg is fertilized, bicoid mRNA is translated into a protein morphogen called bicoid (not italicized) that binds to certain other genes. The products of these genes in turn activate others in a cascade that ultimately causes the production of an anteroposterior gradient. Bicoid is one of about 30 maternal genes that control pattern formation in an embryo. Some of these determine the dorsoventral axis. The gene short gastrulation leads to development of ventral structures, such as the nerve cord. One of the most exciting discoveries in developmental genet-ics has been that the developmental genes of vertebrates and many other animals are similar to those of Drosophila; they are conserved over a wide range of animals. A gene similar to bicoid is also important in pattern formation in vertebrates. In verte-brates, however, the gene, called Pitx2, determines positioning of certain internal organs to either the left or right side of the body. Mutations in Pitx2 in frogs, chicks, and mice can place the heart and stomach on the right instead of the left side. Such mutations may explain a reversal of organ position that sometimes occurs in humans. Pitx2 is in turn activated by a protein produced by the gene sonic hedgehog (Shh), which is similar to a Drosophila gene called hedgehog. (The name hedgehog refers to the bristly appear-ance of fruit fl ies lacking the gene. The “sonic” comes from the video-game character Sonic the Hedgehog.) In vertebrates, sonic hedgehog is active in the left side only at the anterior end of the primitive streak (see Figure 8.13 ). Short gastrulation also has a counterpart in vertebrates—the gene chordin, which produces one of the proteins from the Spemann organizer. In Drosophila, as well as other arthropods, annelid worms, chordates, and a few other groups, one important aspect of pat-tern formation along the anteroposterior axis is segmentation, also called metamerism. Segmentation is a division of the body into discrete segments or metameres (see Fig 9.6, p. 195). The segments are identical early in development, but later activa-tion of different combinations of genes causes each segment to form different structures. For example, the anterior segment of insect embryos will form antennae, eyes, and mouthparts, while segments farther back will form legs. Segments are obvious in insects, but in fi shes segmentation is apparent only in somites that produce such structures as vertebrae and repeated muscle bands (myomeres) (see Figure 24.24, p. 531). In Drosophila the number and orientation of segments is controlled by segmenta-tion genes. There are three classes of segmentation genes: gap, pair-rule, and segment-polarity. Gap genes are activated fi rst and divide an embryo into regions such as head, thorax, and abdomen. Pair-rule genes divide these regions into segments. Finally, segment-polarity genes, such as hedgehog, organize the anterior-to-posterior structures within each segment. Homeotic and Hox Genes Segmentation genes apparently regulate expression of other genes, ensuring that they are active only in appropriate seg-ments. Such segment-specifi c genes are called homeotic genes. Mutations in homeotic genes, called homeotic mutations, place appendages or other structures in the wrong part of the body. For example, in Drosophila the homeotic gene Antennapedia, www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 173 which helps trigger development of legs, is normally active only in the thorax. If the Antennapedia gene is activated by a homeo-tic mutation in the head of a maggot, the adult will have legs in place of antennae ( Figure 8.17 ). Antennapedia and some other homeotic genes, as well as many other genes involved in devel-opment, include a sequence of 180 DNA base pairs, called the homeobox. The homeobox produces the part of a protein that attaches to the DNA of other genes, activating or blocking their expression. Several other homeotic and nonhomeotic genes that are clustered close to Antennapedia on the same chromosome in Drosophila also include a homeobox. Genes in this cluster are called Hom genes. Hom genes do not encode specifi c limbs and organs. Instead, they function by specifying the location in the body along the anteroposterior axis. Intriguingly, the order of the Hom genes within the cluster on the chromosome is the same as the order in which they are expressed along the length of the body ( Figure 8.18 ). One of the most exciting discoveries of the late twentieth century was that genes similar to Hom genes of Drosophila occur in other insects, as well as in chordates and unsegmented animals such as hydra and nematode worms. They also occur in plants and yeasts, and perhaps in all eukaryotes. These genes in organisms other than Drosophila were called Hox genes, but now all such genes are usually called Hox genes. Most Hox genes occur in a cluster on one chromosome. Mam-mals have four clusters, each on a different chromosome, with from 9 to 11 Hox genes each. As in Drosophia, the sequence of Hox genes within a cluster is the same as the front-to-rear order in which they are expressed in the body. Morphogenesis of Limbs and Organs Hox and other homeobox genes also play a role in shaping individual organs and limbs. As shown in Figures 8.18 and 8.19 , for example, regions of the brain and identity of somites are specifi ed by particular Hox and homeobox genes. Many other developmental genes that are also involved in pat-tern formation for the entire body also help shape individual limbs and organs by producing gradients of morphogens. One example, which has been studied by Cheryll Tickle and her coworkers at University College in London, is formation and development of limb buds in chicks. They have found that a new limb bud can be induced to grow from the side of a chick by implanting a bead soaked in fi broblast growth factor (FGF). This result implies that limbs are normally induced to develop by activation of the gene for FGF in appropriate parts of the body. Whether the limb bud develops into a wing or a leg depends on whether the FGF is applied toward the front or the rear of the chick. Figure 8.17 Head of a fruit fl y with a pair of legs growing out of head sockets where antennae normally grow. The Antennapedia homeotic gene normally specifi es the second thoracic segment (with legs), but the dominant mutation of this gene leads to this bizarre phenotype.
Figure 8.18 Homology of Hox genes in insects and mammals. These genes in both insects (fruit fl y) and mammals (mouse) control the subdivision of the embryo into regions of different developmental fates along the anterior-posterior axis. The homeobox-containing genes lie on a single chromosome of the fruit fl y and on four separate chromosomes in the mouse. Clearly defi ned homologies between the two, and the parts of the body in which they are expressed, are shown in color. The open boxes denote areas where it is diffi cult to identify specifi c homologies between the two. The Hox genes shown here are only a small subset of all the homeobox genes.
Fruit fly Fruit fly embryo Mouse embryo Mouse Fly chromosome Mouse chromosomes Anterior end 3′ Posterior end 5′ Hox A Hox B Hox C Hox D pb lab Dfd Scr Antp Abd-B Ubx 174 PART TWO Continuity and Evolution of Animal Life FGF also plays a role in shaping the limb. It is secreted by cells in an apical ectodermal ridge at the end of the limb bud. FGF acts as a morphogen that forms a gradient from the apical ectodermal ridge to the base of a limb bud. This gra-dient helps establish a proximodistal axis—one of three axes that guide development of a limb ( Figure 8.20 ). Fingers or toes develop at the end of the proximodistal axis with the highest level of FGF. An anteroposterior axis is established by a gradi-ent of sonic hedgehog and ensures that fi ngers or toes develop in the appropriate order. Finally, Wnt7a, a protein produced by a gene that is similar to the segment-polarity gene wingless in Drosophila, helps determine the dorsoventral axis. Wnt7a makes the dorsal side of the wing or foot different from the ventral side. Evolutionary Developmental Biology Zoologists have always looked to embryology for clues to the evolutionary history, or phylogeny, of animals. Developmen-tal features such as the number of germ layers and the fate of the blastopore do suggest evolutionary relationships among different phyla. Advances in development genetics, such as those just described in the previous section, have made the relationship between development and evolution even closer and have given rise to an exciting new fi eld called evolutionary Figure 8.19 How the inhibition of a homeodomain regulatory protein alters normal development of the central nervous system of a frog tadpole. When the protein (encoded by a homeobox DNA sequence known as X1Hbox 1) was inactivated by antibodies directed against it, the area that should have become anterior spinal cord transformed into hindbrain instead.
Control tadpole Tadpole injected with antibodies to X1Hbox 1 protein Anterior spinal cord (expresses X1Hbox 1) Hindbrain Figure 8.20 Morphogenesis in a vertebrate limb bud. The skeleton of a mature chicken limb is shown for orientation. Three axes are established in the limb bud: a proximal-distal axis by fi broblast growth factor (FGF) from the apical ectodermal ridge; an anterior-posterior axis by sonic hedgehog protein from the zone of polarizing activity; and a dorsal-ventral axis by Wnt7a protein from dorsal ectoderm.
Ventral Anterior Proximal Dorsal Posterior Distal Zone of polarizing activity Apical ectodermal ridge Dorsal ectoderm Limb bud Developing chick wing www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 175 developmental biology. Evolutionary developmental biology, often nicknamed evo-devo, is based on a realization that evolu-tion is essentially a process in which organisms become differ-ent as a result of changes in the genetic control of development. The fact that the genes that control development are similar in animals as different as fruit fl ies and mice offers hope that we can reconstruct the evolutionary history of animals by under-standing how functioning of those genes came to differ. Evolu-tionary developmental biology has already contributed several exciting concepts to our thinking about animal evolution, but the fi eld is so new that it would be premature to accept these concepts as established. It is best to state them as questions for further study. Are the body plans of all bilaterally symmetric ani-mals fundamentally similar? As noted on page 172, chordin, one of the genes responsible for development of the nervous system in the dorsal part of a frog, is similar to short gastrula-tion, which is necessary for development of the ventral nerve cord in Drosophila. In addition, the gene decapentaplegic pro-motes dorsal development in Drosophila, and the similar gene bone morphogenetic protein-4 promotes ventral development in frogs. In other words, insects and amphibians, whose body plans look so different, actually share a similar control of dor-soventral patterning, except that one is upside down compared with the other. This fi nding has prompted a reappraisal of an idea fi rst proposed by the French naturalist Etienne Geoffroy St. Hilaire in 1822 after he noticed that in a dissected lobster on its back the nerve cord was above the gut, and the heart was below it, as in a vertebrate in its normal position. The idea that a vertebrate is like an inverted invertebrate was quickly rejected, but now biologists are once more considering whether the body plans of protostomes and deuterostomes are simply inverted relative to each other. Can the anatomy of extinct ancestral species be inferred from the developmental genes shared by their descendants? The fact that dorsoventral patterning is simi-lar in protostomes and deuterostomes suggests that the most recent common ancestor of these two branches had a simi-lar dorsoventral patterning with a heart and nervous system separated by the gut. One can also infer from the similarity in Hom / Hox clusters in insects and chordates that the most recent common ancestor of protostomes and deuterostomes may have been segmented and that its segments differentiated by similar genes. It may also have had at least rudimentary eyes, judging from the fact that similar genes, eyeless/Pax-6, are involved in eye formation in a wide range of both protostomes and deuterostomes. Instead of evolution proceeding by the gradual accu-mulation of numerous small mutations, could it proceed by relatively few mutations in a few developmental genes? The fact that formation of legs or eyes can be induced by a muta-tion in one gene suggests that these and other organs develop as modules (see p. 173). If so, then entire limbs and organs could have been lost or acquired during evolution as a result of one or a few mutations, which would challenge Darwin’s theory of gradualism (p. 121). If this is correct, then the apparently rapid evolution of numerous groups of animals during the few mil-lion years of the Cambrian explosion and at other times is more easily explained. Instead of requiring mutations in numerous genes, each with a small effect, evolution of different groups could be a result of changes in timing, number, or expression of relatively few developmental genes. VERTEBRATE DEVELOPMENT The Common Vertebrate Heritage A prominent outcome of shared ancestry of vertebrates is their common pattern of development. This common pattern is best seen in the remarkable similarity of postgastrula vertebrate embryos ( Figure 8.21 ). The likeness occurs at a brief moment in the development of vertebrates when shared chordate hall-marks of dorsal neural tube, notochord, pharyngeal gill pouches with aortic arches, ventral heart, and postanal tail are pres-ent at about the same stage of development. Their moment of similarity—when the embryos seem almost interchangeable—is all the more extraordinary considering the great variety of eggs and widely different types of early development that have con-verged toward a common design. Then, as development con-tinues, the embryos diverge in pace and direction, becoming recognizable as members of their class, then their order, then family, and fi nally their species. The important contribution of early vertebrate development to our understanding of homol-ogy and evolutionary common descent is described in Chapter 6 in the section on Ontogeny, Phylogeny, and Recapitulation, page 116. Amniotes and the Amniotic Egg Reptiles, birds, and mammals form a monophyletic group-ing of vertebrates called amniotes, so named because their embryos develop within a membranous sac, the amnion. The amnion is one of four extraembryonic membranes that compose a sophisticated support system within the amni-otic egg ( Figure 8.22 ), which evolved when the fi rst amniotes appeared in the late Paleozoic era. The amnion is a fl uid-fi lled sac that encloses the embryo and provides an aqueous environment in which the embryo fl oats, protected from mechanical shock and adhesions. Evolution of the second extraembryonic membrane, the yolk sac, actually predates appearance of amniotes many mil-lions of years. The yolk sac with its enclosed yolk is a conspicu-ous feature of all fi sh embryos. After hatching, a growing fi sh larva depends on the remaining yolk provisions to sustain it until it can begin to feed itself ( Figure 8.23 ). The yolk sac func-tions differently in animals that give live birth. In many vivip-arous vertebrates from diverse groups, the yolk sac becomes vascular and intimately associated with the mother’s reproduc-tive tract, allowing transfer of nutrients and respiratory gases 176 PART TWO Continuity and Evolution of Animal Life Figure 8.21 Early vertebrate embryos drawn from photographs. Embryos as diverse as fi sh, salamander, tortoise, bird, and human show remarkable similarity following gastrulation. At this stage (top row) they reveal features common to the entire subphylum Vertebrata. As development proceeds they diverge, each becoming increasingly recognizable as belonging to a specifi c class, order, family, and fi nally, species.
FISH SALAMANDER TORTOISE CHICK HUMAN between mother and fetus. Thus, a yolk sac placenta is formed. The mass of yolk is an extraembryonic structure because it is not a part of the embryo proper, and the yolk sac is an extraembryonic membrane because it is an accessory structure that develops outside the embryo and is discarded after the yolk is consumed. The allantois is a sac that grows out of the hindgut and serves as a repository for metabolic wastes during development. It also functions as a respiratory surface for exchange of oxygen and carbon dioxide. The chorion lies just beneath the eggshell and completely encloses the rest of the embry-onic system. As the embryo grows and its need for oxygen increases, the allantois and chorion fuse to form the chorioallantoic membrane. This double membrane has a rich vascular net-work connected to the embryonic circulation. Figure 8.22 Amniotic egg at an early stage of development showing a chick embryo and its extraembryonic membranes.
Yolk sac Allantois Embryo Shell Shell membrane Amnion Chorion www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 177 Lying just beneath the porous shell, the vascular chorioallantois serves as a provisional “lung” across which oxygen and carbon dioxide can freely exchange. Thus an amniotic egg provides a complete life-support system for the embryo, enclosed by a tough outer shell. The amniotic egg is one of the most important adaptations to have evolved in vertebrates. The evolution of a shelled amniotic egg made internal fer-tilization a reproductive requirement. A male must introduce sperm directly into the female reproductive tract, since sperm must reach and fertilize the egg before the eggshell is wrapped around it. The Mammalian Placenta and Early Mammalian Development Rather than developing within an eggshell like most other vertebrates most mammalian embryos evolved the strategy of developing within the mother’s body. We have already seen that mammalian gastrulation closely parallels that of egg-laying amniotes. The earliest mammals were egg layers, and even today some mammals retain this primitive character; mono-tremes (duck-billed platypus and spiny anteater) lay large yolky eggs that closely resemble bird eggs. In marsupials (pouched mammals such as opossums and kangaroos), embryos develop for a time within the mother’s uterus, but an embryo does not “take root” in the uterine wall, and consequently it receives little nourishment from the mother before birth. The young of marsupials are born at an early stage of development and con-tinue developing sheltered in a pouch in the mother’s abdomi-nal wall, nourished with milk (reproduction in marsupials is described on pp. 628–629). All other mammals, composing 94% of class Mammalia, are placental mammals. These mammals have evolved a placenta, a remarkable fetal structure through which an embryo is nourished. Evolution of this fetal organ required substantial restructuring, not only of extraembryonic mem-branes to form the placenta but also of the maternal ovi-duct, part of which had to expand into long-term housing for embryos, the uterus. Despite these modifications, develop-ment of extraembryonic membranes in placental mammals is remarkably similar to their development in egg-laying amniotes (compare Figures 8.22 and 8.24 ). In fact, for some nonmammalian vertebrates that give live birth, the extraem-bryonic membranes form a placenta. Some viviparous lizards and snakes have either a yolk sac placenta or a chorioallan-toic placenta or both. One of the most intriguing questions the placenta presents is, why is it not immunologically rejected by the mother? Both placenta and embryo are genetically alien to the mother because they contain proteins (called major histocompatibility proteins, p. 775) that differ from those of the mother. We would expect uterine tissues to reject the embryo just as the mother would reject an organ transplanted from her own child. The placenta is a uniquely successful foreign transplant, or allograft, because it has evolved measures for sup-pressing the immune response that normally would be mounted against it and the fetus by the mother. Experiments suggest that the chorion produces proteins and lymphocytes that block the normal immune response by suppressing formation of specifi c antibodies by the mother. Early stages of mammalian cleavage, shown in Fig-ure 8.13E , occur while a blastocyst is traveling down the oviduct toward the uterus, propelled by ciliary action and muscular peristalsis. When a human blastocyst is about six Figure 8.23 Fish larvae showing yolk sac. A, The one-day-old larva of a marine fl ounder has a large yolk sac. B, After 10 days of growth the larva has developed mouth, sensory organs, and a primitive digestive tract. With its yolk supply now exhausted, it must capture food to grow and survive.
A B Yolk sac filled with yolk Figure 8.24 Generalized diagram of extraembryonic membranes of a mammal, showing how their development parallels that of a chick (compare with Figure 8.22). Most extraembryonic membranes of mammals have been redirected to new functions.
Allantoic mesoderm Allantois Future umbilical cord Yolk sac Chorion Villi Amniotic cavity Amnion Extra-embryonic coelom 178 PART TWO Continuity and Evolution of Animal Life days old and composed of about 100 cells, it contacts the uterine endometrium (uterine lining) ( Figure 8.25 ). On con-tact, the trophoblast cells proliferate rapidly and produce enzymes that break down the epithelium of the uterine endo-metrium. These changes allow the blastocyst to implant in the endometrium. By the eleventh or twelfth day the blastocyst is completely buried and surrounded by a pool of maternal blood. The trophoblast thickens, sending out thousands of tiny, fi ngerlike projections, the chorionic villi. These pro-jections sink like roots into the uterine endometrium after the embryo implants. As development proceeds and embry-onic demands for nutrients and gas exchange increase, the great proliferation of chorionic villi vastly increases the total surface area of the placenta. Although a human placenta at term measures only 18 cm (7 inches) across, its total absorb-ing surface is approximately 13 square meters—50 times the surface area of the skin of the newborn infant. Since a mammalian embryo is protected and nourished through the placenta rather than with stored yolk, what happens to the four extraembryonic membranes it has inherited from early amniotes? The amnion remains unchanged, a protective water jacket in which the embryo fl oats. A fl uid-fi lled yolk sac is also retained, although it contains no yolk. It has acquired a new function: during early development it is the source of stem cells that give rise to blood, lymphoid cells, and gametes. These stem cells later migrate into the developing embryo. In organisms such as racoons and mice, a heavily vascularized yolk sac implants in the uterus, along with the typical placenta. The two remaining extraembryonic membranes, allantois and the chorion, are recommitted to new functions. The allantois is no longer needed for storage of metabolic wastes. Instead it contrib-utes to the umbilical cord, which links the embryo physically and functionally with the placenta. The chorion, the outermost membrane, forms most of the placenta itself. The rest of the pla-centa is formed by the adjacent uterine endometrium. The embryo grows rapidly, and in humans all major organs of the body have begun their formation by the end of the fourth week of development. The embryo is now about 5 mm in length and weighs approximately 0.02 g. During the fi rst two weeks of development (germinal period) the embryo is quite resistant to outside infl uences. However, during the next eight weeks, when all major organs are being established and body shape is forming (embryonic period), an embryo is more sen-sitive to disturbances that might cause malformations (such as exposure to alcohol or drugs taken by the mother) than at any other time in its development. The embryo becomes a fetus at approximately two months after fertilization. This ushers in the fetal period, which is primarily a growth phase, although organ systems (especially the nervous and endocrine systems) continue to differentiate. The fetus grows from approximately 28 mm and 2.7 g at 60 days to approximately 350 mm and 3000 g at term (nine months). Figure 8.25 Early development of the human embryo and its extraembryonic membranes.
Inner cell mass Trophoblast Blastocoel 1 week 10 days 5 weeks 4 weeks 2 weeks Placenta Amniotic sac Embryo Endometrium Myometrium Trophoblast Amniotic cavity Embryo Amniotic cavity Early chorionic villi Yolk sac Chorion Implanted embryo Uterine cavity Cervix Hypoblast Epiblast Uterine endometrium www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 179 DEVELOPMENT OF SYSTEMS AND ORGANS During vertebrate gastrulation the three germ layers are formed. These differentiate, as we have seen, fi rst into primordial cell masses and then into specifi c organs and tissues. During this process, cells become increasingly committed to specifi c direc-tions of differentiation. Derivatives of the three germ layers are diagrammed in Figure 8.26 . Assignment of early embryonic layers to specifi c “germ layers” (not to be confused with “germ cells,” which are the eggs and sperm) is for the convenience of embryologists and is of no concern to the embryo. Whereas the three germ layers normally differentiate to form the tissue and organs described here, it is not the germ layer itself that determines differentiation, but rather the precise position of an embryonic cell with relation to other cells. Derivatives of Ectoderm: Nervous System and Nerve Growth The brain, spinal cord, and nearly all outer epithelial structures of the body develop from primitive ectoderm. They are among the earliest organs to appear. Just above the notochord, the ectoderm thickens to form a neural plate. The edges of this plate rise up, fold, and join together at the top to create an elongated, hollow neural tube. The neural tube gives rise to most of the nervous system: anteriorly it enlarges and differ-entiates into the brain and cranial nerves; posteriorly it forms the spinal cord and spinal motor nerves. Much of the rest of the peripheral nervous system is derived from neural crest cells, which pinch off from the neural tube before it closes ( Figure 8.27 ). Among the multitude of different cell types and structures that originate with the neural crest are portions of the cranial nerves, pigment cells, cartilage and bone of most of the skull (including jaws), ganglia of the autonomic nervous Figure 8.26 Derivatives of the primary germ layers in mammals.
ENDODERM MESODERM GASTRULATION ZYGOTE ECTODERM Primordial germ cells of endoderm from extraembryonic membranes Cleavage • Neural tube Brain, spinal cord, motor nerves • Neural crest Sensory ganglia and nerves, adrenal medulla, sympathetic ganglia, skull, gill arches, dentine of teeth • Outer epithelium of body and derivatives Hair, nails, epithelial glands, lining of mouth, enamel of teeth, lens of eye, inner ear, nasal and olfactory epithelium • Epithelium of respiratory tract • Epithelium of the gut • Pharynx Pharyngeal pouches, thyroid, parathyroid • Liver, pancreas • Inner lining of urinary bladder • Smooth muscle and connective tissue of digestive tract • Notochord • Lining of thoracic and abdominal cavities • Circulatory system Blood, bone marrow, endothelium of blood vessels, lymphatics • Somites Skeletal muscle, bone and cartilage of skeleton (except skull), dermis, connective tissues • Organs of urogenital system Ureter, kidney, gonads, reproductive ducts 180 PART TWO Continuity and Evolution of Animal Life system, medulla of the adrenal gland, and contributions to sev-eral other endocrine glands. Neural crest tissue is unique to vertebrates and was probably of prime importance in evolution of the vertebrate head and jaws. How are the billions of nerve axons in the body formed? What directs their growth? Biologists were intrigued with these questions, which seemed to have no easy solutions. Because a single nerve axon may be more than a meter in length (for example, motor nerves running from the spinal cord to the toes), it seemed impossible that a single cell could reach out so far. The answer had to await the development of one of the most powerful tools available to biologists, the cell culture technique. In 1907 embryologist Ross G. Harrison discovered that he could culture living neuroblasts (embryonic nerve cells) for weeks outside the body by placing them in a drop of frog lymph hung from the underside of a cover slip. Watching nerves grow for periods of days, he saw that each axon was an outgrowth of a single cell. As the axon extended outward, materials for growth fl owed down the axon center to the growing tip (growth cone) where they were incorporated into new protoplasm ( Figure 8.28 ). The second question—what directs nerve growth—has taken longer to unravel. An idea held well into the 1940s was that nerve growth is a random, diffuse process. A major hypothesis proposed that the nervous system developed as an equipotential network, or blank slate, that later would be shaped by usage into a functional system. The nervous sys-tem just seemed too incredibly complex for us to imagine that nerve fi bers could fi nd their way selectively to so many prede-termined destinations, yet it appears that this is exactly what they do! Research with invertebrate nervous systems indicated that each of the billions of nerve cell axons acquires a distinct identity that somehow directs it along a specifi c pathway to its destination. Many years ago Harrison observed that a growing nerve axon terminated in a growth cone, from which extend numerous tiny threadlike pseudopodial processes (fi lopodia) ( Figure 8.28 ). Research has shown that the growth cone is steered by an array of guidance molecules secreted along the pathway and by the axon’s target. This chemical guidance sys-tem, which must, of course, be genetically directed, is just one example of the amazing fl exibility that characterizes the entire process of differentiation. Figure 8.27 Development of neural tube and neural crest cells from neural plate ectoderm.
Neural plate Neural plate Neural fold Neural crest Neural crest Neural tube Epidermis Figure 8.28 Growth cone at the growing tip of a nerve axon. Materials for growth fl ow down the axon to the growth cone from which numerous threadlike fi lopodia extend. These serve as a pioneering guidance system for the developing axon. Direction of growth is shown by arrows.
www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 181 The tissue culture technique developed by Ross G. Harrison is now used extensively by scientists in all fi elds of active biomedical research, not just by developmental biologists. The great impact of the technique has been felt only in recent years. Harrison was twice considered for the Nobel Prize (1917 and 1933), but he failed ever to receive the award because, ironically, the tissue culture method was then considered “of rather limited value.” Derivatives of Endoderm: Digestive Tube and Survival of Gill Arches In frog embryos the primitive gut makes its appearance during gastrulation with the formation of the archenteron. From this simple endodermal cavity develop the lining of the digestive tract, lining of the pharynx and lungs, most of the liver and pancreas, the thyroid and parathyroid glands, and the thymus (see Figure 8.26 ). In other vertebrates the alimentary canal develops from the primitive gut and is folded off from the yolk sac by growth and folding of the body wall ( Figure 8.29 ). The ends of the tube open to the exterior and are lined with ectoderm, whereas the rest of the tube is lined with endoderm. Lungs, liver, and pancreas arise from the foregut. Among the most intriguing derivatives of the digestive tract are the pharyngeal pouches, which make their appear-ance in the early embryonic stages of all vertebrates (see Figure 8.21 ). During development the endodermally-lined pharyngeal pouches interact with overlying ectoderm to form gill arches. In fishes, gill arches develop into gills and supportive structures and serve as respiratory organs. When early vertebrates moved onto land, gills were unsuitable for aerial respiration and respiratory function was performed by independently evolved lungs. Why then do gill arches persist in embryos of terrestrial verte-brates? Although gill arches serve no respiratory function in either embryos or adults of terrestrial vertebrates, they are necessary primordia for a variety of other structures. For example, the fi rst arch and its endoderm-lined pouch (the space between adjacent arches) form the upper and lower jaws and inner ear of verte-brates. The second, third, and fourth gill pouches contribute to the tonsils, parathyroid glands, and thymus. We can understand then why gill arches and other fi shlike structures appear in early mam-malian embryos. Their original function has been abandoned, but the structures are retained for new uses. The great conservatism of early embryonic development has conveniently provided us with a telescoped view of the origins of new adaptations. Derivatives of Mesoderm: Support, Movement, and Beating Heart The mesoderm forms most skeletal and muscular tissues, the circulatory system, and urinary and reproductive organs (see Figure 8.26 ). As vertebrates have increased in size and complex-ity, mesodermally derived supportive, movement, and transport structures have become an even greater proportion of the body. Most muscles arise from the mesoderm along each side of the neural tube ( Figure 8.30 ). This mesoderm divides into a lin-ear series of blocklike somites (38 in humans), which by split-ting, fusion, and migration become the axial skeleton, dermis of the dorsal skin, and muscles of the back, body wall, and limbs. Mesoderm gives rise to the fi rst functional organ, the embry-onic heart. Guided by the underlying endoderm, two clusters of precardiac mesodermal cells move amebalike into position on either side of the developing gut. These clusters differentiate into a pair of double-walled tubes, which later fuse to form a single, thin tube (see Figure 8.14 , p. 168). As the cells group together, the fi rst twitchings are evi-dent. In a chick embryo, a favorite animal for experimental embryological studies, the primitive heart begins to beat on the second day of the 21-day incubation period; it begins beating before any true blood vessels have formed and before there is any blood to pump. As the ventricle primordium develops, the spontaneous cellular twitchings become coordinated into a feeble but rhythmical beat. New heart chambers, each with a beat faster than its predecessor, then develop. Figure 8.29 Derivatives of the alimentary canal of a human embryo.
Brain Thyroid gland Notochord Thymus Trachea Stomach Pancreas Intestine Anus Liver Mouth Esophagus Somites Gill pouches Front limb bud Hind limb bud Umbilical cord Post-anal tail Figure 8.30 Human embryo showing somites, which differentiate into skeletal muscles and axial skeleton.
182 PART TWO Continuity and Evolution of Animal Life Early development of the heart and circulation is crucial to continued embryonic development, because without circu-lation an embryo could not obtain materials for growth. Food is absorbed from the yolk and carried to the embryonic body, oxygen is delivered to all tissues, and carbon dioxide and other wastes are carried away. An embryo is totally dependent on these extraembryonic support systems, and the circulation is the vital link between them. S U M M A R Y Developmental biology encompasses the emergence of order and complexity during the development of a new individual from a fertilized egg, and the control of this process. The early prefor-mation concept of development gave way in the eighteenth cen-tury to the theory of epigenesis, which holds that development is the progressive appearance of new structures that arise as the products of antecedent development. Fertilization of an egg by a sperm restores the diploid number of chromosomes and acti-vates the egg for development. Both sperm and egg have evolved devices to promote effi cient fertilization. The sperm is a highly condensed haploid nucleus provided with a locomotory fl agel-lum. Many eggs release chemical sperm attractants, most have surface receptors that recognize and bind only with sperm of their own species, and all have developed devices to prevent polyspermy. During cleavage an embryo divides rapidly and usually syn-chronously, producing a multicellular blastula. Cleavage is greatly infl uenced by quantity and distribution of yolk in the egg. Eggs with little yolk, such as those of many marine invertebrates, divide com-pletely (holoblastic) and usually have indirect development with a larval stage interposed between the embryo and adult. Eggs having an abundance of yolk, such as those of birds, reptiles, and most arthropods divide only partially (meroblastic), and birds and reptiles have no larval stage. Based on several developmental characteristics, bilateral meta-zoan animals are divided into two major groups. The Protostomia have mosaic cleavage and the mouth forms at or near the embryonic blastopore. The Deuterostomia have regulative cleavage and the mouth forms secondarily and not from the blastopore. At gastrulation, cells on an embryo’s surface move inward to form germ layers (endoderm, ectoderm, mesoderm) and the embry-onic body plan. Like cleavage, gastrulation is much infl uenced by the quantity of yolk. Despite the different developmental fates of embryonic cells, every cell contains a complete genome and thus the same nuclear information. Early development through cleavage is governed by cytoplasmic determinants derived from the maternal genome and placed in the egg cortex. As gastrulation approaches, control gradu-ally shifts from maternal to embryonic as an embryo’s own nuclear genes begin transcribing mRNA. Harmonious differentiation of tissues proceeds in three gen-eral stages: pattern formation, determination of position in the body, and induction of limbs and organs appropriate for each position. Each stage is guided by morphogens. Pattern formation refers to determination of the anteroposterior, dorsoventral, and left-to-right body axes. In amphibians the anteroposterior axis is established by morphogens such as chordin from the Spemann organizer in the gray crescent of the zygote. In Drosophila that axis is determined by the morphogen bicoid, which is transcribed from maternal mRNA deposited at the anterior of the egg. In these and other segmented animals, such morphogens activate genes that divide the body into head, thorax and abdomen, and then into correctly oriented segments. The structures appropriate to each segment are then induced by homeotic genes, which are characterized by a particular sequence of DNA bases called the homeobox. Mutations in homeotic genes result in the develop-ment of inappropriate structures on a segment: legs on the head, for example. The anteroposterior axis of an embryo is determined by homeo-tic and other homeobox-containing genes contained in one or more clusters on particular chromosomes. These genes, called Hox genes, occur not only in Drosophila and amphibians, but apparently in all animals. Each Hox gene is active in a particular region of the body, depending on its position within the cluster. Dorsoventral and left-right axes are similarly determined by morphogens that are produced only in the appropriate regions of the embryo. Similarly, morphogens guide the development of limbs along three body axes. Morphogens have been found to be remarkably similar in animals as different as Drosophila and amphibians. This realization has given rise to the fi eld of evolutionary developmental biology, which is based on the idea that the evolution of the enormous variety of ani-mals is the result of changes in the position and timing of relatively few genes that control development. The postgastrula stage of vertebrate development represents a remarkable conservation of morphology when jawed vertebrates from fi sh to humans exhibit features common to all. As development proceeds, species-specifi c characteristics are formed. Amniotes are terrestrial vertebrates that develop extraembry-onic membranes during embryonic life. The four membranes are amnion, allantois, chorion, and yolk sac, each serving a specifi c life-support function for the embryo that develops within a self-contained egg (as in birds and most reptiles) or within the maternal uterus (mammals). Mammalian embryos are nourished by a placenta, a complex fetal-maternal structure that develops in the uterine wall. During pregnancy the placenta becomes an independent nutritive, endo-crine, and regulatory organ for the embryo. Germ layers formed at gastrulation differentiate into tissues and organs. The ectoderm gives rise to skin and nervous system; endoderm gives rise to alimentary canal, pharynx, lungs, and cer-tain glands; and mesoderm forms muscular, skeletal, circulatory, reproductive, and excretory organs. Finally a specialized area of heart muscle called the sino-atrial (SA) node develops and takes command of the entire heartbeat (the role of the SA node in the excitation of the heart is described on p. 694). The SA node becomes the heart’s pri-mary pacemaker. As the heart builds a strong and effi cient beat, vascular channels open within the embryo and across the yolk. Within the vessels are the fi rst primitive blood cells suspended in plasma. www . m hhe . com /hi ckmani pz14e CHAPTER 8 Principles of Development 183 R E V I E W Q U E S T I O N S What is meant by epigenesis? How did Kaspar Friedrich Wolff’s concept of epigenesis differ from the early notion of preformation? How is an egg (oocyte) prepared during oogenesis for fertilization? Why is preparation essential to development? Describe events that follow contact of a spermatozoon with an egg. What is polyspermy and how is it prevented? What is meant by the term “activation” in embryology? How does amount of yolk affect cleavage? Compare cleavage in a sea star with that in a bird. What is the difference between radial and spiral cleavage? What other developmental hallmarks are often associated with spiral or radial cleavage? What is indirect development? Using sea star embryos as an example, describe gastrulation. Explain how the mass of inert yolk affects gastrulation in frog and bird embryos. What is the difference between schizocoelous and enterocoelous origins of a coelom? Describe two different experimental approaches that serve as evidence for nuclear equivalence in animal embryos. 1.
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11. What is meant by “induction” in embryology? Describe the famous organizer experiment of Spemann and Mangold and explain its signifi cance. What are homeotic genes and what is the “homeobox” contained in such genes? What is the function of the homeobox? What are Hox genes? What is the signifi cance of their apparently universal occurrence in animals? What is the embryological evidence that vertebrates form a monophyletic group? What are the four extraembryonic membranes of amniotic eggs of birds and reptiles and what is the function of each membrane? What is the fate of the four extraembryonic membranes in embryos of placental mammals? Explain what the “growth cone” that Ross Harrison observed at the ends of growing nerve fi bers does to infl uence direction of nerve growth. Name two organ system derivatives of each of the three germ layers. What developmental characters are used to divide animals between protostome and deuterostome groups (clades)? 12.
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19. S E L E C T E D R E F E R E N C E S Carroll, S. B., J. K. Grenier, and S. D. Weatherbee. 2005. From DNA to diversity: molecular genetics and the evolution of animal design, ed. 2. Malden, Massachusetts, Blackwell Publishing. Animal body plans develop through a hierarchy of gene interactions. As these interactions are understood, biologists seek commonalities in the “genetic toolkit” across a wide variety of taxa. Cibelli, J. B., R. P. Lanza, and M. D. West. 2002. The fi rst human cloned embryo. Sci. Am. 286: 44–51 (Jan.). Describes the fi rst cloning of human embryos—but only to the 6-cell stage. Many scientists remain skeptical. Degnan, B. M., S. P. Leys, and C. Larroux. 2005. Sponge development and antiquity of animal pattern formation. Integr. Comp. Biol. 45: 335–341. After blastula formation in a demosponge embryo, cell migration produces a two-layered gastrula that develops a third layer before becoming a free-swimming larva. If this pattern is typical, it suggests that both blastula and gastrula stages were present in ancestral metazoans. Gilbert, S. F. 2003. Developmental biology, ed. 7. Sunderland, Massachusetts, Sinauer Associates. Combines descriptive and mechanistic aspects; good selection of examples from many animal groups. Gilbert, S. F., and A. M. Raunio (eds.). 1997. Embryology: constructing the organism. Sunderland, Massachusetts, Sinauer Associates. The embryology of numerous animal groups. Goodman, C. S., and M. J. Bastiani. 1984. How embryonic nerve cells recognize one another. Sci. Am. 251: 58–66 (Dec.). Research with insect larvae shows that developing neurons follow pathways having specifi c molecular labels. Leys, S. P., and D. Eerkes-Medrano. 2005. Gastrulation in calcareous sponges: in search of Haeckel’s gastraea. Integr. Comp. Biol. 45: 342–351. Ingression of cells during embryogenesis produces two germ layers (interpreted as gastrulation) in Sycon, a calcareous sponge. The ancestral pattern of gastrulation may be via ingression rather than by invagination, which occurs much later during larval metamorphosis. Nüsslein-Volhard, C. 1996. Gradients that organize embryo development. Sci. Am. 275: 54–61 (Aug.). An account of the author’s Nobel Prize– winning research. Rosenberg, K. R., and W. R. Trevathan. 2001. The evolution of human birth. Sci. Am. 285: 72–77 (Nov.). Examines reasons why humans are the only primates to seek assistance during childbirth. Wolpert, L. 1991. The triumph of the embryo. Oxford, Oxford University Press. Written for the nonspecialist, this engaging book is rich in detail and insight for all biologists interested in the development of life. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links and more. 3 Diversity of Animal Life P A R T T H R E E A view of coral reef biodiversity. 9 Architectural Pattern of an Animal 10 Taxonomy and Phylogeny of Animals 11 Protozoan Groups 12 Mesozoa and Parazoa 13 Radiate Animals 14 Acoelomate Bilateral Animals 15 Pseudocoelomate Animals 16 Molluscs 17 Annelids and Allied Taxa 18 Smaller Ecdysozoans 19 Trilobites, Chelicerates, and Myriapods 20 Crustaceans 21 Hexapods 22 Chaetognaths, Echinoderms, and Hemichordates 23 Chordates 24 Fishes 25 Early Tetrapods and Modern Amphibians 26 Amniote Origins and Nonavian Reptiles 27 Birds 28 Mammals C H A P T E R 9 Architectural Pattern of an Animal Cnidarian polyps have radial symmetry and cell-tissue grade of organization ( Dendronephthya sp.). New Designs for Living Zoologists today recognize 34 phyla of multicellular animals, each phylum characterized by a distinctive body plan and bio-logical properties that set it apart from all other phyla. All are survivors of perhaps 100 phyla that appeared 600 million years ago during the Cambrian explosion, the most important evolu-tionary event in the geological history of life. Within the space of a few million years, virtually all major body plans that we see today, together with many other novel plans that we know only from the fossil record, were established. Entering a world sparse in species and mostly free of competition, these new life-forms diversified, producing new themes in animal architecture. Later bursts of speciation that followed major extinction events produced mainly variations on established themes. Established themes, in the form of distinctive body plans, are passed down a lineage from an ancestral population to its descen-dants; molluscs carry a hard shell, bird forelimbs make wings. These ancestral traits limit the morphological scope of descendants no matter what their lifestyle. Although penguin bodies are modi-fi ed for an aquatic life, the wings and feathers of their bird ances-tors might never adapt as well as fi sh fi ns and scales. Despite structural and functional evolution, new forms are constrained by the architecture of their ancestors. 186 PART THREE Diversity of Animal Life T he English satirist Samuel Butler proclaimed that the human body was merely “a pair of pincers set over a bellows and a stewpan and the whole thing fi xed upon stilts.” Most people less cynical than Butler would agree that the body is a triumph of intricate, living architecture. Less obvi-ous, perhaps, is that the architecture of humans and most other animals conforms to the same well-defi ned plan. The basic uni-formity of biological organization derives from the common ancestry of animals and from their basic cellular construction. Despite vast differences of structural complexity of organisms ranging from unicellular forms to humans, all share an intrinsic material design and fundamental functional plan. In this intro-duction to the diversity chapters (Chapters 11 through 28), we consider the limited number of body plans that underlie the apparent diversity of animal form and examine some of the common architectural themes that animals share. HIERARCHICAL ORGANIZATION OF ANIMAL COMPLEXITY Among the different unicellular and metazoan groups, we rec-ognize fi ve major grades of organization ( Table 9.1 ). Each grade is more complex than the one preceding, and builds on it in a hierarchical manner. The unicellular groups are the simplest eukaryotic organ-isms and represent the protoplasmic grade of organization. They are nonetheless complete organisms that perform all of the basic functions of life as seen in more complex animals. Within the confi nes of their cell, they show remarkable organization and division of labor, possessing distinct supportive structures, locomotor devices, fi brils, and simple sensory structures. The diversity observed among unicellular organisms is achieved T A B L E 9 . 1 Grades of Organization in Organismal Complexity 1. Protoplasmic grade of organization. Protoplasmic organization characterizes unicellular organisms. All life functions are confi ned within the boundaries of a single cell, the fundamental unit of life. Within a cell, protoplasm is differentiated into organelles capable of performing specialized functions.
2. Cellular grade of organization. Cellular organization is an aggregation of cells that are functionally differentiated. A division of labor is evident, so that some cells are concerned with, for example, reproduction, and others with nutrition. Some fl agellates, such as Volvox, that have distinct somatic and reproductive cells are placed at the cellular level of organization. Many authorities also place sponges at this level.
3. Cell-tissue grade of organization. A step beyond the preceding is an aggregation of similar cells into defi nite patterns or layers and organized to perform a common function, to form a tissue. Sponges are considered by some authorities to belong to this grade, although jellyfi shes and their relatives (Cnidaria) more clearly demonstrate the tissue plan. Both groups are still largely of the cellular grade of organization because most cells are scattered and not organized into tissues. An excellent example of a tissue in cnidarians is the nerve net, in which nerve cells and their processes form a defi nite tissue structure, with the function of coordination.
4. Tissue-organ grade of organization. An aggregation of tissues into organs is a further step in complexity. Organs are usually composed of more than one kind of tissue and have a more specialized function than tissues. This is the organizational level of fl atworms (Platyhelminthes), in which well-defi ned organs such as eyespots, proboscis, and reproductive organs occur. In fl atworms, the reproductive organs transcend the tissue-organ grade and are organized into a reproductive system.
5. Organ-system grade of organization. When organs work together to perform some function, we have the highest level of organization—an organ system. Systems are associated with basic body functions such as circulation, respiration, and digestion. The simplest animals having this type of organization are nemertean worms, which have a complete digestive system distinct from the circulatory system. Most animal phyla demonstrate this type of organization.
www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 187 Figure 9.1 Animal symmetry. Illustrated are animals showing spherical, radial, and bilateral symmetry.
Bilateral symmetry Radial symmetry Spherical symmetry by varying the architectural patterns of subcellular structures, organelles, and the cell as a whole (see Chapter 11). The metazoa, or multicellular animals, evolved greater structural complexity by combining cells into larger units. A metazoan cell is a specialized part of the whole organism and, unlike a unicellular organism, it is not capable of indepen-dent existence. Cells of a multicellular organism are specialized for performing the various tasks accomplished by subcellular elements in unicellular forms. The simplest metazoans show the cellular grade of organization in which cells demonstrate division of labor but are not strongly associated to perform a specifi c collective function ( Table 9.1 ). In the more complex cell-tissue grade of organization, cells are grouped together and perform their common functions as a highly coordinated unit called a tissue. Animals at or beyond the cell-tissue grade of organization are termed eumetazoans. In animals of the tissue-organ grade of organization, tissues are assembled into still larger functional units called organs. Usually one type of tissue carries the burden of an organ’s chief function, as muscle tissue does in the heart; other tissues—epithelial, connective, and nervous—perform supportive roles. The chief functional cells of an organ are called parenchyma (pa-ren´ka-ma; Gr. para, beside, enchyma, infusion). The supportive tissues are its stroma (Gr. bedding). For instance, in the vertebrate pancreas the secreting cells are the parenchyma; capsule and connective tissue framework represent stroma. Most metazoa have an additional level of complexity in which different organs operate together as organ systems. Eleven different kinds of organ systems are observed in meta-zoans: skeletal, muscular, integumentary, digestive, respiratory, circulatory, excretory, nervous, endocrine, immune, and repro-ductive. The great evolutionary diversity of these organ systems is covered in Chapters 14 through 28. ANIMAL BODY PLANS As described in the prologue to this chapter, the ancestral body plan constrains the form of its descendant lineage. Animal body plans differ in the grade of organization, in body symmetry, in the number of embryonic germ layers, and in the number of body cavities. Body symmetry can be generally determined from the external appearance of an animal, but other features of a body plan typically require a more detailed examination. Animal Symmetry Symmetry refers to balanced proportions, or correspondence in size and shape of parts on opposite sides of a median plane. Spherical symmetry means that any plane passing through the center divides a body into equivalent, or mirrored, halves ( Figure 9.1 , left ). This type of symmetry is found chiefl y among some unicellular forms and is rare in animals. Spherical forms are best suited for fl oating and rolling. Radial symmetry ( Figure 9.1 , middle ) applies to forms that can be divided into similar halves by more than two planes pass-ing through the longitudinal axis. These are tubular, vase, or bowl shapes found in some sponges and in hydras, jellyfi sh, sea urchins, and related groups, in which one end of the longitudi-nal axis is usually the mouth (the oral surface). In sessile forms, such as hydra and sea anemones, the basal attachment disc is the aboral surface. A variant form is biradial symmetry in which, because of some part that is single or paired rather than radial, only two planes passing through the longitudinal axis produce mirrored halves. Comb jellies (phylum Ctenophora, p. 282), which are globular but have a pair of tentacles, are an example. Radial and biradial animals are usually sessile, freely fl oating, or weakly swimming. Radial animals, with no anterior or posterior end, can interact with their environment in all directions—an advantage to sessile or free-fl oating forms with feeding struc-tures arranged to snare prey approaching from any direction. The two phyla that are primarily radial as adults, Cnidaria and Ctenophora, have been called the Radiata, although phy-logenetic results suggest that this group is not monophyletic (p. 285). Echinoderms (sea stars and their kin) are primarily bilateral animals (their larvae are bilateral) that have become secondarily radial as adults. 188 PART THREE Diversity of Animal Life Bilateral symmetry applies to animals that can be divided along a sagittal plane into two mirrored portions— right and left halves ( Figure 9.1 , right ). The appearance of bilateral symmetry in animal evolution was a major innovation, because bilateral animals are much better fi tted for directional (forward) movement than are radially symmetrical animals. Bilateral animals form a monophyletic group of phyla called the Bilateria. Bilateral symmetry is strongly associated with cephaliza-tion, differentiation of a head. The concentration of nervous tis-sue and sense organs in a head bestows obvious advantages to an animal moving through its environment head fi rst. This is the most effi cient positioning of organs for sensing the environment and responding to it. Usually the mouth of an animal is located on the head as well, since so much of an animal’s activity is concerned with procuring food. Cephalization is always accom-panied by differentiation along an anteroposterior axis, although the evolution of this axis preceded cephalization. Some convenient terms used for locating regions of bilater-ally symmetrical animals ( Figure 9.2 ) are anterior, used to des-ignate the head end; posterior, the opposite or tail end; dor-sal, the back side; and ventral, the front or belly side. Medial refers to the midline of the body; lateral, to the sides. Distal parts are farther from the middle of the body; proximal parts are nearer. A frontal plane (sometimes called coronal plane) divides a bilateral body into dorsal and ventral halves by run-ning through the anteroposterior axis and the right-left axis at right angles to the sagittal plane, the plane dividing an animal into right and left halves. A transverse plane (also called a cross section) would cut through a dorsoventral and a right-left axis at right angles to both the sagittal and frontal planes and would separate anterior and posterior portions ( Figure 9.2 ). In vertebrates pectoral refers to the chest region or area associ-ated with the anterior pair of appendages, and pelvic refers to the hip region or area associated with the posterior pair of appendages. Body Cavities and Germ Layers A body cavity is an internal space. The most obvious example is a gut cavity, but the vast majority of animals have a second, less obvious, cavity outside the gut. When this second cavity is fl uid-fi lled, it may cushion and protect the gut from forces exerted on the body. In some animals, such as an earthworm, it also forms part of a hydrostatic skeleton used in locomotion. Animals differ in the presence and number of body cavi-ties. Sponges, at the cellular grade of organization, have no body cavities, not even a gut cavity. If sponges share the same developmental sequence as other metazoans, why do they lack a gut cavity? Where in the developmental sequence does a gut form? Sponges, like all metazoans, develop from a zygote to a blastula stage. A typical spherical blastula is composed of a layer of cells surrounding a fl uid-fi lled cav-ity (see Figure 8.9). This cavity, a blastocoel, has no exter-nal opening, so it could not serve as a gut. In sponges, after the formation of a blastula, the cells reorganize to form an adult animal, although some researchers argue that a gastrula stage does occur before reorganization (see Figure 9.5 , upper pathway and Chapter 12). In animals other than sponges, development proceeds from a blastula to a gastrula stage, as one side of the blastula pushes inward, making a depression (see Figure 9.3 ). The depression becomes a gut cavity, also called a gastrocoel or archenteron. The external opening to the depression is the blastopore; it typically becomes the adult mouth or anus. The gut lining is endoderm and the outer layer of cells, surrounding the blas-tocoel, is ectoderm ( Figure 9.3 ). The embryo now has two cavities, a gut and a blastocoel. Animals such as sea anemones and jellyfi sh develop from these two germ layers and are called diploblastic ( Figure 9.5 , upper pathway ). They typically have radial symmetry as adult animals. The fl uid-fi lled blastocoel per-sists in diploblasts, but in others it is fi lled with a third germ layer, mesoderm. Animals that possess ectoderm, mesoderm, and endoderm are termed triploblastic and the majority are bilaterally symmetrical. Methods of Mesoderm Formation Cells forming mesoderm are derived from endoderm, but there are two ways a middle tissue layer of mesoderm can form. In protostomes, mesoderm forms as endodermal cells from near the blastopore migrate into the blastocoel ( Figure 9.3A ). Fol-lowing this event, three different body plans—acoelomate, pseudocoelomate, and coelomate—are possible ( Figure 9.3A ). In the acoelomate plan, mesodermal cells completely fi ll the blastocoel, leaving a gut as the only body cavity ( Figure 9.3A ). The region between the ectodermal epidermis and endodermal digestive tract is fi lled with a spongy mass of space-fi lling cells, the parenchyma ( Figure 9.4 ). Parenchyma is derived from embryonic connective tissue and is important in assimilation and transport of food and in disposal of metabolic wastes. In the pseudocoelomate plan, mesodermal cells line the outer edge of the blastocoel, leaving two body cavities: a per-sistent blastocoel and a gut cavity ( Figures 9.3A and 9.4 ). The Figure 9.2 The planes of symmetry as illustrated by a bilaterally symmetrical animal.
Sagittal plane Dorsal Posterior Frontal plane Ventral Anterior Transverse plane www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 189 blastocoel is now called a pseudocoelom; the name means false coelom in reference to mesoderm only partially surround-ing the cavity, instead of completely surrounding it, as in a true coelom. In the schizocoelous plan, mesodermal cells fi ll the blas-tocoel, forming a solid band of tissue around the gut. Then, through programmed cell death, space opens inside the meso-dermal band ( Figure 9.3A ). This new space is a coelom. The embryo has two body cavities, a gut and a coelom. In deuterostomes, mesoderm forms by an enterocoelous plan, where cells from the central portion of the gut lining begin to grow outward as pouches, expanding into the blastocoel ( Fig-ure 9.3B ). The expanding pouch walls form a mesodermal ring. As the pouches move outward, they enclose a space. The space becomes a coelomic cavity or coelom. Eventually the pouches pinch off from the gut lining, completely enclosing a coelom bounded by mesoderm on all sides. The coelom completely fi lls the blastocoel. The embryo has two body cavities, a gut and a coelom. A coelom made by enterocoely is functionally equiva-lent to a coelom made by schizocoely, and are represented as such in the eucoelomate body plan ( Figure 9.4 ). Both kinds of coelomic cavities are bounded by mesoderm and lined with a peritoneum, a thin cellular membrane derived from mesoderm ( Figure 9.4 ). Mesodermal mesenteries suspend organs in the coelom ( Figure 9.4 ). A pseudocoelom lacks a peritoneum. Developmental Origins of Body Plans in Triploblasts Triploblastic animals follow one of several major developmen-tal pathways to form a blastula from a zygote ( Figure 9.5 ). The most common pathways are by spiral or radial cleavage (see Figure 8.10, p. 167). Radial cleavage is typically accompanied by three other traits: The blastopore becomes an anus and a new opening makes the mouth, the coelom forms via enterocoely, and Figure 9.3 Mesoderm resides in different parts of the gastrula during formation of acoelomate, pseudocoelomate, and schizocoelous body plans (A). Mesoderm and a coelom form together in the enterocoelus plan (B).
Mesoderm lines one side of blastocoel Mesoderm fills blastocoel Band of mesoderm surrounds gut and then splits open ACOELOMATE PLAN PSEUDOCOELOMATE PLAN SCHIZOCOELOUS PLAN ENTEROCOELOUS PLAN Gut cavity Pseudocoelom Coelom Coelom External view of a gastrula Early mesoderm cells Mesodermal pouches surround gut A B Ectoderm Blastocoel Gut cavity Blastopore Endoderm Early mesoderm cells Gastrula Gastrula Longitudinal section Longitudinal section Cross section 190 PART THREE Diversity of Animal Life cleavage is regulative (see Figure 8.10, p. 165). Animals with these features are called deuterostomes ( Figure 9.5 , lower pathway ); this group includes sea urchins and chordates. Spiral cleavage produces an embryo whose developmen-tal patterns contrast with those described for deuterostomes: the blastopore becomes the mouth, cleavage is mosaic (see Figure 8.10, p. 165), and mesoderm forms from a particular cell in the embryo, the 4d cell (see p. 169). The body may become acoelomate, pseudocoelomate, or coelomate, depend-ing on the taxa ( Figure 9.5 , central pathway ). If a coelom is present, it is made via schizocoely. Animals with these features are called lophotrochozoan protostomes; this group includes molluscs, segmented worms, and other taxa ( Figure 9.5 ). Lophotrochozoans are distinguished from ecdysozoan pro-tostomes (not shown in Figure 9.5 ) where a range of cleavage patterns have been described. These include spiral cleavage, a superfi cial cleavage pattern in which nuclei proliferate within common cytoplasm prior to separation by multiple cytoplas-mic divisions (see Figure 8.15, p. 170) and a pattern initially resembling radial cleavage. Ecdysozoans may be coelomate or pseudocoelomate. Insects, crabs, and nematodes are among the ecdysozoans. A Complete Gut Design and Segmentation A few diploblasts and triploblasts have a blind or incomplete gut where food must enter and exit the same opening, but the major-ity of forms possess a complete gut ( Figure 9.5 ). A complete gut makes possible a one-way fl ow of food from mouth to anus. A body constructed in this way is essentially a gut tube within another body tube. A tube-within-a-tube design has proved to be very versatile; members of the most common animal phyla, both invertebrate and vertebrate, have this plan. Segmentation, also called metamerism, is another common feature of metazoans. Segmentation is a serial repetition of similar body segments along the longitudinal axis of the body. Each seg-ment is called a metamere, or somite. In forms such as earth-worms and other annelids ( Figure 9.6 ), in which metamerism is most clearly represented, the segmental arrangement includes both external and internal structures of several systems. There is repetition of muscles, blood vessels, nerves, and setae of locomo-tion. Some other organs, such as those of sex, may be repeated in only a few segments. Evolutionary changes have obscured much of the segmentation in many animals, including humans. The appearance of segmentation in body plans was a highly signifi cant evolutionary event. Segmentation permits greater body mobility and complexity of structure and function. Its potential is amply displayed in phylum Arthropoda, the largest assemblage of animals on earth. Segmentation is found in phylum Chordata in addition to Annelida and Arthropoda ( Figure 9.6 ), although superfi cial segmentation of ectoderm and body wall may appear among diverse groups of animals. The importance and potential of segmentation is discussed in Chapters 17 and 18. COMPONENTS OF METAZOAN BODIES Metazoan bodies consist of cellular components, derived from the three embryonic germ layers—ectoderm, mesoderm, and endoderm—as well as extracellular components. Extracellular Components Metazoan animals contain two important noncellular compo-nents: body fl uids and extracellular structural elements. In all eumetazoans, body fl uids are subdivided into two fl uid “com-partments”: those that occupy intracellular space, within the body’s cells, and those that occupy extracellular space, outside the cells. In animals with closed vascular systems Figure 9.4 Acoelomate, pseudocoelomate, and eucoelomate body plans are shown as cross sections of representative animals. Note the relative positions of parenchyma, peritoneum, and body organs.
Eucoelomate Mesodermal organ Mesodermal peritoneum Mesentery Pseudocoelomate Acoelomate Ectoderm Ectoderm Mesodermal organ Pseudocoel (from blastocoel) Gut (endoderm) Gut (endoderm) Mesoderm (muscle) Gut (endoderm) Mesodermal Ectoderm organ Parenchyma (mesoderm) Coelom www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 191 (such as segmented worms and vertebrates), the extracellular fl uids are subdivided further into blood plasma (the fl uid portion of blood) and interstitial fluid. Interstitial fl uid, also called tissue fl uid, occupies the space surrounding cells. Many invertebrates have open blood systems, however, with no true separation of blood plasma from interstitial fl uid. We explore these relationships further in Chapter 31. The term “intercellular,” meaning “between cells,” should not be confused with the term “intracellular,” meaning “within cells.” Extracellular structural elements are the supportive material of the organism, including loose connective tissue (especially well developed in vertebrates but present in all metazoa), car-tilage (molluscs and chordates), bone (vertebrates), and cuticle (arthropods, nematodes, annelids, and others). These elements provide mechanical stability and protection (see Chapter 29). In some instances, they act also as a depot of materials for exchange between the cells and the interstitial fl uid, and serve as a medium for extracellular reactions. We describe diversity of extracellular structural elements characteristic of different groups of animals in Chapters 15 through 28. Figure 9.5 Different developmental sequences result in diploblastic and triploblastic animals. Of the two major pathways within the triploblasts, one produces acoelomate and pseudocoelomate animals, as well as lophotrochozoan protostomes that form a coelom by schizocoely. Ecdysozoan protostomes are not shown here. The second major triploblastic pathway produces deuterostomes that form a coelom by enterocoely. In chordate deuterostomes, coelom formation is by enterocoely for invertebrate taxa, but by schizocoely for vertebrates.
Blastopore becomes mouth pore Blastopore anus becomes anu Mesoderm fills blastocoel No mesoderm created Mesoderm lines outer edge of blastocoel Mesodermal band fills blastocoel Mesoderm and coelom form together by enterocoely (modified in vertebrates) Coelom opens in mesoderm via schizocoely Blind gut Complete gut Blind gut Complete gut Complete gut Complete gut Complete gut No gut No gut No No distinct No cleavage pattern No germ layers Spiral cleavage Radial cleavage Blastula Blastula Gastrula rula Gastrula D I P L O B L A S T I C T R I P L O B L A S T I C Acoelomate Pseudocoelomate Coelomate Lophotrochozoan Protostome Coelomate Deuterostome 192 PART THREE Diversity of Animal Life Cellular Components: Tissues A tissue is a group of similar cells (together with associated cell products) specialized for performance of a common function. The study of tissues is called histology (Gr. histos, tissue, logos, discourse) or microanatomy. All cells in metazoan animals form tissues. Sometimes cells of a tissue may be of several kinds, and some tissues have much extracellular material. During embryonic development, the germ layers become differentiated into four kinds of tissues. These are epithelial, con-nective, muscular, and nervous tissues ( Figure 9.7 ). This is a sur-prisingly short list of only four basic tissue types that are able to meet the diverse requirements of animal life. Epithelial Tissue An epithelium (pl., epithelia) is a sheet of cells that covers an external or internal surface. Outside the body, epithelium forms a protective covering. Inside, epithelium lines all organs of the body cavity, as well as ducts and passageways through which various materials and secretions move. Thus, ions and molecules must pass through epithelial cells as they move to and from all other cells of the body. Consequently a large variety of transport molecules are located on epithelial cell membranes (see Chapter 3). Epithelial cells are also modifi ed into glands that produce lubricating mucus or specialized products such as hormones or enzymes. Epithelia are classifi ed by cell form and number of cell lay-ers. Simple epithelia (a single layer of cells; Figure 9.8 ) are found in all metazoan animals, while stratifi ed epithelia (many cell lay-ers; Figure 9.9 ) are mostly restricted to vertebrates. All types of epithelia are supported by an underlying basement membrane, which is a condensed region of ground substance of connective tissue, but is secreted by both epithelial and connective tissue cells. Blood vessels never penetrate into epithelial tissues, which depend on diffusion of oxygen and nutrients from underlying tissues. Connective Tissue Connective tissues are a diverse group of tissues that serve various binding and supportive functions. They are so wide-spread in the body that removal of other tissues would still leave the complete form of the body clearly apparent. Con-nective tissue is composed of relatively few cells, a great many extracellular fi bers, and a ground substance, in which the fi bers are suspended (together called matrix ). We recog-nize several different types of connective tissue. Two kinds of connective tissue proper occur in vertebrates. Loose connective tissue is composed of fibers and both fixed and wandering cells suspended in a viscous fl uid ground substance ( Figure 9.10 ). Dense connective tissue, such as tendons and ligaments, is composed largely of densely packed fi bers and little ground substance ( Figure 9.10 ). Much of the fi bers of connective tissue is composed of collagen (Gr. kolla, glue, genos, descent), a protein of great tensile strength. Collagen is the most abundant protein in the animal kingdom, found in animal bodies wherever both fl exibility and resistance to stretching are required. Connective tissue of invertebrates, as in vertebrates, consists of cells, fi bers, and ground substance, and show a wide diversity of structure ranging from highly cellular to acellular histologies. Other types of specialized connective tissue include blood, lymph (collectively considered vascular tissue), adipose (fat) tissue, cartilage, and bone. Vascular tissue is composed of dis-tinctive cells in a fl uid ground substance, the plasma. Vascular tissue lacks fi bers under normal conditions. Blood composition is discussed in Chapter 31. Cartilage is a semirigid form of connective tissue with closely packed fi bers embedded in a gel-like ground substance ( Figure 9.10 ). Bone is a calcifi ed connective tissue containing calcium salts organized around collagen fi bers ( Figure 9.10 ). Structure of cartilage and bone is discussed in the section on skeletons in Chapter 29. Muscular Tissue Muscle is the most abundant tissue in the body of most animals. It originates (with few exceptions) from mesoderm, and its unit is the cell or muscle fi ber, specialized for contraction. When viewed with a light microscope, striated muscle appears trans-versely striped (striated), with alternating dark and light bands ( Figure 9.11 ). In vertebrates we recognize two types of striated Figure 9.6 Segmented phyla. These three phyla have all made use of an important principle in nature: segmentation (also called metamerism), or repetition of structural units. Segmentation brings more varied specialization because segments, especially in arthropods, have become modifi ed for different functions.
Chordata Arthropoda Annelida www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 193 muscle: skeletal and cardiac muscle. In invertebrates, a third striated muscle type called obliquely striated muscle has been described. Smooth (or visceral) muscle, which lacks the char-acteristic alternating bands of the striated type is found in both invertebrates and vertebrates, although major ultrastructural differences have been described between them ( Figure 9.11 ). Unspecialized cytoplasm of muscles is called sarcoplasm, and contractile elements within the fi ber are myofi brils. Muscular movement is covered in Chapter 29. Nervous Tissue Nervous tissue is specialized for reception of stimuli and conduc-tion of impulses from one region to another. Two basic types of cells in nervous tissue are neurons (Gr. nerve), the basic functional unit of nervous systems, and neuroglia (nu-rog´le-a; Gr. nerve, glia, glue), a variety of nonnervous cells that insulate neuron membranes and serve various supportive functions. Figure 9.12 shows functional anatomy of a typical nerve cell. The functional roles of nervous tissue are treated in Chapter 33. COMPLEXITY AND BODY SIZE The most complex grades of metazoan organization permit and to some extent even promote evolution of large body size ( Figure 9.13 ). Large size confers several important physical and ecological consequences for an organism. As animals become larger, the body surface increases much more slowly than body volume because surface area increases as the square of body length (length 2 ), whereas volume (and therefore mass) increases as the cube of body length (length 3 ). In other words, a large animal has less surface area relative to its volume than does a small animal of the same shape. The surface area of a large animal may be inadequate for respiration and nutrition by cells located deep within its body. There are two possible solutions to this problem. One solution is to fold or invaginate the body sur-face to increase the surface area or, as exploited by fl atworms, fl atten the body into a ribbon or disc so that no internal space is far from the surface. This solution allows a body to become large without internal complexity. However, most large animals adopted a second solution; they developed internal transport Figure 9.7 Types of tissues in a vertebrate, showing examples of where different tissues are located in a frog.
Bone tissue Nervous tissue in brain Blood tissue in vascular system Cardiac muscle tissue in heart Columnar epithelium in lining of intestine Smooth muscle tissue in esophagus Skeletal muscle tissue in voluntary muscles Reproductive tissue (testes) Areolar connective tissue in dermis Stratified epithelium in epidermis CONNECTIVE TISSUES NERVOUS TISSUES EPITHELIAL TISSUES MUSCLE TISSUES 194 PART THREE Diversity of Animal Life The tendency for maximum body size to increase within lines of descent is known as “Cope’s law of phyletic increase,” named after nineteenth-century American paleontologist and naturalist Edward Drinker Cope. Cope noted that lineages begin with small organisms that give rise to larger and ultimately to giant forms. Large forms frequently become extinct, providing opportunities for new lin-eages, which in turn evolve larger forms. Cope’s rule holds well for many nonfl ying vertebrates and invertebrate groups, even though Cope’s Lamarckian explanation for the trend—that organisms evolved from an inner urge to attain a higher state of being (and larger size)—was preposterous. Many animal taxa contain lineages that show evolutionary miniaturization, in contrast to Cope’s rule (for example, the insects). systems to shuttle nutrients, gases, and waste products between cells and the external environment. Larger size buffers an animal against environmental fl uc-tuations; it provides greater protection against predation and enhances offensive tactics; and it permits a more effi cient use of metabolic energy. A large mammal uses more oxygen than a small mammal, but the cost of maintaining its body temper-ature is less per gram of weight for a large mammal than for a small one. Large animals also can move at less energy cost than can small animals. For example, a large mammal uses more oxygen in running than does a small mammal, but the energy cost of moving 1 g of its body over a given distance is much less for a large mammal than for a small one ( Fig-ure 9.14 ). For all of these reasons, ecological opportunities of larger animals are very different from those of small ones. In subsequent chapters we describe the extensive adaptive radiations observed in taxa of large animals. Figure 9.8 Types of simple epithelium. A, Simple squamous epithelium, composed of fl attened cells that form a continuous lining of blood capillaries, lungs, and other surfaces where it permits the diffusion of gases and transport of other molecules into and out of cavities. B, Simple cuboidal epithelium is composed of short, boxlike cells. Cuboidal epithelium usually lines small ducts and tubules, such as those of the kidney and salivary glands, and may have active secretory or absorptive functions. C, Simple columnar epithelium resembles cuboidal epithelium, but the cells are taller and usually have elongate nuclei. This type of epithelium is found on highly absorptive surfaces such as the intestinal tract of most animals. The cells often bear minute, fi ngerlike projections called microvilli that greatly increase the absorptive surface. In some organs, such as the female reproductive tract, cells may be ciliated.
ple squamous epithelium Simple squamous epithelial cell Basement membrane Nucleus Free surface A mple cuboidal epithelium Simple cuboidal epithelial cell Basement membrane Lumen (free space) B Epithelial cells Basement membrane Nuclei Microvilli on cell surface mple columnar epithelium C www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 195 Figure 9.9 Types of stratifi ed epithelium.
Free surface Basement membrane Connective tissue Nucleus Transitional epithelial cell Free surface Stratified squamous epithelial cell Nuclei Basement membrane Stratified squamous epithelium Stratifi ed squamous epithelium consists of two to many layers of cells adapted to withstand mild mechanical abrasion and distortion. The basal layer of cells undergoes continuous mitotic divisions, producing cells that are pushed toward the surface where they are sloughed off and replaced by new cells from beneath. This type of epithelium lines the oral cavity, esophagus, and anal canal of many vertebrates, and the vagina of mammals.
Transitional epithelium — unstretched Transitional epithelium — stretched Transitional epithelium is a type of stratifi ed epithelium specialized to accommodate great stretching. This type of epithelium is found in the urinary tract and bladder of vertebrates. In the relaxed state it appears to be four or fi ve cell layers thick, but when stretched it appears to have only two or three layers of extremely fl attened cells. S U M M A R Y From the relatively simple organisms that mark the beginnings of life on earth, animal evolution has produced more intricately orga-nized forms. Organelles are integrated into cells, cells into tissues, tissues into organs, and organs into systems. Whereas a unicellular organism performs all life functions within the confi nes of a single cell, a multicellular animal is an organization of subordinate units united at successive levels. Every organism has an inherited body plan described in terms of body symmetry, number of embryonic germ layers, grade of organization, and number of body cavities. The majority of ani-mals exhibit bilateral symmetry, but spherical and radial symmetry occur in some groups. Most animals are triploblastic and develop from three embryonic germ layers, but cnidarians and a few other forms are diploblastic. Sponges lack germ layers and possess a cel-lular grade of organization. Most animals have the tissue grade of organization. All animals other than sponges have a gut cavity. Most animals have a second cavity that surrounds the gut. The second cavity may be a pseudocoelom or a coelom. There are two taxon-specifi c patterns of coelom formation, schizocoely and enterocoely. Triploblastic animals are divided among deuterostomes and pro-tostomes according to their particular developmental sequence. Pro-tostomes are further divided into lophotrochozoan and ecdysozoan forms on the basis of more detailed features of development. A metazoan body consists of cells, most of which are func-tionally specialized; body fl uids, divided into intracellular and extra-cellular fl uid compartments; and extracellular structural elements, which are fi bers or formless materials that serve various structural functions in the extracellular space. The cells of metazoa develop into various tissues; basic types are epithelial, connective, muscu-lar, and nervous. Tissues are organized into larger functional units called organs, and organs are associated to form systems. One correlate of increased anatomical complexity is an increase in body size, which offers certain advantages such as more effec-tive predation, reduced energy cost of locomotion, and improved homeostasis. 196 PART THREE Diversity of Animal Life Figure 9.10 Types of connective tissue. A, Loose connective tissue, also called areolar connective tissue, is the “packing material” of the body that anchors blood vessels, nerves, and body organs. It contains fi broblasts that synthesize the fi bers and ground substance of connective tissue and wandering macrophages that phagocytize pathogens or damaged cells. The different fi ber types include collagen fi bers (thick and red in micrograph) and thin elastic fi bers (black and branching in micrograph) formed of the protein elastin. B, Dense connective tissue forms tendons, ligaments, and fasciae (fa´sha), the latter arranged as sheets or bands of tissue surrounding skeletal muscle. In a tendon (shown here) the collagenous fi bers are extremely long and tightly packed together. C, Cartilage is a vertebrate connective tissue composed of a fi rm matrix containing cells (chondrocytes) located in small pockets called lacunae, and collagen and /or elastic fi bers (depending on type of cartilage). In hyaline cartilage shown here, both collagen fi bers and matrix are stained uniformly purple and cannot be distinguished one from the other. Because cartilage lacks a blood supply, all nutrients and waste materials must diffuse through the ground substance from surrounding tissues. D, Bone, strongest of vertebrate connective tissues, contains mineralized collagen fi bers. Small pockets (lacunae) within the matrix contain bone cells, called osteocytes. The osteocytes communicate with each other by means of a tiny network of channels called canaliculi. Blood vessels, extensive in bone, are located in larger channels, including central canals. Bone undergoes continuous remodeling during an animal’s life, and can repair itself following even extensive damage.
Collagen fiber Elastic fiber Nucleus Loose connective tissue Bone Dense connective tissue Cartilage Nucleus Fibers Lacuna Matrix Chondrocyte Central canal Osteocytes in lacunae Mineralized matrix A B D C Figure 9.11 Types of muscle tissue.
Nuclei of smooth muscle cells Skeletal muscle fiber Nucleus Striations Nucleus Striations Intercalated discs Smooth muscle is nonstriated muscle found in both invertebrates and vertebrates. Smooth muscle cells are long, and tapering, each containing a single nucleus. Smooth muscle is the most common type of muscle in invertebrates in which it serves as body wall musculature and surrounds ducts and sphincters. In vertebrates, smooth muscle surrounds blood vessels and internal organs such as intestine and uterus. It is called involuntary muscle in vertebrates because its contraction is usually not consciously controlled.
Cardiac muscle is another type of striated muscle found only in the vertebrate heart. The cells are much shorter than those of skeletal muscle and have only one nucleus per cell (uninucleate). Cardiac muscle tissue is a branching network of fi bers with individual cells interconnected by junctional complexes called intercalated discs. Cardiac muscle is considered involuntary muscle because it does not require nerve activity to stimulate contraction. Instead, heart rate is controlled by specialized pacemaker cells located in the heart itself. However, autonomic nerves from the brain may alter pacemaker activity.
Skeletal muscle is a type of striated muscle found in both invertebrates and vertebrates. It is composed of extremely long, cylindrical fi bers, which are multinucleate cells that may reach from one end of the muscle to the other. Viewed through the light microscope, the cells appear to have a series of stripes, called striations, running across them. Skeletal muscle is called voluntary muscle (in vertebrates) because it contracts when stimulated by nerves under conscious central nervous system control.
www . m hhe . com /hi ckmani pz14e CHAPTER 9 Architectural Pattern of an Animal 197 Figure 9.12 Functional anatomy of a neuron. From the nucleated cell body, or soma, extend one or more dendrites (Gr. dendron, tree), which receive electrical signals from receptors or other nerve cells, and a single axon that carries signals away from the cell body to other nerve cells or to an effector organ. The axon is often called a nerve fi ber. Nerves are separated from other nerves or from effector organs by specialized junctions called synapses.
Dendrites: receive stimuli from other neurons Cell body Nucleus Nucleolus Axon hillock Schwann cell: forms insulating sheath around many vertebrate peripheral nerves Nodes of Ranvier: these interruptions in Schwann cell insulation allow action potentials to leap from node to node Synaptic terminals: release neurotransmitter chemicals into synapse when action potential arrives, passing on the electrical signal to the next cell Direction of conduction Axon: transmits electrical signals from cell body to synaptic terminals Figure 9.13 Graph showing the evolution of size (length) increase in organisms at different periods of life on earth. Note that both scales are logarithmic.
Whale Dinosaur Amphibian Fish Eurypterid Trilobite First eukaryote Cyanobacteria Bacteria Origin of earth 4 3 2 1 800 600 400 300 200 100 Present 1 m Length 10 m 100 m 1 mm 10 mm 100 mm 1 m 10 m 100 m Million years ago Billion years ago Figure 9.14 Net cost of running for mammals of various sizes. Each point represents the cost (measured in rate of oxygen consumption) of moving 1 g of body over 1 km. Cost decreases with increasing body size.
White mouse Kangaroo rat Ground squirrel Body weight Dog Human Horse 10 1 0.1 0.01 10 g 100 g 1 kg 10 kg 100 kg 1000 kg ml O2 198 PART THREE Diversity of Animal Life R E V I E W Q U E S T I O N S Name the fi ve grades of organization in organismal complexity and explain how each successive grade is more complex than the one preceding it. Can you suggest why, during the evolutionary history of animals, there has been a tendency for maximum body size to increase? Do you think it inevitable that complexity should increase along with body size? Why or why not? What is the meaning of the terms parenchyma and stroma as they relate to body organs? Body fl uids of eumetazoan animals are separated into fl uid “compartments.” Name these compartments and explain how compartmentalization may differ in animals with open and closed circulatory systems. What are the four major types of tissues in metazoans? How would you distinguish between simple and stratifi ed epithelium? What characteristic of stratifi ed epithelium might explain why it, rather than simple epithelium, is found lining the oral cavity, esophagus, and vagina? What three elements are present in all connective tissue? Give some examples of different types of connective tissue. 1.
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7. What are three muscle tissue types found among animals? Explain how each is specialized for particular functions. Describe the principal structural and functional features of a neuron. Match the animal group with its body plan: Unicellular a. Nematode Cell aggregate b. Vertebrate Blind sac, acoelomate c. Protozoan Tube-within-a-tube, d. Flatworm pseudocoelomate e. Sponge Tube-within-a-tube, f. Arthropod eucoelomate g. Nemertean Distinguish among spherical, radial, biradial, and bilateral symmetry. Use the following terms to identify regions on your body and on the body of a frog: anterior, posterior, dorsal, ventral, lateral, distal, proximal. How would frontal, sagittal, and transverse planes divide your body? What is meant by segmentation? Name three phyla showing segmentation. 8.
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14. S E L E C T E D R E F E R E N C E S Arthur, W. 1997. The origin of animal body plans. Cambridge, U.K., Cambridge University Press. Explores genetic, developmental, and population-level processes involved in the evolution of the 35 or so body plans that arose in the geological past. Baguna, J., and M. Ruitort. 2004. The dawn of bilaterian animals: the case of acoelomorph fl atworms. Bioessays 26: 1046–1057. Goes through the hypotheses for the transition between radial and bilateral body plans and the evidence for them. Cole, A. G., and B. K. Hall. 2004. The nature and signifi cance of invertebrate cartilages revisited: distribution and histology of cartilage and cartilage-like tissues within the Metazoa. Zoology 107: 261–273. Beautifully illustrated and well-written discussion of these tissues. Kessel, R. G. 1998. Basic medical histology: the biology of cells, tissues and organs. New York, Oxford University Press. A current textbook of animal histology. Martindale, M. Q., J. R. Finnerty, and J. Q. Henry. 2002. The Radiata and the evolutionary origins of the bilaterian body plan. Molecular Phylogenetics and Evolution 24: 358–365. Examines germ layers and symmetry in cnidarians and ctenophores to reconstruct the origin of the bilaterally symmetrical animals. McGowan, C. 1999. A practical guide to vertebrate mechanics. New York, Cambridge University Press. Using many examples from his earlier book, Diatoms to dinosaurs, the author describes principles of biomechanics that underlie functional anatomy. Includes practical experiments and laboratory exercises. Royuela, M., B. Fraile, M. I. Arenas, and R. Paniagua. 2000. Characterization of several invertebrate muscle cell types: a comparison with vertebrate muscles. Microsc. Res. Tech. 41: 107–115. Examines the ultrastructure of smooth, striated and oblique striated muscle in order to establish a clearer classifi cation criteria. Welsch, U., and V. Storch. 1976. Comparative animal cytology and histology. London, Sidgwick & Jackson. Comparative histology with good treatment of invertebrates. Willmer, P. 1990. Invertebrate relationships: patterns in animal evolution. Cambridge, U.K., Cambridge University Press. Chapter 2 is an excellent discussion of animal symmetry, developmental patterns, origin of body cavities, and segmentation. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links and more! C H A P T E R and homeothermy, that separate them from even more dissimilar forms such as insects and fl atworms. All human cultures classify familiar animals according to patterns in animal diversity. These classifi cations have many purposes. Some societies classify animals according to their usefulness or destruc-tiveness to human endeavors; others may group animals according to their roles in mythology. Biologists organize animal diversity in a nested hierarchy of groups within groups according to evolution-ary relationships as revealed by ordered patterns in their sharing of homologous features. This ordering is called a “natural system” because it refl ects relationships that exist among animals in nature, outside the context of human activity. A systematic zoologists has three major goals: to discover all species of animals, to reconstruct their evolutionary relationships, and to communicate those relation-ships by constructing an informative taxonomic system. Order in Diversity Evolution has produced a great diversity of species in the animal kingdom. Zoologists have named more than 1.5 million species of animals, and thousands more are described each year. Some zoologists estimate that species named so far constitute less than 20% of all living animals and less than 1% of all those that have existed. Despite its magnitude, the diversity of animals is not without limits. Many conceivable forms do not exist in nature, as our myths of minotaurs and winged horses show. Animal diversity is not ran-dom but has defi nite order. Characteristic features of humans and cattle never occur together in a single organism as they do in the mythical minotaurs; nor do characteristic wings of birds and bodies of horses occur together naturally as they do in the mythical horse, Pegasus. Humans, cattle, birds, and horses are distinct groups of ani-mals, yet they do share some important features, including vertebrae 10 Taxonomy and Phylogeny of Animals Molluscan shells from the collection of Jean Baptiste de Lamarck (1744 to 1829). 200 PART THREE Diversity of Animal Life D arwin’s theory of common descent (Chapters 1 and 6) is the underlying principle that guides our search for order in the diversity of animal life. Our science of taxonomy (“arrangement law”) produces a formal system for naming and grouping species to communicate this order. Animals that have very recent common ancestry share many features in common and are grouped most closely in our taxonomic classifi cation. Taxon-omy is part of the broader science of systematics, or comparative biology, in which studies of variation among animal populations are used to understand their evolutionary relationships. The study of taxonomy predates evolutionary biology, however, and many taxonomic practices are remnants of a pre-evolutionary world view. Adjusting our taxonomic system to accommodate evolu-tion has produced many problems and controversies. Taxonomy has reached an unusually active and controversial point in its development in which several alternative taxonomic systems are competing for use. To understand this controversy, it is necessary fi rst to review the history of animal taxonomy. LINNAEUS AND TAXONOMY The Greek philosopher and biologist Aristotle was the fi rst to classify organisms according to their structural similarities. The fl owering of systematics in the eighteenth century culminated in the work of Carolus Linnaeus ( Figure 10.1 ), who designed our current scheme of classifi cation. Linnaeus was a Swedish botanist at the University of Uppsala. He had a great talent for collecting and classifying objects, espe-cially fl owers. Linnaeus produced an extensive system of classifi -cation for both plants and animals. This scheme, published in his great work, Systema Naturae, used morphology (the comparative study of organismal form) for arranging specimens in collections. He divided the animal kingdom into species and gave each one a distinctive name. He grouped species into genera, genera into orders, and orders into “classes” (we use quotation marks or a capital letter to distinguish “class” as a formal taxonomic rank from its broader meaning as a group of organisms that share a common essential property). Because his knowledge of animals was limited, his lower categories, such as genera, often were very broad and included animals that are only distantly related. Much of his classifi cation is now drastically altered, but the basic principle of his scheme is still followed. Linnaeus’s scheme of arranging organisms into an ascending series of groups of ever-increasing inclusiveness is a hierarchi-cal system of classifi cation. Major taxa (sing., taxon ), into which organisms are grouped were given one of several standard taxo-nomic ranks to indicate the general degree of inclusiveness of the group. The hierarchy of taxonomic ranks has been expanded considerably since Linnaeus’s time ( Table 10.1 ). It now includes seven mandatory ranks for the animal kingdom, in descending Figure 10.1 Carolus Linnaeus (1707 to 1778). This portrait was made of Linnaeus at age 68, three years before his death.
T A B L E 1 0 . 1 Examples of Taxonomic Categories to Which Representative Animals Belong Linnaean Rank Human Gorilla Southern Leopard Frog Katydid Kingdom Animalia Animalia Animalia Animalia Phylum Chordata Chordata Chordata Arthropoda Subphylum Vertebrata Vertebrata Vertebrata Uniramia Class Mammalia Mammalia Amphibia Insecta Subclass Eutheria Eutheria — Pterygota Order Primates Primates Anura Orthoptera Suborder Anthropoidea Anthropoidea — Ensifera Family Hominidae Hominidae Ranidae Tettigoniidae Subfamily — — Raninae Phaneropterinae Genus Homo Gorilla Rana Scudderia Species Homo sapiens Gorilla gorilla Rana sphenocephala Scudderia furcata Subspecies — — — Scudderia furcata furcata The hierarchical taxonomy of four species (human, gorilla, Southern leopard frog, and katydid). Higher taxa generally are more inclusive than lower-level taxa, although taxa at two different levels may be equivalent in content. Closely related species are united at a lower point in the hierarchy than are distantly related species. For example, humans and gorillas are united at the level of the family (Hominidae) and above; they are united with the Southern leopard frog at the subphylum level (Vertebrata) and with the katydid at the kingdom level (Animalia). Mandatory Linnaean ranks are shown in bold type.
www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 201 series: kingdom, phylum, “class,” order, family, genus, and spe-cies. All organisms must be placed into at least seven taxa, one at each of the mandatory ranks. Taxonomists have the option of subdividing these seven ranks further to recognize more than seven taxa (superfamily, subfamily, superorder, suborder, etc.) for any particular group of organisms. In all, more than 30 taxonomic ranks are recognized. For very large and complex groups, such as fi shes and insects, these additional ranks are needed to express different degrees of evolutionary divergence. Unfortunately, they also make the system more complex. Introduction of evolutionary theory into animal taxonomy has changed the taxonomist’s role from one of classifi cation to systematization. Classifi cation denotes the construction of classes, groupings of organisms that possess a common feature, called an essence, used to defi ne the class. Organisms that pos-sess the essential feature are members of the class by defi nition, and those that lack it are excluded. Because evolving species are subject always to change, the static nature of classes makes them a poor basis for a taxonomy of living systems. The activity of a taxonomist whose groupings of species represent units of com-mon evolutionary descent is systematization, not classifi cation. Species placed into a taxonomic group include the most recent common ancestor of the group and its descendants and thus form a branch of the phylogenetic tree of life. The species of a group thus formed represent a system of common descent, not a class defi ned by possession of an essential characteristic. Because organismal characteristics are inherited from ances-tral to descendant species, character variation is used to diag-nose systems of common descent, but there is no requirement that an essential character be maintained throughout the system for its recognition as a taxon. The role of morphological or other features in systematization is therefore fundamentally different from the role of such characters in classifi cation. In classifi cation, a taxonomist asks whether a species being classifi ed contains the defi ning feature(s) of a particular taxonomic class; in systemati-zation, a taxonomist asks whether the characteristics of a species confi rm or reject the hypothesis that it descends from the most recent common ancestor of a particular taxon. For example, tet-rapod vertebrates descend from a common ancestor that had four limbs, a condition retained in most but not all of its descen-dants. Although they lack limbs, caecilians (p. 548) and snakes (p. 575) are tetrapods because they are parts of this system of common descent; other morphological and molecular characters group them respectively with living amphibians and lizards. Although the hierarchical structure of Linnaean classifi cation is retained in current taxonomy, the taxa are groupings of species related by evolutionary descent with modifi cation, as diagnosed by sharing of homologous characters. As one moves up the taxo-nomic hierarchy from a species toward more inclusive groups, each taxon represents the descendants of an earlier ancestor, a larger branch of the tree of life. Linnaeus’s system for naming species is known as binomial nomenclature. Each species has a latinized name composed of two words (hence binomial) printed in italics (or underlined if handwritten or typed). The fi rst word names the genus, which is capitalized; the second word is the species epithet, which is peculiar to the species within the genus and is written in lower-case (see Table 10.1 ). The great communicative value of Latin species names is that they are used consistently by scientists in all countries and languages; they are much more precise than “common names,” which vary culturally and geographically. The genus name is always a noun, and the species epithet is usually an adjective that must agree in gender with the genus. For instance, the scientifi c name of the common robin is Turdus migratorius (L. turdus, thrush; migratorius, of migratory habit). The species epithet never stands alone; the complete binomial must be used to name a species. Names of genera must refer only to single groups of organisms; the same name cannot be given to two different genera of animals. The same species epithet may be used in different genera, however, to denote different species. For example, the scientifi c name of the white-breasted nuthatch is Sitta carolinensis. The species epithet “carolinensis” is used in other genera for the species Poecile carolinensis (Caro-lina chickadee) and Anolis carolinensis (green anole, a lizard) to mean “of Carolina.” All ranks above the species are designated using uninomial nouns, written with a capital initial letter. Sometimes a species is divided into subspecies using a trinomial nomenclature (see katydid example, Table 10.1 , and salaman-der example, Figure 10.2 ); such species are called polytypic. The generic, specifi c, and subspecifi c names are printed in italics (underlined if handwritten or typed). A polytypic species con-tains one subspecies whose subspecifi c name is a repetition of the species epithet and one or more additional subspecies whose names differ. Thus, to distinguish geographic variants of Ensatina eschscholtzii, one subspecies is named Ensatina eschscholtzii eschscholtzii, and different subspecies names are used for each of six other subspecies ( Figure 10.2 ). Both the genus name and spe-cies epithet may be abbreviated as shown in Figure 10.2 . Formal recognition of subspecies has lost popularity among taxonomists because subspecies are often based on minor differences in appear-ance that do not necessarily diagnose evolutionarily distinct units. When further study reveals that named subspecies are distinct evolutionary lineages, the subspecies are then often recognized as full species; indeed, many authors argue that the subspecies of Ensatina eschscholtzii are in fact separate species. Subspecies designations, therefore, should be viewed as tentative statements indicating that the species status of the populations needs further investigation. SPECIES While discussing Darwin’s book, On the Origin of Species, in 1859, Thomas Henry Huxley asked, “In the fi rst place, what is a species? The question is a simple one, but the right answer to it is hard to fi nd, even if we appeal to those who should know most about it.” We have used the term “species” so far as if it had a simple and unambiguous meaning. Actually, Huxley’s commentary is as valid today as it was over 140 years ago. Our concepts of species have become more sophisticated, but the 202 PART THREE Diversity of Animal Life diversity of different concepts and disagreements surrounding their use are as evident now as in Darwin’s time. Despite widespread disagreement about the nature of spe-cies, biologists have repeatedly used certain criteria for identi-fying species. First, common descent is central to nearly all modern concepts of species. Members of a species must trace their ancestry to a common ancestral population although not necessarily to a single pair of parents. Species are thus historical entities. A second criterion is that species must be the smallest distinct groupings of organisms sharing patterns of ancestry and descent; otherwise, it would be diffi cult to separate species from higher taxa whose members also share common descent. Morphological characters traditionally have been important in identifying such groupings, but chromosomal and molecular characters now are used extensively for this purpose. A third important criterion is that of reproductive community. Mem-bers of a species must form a reproductive community that excludes members of other species. For sexually reproducing populations, interbreeding is critical for maintaining a repro-ductive community. For organisms whose reproduction is strictly asexual, reproduc-tive community entails occupation of a particular ecological habitat in a particu-lar place so that a reproducing population responds as a unit to evolutionary forces such as natural selection and genetic drift. Any species has a distribution through space, its geographic range, and a dis-tribution through time, its evolutionary duration. Species differ greatly from each other in both dimensions. Species having very large geographic ranges or worldwide distributions are called cosmopolitan, whereas those with very restricted geo-graphic distributions are called endemic. If a species were restricted to a single point in space and time, we would have little diffi culty recognizing it, and nearly every species concept would lead us to the same decision. We have little difficulty distin-guishing from each other the different spe-cies of animals that we can fi nd living in our local park or woods. However, when we compare a local population to similar but not identical populations located hun-dreds of miles away, it may be hard to determine whether these populations rep-resent parts of a single species or different species ( Figure 10.2 ). Throughout the evolutionary dura-tion of a species, its geographic range can change many times. A geographic range may be either continuous or disjunct, the latter having breaks within it where the species is absent. Suppose that we fi nd two similar but not identical populations living 300 miles apart with no related populations between them. Are we observing a single species with a disjunct distribution or two different but closely related species? Suppose that these popula-tions have been separated historically for 50,000 years. Is this enough time for them to have evolved separate reproductive communities, or can we still view them as parts of the same reproductive community? Clear answers to such questions are very hard to fi nd. Differences among species concepts relate to solving these problems. Typological Species Concept Before Darwin, a species was considered a distinct and immu-table entity. Species were defi ned by fi xed, essential features (usually morphological) considered as a divinely created pattern or archetype. This practice constitutes the typological (or mor-phological ) species concept. Scientists recognized species for-mally by designating a type specimen that was labeled and Figure 10.2 Geographic variation of color patterns in the salamander genus Ensatina. The species status of these populations has puzzled taxonomists for generations and continues to do so. Current taxonomy recognizes only a single species (Ensatina eschscholtzii) divided into subspecies as shown. Hybridization is evident between most adjacent populations, but studies of variation in proteins and DNA show large amounts of genetic divergence among populations. Furthermore, populations of the subspecies E. e. eschscholtzii and E. e. klauberi can overlap geographically without interbreeding.
E. e. picta E. e. oregonensis E. e. platensis E. e. xanthoptica E. e. croceater E. e. eschscholtzii E. e. klauberi Area in which two subspecies occupy the same territory but do not interbreed Area of smooth intergradation between subspecies Area in which two closely adjacent subspecies hybridize frequently www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 203 deposited in a museum to represent the ideal form or morphol-ogy for the species ( Figure 10.3 ). When scientists obtained addi-tional specimens and wanted to assign them to a species, the type specimens of described species were consulted. The new specimens were assigned to a previously described species if they possessed the essential features of its type specimen. Small differences from the type specimen were considered accidental imperfections. Large differences from existing type specimens would lead a scientist to describe a new species with its own type specimen. In this manner, the living world was categorized into species. Evolutionists discarded the typological species concept, but some of its traditions remain. Scientists still name species by describing type specimens deposited in museums, and the type specimen formally bears the name of the species. Organismal morphology is likewise still important in recognizing species; however, species are no longer viewed as classes of organisms defi ned by possession of certain morphological features. The basis of the evolutionary world view is that species are historical entities whose properties are subject always to change. Varia-tion that we observe among organisms within a species is not an imperfect manifestation of an eternal “type”; the type itself is only an abstraction taken from the very real and important varia-tion present within the species. A type is at best an average form that changes as organismal variation is sorted through time by natural selection. A type specimen serves only as a guide to the general morphological features that one may expect to fi nd in a particular species as we observe it today. The person who fi rst describes a type specimen and publishes the name of a species is called the authority. This person’s name and date of publication are often written after the species name. Thus , Didelphis marsupialis Linnaeus, 1758, tells us that Linnaeus was the fi rst person to publish the species name of the opossum. Sometimes, the generic status of a species is revised following its initial description. In this case, the name of the authority is pre-sented in parentheses. The Nile monitor lizard is denoted Varanus niloticus (Linnaeus, 1766) because the species originally was named by Linnaeus as Lacerta nilotica, and subsequently placed into a different genus. Biological Species Concept The most infl uential concept of species inspired by Darwinian evolutionary theory is the biological species concept formu-lated by Theodosius Dobzhansky and Ernst Mayr. This concept emerged during the evolutionary synthesis of the 1930s and 1940s from earlier ideas, and it has been refi ned and reworded several times since then. In 1982, Mayr stated the biological spe-cies concept as follows: “A species is a reproductive community of populations (reproductively isolated from others) that occupies a specifi c niche in nature.” Note that a species is identifi ed here according to reproductive properties of populations, not accord-ing to possession of any specifi c organismal characteristics. A species is an interbreeding population of individuals hav-ing common descent and sharing intergrading characteristics. Studies of populational variation in organismal morphology, chromosomal structure, and molecular genetic features are very useful for evaluating the geographical boundaries of interbreed-ing populations in nature. The criterion of the “niche” (see Chapter 38) recognizes that members of a reproductive commu-nity are expected also to have common ecological properties. Because a reproductive community should maintain genetic cohesiveness, we expect organismal variation to be relatively smooth and continuous within species and discontinuous between them. Although the biological species is based on reproductive properties of populations rather than organismal morphology, morphology nonetheless can help us to diagnose biological species. Sometimes species status can be evaluated directly by conducting breeding experiments. Controlled breed-ing is practical only in a minority of cases, however, and our decisions regarding species membership usually are made by studying character variation. Variation in molecular characters is very useful for identifying geographical boundaries of repro-ductive communities. Molecular studies have revealed the occur-rence of cryptic or sibling species (p. 119), which are too similar in morphology to be diagnosed as separate species by morphological characters alone. The biological species concept has received strong criti-cism because of several perceived problems. First, the concept lacks an explicit temporal dimension. It provides a means for diagnosing species status of contemporary populations but gives little guidance regarding the species status of ancestral popula-tions relative to their evolutionary descendants. Proponents of the biological species concept often disagree on the degree of reproductive isolation necessary for considering two populations separate species, thereby revealing some ambiguity in the con-cept. For example, should occurrence of limited hybridization between populations in a small geographic area cause them to be considered a single species despite evolutionary differences Figure 10.3 Specimens of birds from the Smithsonian Institution (Washington D.C.), including birds originally collected by John J. Audubon, Theodore Roosevelt, John Gould, and Charles Darwin.
204 PART THREE Diversity of Animal Life between them? Another problem is that because the biological species concept emphasizes interbreeding as the criterion of reproductive community, it denies the existence of species in groups of organisms that reproduce only asexually. It is common systematic practice, however, to describe species in all groups of organisms, regardless of whether reproduction is sexual or asexual. Evolutionary Species Concept The time dimension creates obvious problems for the biologi-cal species concept. How do we assign fossil specimens to bio-logical species that are recognized today? If we trace a lineage backward through time, how far must we go before we have crossed a species boundary? If we could follow the unbroken genealogical chain of populations backward through time to the point where two sister species converge on their common ances-tor, we would need to cross at least one species boundary some-where. It would be very hard to decide, however, where to draw a sharp line between the two species. To address this problem, the evolutionary species concept was proposed by Simpson in the 1940s to add an evolutionary time dimension to the biological species concept. This concept persists in a modifi ed form today. A current defi nition of the evo-lutionary species is a single lineage of ancestor-descendant popula-tions that maintains its identity from other such lineages and that has its own evolutionary tendencies and historical fate. Note that the criterion of common descent is retained here in the need for a lineage to have a distinct historical identity. Reproductive cohesion is the means by which a species maintains its identity from other such lineages and keeps its evolutionary fate separate from other species. The same kinds of diagnostic features discussed for the biological species concept are relevant for identifying evolutionary species, although in most cases only morphological features are available from fossils. Unlike the biological species concept, the evolutionary species concept applies both to sexually and asexu-ally reproducing forms. As long as continuity of diagnostic features is maintained by the evolving lineage, it is recognized as a spe-cies. Abrupt changes in diagnostic features mark the boundaries of different species in evolutionary time. Phylogenetic Species Concept The last concept that we present is the phylogenetic species concept. The phylogenetic species concept is defi ned as an irreducible (basal) grouping of organisms diagnosably distinct from other such groupings and within which there is a paren-tal pattern of ancestry and descent. This concept emphasizes most strongly the criterion of common descent. Both asexual and sexual groups are covered. A phylogenetic species is a single population lineage with no detectable branching. The main difference in practice between the evolutionary and phylogenetic species concepts is that the latter emphasizes recognizing as separate species the small-est groupings of organisms that have undergone independent evolutionary change. The evolutionary species concept would group into a single species geographically disjunct populations that demonstrate some phylogenetic divergence but are judged similar in their “evolutionary tendencies,” whereas the phylo-genetic species concept would treat them as separate species. In general, a greater number of species would be described using the phylogenetic species concept than any other species concept, and many taxonomists consider it impractical for this reason. For strict adherence to cladistic systematics (p. 209), the phylogenetic species concept is ideal because only this concept guarantees strictly monophyletic units at the species level. The phylogenetic species concept intentionally disregards details of evolutionary process and gives us a criterion that allows us to describe species without fi rst needing to conduct detailed studies on evolutionary processes. Advocates of the phyloge-netic species concept do not necessarily disregard the impor-tance of studying evolutionary process. They argue, however, that the fi rst step in studying evolutionary process is to have a clear picture of life’s history. To accomplish this task, the pattern of common descent must be reconstructed in the greatest detail possible by starting with the smallest taxonomic units that have a history of common descent distinct from other such units. Dynamism of Species Concepts Current disagreements concerning concepts of species should not be considered discouraging. Whenever a fi eld of scientifi c investigation enters a phase of dynamic growth, old concepts are reevaluated and either refi ned or replaced with newer, more progressive ones. The active debate occurring within systematics shows that this fi eld has acquired unprecedented activity and importance in biology. Just as Thomas Henry Huxley’s time was one of enormous advances in biology, so is the present time. Both times are marked by fundamental reconsiderations of the meaning of species. We cannot predict which concepts of spe-cies will remain useful 10 years from now. Researchers whose main interests are branching of evolutionary lineages, evolution of reproductive barriers among populations (p. 118), or ecologi-cal properties of species may favor different species concepts. The confl icts among the current concepts will lead us into the future. In many cases, different concepts agree on the locations of species boundaries, and disagreements identify particularly interesting cases of evolution in action. Understanding the con-fl icting perspectives, rather than learning a single species con-cept, is therefore of greatest importance for people now entering the study of zoology. DNA Barcoding of Species DNA barcoding is a technique for identifying organisms to spe-cies using sequence information from a standard gene present in all animals. The mitochondrial gene encoding cytochrome c oxidase subunit 1 ( COI ), which contains about 650 nucleo-tide base pairs, is a standard ”barcode” region for animals. DNA sequences of COI usually vary among individuals of the same www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 205 species but not extensively, so that variation within a species is much smaller than differences among species. DNA barcod-ing is applied to specimens in nature by taking a small DNA sample from blood or another expendable tissue. The method is useful also for specimens in natural-history museums, zoos, aquaria, and frozen-tissue collections. DNA sequences from such sources are checked against a public reference library of spe-cies identifi ers to assign unknown specimens to known species. DNA barcoding does not solve the controversies regarding use of different species concepts, but it often permits the origin of a specimen to be identifi ed to a particular local population, which is valuable information regardless of the species status that a taxonomist assigns to that population. TAXONOMIC CHARACTERS AND PHYLOGENETIC RECONSTRUCTION A major goal of systematics is to infer the evolutionary tree or phylogeny that relates all extant and extinct species. This task is accomplished by identifying organismal features, formally called characters, that vary among species. A character is any feature that the taxonomist uses to study variation within and among species. Taxonomists fi nd characters by observing patterns of similarity among organisms in morphological, chromosomal, and molecular features (see p. 206), and less frequently in behavioral and ecological ones. Phylogenetic analysis depends upon fi nd-ing among organisms shared features that are inherited from a common ancestor. Character similarity that results from common ancestry is called homology (see Chapter 6). Similarity does not always refl ect common ancestry, however. Independent evolu-tionary origin of similar features on different lineages produces patterns of similarity among organisms that do not refl ect com-mon descent; this occurrence complicates the work of taxono-mists. Character similarity that misrepresents common descent is called nonhomologous similarity or homoplasy. Endothermy of birds and mammals is an example of homoplasy; this condi-tion arose separately in ancestral lineages of birds and mammals. Variation in other characters shows that birds and mammals are not each other’s closest relatives (p. 499). For an example of molecular homoplasy, see the interpretation of character 41 (p. 211) in the boxed essay, Phylogenies from DNA Sequences. Using Character Variation to Reconstruct Phylogeny To infer the phylogeny of a group using characters that vary among its members, the fi rst step is to determine which variant form of each character was present in the common ancestor of the entire group. This character state is called ancestral for the group as a whole. We presume that all other variant forms of the character arose later within the group, and these are called evo-lutionarily derived character states. Determining the polarity of a character refers to identifying which one of its contrasting states is ancestral and which one(s) derived. For example, if we consider as a character the dentition of amniotic vertebrates (reptiles, birds, and mammals), presence versus absence of teeth in the jaws constitute alternative character states. Teeth are absent from modern birds but present in the other amniotes. To evaluate the polarity of this character, we must determine which character state, presence or absence of teeth, characterized the most recent common ancestor of amniotes and which state was derived subsequently within amniotes. The method used to examine the polarity of a variable char-acter is called outgroup comparison. We consult an additional group of organisms, called an outgroup, that is phylogenetically close but not within the group being studied. We infer that any character state found both within the group being studied and in the outgroup is ancestral for the study group. Amphibians and different groups of bony fi shes constitute appropriate outgroups to the amniotes for polarizing variation in dentition of amniotes. Teeth are usually present in amphibians and bony fi shes; there-fore, we infer that presence of teeth is ancestral for amniotes and absence of teeth is derived. The polarity of this character indi-cates that teeth were lost in the ancestral lineage of all modern birds. Polarity of characters is evaluated most effectively when several different outgroups are used. All character states found in the study group that are absent from appropriate outgroups are considered derived. Organisms or species that share derived character states form subsets within the study group called clades (Gr. klados, branch). A derived character shared by the members of a clade is formally called a synapomorphy (Gr. synapsis, joining together, morph¯ e, form) of that clade. Taxonomists use synapomorphies as evidence of homology to infer that a particular group of organ-isms forms a clade. Among extant amniotes, absence of teeth and presence of feathers are synapomorphies that identify the birds as a clade. A clade corresponds to a unit of evolutionary com-mon descent; it includes all descendants of a particular ancestral lineage. The pattern formed by the derived states of all characters within our study group takes the form of a nested hierarchy of clades within clades. The goal is to identify all of the different clades nested within the study group, which would give a com-plete account of the patterns of common descent among species in the group. Character states ancestral for a taxon are often called ple-siomorphic for that taxon, and the sharing of ancestral states among organisms is termed symplesiomorphy. Unlike synapo-morphies, however, symplesiomorphies do not provide useful information on nesting of clades within clades. In the example just given, we found that presence of teeth in jaws was ple-siomorphic for amniotes. If we grouped together mammalian and reptilian groups, which possess teeth, to the exclusion of modern birds, we would not obtain a valid clade. Birds also descend from all common ancestors of reptiles and mammals and must be included within any clade that includes all reptiles and mammals. Errors in determining polarity of characters there-fore clearly can produce errors in inference of phylogeny. It is important to note, however, that character states that are ple-siomorphic at one taxonomic level can be synapomorphies at a 206 PART THREE Diversity of Animal Life more inclusive level. For example, the presence of jaws bearing teeth is a synapomorphy of gnathostome vertebrates (p. 511), a group that includes amniotes plus amphibians, bony fi shes, and cartilaginous fi shes, although teeth have been lost in birds and some other gnathostomes. The goal of phylogenetic analy-sis therefore can be restated as one of fi nding the appropriate taxonomic level at which any given character state is a synapo-morphy. The character state is then used at that level to identify a clade. The nested hierarchy of clades is presented as a branching diagram called a cladogram ( Figure 10.4 ; see also Figure 6.16, and try to reconstruct this cladogram using only the sharing of numbered synapomorphies among the bird species). Tax-onomists often make a technical distinction between a clado-gram and a phylogenetic tree. The branches of a cladogram are only a formal device for indicating the nested hierarchy of clades within clades. The cladogram is not strictly equivalent to a phylogenetic tree, whose branches represent real lineages that occurred in the evolutionary past. To obtain a phylogenetic tree, we must add to the cladogram important additional inter-pretations concerning ancestors, the durations of evolutionary lineages, or the amounts of evolutionary change that occurred on the lineages. A cladogram is often used, however, as a fi rst approximation of the branching structure of the corresponding phylogenetic tree. Sources of Phylogenetic Information We fi nd characters used to construct cladograms in comparative morphology (including embryology), comparative cytology, and comparative biochemistry. Comparative morphology examines the varying shapes and sizes of organismal structures, includ-ing their developmental origins. Both macroscopic and micro-scopic characters are used, including details of cellular structure revealed by histology. As seen in Chapters 23 through 28, the variable structures of skull bones, limb bones, and integument (scales, hair, feathers) are particularly important for reconstruct-ing the phylogeny of vertebrates. Comparative morphology uses specimens obtained from both living organisms and fossilized remains. Comparative biochemistry uses sequences of amino acids in proteins and the sequences of nucleotides in nucleic acids (see Chapter 5) to identify variable characters for constructing a cladogram ( Figure 10.5 ). Direct sequencing of DNA is regularly Figure 10.4 A cladogram as a nested hierarchy of taxa among fi ve sampled chordate groups (Amphioxus, bass, lizard, horse, monkey). Amphioxus is the outgroup, and the study group comprises the four vertebrates. Four characters that vary among vertebrates are used to generate a simple cladogram: presence versus absence of four legs, amniotic eggs, hair, and mammary glands. For all four characters, absence is the ancestral state in vertebrates because this is the condition found in the outgroup, Amphioxus; for each character, presence is the derived state in vertebrates. Because they share presence of four legs and amniotic eggs as synapomorphies, the lizard, horse, and monkey form a clade relative to the bass. This clade is subdivided further by two synapomorphies (presence of hair and mammary glands) that unite the horse and monkey relative to the lizard. We know from comparisons involving even more distantly related animals that presence of vertebrae and jaws constitute synapomorphies of vertebrates and that Amphioxus, which lacks these features, falls outside the vertebrate clade.
Monkey Horse Lizard Bass Amphioxus Vertebrae, jaws Four legs, amniotic eggs Hair, mammary glands OUTGROUP Figure 10.5 An early phylogenetic tree of representative amniotes based on inferred base substitutions in the gene that encodes the respiratory protein, cytochrome c. Numbers on the branches are the expected numbers of mutational changes that occurred in this gene along the different evolutionary lineages. Publication of this tree by Fitch and Margoliash in 1967 was infl uential in convincing systematists that molecular sequences contain phylogenetic information. Subsequent work confi rms some hypotheses, including monophyly of mammals (blue) and birds (red) while rejecting others; kangaroo, for example, should be outside a branch containing all other mammals sampled.
Human Monkey Dog Horse Donkey Pig Rabbit Kangaroo Pigeon Duck Chicken Penguin Turtle Snake 0.8 0.2 1.0 1.1 1.0 1.1 5 1.6 4.6 4.9 3.0 1.4 0.6 1.4 1.2 16.5 3.3 6.9 1.1 3.3 1.7 2.9 1.3 0.9 0.1 2.7 www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 207 applied to phylogenetic studies; however, comparisons of protein sequences are usually indirect, involving immunological or allozy-mic (see Figure 6.31) methods, or inferences from DNA sequences of protein-coding genes. Recent studies show that comparative biochemistry can be applied to some fossils in addition to living organisms. Comparative cytology uses variation in the numbers, shapes, and sizes of chromosomes and their parts (see Chapter 3 and p. 114) to obtain variable characters for constructing clado-grams. Comparative cytology is used almost exclusively on living rather than fossilized organisms. To add an evolutionary timescale necessary for producing a phylogenetic tree, we must consult the fossil record. We can look for the earliest appearance in fossils of derived morpho-logical characters to estimate the ages of clades distinguished by those characters. The age of a fossil showing the derived char-acters of a particular clade is determined by radioactive dating (p. 109). An example of a phylogenetic tree constructed using these methods is Figure 25.3, page 547. We can use comparative biochemical data to estimate the ages of different lineages on a phylogenetic tree. Some pro-tein and DNA sequences undergo approximately linear rates of divergence through evolutionary time. The age of the most recent common ancestor of two species is therefore propor-tional to the differences measured between their proteins and DNA sequences. We calibrate evolution of proteins and DNA sequences by measuring their divergence between species whose most recent common ancestor has been dated using fossils. We then use the molecular evolutionary calibration to estimate ages of other branches on the phylogenetic tree. THEORIES OF TAXONOMY A theory of taxonomy establishes the principles that we use to recognize and to rank taxonomic groups. There are two cur-rently popular theories of taxonomy: (1) traditional evolutionary taxonomy and (2) phylogenetic systematics (cladistics). Both are based on evolutionary principles. These two theories differ, however, on how evolutionary principles are used. These differ-ences have important implications for how we use a taxonomy to study evolutionary processes. The relationship between a taxonomic group and a phylo-genetic tree or cladogram is important for both theories. This relationship can take one of three forms: monophyly, para-phyly, or polyphyly ( Figure 10.6 ). A taxon is monophyletic if it includes the most recent common ancestor of the group and all descendants of that ancestor ( Figure 10.6A ). A taxon is paraphyletic if it includes the most recent common ancestor of all members of a group and some but not all descendants of that ancestor ( Figure 10.6B ). A taxon is polyphyletic if it does not include the most recent common ancestor of all members of a group; this condition requires that the group has had at least two separate evolutionary origins, usually requiring indepen-dent evolutionary acquisition of similar features ( Figure 10.6C ). Both evolutionary and cladistic taxonomy accept monophyletic groups and reject polyphyletic groups. They differ on accep-tance of paraphyletic groups, however, and this difference has important evolutionary implications. Traditional Evolutionary Taxonomy Traditional evolutionary taxonomy incorporates two different evolutionary principles for recognizing and ranking higher taxa: (1) common descent and (2) amount of adaptive evolutionary change, as shown on a phylogenetic tree. Evolutionary taxa must have a single evolutionary origin, and must show unique adaptive features. The mammalian paleontologist George Gaylord Simpson ( Figure 10.7 ) and Ernst Mayr (see Figure 6.19) were highly infl u-ential in developing and formalizing the procedures of evolu-tionary taxonomy. According to Simpson and Mayr, a particular Figure 10.6 Relationships between phylogeny and taxonomic groups illustrated for a hypothetical phylogeny of eight species (A through H). A, Monophyly—a monophyletic group contains the most recent common ancestor of all members of the group and all of its descendants. B, Paraphyly—a paraphyletic group contains the most recent common ancestor of all members of the group and some but not all of its descendants. C, Polyphyly—a polyphyletic group does not contain the most recent common ancestor of all members of the group, thereby requiring that the group have at least two separate phylogenetic origins.
Monophyletic grouping A B C Paraphyletic grouping D E F G H C B A D E F G H C B A Polyphyletic grouping D E F G H C B A 208 PART THREE Diversity of Animal Life taxonomy places humans (genus Homo ) and their immediate fos-sil ancestors in the family Hominidae, and it places the chim-panzees (genus Pan ), gorillas (genus Gorilla ), and orangutans (genus Pongo ) in the family Pongidae. However, the pongid gen-era Pan and Gorilla share more recent common ancestry with the Hominidae than they do with the remaining pongid genus, Pongo. This arrangement makes the family Pongidae paraphyletic because it does not include humans, who also descend from the most recent common ancestor of all pongids ( Figure 10.9 ). Evo-lutionary taxonomists nonetheless recognize the pongid genera as a single, family-level grade of arboreal, herbivorous primates having limited mental capacity; in other words, they show the same family-level adaptive zone. Humans are terrestrial, omnivo-rous primates who have greatly expanded mental and cultural attributes, thereby forming a distinct adaptive zone at the taxo-nomic level of the family. Unfortunately, if we want our taxa to constitute adaptive zones, we compromise our ability to present common descent effectively. Traditional evolutionary taxonomy has been challenged from two opposite directions. One challenge states that because phylogenetic trees can be very diffi cult to obtain, it is impractical to base our taxonomic system on common descent and adap-tive evolution. We are told that our taxonomy should represent a more easily measured feature, the overall similarity of organ-isms evaluated without regard to phylogeny. This principle is called phenetic taxonomy. Phenetic taxonomy contributed some useful analytical methods but did not have a strong impact on animal taxonomy, and scientifi c interest in this approach has declined. Despite the diffi culties of reconstructing phylogeny, zoologists still consider this endeavor a central goal of their sys-tematic work, and they are unwilling to compromise this goal for methodological purposes. Figure 10.7 George Gaylord Simpson (1902 to 1984) formulated the principles of evolutionary taxonomy.
branch on an evolutionary tree is considered a higher taxon if it represents a distinct adaptive zone. Simpson describes an adaptive zone as “a characteristic reaction and mutual rela-tionship between environment and organism, a way of life and not a place where life is led.” By entering a new adap-tive zone through a fundamental change in organismal structure and behavior, an evolving population can use environmental resources in a completely new way. A taxon that constitutes a distinct adaptive zone is termed a grade. Simpson gives the example of penguins as a distinct adaptive zone within birds. The lineage imme-diately ancestral to all penguins underwent fundamental changes in the form of the body and wings to switch from aerial to aquatic locomotion ( Figure 10.8 ). Aquatic birds that can fl y both in air and underwater are some-what intermediate in habitat, morphology, and behavior between aerial and aquatic adaptive zones. Nonetheless, the obvious modifi cations of the wings and body of penguins for swim-ming represent a new grade of organization. Penguins are therefore recognized as a dis-tinct taxon within birds, the family Sphenis-cidae. The broader the adaptive zone when fully occupied by a group of organisms, the higher the rank given to the corresponding taxon. Evolutionary taxa may be either monophy-letic or paraphyletic. Recognition of paraphyletic taxa requires, however, that our taxonomies distort patterns of common descent. An evolu-tionary taxonomy of the anthropoid primates provides a good example ( Figure 10.9 ). This Figure 10.8 A, Penguin. B, Diving petrel. Penguins (avian family Spheniscidae) were recognized by George G. Simpson as a distinct adaptive zone within birds because of their adaptations for submarine fl ight. Simpson believed that the adaptive zone ancestral to penguins resembled that of diving petrels, which display adaptations for combined aerial and aquatic fl ight. Adaptive zones of penguins and diving petrels are distinct enough to be recognized taxonomically as different families within a common order (Ciconiiformes).
A B www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 209 Phylogenetic Systematics/Cladistics A second and stronger challenge to evolutionary taxonomy is one known as phylogenetic systematics or cladistics. As the fi rst name implies, this approach emphasizes the criterion of common descent and, as the second name implies, it is based on the cladogram of the group being classifi ed. This approach to taxonomy was fi rst proposed in 1950 by the German entomol-ogist, Willi Hennig ( Figure 10.10 ), and therefore is sometimes called “Hennigian systematics.” All taxa recognized by Hennig’s cladistic system must be monophyletic. We saw on Figure 10.9 how evolutionary taxonomists’ recognition of the primate fami-lies Hominidae and Pongidae distorts genealogical relationships to emphasize adaptive uniqueness of the Hominidae. Because the most recent common ancestor of the paraphyletic family Pongidae is also an ancestor of the Hominidae, recognition of the Pongidae is incompatible with cladistic taxonomy. To avoid paraphyly, cladistic taxonomists have discontinued use of the traditional family Pongidae, placing chimpanzees, gorillas, and orangutans with humans in the family Hominidae. We adopt the cladistic classifi cation in this book. Disagreement on the validity of paraphyletic groups may seem trivial at fi rst, but its important consequences become clear when we discuss evolution. For example, claims that amphibians evolved from bony fi sh, that birds evolved from reptiles, or that humans evolved from apes might be made by an evolutionary taxonomist but are meaningless to a cla-dist. We imply by these statements that a descendant group (amphibians, birds, or humans) evolved from part of an ancestral group (bony fi sh, reptiles, and apes, respectively) to which the descendant does not belong. This usage auto-matically makes the ancestral group paraphyletic, and indeed bony fi sh, reptiles, and apes as traditionally recognized are paraphyletic groups. How are such paraphyletic groups recog-nized? Do they share distinguishing features not shared by the descendant group? Figure 10.9 Phylogeny and family-level classifi cation of anthropoid primates. Evolutionary taxonomy groups the genera Gorilla, Pan, and Pongo into a paraphyletic family Pongidae because they share the same adaptive zone or grade of organization. Humans (genus Homo) are phylogenetically closer to Gorilla and Pan than any of these genera are to Pongo, but humans are placed in a separate family (Hominidae) because they represent a new grade of organization. Cladistic taxonomy discontinues recognition of paraphyletic family Pongidae, consolidating Pongo, Gorilla, Pan, and Homo in family Hominidae.
Hominidae Hylobatidae Family-level classification according to evolutionary taxonomy, based principally on unique adaptive zones Pongidae Hylobates Pongo Gorilla Pan Homo Figure 10.10 Willi Hennig (1913 to 1976), German entomologist who formulated the principles of phylogenetic systematics/cladistics.
210 PART THREE Diversity of Animal Life A simple example illustrates cladistic analysis of DNA sequence data to examine phylogenetic relationships among species. The study group in this example contains three species of chameleons, two from the island of Madagascar (Brookesia theili and B. brygooi) and one from Equatorial Guinea (Chamaeleo feae). The outgroup is a lizard of genus Uromastyx, which is a distant relative of chameleons. Do the molecular data in this example confi rm or reject the prior taxonomic hypothesis that the two Madagascan chameleons are more closely related to each other than either one is to the Equatorial Guinean species?
The molecular information in this example comes from a piece of the mitochondrial DNA sequence (57 bases) for each species. Each sequence encodes amino acids 221–239 of a protein called “NADH dehydrogenase subunit 2” in the species from which it was obtained. These DNA base sequences are aligned and numbered as: Phylogenies from DNA Sequences 10 20 30 40 50 | | | | | Uromastyx AAACCTTAAAAGACACCACAACCATATGAACAACAACACCAACAATCAGCACACTAC B. theili AAACACTACAAAATATAACAACTGCATGAACAACATCAACCACAGCAAACATTTTAC B. brygooi AAACACTACAAGACATAACAACAGCATGAACTACTTCAACAACAGCAAATATTACAC C. feae AAACCCTACGAGACGCAACAACAATATGATCCACTTCCCCCACAACAAACACAATTT Each column in the aligned sequences constitutes a character that takes one of four states: A, C, G, or T (a fi fth possible state, absence of a base, is not observed in this example). Only characters that vary among the three chameleon species potentially contain information on which pair of species is most closely related. Twenty-three of the 57 aligned bases show variation among chameleons, as shown here in bold letters: 10 20 30 40 50 | | | | | Uromastyx AAACCTTAAAAGACACCACAACCATATGAACAACAACACCAACAATCAGCACACTAC B. theili AAACACTACAAAATATAACAACTGCATGAACAACATCAACCACAGCAAACATTTTAC B. brygooi AAACACTACAAGACATAACAACAGCATGAACTACTTCAACAACAGCAAATATTACAC C. feae AAACCCTACGAGACGCAACAACAATATGATCCACTTCCCCCACAACAAACACAATTT To be useful for constructing a cladogram, a character must dem-onstrate sharing of derived characters (=synapomorphy). Which of these 23 characters demonstrate synapomorphies for chameleons? For each of the 23 variable characters, we must ask whether one of the states observed in chameleons is shared with the outgroup, Uromas-tyx. If so, this state is judged ancestral for chameleons and the alterna-tive state(s) derived. Derived characters are identifi ed for 21 of the 23 characters just identifi ed; derived states are shown in blue: 10 20 30 40 50 | | | | | Uromastyx AAACCTTAAAAGACACCACAACCATATGAACAACAACACCAACAATCAGCACACTAC B. theili AAACACTACAAAATATAACAACTGCATGAACAACATCAACCACAGCAAACATTTTAC B. brygooi AAACACTACAAGACATAACAACAGCATGAACTACTTCAACAACAGCAAATATTACAC C. feae AAACCCTACGAGACGCAACAACAATATGATCCACTTCCCCCACAACAAACACAATTT Note that polarity is ambiguous for two variable characters (at positions 23 and 54) whose alternative states in chameleons are not observed in the outgroup.
Of the characters showing derived states, 10 of them show synapo-morphies among chameleons. These characters are marked here with numbers 1, 2, or 3 below the appropriate column. 10 20 30 40 50 | | | | | Uromastyx AAACCTTAAAAGACACCACAACCATATGAACAACAACACCAACAATCAGCACACTAC B. theili AAACACTACAAAATATAACAACTGCATGAACAACATCAACCACAGCAAACATTTTAC B. brygooi AAACACTACAAGACATAACAACAGCATGAACTACTTCAACAACAGCAAATATTACAC C. feae AAACCCTACGAGACGCAACAACAATATGATCCACTTCCCCCACAACAAACACAATTT 1 1 1 1 2 1 3 1 1 1 Paraphyletic groups are usually defi ned in a negative man-ner. They are distinguished only by lacking features found in a particular descendant group, because any traits that they share from their common ancestry are symplesiomorphies present also in the excluded descendants (unless secondarily lost). For example, apes are those “higher” primates that are not humans. Likewise, fi sh are those vertebrates that lack the distinguishing characteristics of tetrapods (amphibians and amniotes). What does it mean then to say that humans evolved from apes? To an evolutionary taxonomist, apes and humans are different adaptive zones or grades of organization; to say that humans evolved from apes states that bipedal, tailless organisms of large brain capacity evolved from arboreal, tailed organisms of smaller brain capacity. To a cladist, however, the statement www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 211 that humans evolved from apes says essentially that humans evolved from an arbitrary grouping of species that lack the distinctive characteristics of humans, a trivial statement that conveys no useful information. To a cladist, any statement that a particular monophyletic group descends from a paraphyletic one is nothing more than a claim that the descendant group evolved from something that it is not. Extinct ancestral groups are always paraphyletic because they exclude a descendant that shares their most recent common ancestor. Although many such groups have been recognized by evolutionary taxonomists, none are recognized by cladists. Zoologists often construct paraphyletic groups because they are interested in a terminal, monophyletic group (such as humans), and they want to ask questions about its ancestry. The eight characters marked 1 show synapomorphies grouping the two Madagascan species (Brookesia theili and B. brygooi) to the exclusion of the Equatorial Guinean species, Chamaeleo feae. We can represent these relationships as a cladogram: We can explain evolution of all characters favoring this clado-gram by placing a single mutational change on the branch ancestral to the two Brookesia species. This is the simplest explanation for evolutionary change of these characters.
Characters marked 2 and 3 disagree with our cladogram and favor alternative relationships as shown here: To explain evolutionary changes in characters favoring clado-grams 2 or 3 using cladogram 1, we need at least two changes per character. Likewise, if we try to explain evolution of characters favoring cladogram 1 on cladograms 2 or 3, we need at least two changes for each of these characters. These two diagrams show the minimum numbers of changes required for character 5 (which favors cladogram 1) and character 41 (which favors cladogram 3) on clado-gram 1; the ancestral state of each character is shown at the root of the tree and the states observed in each species at the tips of the branches: Systematists often use a principle called parsimony to resolve confl icts among taxonomic characters, as seen here. We choose as our best working hypothesis the cladogram that requires the small-est total amount of character change. In our example, cladogram 1 is favored by parsimony. For all 10 phylogenetically informative characters, cladogram 1 requires a total of 12 changes of character state (one for each of the 8 characters favoring it and two for each of the other 2 characters). Cladograms 2 and 3 each require at least 19 character-state changes, 7 steps longer than cladogram 1. By choos-ing cladogram 1, we claim that characters favoring cladograms 2 and 3 show homoplasy in their evolution.
The molecular sequences shown in this example therefore con-fi rm predictions of the prior hypothesis, based on appearance and geography of these chameleons, that the Brookesia species shared a common ancestor with each other more recently than either one did with Chamaeleo feae.
As a further exercise, you should convince yourself that the 12 characters that vary among chameleons but which do not demon-strate unambiguous sharing of derived states are equally compatible with each of the three possible cladograms. For each character, fi nd the minimum total number of changes that must occur to explain its evolution on each cladogram. You will see, if you do this exercise correctly, that the three cladograms do not differ in minimum num-bers of changes required for each of these characters. For this reason, the characters are phylogenetically uninformative by the parsimony criterion.
Data from Townsend, T., and A. Larson. 2002. Molecular phylogenetics and mitochondrial genomic evolution in the Chamaeleonidae (Reptilia, Squamata). Molecular Phylogenetics and Evolution 23:22–36.
Brookesia theili Brookesia brygooi Chamaeleo feae 1 Brookesia brygooi Chamaeleo feae Brookesia theili 2 Brookesia theili Chamaeleo feae Brookesia brygooi 3 Brookesia brygooi Chamaeleo feae Brookesia theili A C C character 5 (1 change) C A A A C A C Brookesia brygooi Chamaeleo feae Brookesia theili A C A 1 1 character 41 (2 changes) C 212 PART THREE Diversity of Animal Life It is often convenient to lump together organisms whose fea-tures are considered approximately equally distant from the group of interest and to ignore their own unique features. It is signifi cant in this regard that humans have never been placed in a paraphyletic group, whereas most other organ-isms have been. Apes, reptiles, fi shes, and invertebrates are all terms that traditionally designate paraphyletic groups formed by combining various “side branches” found when human ancestry is traced backward through the tree of life. Such a taxonomy can give the erroneous impression that all of evolu-tion is a progressive march toward humanity or, within other groups, a progressive march toward whatever species humans designate most “advanced.” Such thinking is a relic of pre-Darwinian views that there is a linear scale of nature having “primitive” creatures at the bottom and humans near the top just below angels. Darwin’s theory of common descent states, however, that evolution is a branching process with no linear scale of increasing perfection along a single branch. Nearly every branch contains its own combination of ancestral and derived features. In cladistics, this perspective is emphasized by recognizing taxa only by their own unique properties and not grouping organisms only because they lack the unique properties found in related groups. Fortunately, there is a convenient way to express the com-mon descent of groups without constructing paraphyletic taxa. It is done by fi nding what is called the sister group of the taxon of interest to us. Two different monophyletic taxa are each other’s sister group if they share common ancestry more recently than either one does with any other taxa. The sister group of humans appears to be chimpanzees, with gorillas forming the sister group to humans and chimpanzees com-bined. Orangutans are the sister group of a clade that includes humans, chimpanzees, and gorillas; gibbons form the sister group of the clade that includes orangutans, chimpanzees, gorillas, and humans (see Figure 10.9 ). Current State of Animal Taxonomy The formal taxonomy of animals that we use today was estab-lished using the principles of evolutionary systematics and has been revised recently in part using the principles of cladistics. Introduction of cladistic principles initially replaces paraphyletic groups with monophyletic subgroups while leaving the remain-ing taxonomy mostly unchanged. A thorough revision of tax-onomy along cladistic principles, however, will require profound changes, one of which almost certainly will be abandonment of Linnaean ranks. A new taxonomic system called PhyloCode is being developed as an alternative to Linnaean taxonomy; this system replaces Linnaean ranks with codes that denote the nested hierarchy of monophyletic groups conveyed by a clado-gram. In our coverage of animal taxonomy, we try to use taxa that are monophyletic and therefore consistent with criteria of both evolutionary and cladistic taxonomy. We continue, how-ever, to use Linnaean ranks. For familiar taxa that are clearly paraphyletic grades, we note this fact and suggest alternative taxonomic schemes that contain only monophyletic taxa. In discussing patterns of descent, we avoid statements such as “mammals evolved from reptiles” that imply paraphyly and instead specify appropriate sister-group relationships. We avoid referring to groups of organisms as being primitive, advanced, specialized, or generalized because all groups of animals con-tain combinations of primitive, advanced, specialized, and gen-eralized features; these terms are best restricted to describing specifi c characteristics and not an entire group. Revision of taxonomy according to cladistic principles can cause confusion. In addition to new taxonomic names, we see old ones used in unfamiliar ways. For example, cladistic use of “bony fi shes” includes amphibians and amniotes (including reptilian groups, birds, and mammals) in addition to fi nned, aquatic animals that we normally term “fi sh.” Cladistic use of “reptiles” includes birds in addition to snakes, lizards, turtles, and crocodilians; however, it excludes some fossil forms, such as synapsids, that were traditionally placed in Reptilia (see Chapters 26 through 28). Taxonomists must be very careful to specify when using these seemingly familiar terms whether the traditional evolutionary taxa or newer cladistic taxa are being referenced. MAJOR DIVISIONS OF LIFE From Aristotle’s time to the late 1800s, every living organism was assigned to one of two kingdoms: plant or animal. However, the two-kingdom system had serious problems. Although it was easy to place rooted, photosynthetic organisms such as trees and herbs among the plants and to place food-ingesting, motile forms such as insects, fi shes, and mammals among the animals, uni-cellular organisms presented diffi culties (see Chapter 11). Some forms were claimed both for the plant kingdom by botanists and for the animal kingdom by zoologists. An example is Euglena (p. 225), which is motile, like animals, but has chlorophyll and photosynthesis, like plants. Other groups, such as bacteria, were assigned rather arbitrarily to the plant kingdom. Several alternative systems have been proposed to solve the problem of classifying unicellular forms. In 1866 Haeckel proposed the new kingdom Protista to include all single-celled organisms. At first bacteria and cyanobacte-ria (blue-green algae), forms that lack nuclei bounded by a membrane, were included with nucleated unicellular organisms. Finally, important differences were recognized between the anucleate bacteria and cyanobacteria (prokary-otes) and all other organisms that have membrane-bound nuclei (eukaryotes). In 1969 R. H. Whittaker proposed a fi ve-kingdom system that incorporated the basic prokaryote-eukaryote distinction. The kingdom Monera contained the prokaryotes. The kingdom Protista contained the unicellular eukaryotic organisms (protozoa and unicellular eukaryotic algae). Multicellular organisms were split into three kingdoms by mode of nutrition and other fundamental differences in organization. The kingdom Plantae included multicellular photosynthesizing organisms, higher plants, and multicellu-lar algae. Kingdom Fungi contained molds, yeasts, and fungi that obtain their food by absorption. Invertebrates (except the www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 213 protozoa) and vertebrates compose the kingdom Animalia. Most of these forms ingest their food and digest it internally, although some parasitic forms are absorptive. These different systems were proposed without regard to the phylogenetic relationships needed to construct evolutionary or cladistic taxonomies. The oldest phylogenetic events in the his-tory of life have been obscure because the different forms of life share very few characters that can be compared among them to reconstruct phylogeny. Recently, however, a cladistic classifi ca-tion of all life-forms has been proposed based on phylogenetic information obtained from molecular data (the nucleotide base sequence of DNA encoding ribosomal RNA). According to this tree ( Figure 10.11 ), Woese, Kandler, and Wheelis (1990) recog-nized three monophyletic domains above the kingdom level: Eucarya (all eukaryotes), Bacteria (the true bacteria), and Archaea (prokaryotes differing from bacteria in membrane structure and ribosomal RNA sequences). They did not divide Eucarya into kingdoms, although if we retain Whittaker’s kingdoms Plantae, Animalia, and Fungi, Protista becomes a paraphyletic group ( Fig-ure 10.11 ). To maintain a cladistic classifi cation, Protista must be discontinued by recognizing as separate kingdoms all of the labelled branches of Eucarya as shown in Figure 10.11 . Until a few years ago, animal-like protistans were tradi-tionally studied in zoology courses as animal phylum Protozoa. Given current knowledge and the principles of phylogenetic systematics, this taxonomy commits two errors; “protozoa” are neither animals nor are they a valid monophyletic taxon at any level. Kingdom Protista is likewise invalid because it is not monophyletic. Animal-like protistans, now divided into seven or more phyla, are nonetheless of interest to students of zool-ogy because they provide an important phylogenetic context for the study of animal diversity. MAJOR SUBDIVISIONS OF THE ANIMAL KINGDOM The phylum is the largest formal taxonomic category in the Lin-naean classifi cation of the animal kingdom. Metazoan phyla are often grouped together to produce additional, informal taxa intermediate between the phylum and the animal king-dom. Phylogenetic relationships among metazoan phyla have been particularly diffi cult to resolve both by morphological and molecular characters. Traditional groupings based on embryo-logical and anatomical characters that may reveal phylogenetic affi nities are: Branch A (Mesozoa): phylum Mesozoa, the mesozoa Branch B (Parazoa): phylum Porifera, the sponges, and phylum Placozoa Branch C (Eumetazoa): all other phyla Grade I (Radiata): phyla Cnidaria, Ctenophora Grade II (Bilateria): all other phyla Figure 10.11 Phylogenetic overview of the three domains of life, Archaea, Bacteria and Eucarya, based on analysis of genes encoding ribosomal RNA. Because of their endosymbiotic origin (p. 34), organellar genomes of domain Eucarya (mitochondria, chloroplasts) are phylogenetically within the Bacteria rather than the clade that includes all eukaryotic nuclear genomes. Organisms of domain Eucarya therefore include cellular components of disparate evolutionary origins.
Fungi Plants Ciliates Flagellates Trichomonads Microsporidia Diplomonads Animals Slime molds Entamoebae Halophiles Methanosarcina Methanobacterium Methanococcus T. celer Thermoproteus Pyrodicticum Spirochetes Proteobacteria Cyanobacteria Planctomyces Aquifex Thermotoga Bacteroides, Cytophaga Chloroplasts Mitochondria Gram positive-bacteria Green filamentous bacteria Bacteria Archaea Eucarya (nuclear genome) 214 PART THREE Diversity of Animal Life Division A (Protostomia): characteristics in Figure 10.12 Acoelomates: phyla Platyhelminthes, Gnathostomulida, Nemertea Pseudocoelomates: phyla Rotifera, Gastrotricha, Kinorhyncha, Nematoda, Nematomorpha, Acanthocephala, Entoprocta, Priapulida, Loricifera Eucoelomates: phyla Mollusca, Annelida, Arthropoda, Echiurida, Sipunculida, Tardigrada, Onychophora Division B (Deuterostomia): characteristics in Figure 10.12 phyla Phoronida, Ectoprocta, Chaetognatha, Brachiopoda, Echinodermata, Hemichordata, Chordata Molecular phylogenetic results have challenged the placement of mesozoans (Branch A), suggesting that they might be derived from protostomia (Branch C, Grade II, Division A). As in the Figure 10.12 Basis for the distinction between divisions of bilateral animals. Traditional classifi cations often place phyla Brachiopoda, Ectoprocta, and Phoronida with deuterostomes, but recent molecular phylogenetic analyses place them with protostomes as shown here. Phylum Chaetognatha is of uncertain phylogenetic affi nity and might lie outside these two groups.
Spiral cleavage Cell from which mesoderm will derive Primitive gut Mesoderm Coelom Anus Annelid (earthworm) Mouth Blastopore 4d Includes phyla Platyhelminthes, Nemertea, Annelida, Mollusca, Arthropoda, Phoronida, Ectoprocta, Brachiopoda, minor phyla Embryology mostly determinate (mosaic) Mouth forms from or near blastopore; anus a new formation In coelomate protostomes the coelom forms as a split in mesodermal bands (schizocoelous) Endomesoderm usually from a particular blastomere designated 4d Cleavage mostly spiral Includes phyla Echinodermata, Hemichordata, Chordata Embryology usually indeterminate (regulative) Anus forms from or near blastopore, mouth a new formation All coelomate, coelom from fusion of enterocoelous pouches (except vertebrates, which are schizocoelous) Endomesoderm from enterocoelous pouching (except vertebrates) Endomesoderm from pouches from primitive gut Cleavage mostly radial Mouth Anus Radial cleavage Primitive gut Mesoderm Blastopore Coelom PROTOSTOMES DEUTEROSTOMES outline, bilateral animals are customarily divided into proto-stomes and deuterostomes by their embryological development ( Figure 10.12 ). However, some of the phyla are diffi cult to place into one of these two categories because they possess some characteristics of each group (see Chapter 15). Molecular phylogenetic studies have challenged traditional classification of the Bilateria, but results are not yet strong enough to present a precise hypothesis of phylogenetic relation-ships among metazoan phyla. Molecular phylogenetic results place four phyla classifi ed above as deuterostomes (Brachiop-oda, Chaetognatha, Ectoprocta, and Phoronida) in the Proto-stomia. This is the scheme shown in Figure 10.12 . Furthermore, the traditional major groupings of protostome phyla (acoelo-mates, pseudocoelomates, and eucoelomates) appear not to be monophyletic. Instead, protostomes are divided into two major www . m hhe . com /hi ckmani pz14e CHAPTER 10 Taxonomy and Phylogeny of Animals 215 S U M M A R Y Animal systematics has three major goals: (1) to identify all species of animals, (2) to evaluate evolutionary relationships among animal species, and (3) to group animal species in a hierarchy of taxonomic groups (taxa) that conveys evolutionary relationships. Taxa are ranked to denote increasing inclusiveness as follows: species, genus, family, order, “class,” phylum, and kingdom. All of these ranks can be subdivided to signify taxa that are intermediate between them. Names of species are binomial, with the fi rst name designating the genus to which the species belongs (capitalized) followed by a spe-cies epithet (lowercase), both written in italics. Taxa at all other ranks are given single capitalized but nonitalicized names. The biological species concept has guided the recognition of most animal species. A biological species is defi ned as a reproduc-tive community of populations (reproductively isolated from others) that occupies a specifi c niche in nature. It is not immutable through time but changes during the course of evolution. Because the bio-logical species concept may be diffi cult to apply in spatial and tem-poral dimensions, and because it excludes asexually reproducing forms, alternative concepts have been proposed. These alternatives include the evolutionary species concept and the phylogenetic spe-cies concept. No single concept of species is universally accepted by all zoologists, but zoologists agree that a species should constitute a population lineage with a history of evolutionary descent separate from other such lineages. Because species lineages are expected to differ from each other in the DNA sequence of the rapidly evolving mitochondrial gene COI, this gene sequence is used as a diagnostic “barcode” to assign specimens to species. Two major schools of taxonomy are currently active. Traditional evolutionary taxonomy groups species into higher taxa according to the joint criteria of common descent and adaptive evolution; such taxa have a single evolutionary origin and occupy a distinctive adap-tive zone. A second approach, called phylogenetic systematics or cladistics, emphasizes common descent exclusively in grouping spe-cies into higher taxa. Only monophyletic taxa (those having a single evolutionary origin and containing all descendants of the group’s most recent common ancestor) are used in cladistics. In addition to monophyletic taxa, evolutionary taxonomy recognizes some taxa that are paraphyletic (having a single evolutionary origin but exclud-ing some descendants of the most recent common ancestor of the group). Both schools of taxonomy exclude polyphyletic taxa (those having more than one evolutionary origin). Both evolutionary taxonomy and cladistics require that pat-terns of common descent among species be assessed before higher taxa are recognized. Comparative morphology (including develop-ment), cytology, and biochemistry are used to reconstruct nested hierarchical relationships among taxa that refl ect the branching of evolutionary lineages through time. The fossil record provides esti-mates of the ages of evolutionary lineages. Comparative studies and the fossil record jointly permit us to reconstruct a phylogenetic tree representing the evolutionary history of the animal kingdom. Traditionally, all living forms were placed into two kingdoms (animal and plant) but more recently, a fi ve-kingdom system (ani-mals, plants, fungi, protistans, and monerans) has been followed. Neither of these systems conforms to the principles of evolutionary or cladistic taxonomy because they place single-celled organisms into either paraphyletic or polyphyletic groups. Based on our cur-rent knowledge of the phylogenetic tree of life, “protozoa” do not form a monophyletic group and they do not belong within the animal kingdom. Phylogenetic relationships among animal phyla have been clarifi ed by molecular phylogenetic studies, although many of these higher-level groupings remain tentative. Particularly controversial is the grouping of bilaterally symmetrical animals into clades Deuterostomia, Protostomia, Ecdysozoa, and Lophotrochozoa. R E V I E W Q U E S T I O N S List in order, from most inclusive to least inclusive, the principal categories (taxa) in Linnaean classifi cation as currently applied to animals. Explain why the system for naming species that originated with Linnaeus is “binomial.” How does the biological species concept differ from earlier typological concepts of a species? Why do evolutionary biologists prefer it to typological species concepts? 1.
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3. What problems have been identifi ed with the biological species concept? How do other species concepts attempt to overcome these problems? How are taxonomic characters recognized? How are such characters used to construct a cladogram? How do monophyletic, paraphyletic, and polyphyletic taxa differ? How do these differences affect the validity of such taxa for both evolutionary and cladistic taxonomies? 4.
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monophyletic groups called the Lophotrochozoa and Ecdysozoa. Reclassifi cation of the Bilateria is summarized: Grade II: Bilateria Division A (Protostomia): Lophotrochozoa: phyla Platyhelminthes, Nemertea, Rotifera, Gastrotricha, Acanthocephala, Mollusca, Annelida, Echiurida, Sipunculida, Phoronida, Ectoprocta, Entoprocta, Gnathostomulida, Chaetognatha, Brachiopoda Ecdysozoa: phyla Kinorhyncha, Nematoda, Nematomorpha, Priapulida, Arthropoda, Tardigrada, Onychophora, Loricifera Division B (Deuterostomia): phyla Chordata, Hemichordata, Echinodermata Although further study is needed to confi rm these new groupings, we use them to organize our survey of animal diversity. 216 PART THREE Diversity of Animal Life S E L E C T E D R E F E R E N C E S Aguinaldo, A. M. A., J. M. Turbeville, L. S. Linford, M. C. Rivera, J. R. Garey, R. A. Raff, and J. A. Lake. 1997. Evidence for a clade of nematodes, arthropods and other moulting animals. Nature 387: 489–493. This molecular phylogenetic study challenges traditional classifi cation of the Bilateria. Avise, J. C. 2006. Evolutionary pathways in nature: a phylogenetic approach. Cambridge, U.K., Cambridge University Press. A current treatment of phylogenetic knowledge. Ereshefsky, M. (ed.). 1992. The units of evolution. Cambridge, Massachusetts, MIT Press. A thorough coverage of concepts of species, including reprints of important papers on the subject. Ereshefsky, M. 2001. The poverty of the Linnaean hierarchy. Cambridge, U.K., Cambridge University Press. A philosophical critique of Linnaean taxonomy illustrating its problems with cladistic taxonomy. Felsenstein, J. 2002. Inferring phylogenies. Sunderland, Massachusetts, Sinauer Associates. A thorough coverage of phylogenetic methods. Hall, B. K. 1994. Homology: the hierarchical basis of comparative biology. San Diego, Academic Press. A collection of papers discussing the many dimensions of homology, the central concept of comparative biology and systematics. Hull, D. L. 1988. Science as a process. Chicago, University of Chicago Press. A study of the working methods and interactions of systematists, containing a thorough review of the principles of evolutionary, phenetic, and cladistic taxonomy. Maddison, W. P., and D. R. Maddison. 2003. MacClade version 4.06. Sunderland, Massachusetts, Sinauer Associates, Inc. A computer program for the MacIntosh that conducts phylogenetic analyses of systematic characters. The instruction manual stands alone as an excellent introduction to phylogenetic procedures. The computer program is user-friendly and excellent for instruction in addition to serving as a tool for analyzing real data. Mayr, E., and P. D. Ashlock. 1991. Principles of systematic zoology. New York, McGraw-Hill. A detailed survey of systematic principles as applied to animals. Panchen, A. L. 1992. Classifi cation, evolution, and the nature of biology. New York, Cambridge University Press. Excellent explanations of the methods and philosophical foundations of biological classifi cation. Swofford, D. 2002. Phylogenetic analysis using parsimony (and other methods) PAUP version 4. Sunderland, Massachusetts, Sinauer Associates. A powerful computer package for constructing phylogenetic trees from data. Valentine, J. W. 2004. On the origin of phyla. Chicago, Illinois, University of Chicago Press. Phylogenetic analysis of animal phyla in a paleontological perspective. Wagner, G. P. (ed.). 2001. The character concept in evolutionary biology. San Diego, Academic Press. A thorough coverage of evolutionary character concepts. Woese, C. R., O. Kandler, and M. L. Wheelis. 1990. Towards a natural system of organisms: proposal for the domains Archaea, Bacteria, and Eucarya. Proceedings of the National Academy of Sciences, USA, 87: 4576–4579. Proposed cladistic classifi cation for the major taxonomic divisions of life. O N L I N E L E A R N I N G C E N T E R Visit www.mhhe.com/hickmanipz14e for chapter quizzing, key term fl ash cards, web links and more! How many different clades of two or more species are possible for species A–H shown in Figure 10.6A ? What is the difference between a cladogram and a phylogenetic tree? Given a cladogram for a group of species, what additional information is needed to obtain a phylogenetic tree? How would cladists and evolutionary taxonomists differ in their interpretations of the statement that humans evolved from apes, which evolved from monkeys? 7.
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9. What taxonomic practices based on the typological species concept are retained in systematics today? How has their interpretation changed? What are the fi ve kingdoms distinguished by Whittaker? How does their recognition confl ict with the principles of cladistic taxonomy? 10.
11. C H A P T E R 11 Protozoan Groups UNICELLULAR EUKARYOTES • Emergence of Eukaryotes and a New Life Pattern The fi rst reasonable evidence for life on earth dates from approxi-mately 3.5 billion years ago. The fi rst cells were prokaryotic bac-teria-like organisms. The early prokaryotes diversifi ed greatly over an enormous time span; their prokaryotic descendants now belong to two groups, the Eubacteria and the Archaea. One lineage within the ancient prokaryotes also gave rise to the fi rst eukaryotic form. The key steps in the evolution of a eukaryotic cell from a prokary-otic ancestor involved symbiogenesis, a process whereby one pro-karyote engulfed, but did not digest, another. The engulfed cell was eventually reduced to an organelle inside the host cell. The eukary-otic products of symbiogenesis include mitochondria and plastids. A mitochondrion originated from an aerobic prokaryote capa-ble of deriving energy in the presence of environmental oxygen. An anaerobic bacterium that engulfed such an aerobic form gained the capacity to grow in an oxygen-rich environment. The engulfed aerobic bacterium persisted inside the cell as a mitochondrion with its own genetic material. Over evolutionary time, most, but not all, genes from the mitochondrion came to reside in the host cell nucleus. Almost all present-day eukaryotes have mitochondria and are aerobic. The eukaryotic plastid originated when a cell engulfed a photo-synthetic bacterium. When a prokaryote is engulfed and modifi ed to become a eukaryotic organelle, we say the organelle developed via primary endosymbiosis. The chloroplasts in red algae, and in green algae and multicellular plants, arose this way. However, in some cases, a eukaryotic cell may obtain plastids from another eukaryote. This is secondary endosymbiosis. Two similar cells may have formed very differently, so it is not easy to untangle the evolutionary rela-tionships among the diverse array of unicellular forms we now see. The assemblage of eukaryotic unicellular organisms is collec-tively called protozoa. The inclusion of “zoa” in the name refers to two animal-like features: the absence of a cell wall, and the presence of at least one motile stage in the life cycle. However, the plant-animal distinction is not easily made in unicellular forms because many motile unicells carry photosynthetic plastids. The myriad of ways to live as a unicellular organism is fascinating, beguiling, and a little bewildering. A paramecium. 218 PART THREE Diversity of Animal Life A protozoan, or unicellular eukaryote, is a complete organ-ism in which all life activities occur within the limits of a single plasma membrane. Unicellular eukaryotes are found wherever life exists. They are highly adaptable and easily distributed from place to place. They require moisture, whether they live in marine or freshwater habitats, soil, decaying organic matter, or plants and animals. They may be sessile or free swim-ming, and they form a large part of the fl oating plankton. The same species are often found widely separated in time as well as in space. Some species may have spanned geological eras exceeding 100 million years. Despite their wide distribution, many protozoa can live successfully only within narrow environmental ranges. Species adaptations vary greatly, and successions of species frequently occur as environmental conditions change. Protozoa play an enormous role in the economy of nature. Their fantastic numbers are attested by the gigantic ocean soil deposits formed over millions of years by their skeletons. About 10,000 species of unicellular eukaryotes are symbiotic in or on animals or plants, sometimes even other protozoa. The relation-ship may be mutualistic (both partners benefi t), commensal-istic (one partner benefi ts, no effect on the other), or parasitic (one partner benefi ts at the expense of the other), depending on the species involved. Parasitic forms cause some of the most important diseases of humans and domestic animals. HOW DO WE DEFINE PROTOZOAN GROUPS? For many years, all protozoans were placed within a single phy-lum comprised of eukaryotic unicells, but phylogenetic studies showed this group was not monophyletic. Evidence suggests that the origin of the fi rst eukaryote was followed by great diver-sifi cation, leading some biologists to predict that more than 60 monophyletic eukaryotic clades will eventually emerge. The well-supported clade Opisthokonta ( Figure 11.1 ) includes unicellular choanofl agellates, multicellular animals (metazoans) and fungi, among others (see p. 228). Like the Opisthokonta, the clade Viri-diplantae has both unicellular and multicellular members; this group contains green algae, the bryophytes, and the vascular plants. The remaining eukaryotic clades contain less well-known organisms, many of which were considered protozoans. Protozoans and their relatives have been given several names. Protozoans are usually unicellular, so the name Protoc-tista was initiated to include unicellular and closely related multi-cellular organisms in one group. However, protoctistan is far less commonly used than the names protist and protozoan. Protist is a general term that does not distinguish between plantlike and animal-like unicells, whereas protozoan was intended for a sub-set of animal-like unicellular organisms. Figure 11.1 Cladogram showing two major prokaryotic branches and diversifi cation of eukaryotes. Major eukaryotic clades containing protists are shown, but several clades of amebas and other forms are not shown. The order of branching remains to be determined for most clades. The very large opisthokont clade contains choanofl agellates, fungi, and all multicellular animals.
Bacteria Archaea Parabasalids Opisthokonta Amoebozoa (= Ramicrisates) Viridiplantae Stramenopiles (= Heterokonts) Euglenozoa Heterolobosea Alveolata Granuloreticulosa Diplomonads Retortamonads Prokaryotic cell DNA with histone proteins Membrane-bound nucleus Symbiogenesis leading to mitochondria Eucarya www . m hhe . com /hi ckmani pz14e CHAPTER 11 Protozoan Groups 219 The two concepts, plantlike and animal-like, refer in part to the way that food is gathered. Plants are typically autotrophic, meaning that they synthesize their own organic constituents from inorganic substrates. Photosynthesis is one kind of autotrophy. Animals are typically heterotrophic, meaning that they obtain organic molecules synthesized by other organisms. Heterotrophic protozoa may ingest their food in a soluble form or in a particu-late form. Particulate food is acquired by phagocytosis via an infolding or invagination of the cell membrane to surround a vis-ible food particle ( Figure 11.2 ). Heterotrophs that feed on visible particles are phagotrophs or holozoic feeders, whereas those that ingest soluble food are osmotrophs or saprozoic feeders. A distinction between plants and animals on the basis of nutrition works well for multicellular forms, but the plant-animal distinction is not so clear among unicells. Autotrophic protozoa (phototrophs) use light energy to synthesize their organic mol-ecules, but they often practice phagotrophy and osmotrophy as well. Even among heterotrophs, few are exclusively either phago-trophic or osmotrophic. A single class Euglenoidea (phylum Euglenozoa) contains some forms that are mainly phototrophs, some that are mainly osmotrophs, and some that are mainly phagotrophs. Species of Euglena show considerable variety in nutritional capability. Some species require certain preformed organic molecules, even though they are autotrophs, and some lose their chloroplasts if maintained in darkness, thus becom-ing permanent osmotrophs. The mode of nutrition employed by unicellular organisms is opportunistic and highly variable, even within a single species, so nutritional features have proved unre-liable for defi ning protozoans, or protozoan subgroups. Originally, the means of locomotion was used to distinguish three of the four classes in the traditional phylum Protozoa. Mem-bers of a parasitic class, once called Sporozoa, lack a distinct locomotory structure, but share an organelle capable of invad-ing host cells. Members of the other three traditional protozoan classes differ in means of locomotion: fl agellates ( Figure 11.3 ) use fl agella, ciliates ( Figure 11.4 ) travel via a ciliated body surface, and amebas extend their pseudopodia ( Figure 11.5 ) to move. Typically, a fl agellate has a few long fl agella, and a ciliate has many short cilia, but no real morphological distinction exists between cilia and fl agella. Some investigators have preferred to Figure 11.2 Some feeding methods among protozoa. Amoeba surrounds a small fl agellate with pseudopodia. Leidyopsis, a fl agellate living in the intestine of termites, forms pseudopodia and ingests wood chips. Didinium, a ciliate, feeds only on Paramecium, which it swallows through a temporary cytostome in its anterior end. Sometimes more than one Didinium feed on the same Paramecium. Podophrya is a suctorian ciliophoran. Its tentacles attach to its prey and suck prey cytoplasm into the body of the Podophrya, where it is pinched off to form food vacuoles. Codonosiga, a sessile fl agellate with a collar of microvilli, feeds on particles suspended in the water drawn through its collar by the beat of its fl agellum. Technically, all of these methods are types of phagocytosis.
Amoeba Didinium Podophrya Codonosiga Leidyopsis Figure 11.3 One fl agellum is clearly visible in the lower left of this photograph of Euglena.
Figure 11.4 Scanning electron micrograph of a free-living ciliate Tetrahymena thermophila showing rows of cilia (2000). Beating of fl agella either pushes or pulls the organism through its medium, while cilia propel the organism by a “rowing” mechanism. Their structure is similar, whether viewed by scanning or transmission electron microscopy. |
6583 | https://www.grammarly.com/commonly-confused-words/censor-vs-censure | Censor vs. Censure: What's the Difference?
Censor and censure may sound similar, but they have distinct meanings and uses. A censor is someone who examines materials such as books, movies, or news articles and suppresses parts deemed offensive, inappropriate, or a threat to security. The action of suppression is also referred to as censoring. On the other hand, censure is a formal expression of disapproval or criticism, often used in official contexts. It does not involve suppression or removal of content, but rather the condemnation of actions or behavior.
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How do you use the word censor in a sentence?
Use the word censor when you are discussing the act of suppressing material that is considered inappropriate or sensitive. It can refer to the actions taken by individuals or authorities to control information and prevent certain ideas or images from being disseminated. Censor primarily functions as a noun but can also be a verb when someone is carrying out the act of censorship.
Examples of censor in a sentence
The government appointed a censor to review all foreign films before they could be shown in domestic cinemas.
Artists often find creative ways to express themselves despite the strict censoring of their work.
Parents sometimes act as censors within the home, restricting what their children can watch on television.
How do you use the word censure in a sentence?
The word censure is used to signify formal disapproval or criticism. It is most often encountered in political, legal, and professional contexts. Censure can serve as a noun referencing the act of expressing disapproval or as a verb to describe the act of delivering this criticism. It does not imply restriction of speech but rather a reprimand or condemnation of certain actions or statements.
Examples of censure in a sentence
The senator faced censure from her peers over her controversial remarks on the senate floor.
Despite the censure from international communities, the country proceeded with its plans.
The committee issued a formal censure against the company for its unethical business practices.
Censor and censure definition, parts of speech, and pronunciation
Censor definition:
Censor (noun) - An official who examines materials like books, films, or news to suppress any content deemed objectionable on moral, political, or other grounds. Censor (verb) - To examine and suppress unacceptable parts of a work, communication, or information.
Censor parts of speech:
As a noun, censor refers to the person performing the act of censorship: 'The film board's censor reviewed the movie ratings.'
As a verb, censor describes the action of removing objectionable content: 'The editor had to censor the manuscript before publication.'
Censor pronunciation:
Censor is pronounced as /ˈsen.sər/.
Censure definition:
Censure (noun) - The expression of formal disapproval. Censure (verb) - To express severe disapproval of (someone or something), typically in a formal statement.
Censure parts of speech:
As a noun, censure represents a formal expression of disapproval: 'The council passed a vote of censure against the mayor.'
As a verb, censure is the act of formally reprimanding someone: 'The ethics board decided to censure the lawyer for misconduct.'
Censure pronunciation:
Censure is pronounced as /ˈsen.ʃər/.
Censor vs. censure in a nutshell
Understanding the distinction between censor and censure is key to using them correctly. A censor suppresses content that may be offensive or harmful, whereas censure is a formal rebuke for behavior or activities deemed improper. While censor often deals with the control of information and media, censure relates to expressions of disapproval without direct interference with freedom of speech. Grasping these nuances helps ensure precise communication, particularly in discussions about media regulation or formal condemnations.
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6584 | https://www.gauthmath.com/solution/1795513720161286/cos-2-cos-sin-cos-2-sin-2-2cos-2-1-2sin-2-1 | Solved: cos (2θ )= cos θ -sin θ cos^2θ +sin^2θ 2cos^2θ -1 2sin^2θ -1 [Calculus]
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cos (2θ )= cos θ -sin θ cos^2θ +sin^2θ 2cos^2θ -1 2sin^2θ -1
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$$2\cos^{2}(\theta) - 1$$2 cos 2(θ)−1
Explanation
Apply the double-angle identity for cosine: $$\cos(2\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)$$cos(2 θ)=cos 2(θ)−sin 2(θ)
The correct expression is $$2\cos^{2}(\theta) - 1$$2 cos 2(θ)−1
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6585 | https://www.cuemath.com/algebra/square-root-of-62/ | Square Root of 62
The square root of 62 is a number which when multiplied by itself, results in the number 62. The square root is an important part of mathematics which one will use at some point in their life. Without saying more, let us see how to calculate the value of the square root of 62 and see some problems to help us understand this topic.
Square Root of 62: √62 =7.87400787...
Square of 62: 622= 3844
| | |
--- |
| 1. | What is the Square Root of 62? |
| 2. | Is Square Root of 62 Rational or Irrational? |
| 3. | How to Find the Square Root of 62? |
| 4. | FAQs on Square Root of 62 |
What is the Square Root of 62?
The square root of a number n is written as √n. This number when squared or multiplied by itself results in the original number n. The square root of 62 can be written in multiple ways
Is Square Root of 62 Rational or Irrational?
How to Find the Square Root of 62?
There are multiple ways to find the square root of 62-
One can find out more about these methods by clicking here
Long Division Method
The square root of 62 by long division method consists of the following steps:
Therefore, the square root of 62 = 7.874
Estimation and Approximation
The estimation method gives us an approximate answer and is usually not accurate to more than 1 decimal place. However, it is easy to perform as can be seen under.
Therefore, we can estimate the square root of 62 ≅ 7.85
Explore Square roots using illustrations and interactive examples
Important Notes
Challenging Questions
Solved Examples
Example 1: Jake wants to calculate the side length of his square courtyard in his home. It has an area of 62 square feet.
Solution:
To find the side of the courtyard, we will have to find the square root of 62. The square root is √62 = 7.874.
Hence, the side length of the courtyard is approximately 7.9 feet.
Example 2: Gary was asked to solve an equation x2 - 62 = 0. Help him find all values of x
Solution:
x2 - 62 = 0
x2 = 62.
Taking the square root on both sides, we get
x = ±7.874
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Interactive Questions
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FAQs on Square Root of 62
What is the prime factorization of 62?
The prime factorization of 62 is 2 × 31.
Is 62 a perfect square root?
62 is not a perfect square. 62 is a natural number, but since there is no other natural number that can be squared to result in the number 62, it is NOT a perfect square.
What is the decimal value of the square root of 62?
The decimal value of the square root of 62 is 7.874.
Is the square root of 62 a rational number?
No, the square root of 62 is not a rational number since the square root of 62 is non-terminating and cannot be represented in the form of p/q.
Can we find the square root of 62 by the repeated subtraction method?
No, we can’t find the square root of 62 by repeated subtraction method as it can be used only for perfect squares. 62 is not a perfect square. |
6586 | https://concepto.de/sustantivo/ | Published Time: 2020-06-12T04:17:14+00:00
Sustantivo - Qué es, tipos, ejemplos y oraciones
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¿Qué es un sustantivo?
Tipos de sustantivos
Flexión del sustantivo: género y número
Ejemplos de sustantivos
El sustantivo dentro de la oración
Relación entre sustantivos y adjetivos
Te explicamos qué es un sustantivo y su clasificación (propio, común, concreto, abstracto). Además, su género y número, y ejemplos y oraciones.
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¿Qué es un sustantivo?
Un sustantivo es una categoría gramatical o clase de palabra que se utiliza para nombrar un objeto, sujeto, lugar, concepto. Por ejemplo: Juan, auto, casa, Madrid.
Ver además: Categorías gramaticales
Tipos de sustantivos
El sustantivo puede clasificarse según lo que nombra en:
Sustantivos propios.Nombran a un objeto o sujeto de forma concreta y de manera particular y siempre se escriben con mayúscula. Se utilizan para nombrar: personas (María, Juan), países, ciudades y continentes (Colombia, Montevideo, Asia), marcas (Bimbo, Sony), organizaciones o instituciones (Universidad Autónoma de Barcelona, Greenpeace), festividades (Navidad, Pesaj), grupos musicales (Nirvana, The Beatles), obras literarias (El Quijote de la Mancha, La Gioconda).
Sustantivos comunes.Designan una persona, animal o cosa de manera general. Por ejemplo: mesa, silla, vaca, perro, montaña, amor. Los sustantivos comunes siempre se escriben con minúscula y se clasifican de diferentes maneras.
Los sustantivos comunes se clasifican en:
Sustantivos abstractos. Designan ideas, conceptos y sentimientos que no pueden ser percibidos por los sentidos. Por ejemplo:libertad,felicidad, odio, compasión.
Sustantivos concretos. Nombran cosas y objetos que sí pueden ser percibidas por los sentidos. Por ejemplo: casa, gato, árbol, pelota.
Los sustantivos concretos se clasifican en sustantivos contables e incontables:
Sustantivos contables.Pueden numerarse. Por ejemplo: manzana, mesa, amigo.
Sustantivos no contables.No se pueden numerar, pero sí medir. Por ejemplo: agua, azúcar. En este último caso, no se puede decir dos azúcares, pero sí un kilo de azúcar. Los sustantivos no contables no tienen plural.
Los sustantivos contables se pueden clasificar en:
Sustantivos individuales.Nombran un ser en particular en singular. Por ejemplo: pez, perro.
Sustantivos colectivos.Nombran de manera global o grupal a un conjunto de personas, animales, objetos. Por ejemplo: “pez” es individual y “cardumen” es colectivo, ya que hace referencia a un conjunto de peces.
Otra clasificación que se puede emplear dentro de los sustantivos comunes es la de dividir entre sustantivos primitivos y derivados(derivan de otro sustantivo, como por ejemplo “zapatería”, “heladería”, “panadería”). También es posible indicar si un sustantivo es simple ocompuesto(en el caso de “lavarropas”, “paraguas” por ejemplo).
Flexión del sustantivo: género y número
La flexión de los sustantivos consta de género y número, que podemos explicar de la siguiente manera:
Género: femenino o masculino. La mayoría de los sustantivos femeninos terminan con la vocal -a mientras que los masculinos con la vocal -o. Por eso es que podemos, por ejemplo, decir gato y gata, perro y perra. Sin embargo, en algunos casos, el significado se modifica, como por ejemplo canasto y canasta (cambia el tamaño del objeto).
Número: singular o plural. Algunos sustantivos se escriben igual en singular y plural como: tórax, clímax, caries.
La mayoría de los sustantivos pueden modificarse de masculino a femenino y viceversa, así como de singular a plural y viceversa. Algunos femeninos son completamente diferentes a su masculino, tal es el caso de: caballo y yegua o actor y actriz.
Ejemplos de sustantivos
Sustantivos propios
Nombres propios: Mateo Olivero, Karina Sánchez, Osvaldo Cortez.
Lugares y espacios urbanos: Argelia, Tijuana, Oceanía, Avenida del Libertador.
Organizaciones o instituciones: Organización Mundial de la Salud, Club Deportivo Guadalajara.
Obras de arte:Las meninas, Cien años de soledad.
Festividades: Año Nuevo, Ramadán, Día de los Muertos.
Sustantivos comunes
Mesa, perro, colchón, automóvil, taladro, madre, plato, camiseta, saxofón, arroz, teléfono.
Sustantivos abstractos
Amor, pasión, lujuria, temor, libertad, amistad, dolor, ansiedad, religión, justicia, mentira, compasión, tiempo.
Sustantivos concretos
Tierra, botella, pulgar, biblioteca, pelota, arena, reloj, pincel, edificio, mono, aceite, vela, agua.
Sustantivos individuales y colectivos
| Individual | Colectivo |
--- |
| Libro | Biblioteca |
| Perro | Jauría |
| Ave | Bandada |
| Pez | Cardumen |
| País | Continente |
| Familia | Clan |
| Estrella | Constelación |
| Letra | Abecedario |
Sustantivos contables
Libro, pelota, banana, niño, pato.
Sustantivos no contables
Harina, azúcar, sangre, dinero, basura, gasolina
El sustantivo dentro de la oración
El sustantivoes el núcleo del sintagma nominal (conjunto de palabras con un núcleo), principalmente del sujeto de laoración.
Junto al núcleo encontramos: los modificadores directos (adjetivos y artículos), el modificador indirecto (preposición más sustantivo o construcción sustantiva), la aposición (sustantivo o construcción sustantiva que aclara el núcleo o lo repite) y la construcción comparativa (introducida por “como” o “cual”).
Por ejemplo:
Juan, mi hermano, se mudó hace poco. (“Juan” es núcleo del sujeto y “hermano” es núcleo de la aposición explicativa)
La profesora del colegio nos enseña muy bien. (“profesora” es núcleo del sujeto y “colegio” es núcleo del modificador indirecto)
En una oración puede haber uno o más núcleos sustantivosdentro de un sujeto (simple o compuesto) y deben concordar con el verbo conjugado del predicado (singular o plural). Lo mismo ocurre con los adjetivos, que deben concordar con el sustantivo en género y número para que la cohesión de la oración sea correcta.
Por ejemplo:
Clara y Andrea corren todas las mañanas. (“Clara” y “Andrea” son los dos núcleos del sujeto compuesto)
Los empleados y los directivos se reunieron en el salón. (“empleados” y “directivos” son los dos núcleos del sujeto compuesto)
Además, el sustantivo puede estar presente en construcciones nominales del predicado, dentro de un objeto directo, objeto indirecto, circunstancial, complemento o predicativo.
Por ejemplo:
Estamos preparando una sorpresa para Javier. (“Javier” es núcleo del objeto indirecto y “sorpresa” es núcleo del objeto directo)
El show será en el escenario principal. (“escenario” es núcleo del complemento circunstancial de lugar)
El misterio fue descubierto por el detective. (“detective” es el núcleo del complemento agente)
Ver también: Oración
Ejemplos de oraciones con sustantivos
Oraciones con sustantivos propios
Mateo Olivero es primo de mi mamá.
Ayer me encontré en la calle con Karina Sánchez.
Le dieron el alta médica a Osvaldo Cortez.
Su hermana se fue a vivir a Argelia.
En Tijuana visitamos la Avenida Revolución.
Oceanía es el continente más pequeño del planeta Tierra.
La tienda de regalos queda sobre Avenida del Libertador.
La Organización Mundial de la Salud prohibió el uso de la vacuna.
El Club Deportivo Guadalajara ganó el campeonato zonal.
Diego Velázquez es el autor de Las meninas.
María está leyendo Cien años de soledad del escritor colombiano, Gabriel García Márquez.
Oraciones con sustantivos comunes
Sonia compró la mesa en la tienda del pueblo.
Mi perro ladra cuando escucha truenos.
El colchón de Marisa no entró en el camión de mudanza.
El automóvil carga 50 litros de gasolina.
No se puede utilizar el taladro en horarios nocturnos.
La madre de Pedro lo visitó para Navidad.
¡No tires toda la comida fuera del plato!
El directorio cambió el diseño de la camiseta del club.
Luis se compró un saxofón de segunda mano.
No hace falta que compres más arroz.
El teléfono sonó justo a la hora de la cena.
Oraciones con sustantivos abstractos
El libro habla sobre el amor y la pasión entre dos enamorados.
La lujuria es uno de los siete pecados capitales.
No te dejes llevar solo por el temor.
La libertad es un deseo de todos los hombres y mujeres.
Nada podrá romper esta amistad.
El dolor en su pierna persiste a pesar de los medicamentos que le dieron.
Su partida le generó mucha ansiedad.
Miguel Ángel es fiel a su religión.
El tribunal hizo justicia con Paola y falló a su favor.
Pedro no se deja llevar por las mentiras de su supervisora.
Tuvo compasión con su vecino y le perdonó la deuda.
Es solo una cuestión de tiempo.
Oraciones con sustantivos colectivos
Guardé el libro que me regalaste en la biblioteca.
Ella no se amedrentó y enfrentó a la jauría que la perseguía.
Una bandada cruzó los Andes al amanecer.
El cardumen recorrió las costas del Mar Muerto.
La solidaridad no reconoce fronteras de países en este continente.
Su familia y los Gómez son un clan.
Julián se quedó despierto contemplando la constelación.
¿Están listos? ¡Llegó la hora de aprender el abecedario!
Relación entre sustantivos y adjetivos
El adjetivo es una clase de palabra que tiene la función de calificar al sustantivo. El adjetivo expresa características o propiedades atribuidas a un sustantivo y aporta información sobre él.
Ambas son categorías de palabras que se encuentran estrechamente vinculadas, ya que el sustantivo nombra a una cosa (objeto, lugar, animal, persona, emoción) y el adjetivo la caracteriza. Por ejemplo: El mar (sustantivo) es profundo (adjetivo). Sustantivo y adjetivo deben concordar en género y número.
Los adjetivos se clasifican en dos categorías principales:
Adjetivos calificativos.Son los que detallan características o propiedades del sustantivo al que califican, y pueden ser especificativos o explicativos.
Adjetivos determinantes.Indican cantidad, posición o número del sustantivo, y pueden ser demostrativos, indefinidos y posesivos.
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Toda la información que ofrecemos está respaldada por fuentes bibliográficas autorizadas y actualizadas, que aseguran un contenido confiable en línea con nuestros principios editoriales.
“Sustantivos o nombres”
“Clasificación de sustantivos” en Universidad Complutense de Madrid.
“Sustantivo”
“Los adjetivos en español”
¿Cómo citar?
Citar la fuente original de donde tomamos información sirve para dar crédito a los autores correspondientes y evitar incurrir en plagio. Además, permite a los lectores acceder a las fuentes originales utilizadas en un texto para verificar o ampliar información en caso de que lo necesiten.
Para citar de manera adecuada, recomendamos hacerlo según las normas APA, que es una forma estandarizada internacionalmente y utilizada por instituciones académicas y de investigación de primer nivel.
Raffino, Equipo editorial, Etecé (13 de julio de 2025). Sustantivo. Enciclopedia Concepto. Recuperado el 29 de agosto de 2025 de
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6587 | https://www.seventransistorlabs.com/tmoranwms/Elec_Inductor.html | Inductors
Testing Inductors, and Inductor Theory
Alright, so, you've got an inductor. You know it makes a magnetic field, but beyond that, so what!? Well I'll tell you what...
Oh noes, teh math!1
No, not really- the worst thing I have is a tinge of differential calculus. Hey wait, come back here, it's a lot easier than you think! The fundamental inductor equation is:
V = L dI/dt
If you're much familiar with capacitors, you'll note the striking resemblance to the capacitor equation, I = C dV/dt. Indeed, the capacitor and inductor are mirror images of each other (not opposites, for reasons I'll get to later).
How to apply this equation? Well, when V and L are relatively constant, you can change dees to deltas and use it to find how much current rises in some period of time, when you apply a voltage. If you feel like solving differential equations, you can do all sorts of stuff, too.
Take an inductor, ground one end and apply a square wave to the loose end. If this is what voltage looks like, this is what current will look like. Where for a capacitor you charge it with a constant current to get a voltage slope, when you charge an inductor with a constant voltage, you get a current slope.
Note that, when voltage reverses, current stays put. It starts moving, but it doesn't change position. Compare to a resistor, where current is always in the direction of voltage. This can cause trouble for circuits not designed to handle reverse currents, such as audio amplifiers. Actually, the average audio amplifier, probably using an emitter follower output, wouldn't handle it too badly, but as flyback current decays, the amp has to "catch" it in time as the current flow reverses polarity. I have an example of this from my induction heater drive transformer:
If you check the page I originally posted this image on, you'll see I didn't know what the bumps were after the transitions. It turns out the inductor equation describes it perfectly:
When voltage rises, finite primary inductance causes a dI/dt.
When voltage falls to zero, primary current holds constant (no change -- dI/dt = 0). (Remember, an inductor holds its charge at 0V terminal voltage, just as a capacitor holds its charge (i.e., dV/dt = 0) at 0A terminal current!)
When voltage goes negative, current remains positive for a moment. The class C emitter follower driving this is conducting through the NPN transistor (at a relatively large cost of power, since it has full supply voltage across it).
As "flyback" current decays, less current flows through the NPN. Gradually, the PNP is turned on instead. At this point, the reverse of the cycle starts.
The crunchy part about this is, to turn on MOSFETs I need a quick burst of dV/dt. When the driver circuit starts a pulse, the forward transistor might conduct briefly due to the capacitance, but then as the voltage comes to a stop, the flyback current takes over and the opposite transistor is needed. Bipolar transistors take time to shut off, so this could create a situation where shoot-through occurs, seriously blasting the dissipation. The original circuit I built, using TIP31 output transistors, almost certainly suffered from this, as well as general slowness of the transistors.
Testing -------
Let's say you have an inductor laying around, and you want to know what value it is. You can estimate current rating by the wire gauge (if you can see it) and you can find DC resistance with an ohmmeter. Together you can estimate how much copper wire is on the thing, and if you can see the geometry, even calculate a ballpark inductance. But that's not accurate, and all bets are off if a core is used.
Here's what I use to measure inductors. You'll need a floating signal generator or 'scope since they have 20V or so between grounds. Alternately, you could use a current transformer instead of the resistor, but that's perhaps another web page. Anyway, when the MOSFET is switched on (remember to drive it with around +10V / 0V square wave, and start with a low duty cycle in case things want to burn!), voltage appears on the inductor and current rises blah blah. Magnetic field is proportional to current. Magnetic materials can only help out so much, so if there is a ferrite core, you can test saturation current by measuring at what current the inductance starts to increase. This is labeled I sat on the diagram. You may want to check voltages in the circuit, power supply sag for instance, so you know exactly how much voltage is on the inductor. Since you know voltage, peak current and what time it took to reach that current, you can use V = L ΔI/Δt, er, L = V Δt/ΔI to find inductance.
You can also find inductance from the falling (flyback) curve (which is actually rising, because the voltage was negative, but you know what I mean), but voltage is usually more variable so I prefer to take it from the front slope. A note: it helps to have an extra source of current for the flyback load. That way, inductor current falls to zero. As shown, current won't actually go to zero -- duty cycle mostly determines flyback voltage and current will be continuous, which can make ugly displays.
Here's a passable equivalent circuit. The capacitor C represents the parasitic capacitance between every turn. Resistor is the DC resistance. You could add a series RC to represent loss at HF, but I don't know that that would be accurate. L of course represents inductance. Although this is a parallel resonant circuit, it looks very inductive for low frequencies and impedances, which is the point. However, when you turn off the transistor, the impedance goes through the roof, and you can observe the ringing at the end of the cycle. In fact, since you calculated inductance from the first half of the cycle, you can use this opportunity to measure the capacitance of the inductor. The ringing will proceed around the resonant frequency, which can be measured by counting divs, and used to calculate C = 1 / (4π 2LF 2). Just don't forget to subtract MOSFET drain capacitance!
Winding Tips
When winding your own coils, it helps to know what you're winding. For single layer solenoids, I use COIL.EXE, written by Brian Beezley, which is very accurate, from small coils to large and across the frequency spectrum. For larger or stockier inductances, though, I need a ferrite core, so what I do is put down say 20 turns of 18AWG on an ungapped core picked from my drawer and hook it up to this circuit. I determine inductance and, if resistance is low enough, saturation current as well. (Remember, the DCR of the inductor causes a voltage drop inside the wire itself which subtracts from the pure inductance's voltage). Let's say I found 50uH. Since inductance is proportional to turns squared, I divide inductance by square of the turns (400 in this case) to find my core does 125nH/T 2. That means one turn would be 125nH, two would be 0.5uH, 200 turns would be 5mH, and so on. Likewise, the magnetic field corresponds to the total rate of electrons spinning around the core, or amp-turns. If saturation happens at 5A, I have 520 = 100A·T saturation on this core. The maximum energy storage is E = ½LI 2 (compare to the equation for a capacitor's stored energy, or Newtonian kinetic energy for that matter!), or 0.625mJ. If I find this is insufficient for my application, I might add paper or cardboard gaps to the core, measure again, and if it's still no good despite a huge gap, I'll just look for a different core (or come up with some creative combinations!). When you're doing this, remember inductance drops off roughly in proportion to the increase in amp-turns. Since amps are squared and L is linear, you can hold more energy with a greater gap, but since turns are higher for the same inductance, you need bigger wire for the same resistive loss. Pretty quickly you're fighting a losing battle as you discover you have pounds of wire wrapped around this poor little core that's gapped to nothingness. ;-)
Differential Equations!
HTML is probably going to suck at this, but meh. Let's say you connect an inductor L in series with a resistor R, and connect this to a power supply V. What happens? Well, first of all let's start with basics. Kirchoff says:
V R + V L = V so V R = V - V L
That is, the voltage across the resistor and inductor respectively sum to the supply voltage, and:
I R = I L = I,
I being the current in the loop, being that the circuit is only one loop.
By Ohm's law, V R = I R, and by the inductor equation, V L = L dI/dt.
So, IR = V - LdI/dt or dividing by R, I = V/R - L/R dI/dt.
(Note that I'm using the asterisk to indicate multiplication, and strategically placed spaces to group fractions and coefficients. Hopefully, this will remain clear!)
What kind of function is that? Well let's break it down to its fundamental form.
Let: C = V/R, and τ = -L/R. Now it looks like:
I = τdI/dt + C,
that is to say, I equals the time derivative I, times a constant. I need a function where, when I take the derivative of it, I get itself times a constant. Remember way back in calculus class that d/dx e kx = ke kx, and it should be realized that this is eerily similar to the current situation (ha! ha!). So on a hunch, I'm going to let I = e τt. Thus, dI/dt = τ e τt = τ I. Substituting,
e τt = τ 2e τt + C
...This is where I get stuck... I have yet to take differential equations. It does yield the familiar RL decay equation, I = V/R e-tR/L.
LC Circuit
Likewise, you can hook up an inductor and capacitor and get a differential equation.
For the capacitor, I = CdV/dt, and for the inductor, V = LdI/dt.
If I take the derivative of the C equation,
dI/dt = Cd 2 V/dt 2, which can be substituted into the L equation:
V = LCd 2 V/dt 2. (Note that I haven't used Kirchoff's rules, other than calling the dI/dt's equal. V should actually be negative!) The function that fits this equation is a trig function, simple, so that the second derivative equals the opposite of the function. Sine and cosine work, and for convienience I'll pick cosine. Something like V = Vi cos(ωt), where L and C determine ω (ω = 1 / √LC), as L and R determine τ above. So that, at t = 0, V = Vi, which C is charged to before the switch is closed. The fun thing about this is you get a repeating function...funky. This implies, nay, describes resonance, significantly important for selecting frequencies and such.
Of course, if you include a resistor, either in series or in parallel, the oscillation decays. Everything real has resistance, so it's important and inescapable. But since I suck at differential equations, I'll leave it to other people to calculate it and present the transition from these differential equations to the equations of reactance, and some creative and very useful ways of using them.
Return to Electronics
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Tools & Reference>General Surgery
Peritonsillar Abscess
Updated: Sep 03, 2024
Author: Benoit J Gosselin, MD, FRCSC; Chief Editor: John Geibel, MD, MSc, DSc, AGAF more...;)
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Peritonsillar Abscess
Sections Peritonsillar Abscess
Overview
Practice Essentials
Anatomy
Pathophysiology
Etiology
Epidemiology
Prognosis
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Presentation
History
Physical Examination
Complications
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Treatment
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Overview
Practice Essentials
Peritonsillar abscess (PTA) was first described as early as the 14th century; however, it is only since the advent of antibiotics in the 20th century that the condition has been described more extensively. A PTA is a localized accumulation of pus in the peritonsillar tissues that forms as a result of suppurative tonsillitis. An alternative explanation is that a PTA is an abscess formed in a group of salivary glands in the supratonsillar fossa, known as Weber glands.
The nidus of accumulation is located between the capsule of the palatine tonsils and the constrictor muscles of the pharynx. The anterior and posterior pillars, torus tubarius (superior), and pyriform sinus (inferior) form the boundaries of this potential peritonsillar space. Because this area is composed of loose connective tissue, severe infection may rapidly lead to formation of purulent material. Progressive inflammation and suppuration may extend to directly involve the soft palate, the lateral wall of the pharynx, and, occasionally, the base of the tongue.
PTA is usually a complication of an acute tonsillitis. Inflammatory edema may lead to significant difficulty in swallowing. Dehydration frequently occurs secondary to the patient's avoidance of painful ingestion of food and liquids. Expansion of the abscess may lead to extension of the inflammation into adjacent fascial compartments of the head and neck, potentially resulting in airway obstruction.
Antipyretics and analgesics are used to alleviate fever and discomfort. Antibiotic therapy should begin after cultures are obtained from the abscess. The use of steroids has been controversial. Surgical treatment measures include needle aspiration and incision and drainage (I&D). There continues to be controversy regarding the relative merits of needle aspiration and I&D as definitive therapeutic modalities. (See Treatment.)
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Anatomy
The palatine tonsils are paired lymphoid organs found between the palatoglossal and palatopharyngeal folds of the oropharynx. They are surrounded by a thin capsule that separates the tonsil from the superior and middle constrictor muscles.
The anterior and posterior pillars form the front and back limits of the peritonsillar space. Superiorly, this potential space is related to the torus tubarius, while inferiorly it is bounded by the pyriform sinus. Composed solely of loose connective tissue, a severe infection may rapidly result in pus formation. The inflammation and suppurative process may extend to involve the soft palate, the lateral wall of the pharynx, and, occasionally, the base of the tongue.
The tonsillar fossa has a rich network of lymphatic vessels leading to the parapharyngeal space and the upper cervical lymph nodes, which explains the pattern of adenopathy observed clinically. Ipsilateral upper cervical lymphadenopathy is the result of the spread of infection to the regional lymphatics. Occasionally, the severity of the suppurative process may lead to a cervical abscess, especially in very fulminant or rapidly progressive cases.
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Pathophysiology
The pathophysiology of PTA is unknown. The most widely accepted theory involves the progression of an episode of exudative tonsillitis first into peritonsillitis and then into frank abscess formation. Extension of the inflammatory process may occur in both treated and untreated populations. PTA also has been documented to arise de novo without any prior history of recurrent or chronic tonsillitis. A PTA also can be the presentation of an Epstein-Barr virus (ie, mononucleosis) infection.
Another theory proposes the origin of PTA in Weber glands. These minor salivary glands are found in the peritonsillar space and are thought to help in clearing debris from the tonsils. Should obstruction as a result of scarring from infection occur, tissue necrosis and abscess formation result, leading to PTA.
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Etiology
Any of the microorganisms that cause acute or chronic tonsillitis may be the cause of a PTA. Most commonly, aerobic and anaerobic gram-positive organisms are identified by means of culture.
Cultures of affected patients reveal group A beta-hemolytic streptococci as most prevalent. Next most commonly, staphylococci, pneumococci, and Haemophilus organisms are found. Finally, other microorganisms that can be cultured include lactobacilli, filamentous forms such as Actinomyces species, micrococci, Neisseria species, diphtheroids, Bacteroides species, and nonsporulating bacteria. Some evidence indicates that anaerobic bacteria frequently cause these infections.
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Epidemiology
The incidence of PTA in the United States is about 30 cases per 100,000 people per year, representing about 45,000 new cases each year. No accurate data are available internationally.
Although tonsillitis is a disease of childhood, only one third of PTA cases are found in this age group. The age of patients with the condition is variable, ranging from 1 to 76 years, with the highest incidence in persons aged 15-35 years. No sexual or racial predilection has been established.
In a retrospective cohort study of 427 patients with PTA, Marom et al investigated how the characteristics of PTA may have changed over time. The results led the authors to conclude that PTA currently tends to affect an older population than it once did, that its course in older individuals has become longer and worse, and that smoking may be a predisposing factor in its development.
A study of 685 patients by Kordeluk et al looked at the relation between peritonsillar cellulitis and abscess and outbreaks of acute tonsillitis. The authors found peaks with seasonal variation for presentations with acute tonsillitis but no association with PTA, which occurred at similar rates throughout the calendar year.
A study from Spain found that the institution of protective measures as a result of the COVID-19 pandemic led to a 43% reduction in the incidence of peritonsillar infection from the 2017-2019 period to 2020; however, the cases that did occur tended to have a longer recovery period, and there were more PTAs.
A study from the United States did not find the pandemic to have a significant effect on the incidence of PTA in pediatric otolaryngology practice.
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Prognosis
Most patients treated with antibiotics and adequate drainage of their abscess cavity recover within a few days. A small number present with another abscess later, requiring tonsillectomy. If patients continue to report recurring or chronic sore throats after proper incision and drainage (I&D), a tonsillectomy may be indicated.
In a nationwide retrospective cohort study from Taiwan, Wang et al found that the risk of PTA recurrence rose with higher degrees of previous tonsillitis in patients of all ages, whereas such a rise was associated with management by needle aspiration only in pediatric patients. The risk of PTA recurrence was highest in patients who were younger than 30 years and had experienced more than five prior episodes of tonsillitis.
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Clinical Presentation
References
Repanos C, Mukherjee P, Alwahab Y. Role of microbiological studies in management of peritonsillar abscess. J Laryngol Otol. 2009 Aug. 123 (8):877-9. [QxMD MEDLINE Link].
Marom T, Cinamon U, Itskoviz D, Roth Y. Changing trends of peritonsillar abscess. Am J Otolaryngol. 2010 May-Jun. 31 (3):162-7. [QxMD MEDLINE Link].
Kordeluk S, Novack L, Puterman M, Kraus M, Joshua BZ. Relation between peritonsillar infection and acute tonsillitis: myth or reality?. Otolaryngol Head Neck Surg. 2011 Dec. 145 (6):940-5. [QxMD MEDLINE Link].
García-Callejo FJ, Alba-García JR, Orozco-Núñez S, Martínez-Giménez L, Balaguer-García R, Ruescas-Gómez L. [Healthcare aspects of peritonsillar infection before and during the COVID-19 pandemic]. Acta Otorrinolaringol Esp. 2023 Mar-Apr. 74 (2):108-115. [QxMD MEDLINE Link]. [Full Text].
Allen DZ, Challapalli S, Lee KH, Bell CS, Roy S, Bowe S, et al. Impact of COVID-19 on nationwide pediatric otolaryngology practice: Adenotonsillectomies (TA) and tonsil-related diagnoses trends. Am J Otolaryngol. 2022 Jun 11. 43 (5):103526. [QxMD MEDLINE Link].
Wang YP, Wang MC, Lin HC, Chou P. The impact of prior tonsillitis and treatment modality on the recurrence of peritonsillar abscess: a nationwide cohort study. PLoS One. 2014. 9 (10):e109887. [QxMD MEDLINE Link]. [Full Text].
Kilty SJ, Gaboury I. Clinical predictors of peritonsillar abscess in adults. J Otolaryngol Head Neck Surg. 2008 Apr. 37 (2):165-8. [QxMD MEDLINE Link].
Thapar A, Tassone P, Bhat N, Pfleiderer A. Parapharyngeal abscess: a life-threatening complication of quinsy. Clin Anat. 2008 Jan. 21 (1):23-6. [QxMD MEDLINE Link].
da Silva PS, Waisberg DR. Internal carotid artery pseudoaneurysm with life-threatening epistaxis as a complication of deep neck space infection. Pediatr Emerg Care. 2011 May. 27 (5):422-4. [QxMD MEDLINE Link].
Roccia F, Pecorari GC, Oliaro A, Passet E, Rossi P, Nadalin J, et al. Ten years of descending necrotizing mediastinitis: management of 23 cases. J Oral Maxillofac Surg. 2007 Sep. 65 (9):1716-24. [QxMD MEDLINE Link].
Ehrenfried Berthelsen R, Hein L. [Lemierre's syndrome following peritonsillar abscess]. Ugeskr Laeger. 2012 May 28. 174 (22):1534-5. [QxMD MEDLINE Link].
Ramirez-Schrempp D, Dorfman DH, Baker WE, Liteplo AS. Ultrasound soft-tissue applications in the pediatric emergency department: to drain or not to drain?. Pediatr Emerg Care. 2009 Jan. 25 (1):44-8. [QxMD MEDLINE Link].
Kim DJ, Burton JE, Hammad A, Sabhaney V, Freder J, Bone JN, et al. Test characteristics of ultrasound for the diagnosis of peritonsillar abscess: A systematic review and meta-analysis. Acad Emerg Med. 2023 Aug. 30 (8):859-869. [QxMD MEDLINE Link]. [Full Text].
Ozbek C, Aygenc E, Tuna EU, Selcuk A, Ozdem C. Use of steroids in the treatment of peritonsillar abscess. J Laryngol Otol. 2004 Jun. 118 (6):439-42. [QxMD MEDLINE Link].
Tsikopoulos A, Fountarlis A, Tsikopoulos K, Dilmperis F, Garefis K, Tsikopoulos I, et al. Immediate or interval abscess tonsillectomy? A systematic review and meta-analysis. Eur Arch Otorhinolaryngol. 2022 May. 279 (5):2245-2257. [QxMD MEDLINE Link].
Marom T, Cinamon U, Itskoviz D, Roth Y. Changing trends of peritonsillar abscess. Am J Otolaryngol. 2010 May-Jun. 31 (3):162-7. [QxMD MEDLINE Link].
Heidemann CH, Wallén M, Aakesson M, Skov P, Kjeldsen AD, Godballe C. Post-tonsillectomy hemorrhage: assessment of risk factors with special attention to introduction of coblation technique. Eur Arch Otorhinolaryngol. 2009 Jul. 266 (7):1011-5. [QxMD MEDLINE Link].
Media Gallery
Right peritonsillar abscess. Soft palate, which is erythematous and edematous, is displaced anteriorly. Patient has "hot potato–sounding" voice.
Pus is aspirated from right peritonsillar abscess through large-bore needle. Additional incision will be made to drain any other pus pockets.
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Contributor Information and Disclosures
Author
Benoit J Gosselin, MD, FRCSC Associate Professor of Surgery, Dartmouth Medical School; Director, Comprehensive Head and Neck Oncology Program, Norris Cotton Cancer Center; Staff Otolaryngologist, Division of Otolaryngology-Head and Neck Surgery, Dartmouth-Hitchcock Medical Center
Benoit J Gosselin, MD, FRCSC is a member of the following medical societies: American Head and Neck Society, American Academy of Facial Plastic and Reconstructive Surgery, North American Skull Base Society, American Academy of Otolaryngology-Head and Neck Surgery, American Medical Association, American Rhinologic Society, Canadian Medical Association, Canadian Society of Otolaryngology-Head & Neck Surgery, College of Physicians and Surgeons of Ontario, New Hampshire Medical Society, Ontario Medical Association
Disclosure: Nothing to disclose.
Specialty Editor Board
Francisco Talavera, PharmD, PhD Adjunct Assistant Professor, University of Nebraska Medical Center College of Pharmacy; Editor-in-Chief, Medscape Drug Reference
Disclosure: Received salary from Medscape for employment. for: Medscape.
Chief Editor
John Geibel, MD, MSc, DSc, AGAF Vice Chair and Professor, Department of Surgery, Section of Gastrointestinal Medicine, Professor, Department of Cellular and Molecular Physiology, Yale University School of Medicine; Director of Surgical Research, Department of Surgery, Yale-New Haven Hospital; American Gastroenterological Association Fellow; Fellow of the Royal Society of Medicine
John Geibel, MD, MSc, DSc, AGAF is a member of the following medical societies: American Gastroenterological Association, American Physiological Society, American Society of Nephrology, Association for Academic Surgery, International Society of Nephrology, New York Academy of Sciences, Society for Surgery of the Alimentary Tract
Disclosure: Nothing to disclose.
Additional Contributors
Amy L Friedman, MD Professor of Surgery, Director of Transplantation, State University of New York Upstate Medical University
Amy L Friedman, MD is a member of the following medical societies: American College of Surgeons, American Medical Association, American Medical Women's Association, American Society for Artificial Internal Organs, American Society of Transplant Surgeons, American Society of Transplantation, Association for Academic Surgery, Association of Women Surgeons, International College of Surgeons, International Liver Transplantation Society, New York Academy of Sciences, Pennsylvania Medical Society, Philadelphia County Medical Society, Society of Critical Care Medicine, Transplantation Society
Disclosure: Nothing to disclose.
Brian J Daley, MD, MBA, FACS, FCCP, CNSC Professor and Program Director, Department of Surgery, Chief, Division of Trauma and Critical Care, University of Tennessee Health Science Center College of Medicine
Brian J Daley, MD, MBA, FACS, FCCP, CNSC is a member of the following medical societies: American Association for the Surgery of Trauma, Eastern Association for the Surgery of Trauma, Southern Surgical Association, American College of Chest Physicians, American College of Surgeons, American Medical Association, Association for Academic Surgery, Association for Surgical Education, Shock Society, Society of Critical Care Medicine, Southeastern Surgical Congress, Tennessee Medical Association
Disclosure: Nothing to disclose.
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Sections Peritonsillar Abscess
Overview
Practice Essentials
Anatomy
Pathophysiology
Etiology
Epidemiology
Prognosis
Show All
Presentation
History
Physical Examination
Complications
Show All
Workup
Laboratory Studies
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Treatment
Approach Considerations
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6589 | https://hellothinkster.com/math-tutor/fractions/what-is-0.46-as-a-fraction | Math Tutor Explains: What is 0.46 as a fraction? | Thinkster Math
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What Is 0.46 As A Fraction
Math Tutor Explains: What is 0.46 as a fraction?
Numbers can be represented in various forms, including decimals, percentages, and fractions. With guidance from a math tutor, mastering these conversions becomes even easier. In this article, we’ll show you how to convert a decimal to a fraction, helping you build confidence and accuracy in your math skills!
Math Tutor's Solution: 0.46 as a fraction is 23/50
Methods
Converting 0.46 to a fraction, Step-by-Step from a Math Tutor
Step 1:
The first step to converting 0.46 to a fraction is to re-write 0.46 in the form p/q where p and q are both positive integers. To start with, 0.46 can be written as simply 0.46/1 to technically be written as a fraction.
Step 2:
Next, we will count the number of fractional digits after the decimal point in 0.46, which in this case is 2. For however many digits after the decimal point there are, we will multiply the numerator and denominator of 0.46/1 each by 10 to the power of that many digits. For instance, for 0.45, there are 2 fractional digits so we would multiply by 100; or for 0.324, since there are 3 fractional digits, we would multiply by 1000. So, in this case, we will multiply the numerator and denominator of 0.46/1 each by 100:
0.46×100 1×100=46 100\frac{0.46 × 100}{1 × 100} = \frac{46}{100}1×100 0.46×100=100 46
Step 3:
Now the last step is to simplify the fraction (if possible) by finding similar factors and cancelling them out, which leads to the following answer:
46 100=23 50\frac{46}{100} = \frac{23}{50}100 46=50 23
Math Tutor Suggests: Try to Convert Other Values to Fractions
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6590 | https://www.cuemath.com/algebra/floor-and-ceiling-function/ | LearnPracticeDownload
Floor And Ceiling Function
Floor function and ceiling function help in consolidating the function output as an integral value. The floor function gives an integer number value which is a numeric value lesser than the value of the function, and a ceiling function gives an integer number value which is a numeric value greater than the value of the function.
Let us check out the graph of floor and ceiling function, their properties, differences, with the help of examples, FAQs.
| | |
--- |
| 1. | What Is Floor Function And Ceiling Function? |
| 2. | Graph of Floor Function And Ceiling Function |
| 3. | Properties of Floor Function And Ceiling Function |
| 4. | Examples on Floor Function And Ceiling Function |
| 5. | Practice Questions |
| 6. | FAQs on Floor Function And Ceiling Function |
What Is Floor Function And Ceiling Function?
The floor function or a ceiling function helps in consolidating the value of a function into a simple integer number value. This helps in simplifying a numeric value in decimal into a simple integer number value.
Floor Function: It is a function that takes an input as a real number and gives an output that is an integral value less than the input real number. The floor function gives the greatest integer output which is lesser than or equal to a given number. The floor function is denoted by floor(x) or (\lfloor x \rfloor). Also sometimes the floor function is represented using double brackets and is written as . An example of floor function is (\lfloor 2.3 \rfloor) = 2, and (\lfloor -3.4 \rfloor ) = -4.
Ceiling Function: It is a function that takes an input as a real number and gives an output that is an integral value greater than the input real number. The ceiling function gives the least integer output which is greater than or equal to the given number. The ceiling function is denoted by ceil(x) or (\lceil x \rceil). Further, the ceiling function is sometimes written using reverse brackets or ]] x [[ or ] x [. An example of a ceiling function is (\lceil 4.7 \rceil) = 5, or (\lceil -2.8 \rceil) = -2.
Graph Of Floor Function And Ceiling Function
The graph of a floor function or a ceiling function can be taken as a step function or a break function, which are lines parallel to the x-axis. The range of input values is represented as a line and the final floor or the ceiling value is represented as a dot, in the graph of the floor function or the ceiling function.
The floor function for a range of values gives the output for only a minimum integer number value which is represented by a darkened dot. The ceiling function also takes a range of values as input but gives an output which is a maximum integer number value and is represented by a darkened dot.
Properties Of Floor Function And Ceiling Function
The following are some of the important properties of a floor function and ceiling function.
The floor function and the ceiling function give the output as an integral value.
The floor function has an integer number value less than the domain value of the function.
The ceiling function has an integer number value greater than the domain value of the function.
The floor function can be represented as (\lfloor x \rfloor ) = max {m ∈ Z | m < x}, and the ceiling function is represented as (\lceil x \rceil) = min{n ∈ Z | n > x}.
If (\lfloor x \rfloor) = m then we have m < x < m + 1 and if (\lceil x \rceil ) = n then n - 1 < x < n.
If (\lfloor x \rfloor) = m then x - 1 < m < x, and if (\lceil x \rceil) = n, then x < n < x + 1.
For a floor function n < x if and only if n < (\lfloor x \rfloor ).and for a ceiling function x < n, if and only if we have (\lceil x \rceil) < n.
For a floor function n < x, if and only if n < (\lfloor x \rfloor), and for a ceiling function x < n if and only if (\lceil x \rceil) < n.
The sum of a floor function and an integer can be represented as (\lfloor x + n\rfloor) = (\lfloor x \rfloor) + n, and the sum of a ceiling function and an integer can be represented as (\lceil x + n \rceil) = (\lceil x \rceil) + n.
The sum of two floor functions can be represented as (\lfloor x \rfloor ) + (\lfloor y \rfloor) < (\lfloor x + y \rfloor ) < (\lfloor x \rfloor ) + (\lfloor y \rfloor) + 1, and the sum of two ceiling functions can be represented as (\lceil x \rceil ) + (\lceil y \rceil ) - 1 < (\lceil x + y\rceil ) < (\lceil x \rceil ) + (\lceil y \rceil).
☛ Related Topics
Objective Function
Piecewise Function
Cube Root Function
Fractional Part Function
Into Function
Discontinuous Function
Examples on Floor Function And Ceiling Function
Example 1: What is the value for the floor function f(x) = (\lfloor 2x + 1 \rfloor ), given that x = 2.8?
Solution:
The given floor function is as follows.
f(x) = (\lfloor 2x + 1 \rfloor )
Let us substitute x = 2.8
f(2.8) = (\lfloor 2(2.8) + 1 \rfloor )
f(2.8) = (\lfloor 5.6 + 1 \rfloor )
f(2.8) = (\lfloor 6.6 \rfloor )
f(2.8) = 6
Therefore the value of the floor function is 6.
2. Example 2: What is the value for the ceiling function for f(x) = (\lceil 3x - 5 \rceil) given that x = 3.9?
Solution:
The given ceiling function is as follows.
f(x) = (\lceil 3x - 5 \rceil)
f(3.9) = (\lceil 3(3.9) - 5 \rceil )
f(3.9) = (\lceil 11.7 - 5 \rceil )
f(3.9) = (\lceil 6.7 \rceil )
f(3.9) = 7
Therefore the value of the ceiling function is 7.
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Practice Questions on Floor Function And Ceiling Function
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FAQs on Floor Function And Ceiling Function
What Is Floor And Ceiling Function in Maths?
The floor function and ceiling function gives an integer value for the decimal answer values of the function. The floor function gives a maximum value that is lesser than or equal to the functional numeric value, and the ceiling function gives a minimum value that is greater than or equal to the functional numeric value.
What Is The Notation Used To Write A Floor Function And Ceiling Function?
The floor function is represented as f(x) = (\lfloor x \rfloor), or as . And the ceiling function is represented as f(x) = (\lceil x \rceil ) or as ]] x [[.
What Is The Domain Of Floor Function And Ceiling Function?
The domain of floor function and the ceiling function can be any real number value, which can include a decimal or a fraction value.
What Is The Range Of Floor And Ceiling Function?
The range of a floor function and a ceiling function can be any integer value. The integral value can be a positive value or a negative integer value.
What Is The Difference Between Floor And Ceiling Function?
The floor function gives a value which is lesser than the f(x) value, and the ceiling function gives a value which is greater than the f(x). For a floor function we have (\lfloor x \rfloor ) < x and for the ceiling function we have x < (\lceil x \rceil ).
What Is The Target Of Floor Function And Ceiling Functions?
The target of a floor function is an integral value that is lesser than the value of the function, and the target of a ceiling function is an integral value that is greater than the value of the function.
How To Measure Floor Function And Ceiling Function?
The measure of the floor function and the ceiling function is based on the output value of the function. For a function x = 5.6, we have the floor function value of (\lfloor x \rfloor ) = 5, and the ceiling function value as (\lceil x \rceil ) = 6.
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6591 | https://blog.csdn.net/weixin_53463894/article/details/144002278 | Published Time: 2024-11-28T09:49:54+08:00
供应链 | OR论文解读：基于批量的差异定价的最优定价和库存管理策略_base stock 订货-CSDN博客
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供应链 | OR论文解读：基于批量的差异定价的最优定价和库存管理策略
最新推荐文章于 2025-09-15 14:25:36 发布
原创于 2024-11-28 09:49:54 发布·1.4k 阅读
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CC 4.0 BY-SA版权
版权声明：本文为博主原创文章，遵循CC 4.0 BY-SA版权协议，转载请附上原文出处链接和本声明。
文章标签：
#机器学习#人工智能#算法
编者按
本次解读的文章发表于 Operations Research，原文信息：Lu, Y., Chen, Y., Song, M., & Yan, X. (2014). Optimal pricing and inventory control policy with quantity-based price differentiation. Operations Research, 62(3), 512-523.
关键词：随机库存系统；动态定价；多种销售模式；动态规划
问题背景
“买二赠一”、“第二件半价”这类标语是生活中非常简单而常见的促销折扣，本文将这类，对不同购买件数的商品区分定价的销售方案称为批量销售（模式） (quantity sale mode). 将所有单位数量的产品以全价（或正常价格）销售的方案称则为单件销售（模式）(unit sales mode).
此外，动态定价(dynamic pricing)是过去十年中愈发常见的定价方式，批量销售模式则是动态定价的一种新的工具。具体来讲，本文将标准动态定价称为统一定价（策略） (uniform pricing)，将动态定价与基于批量销售的组合称为基于批量的定价（策略） (quantity-based pricing ).
本文研究的主题为基于批量的定价和库存控制之间的相互作用，并在以下两个条件的情景中讨论：
需求是随机的，并且依赖于价格。
补货订单可以在任何时期下达，并产生线性采购成本。
在每个周期初，公司需要完成三项决策：补货批量，销售模式，如果选择了批量销售模式，则需要进一步决定折扣的数额。而在周期末，所有的需求都由库存满足，同时会产生库存成本 (inventory cost) 或者短缺成本 (shortage cost). 公司的最终目标是最大化有限或无限时间跨度下的总利润期望。
那么，本文本质上需要回答三个问题：
首先，在动态补货的设置下，最优库存控制策略和最优销售模式是什么？同时采用单件和批量销售模式，还是选择其中之一？
其次，如果同时提供两种销售模式，顾客可以通过在批量销售模式中购买m个单位来节省一些钱。我们将这个节省的金额称为批量折扣金额(quantity-discound amount). 批量折扣金额如何依赖于最优库存水平？
第三，零售商可能通过提供批量销售模式获得额外利润。基于批量的价格差异化的收益如何依赖于系统参数？
本文为了解答以上问题，构建了两个模型：
作者发现，最优库存决策遵循基础库存策略 (base stock policy)，而如图1所示的最优销售策略取决于 tt t 期的最优订货至水平 yt∗y_t^∗y t∗.
图一：销售模式选择
使用随机效用（或估值）模型来刻画需求，可以进一步推导出销售策略的使用条件、最佳价格的单调性性质。
只要需要补货，当期最佳策略则一定不是批量销售。
q1∗q_1^q 1∗表示商品单价，q2∗q_2^q 2∗表示批量销售模式下每 mm m 个单位商品的总价，则批量销售模式的折扣为 mq1∗−p2∗mq_1^ - p_2^m q 1∗−p 2∗，可能随着库存水平上升而增加或减少。单调性的方向由效用函数决定。
q1∗q_1^q 1∗和 q2∗q_2^q 2∗都随着库存水平上升而减少，同时 q2∗−q1∗q_2^-q_1^q 2∗−q 1∗也随着库存水平上升而减少。
是否采用批量销售模式的两种模型的利润差异被称为基于批量的定价策略的价值 (value of quantity-based pricing). 本文证明该值随订购成本的增加而减少，但随着额外(m−1)(m-1)(m−1)件产品为顾客产生的增量效用（价值）的增加而增加。
文献综述
过往文献研究主题集中在：库存控制和批量销售模式的折扣(quantity discounts)之间的关系、库存和定价协同问题、动态定价问题。
库存控制：现有文献分为两条研究主线，研究买方的经济订货量(economic order quantity), 研究卖方的最佳基于批量的定价策略。本文是后者研究方向的拓展，重要参考文献包括 Altintas, Erhun, and Tayur (2008).
库存和定价协同：这类文献研究随机且依赖价格的需求，在有限/无限时间范围内动态地做出定价和库存决策，具体来讲，即需要决策订货量、单位价格。
而经典动态定价问下仅研究不同周期的定价策略，不允许补货、基于批量的差异定价。
相比于过去的研究，本文对于动态定价模型拓展以及理论研究有以下贡献：
首先，本文将卖方库存控制 (seller’s inventory control) 模型扩展到在随机需求、周期性检查的库存环境中，并对基于批量的定价策略进行建模。
其次，本文证明了需求模型所做的三个假设可以通过简单的规律性假设（递增的虚拟价值函数）来统一。
基础模型构建
为了简化参数表达，文章考虑参数稳定的系统，即固定参数不随周期变化的情况（该系统可以扩展到参数不固定的场景）。
对于每一个周期 t=1,2,⋯ ,nt=1, 2, \cdots, n t=1,2,⋯,n 公司在观测到周期初库存水平 xtx_t x t后，确定订单的商品数量，且单位成本为 cc c. 不考虑前置时间 (lead time), 库存立即变为 yt≥xty_t \geq x_t y t≥x t。同时，公司可以选择两种定价 pt=(p1t,p2t)p_t = (p_{1t}, p_{2t})p t=(p 1 t,p 2 t)：
单件销售模式的价格 p1tp_{1t}p 1 t,
批量销售模式价格 p2tp_{2t}p 2 t.
此后，需求被观测到。tt t 周期末，未满足的需求（剩余库存）或积压库存被结转至下一周期。每一周期都由以下几种参数刻画：
市场规模 DtD_t D t. 假设 DtD_t D t 随时间独立同分布，且 μ1=E(Dt)>0\mu_1 = \mathbb{E}(D_t) > 0 μ 1=E(D t)>0.
随机市场需求 ξt=[λ1(pt)+mλ2(pt)]Dt+εt\xi_t= [\lambda_1(p_t)+ m\lambda_2(p_t)]D_t + \varepsilon_t ξ t=[λ 1(p t)+m λ 2(p t)]D t+ε t, 其中 λ1(pt)\lambda_1(p_t)λ 1(p t), λ2(pt)\lambda_2(p_t)λ 2(p t)分别代表单件销售，批量销售的市场份额 λ1+λ2≤1\lambda_1 + \lambda_2 \leq 1 λ 1+λ 2≤1,
由于这两种模式基本上销售相同的产品，因此它们的需求首先取决于需要该产品的客户数量，然后取决于客户如何在两种模式之间分配需求。因此，销售单位和数量的需求必须具有某种强相关性，这一点由该需求模型的第一部分 λ1(pt)+mλ2(pt)\lambda_1(p_t)+ m\lambda_2(p_t)λ 1(p t)+m λ 2(p t)刻画。
除价格外，其他因素也会影响需求，例如天气和社交活动。同时对价格不敏感的买家数量也是影响需求的不确定因素。因此，模型中采用 εt=ε1t+mε2t\varepsilon_t = \varepsilon_{1t} + m \varepsilon_{2t}ε t=ε 1 t+m ε 2 t来分别表示两种销售模式的需求的不确定性，由此二者也不一定完全相关了。为简化符号表达，价格表达中去掉周期 tt t 的下标。
根据常规产品定价的一般规律，λi(pi,pj)\lambda_i(p_i, p_j)λ i(p i,p j)随 pip_i p i减少，随 pjp_j p j增加。文章模仿过往文献进一步假设价格向量 ptp_t p t与市场份额 λt\lambda_t λ t之间存在一一对应关系，因此本文可以将市场份额而不是价格作为决策变量，并且定义出市场份额、价格的可行域：
举例说明
基于价值的需求模型的收益计算
收益期望
假设 VV V 表示一单位产品的价值（即效用），其累积分布函数为 G(p)G(p)G(p)，支撑集为[0,vˉ][0, v̄][0,v ˉ]. 假设 mm m 单位产品的价值为 aVaV aV, a∈(1,m)a \in(1,m)a∈(1,m). 因此，单位销售模式在 p1p_1 p 1 的市场份额和批量销售模式在 p2p_2 p 2 的市场份额可分别推导如下：
其中 V−p1>0V-p_1 >0 V−p
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【P or teus】S-policy 和 (s,S)-policy
zte10096334的博客
11-16 3024
1. 背景 考虑单产品多周期的 库存 管理问题，寻求 最优 的 订货 策略，使得总体成本最低。在每个周期，如果 订货 过多，导致 库存 多余，会产生 库存 持有成本；如果 订货 较少，导致缺货，会有延迟交货成本（这里假设缺货的情况下不会丢失订单）。 2. 模型 参数 含义 cHc_HcH 单位产品 库存 持有成本 cPc_PcP 单位产品延迟交付成本 ccc 单位产品 订货 成本 DDD 单周期内需求...
base-stock 策略的相关证明
天天向上的专栏
08-10 3457
base-stock 策略适用一个多阶段报童模型，构建动态规划模型： 状态变量：每阶段初始 库存 水平 xt−1xt−1x_{t-1} 决策变量：补货上限 ytyty_t, （yt=xt−1+Qtyt=xt−1+Qty_t=x_{t-1}+Q_t，QtQtQ_t 为 订货 量） 状态转移方程： xt+1=yt−ξtxt+1=yt−ξtx_{t+1}=y_t-\xi_t 递推函数：ft(xt−1)f...
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base-stock 策略的相关证明
8-23
3. 根据性质 : 最小化一个联合凸函数形成的单变量函数仍是凸函数,可以证明第三个命题。 通过从最后一阶段递推,可以得出每个阶段的 最优 订货 策略都为 : base-stock 策略,这与 (s, S) 策略的最大区别在于 : 多阶段报童模型里面没有固定 订货 成本,若含有固定 订货 成本,则需要证明递推函数为 K-凸,然后得出 最优 订货 策略...
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在这些设定下,Huh et al.(2009) 与 Goldberg et al.(2016) 分别证明了 : 当惩罚成本高且使用固定 订货 策略时,基本 库存 策略在提前期较大时具有渐近 最优 性 (asymptotic optimality). Xin (2021) 将上述结果结合,提出了一个截断基本 库存 策略 (capped base stock policy), 在提前期增长时保持渐近 最优,且在提前期较...
base stock 订货, （s, S）订货
天天向上的专栏
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多阶段 库存 管理中经常看到这个策略，好几年一直不太理解其内涵。这几天看了赵修利教授的两篇 论文，大致明白了这个策略是怎么操作的。 这个策略有一个 库存 水平临界点 和 补货上限，当前 库存 水平小于 库存 临界点时，不补货。当前 库存 水平大于临界点时，补货到补货上限。用下面公式表示： 要证明 base stock 策略是 最优 订货 策略时，需要证明动态规划的递推函数的凹凸性。
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天天向上的专栏
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最近看赵修利于龚希亭的两篇 论文，都是用多阶段报童模型建模，只考虑lost sales的情况，通过证明 最优 订货 策略是 base stock 策略，进而得出一系列性质。 文章中大部分是证明，不过读了受益匪浅，我计划将数学证明自己推一遍。 状态变量： 初始 库存 水平 初始资金 补货上限 其他参数： ...
【论文 阅读】Research Handbook on Invent or y ManagemenT --- Ch...
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1.2.3 Or der-Up-To Policies (base-stock policy) OUT 策略,即我们平时常说的 base stock policy。 OUT level S 的定义为 : 在任意周期 t 内,给定状态 x ,订货 量为 : 我们需要回答的是以下两个问题 : 在OUT策略下,我们应该如何进行优化? 如果使用这一策略,会造成多少损失?损失的gap是多少?在哪些条件下可以最小...
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穷举/动态规划 求解商品 最优 价格问题
pyb的博客
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问题大意 : 商店进行优惠活动,每一件商品的价格达到500以上将对下一件商品减少200,达到1000减少500,达到2000减少1000.也可以将前几件商品看作一个组合(比如299，399，599),那么下一件商品就会减少500元,但是前三个商品不优惠.现在需要购买一系列商品,如何实现 最优 的购买策略?此题穷举法可解 : 首先对所有商品进行排序,按照价格从高到低进行排序.然后寻找一个商品区间[0，k1],然后
OM | 电子商务平台中的合约选择：批发合约 or 代理合约？
运筹OR帷幄
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本次 解读 的文章发表于European Journal of Operational Research，原文摘要总结如下：本文旨在分析一个垄断电商平台与其供应商之间的战略合约问题。本文考虑了两个相互竞争的供应商：一个大 批量 供应商（具有较大的潜在需求），一个小 批量 供应商（具有较小的潜在需求）。每个供应商可以选择以下两种合约：批发合约（Wholesale Contract）或代理合约（Agency Contract）。.........
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defadd_product(name, price, stock, categ or y): product_id =len(data base) +1 product = { 'product_id': product_id, 'name': name, 'price': price, 'stock': stock, 'categ or y': categ or y } data base[product_id] = product print(f"商品{product_id}已添加到数据库。") ...
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base,stock,和 bonus三个方面。一般来说,base 在glassdo or 等网站、论坛上都是比较容易找到的,而且,同等水平的engineer,一般 base 不会差别太大。但是 stock 和 bonus 则处于一个晦暗不明的状态,高低之间或许是几倍的差距。今天我以google为例,来教大家什么才是跟 ...
AI应用架构师如何优化智能产品 定价 AI平台
SuperAGI2025
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{ // Fetch or der details from data base by or der ID // Return the or der details } function validate Or derDetails(or derDetails) { // Check if the details in the or der match the stock inf or mation // If there's a mismatch, raise an err or } function generateShipmentList(or derDetails) { // ...
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whaosoft~aiotの开发板商城
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通过Python代码实现了模型构建、求解 和 可视化，验证了跨渠道退货能通过成本节约、需求刺激 和 渠道协同三重机制提升 供应链 利润，并提出了 最优 跨渠道退货策略及收益共享契约设计，为双渠道 供应链 管理提供了理论 和 实践...
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2025 年 1 月 7 日，弗吉尼亚联邦大学与纽约州立大学 Albany 分校的 Reshchikov 等人在《Nanotechnology》期刊发表文章，基于光致发光（PL）实验与第一性原理 HSE 计算，系统研究了 GaN 中 BeGa、CN 及 MgGa 受主与氢的复合、钝化 和 再激活过程；结果表明，350–800 °C 退火释放的氢会钝化 BeGa 与 CN 受主，导致其特征黄光带强度骤降，而 900 °C 以上退火则通过氢脱附恢复受主发光，且 BeGaHi 复合体解离能高达 2.2 eV，显著高于
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本文介绍了 机器学习 和 大模型中的核心数学概念——向量及其相关运算。向量是具有方向 和 大小的量，可以理解为空间中的"箭头"。重点讲解了三个关键概念：1）线性组合，即通过加权向量相加生成新向量，在词向量运算 和 神经网络中广泛应用；2）基向量，作为生成整个空间的最小向量集，类似于RGB三原色混合原理；3）维数，表示空间的独立方向数量，决定了数据的表示自由度。这些概念构成了 机器学习 的基础，影响着模型参数更新、特征提取 和 高维表示等关键环节。理解这些向量运算原理，是掌握大模型开发 和 优化的重要前提。
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6592 | https://www.superprof.es/apuntes/escolar/fisica/mecanica-clasica/cinematica/ejercicios-resueltos-de-caida-libre.html | Problemas de caida libre
RecursosEscolarFísicaMecánica ClásicaCinemáticaEjercicios resueltos de Caída libre
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La caída libre es un fenómeno físico fundamental que ocurre cuando un objeto se mueve bajo la influencia exclusiva de la gravedad, sin resistencia del aire u otras fuerzas externas. Este concepto, central en la mecánica clásica, permite comprender cómo los objetos caen hacia la Tierra y cómo varían sus propiedades cinemáticas durante este proceso.
Los ejercicios de caída libre constituyen una herramienta esencial en el estudio de la física, ya que permiten explorar y aplicar los principios de la cinemática en un contexto simple pero crucial. en los siguientes ejercicios no consideramos la resistencia del aire.
1Se deja caer una pelota desde la parte alta de un edificio, ¿cuál es la velocidad de la pelota a los dos segundos de comenzar a caer?
1 Elegimos la dirección hacia abajo como positiva y como la pelota se deja caer, su velocidad inicial es cero
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es positiva ya que elegimos al inicio que la dirección hacia abajo es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
2Se lanza una pelota verticalmente hacia arriba con velocidad , ¿qué altura alcanza la pelota?, ¿cuál es el tiempo empleado por la pelota hasta caer al piso?
1 Elegimos la dirección hacia arriba como positiva y notamos que su velocidad final es cero
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos y despejamos
Así, la altura es
6Para calcular el tiempo empleado hasta regresar al piso, notamos que . Utilizamos la fórmula
7Sustituimos los datos conocidos y despejamos
Así, se tiene que y . Ambas soluciones corresponden al tiempo en que la pelota se encuentra en el piso. La solución pedida es
3Se deja caer verticalmente un objeto desde una altura de . ¿Con que velocidad toca el objeto el piso y cuánto tiempo le lleva realizarlo?
1 Elegimos la dirección hacia abajo como positiva y notamos que la velocidad inicial es cero
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es positiva ya que elegimos al inicio que la dirección hacia abajo es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
6Para encontrar el tiempo empleado, utilizamos la fórmula
7Sustituimos los datos conocidos
4Una pelota se lanza verticalmente hacia arriba desde el suelo a una velocidad de . ¿Cuánto tiempo le tomará estar a 30 metros sobre el suelo?
1 Elegimos la dirección hacia arriba como positiva
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
Utilizando la fórmula cuadrática obtenemos y que corresponde al tiempo en que va subiendo y bajando respectivamente.
5Un objeto se lanza verticalmente hacia arriba desde el suelo. Si alcanza una altura de 18 metros en 3 segundos. ¿A qué velocidad se lanzó el objeto?
1 Elegimos la dirección hacia arriba como positiva
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
6Un objeto se deja caer verticalmente hacia abajo y llega al piso con una velocidad de . ¿Cuál es la distancia recorrida por el objeto?
1 Elegimos la dirección hacia abajo como positiva y notamos que la velocidad inicial es cero
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es positiva ya que elegimos al inicio que la dirección hacia abajo es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
7Una pelota se lanza verticalmente hacia arriba a una velocidad de . ¿En cuánto tiempo alcanza el punto más alto de su recorrido? ¿Cuál es la velocidad cuando regresa al punto de partida?
1 Elegimos la dirección hacia arriba como positiva y notamos que en el punto más alto su velocidad es cero.
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
La pelota alcanza su punto más alto en 1.02 segundos
6Para calcular la velocidad cuando vuelve al punto de partida, no tamos que la posición final es igual a la inicial. Empleamos la fórmula
7Sustituimos los datos conocidos
La velocidad tiene las misma magnitud que la inicial, pero con dirección contraria
8Una pelota se lanza verticalmente hacia arriba a de manera que alcanza su punto más alto y regresa a su punto de inicio. ¿Cómo es la velocidad de regreso de la pelota respecto a su velocidad durante su ascenso cuando pasa por la misma posición?
1 Elegimos la dirección hacia arriba como positiva y notamos que en el punto más alto su velocidad es cero.
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Como , luego
Entonces la magnitud de ambas velocidades es la misma, pero sus direcciones son contrarias. De hecho, lo mismo ocurre en cada punto del recorrido de la pelota.
9¿A qué velocidad debe lanzarse una pelota verticalmente hacia arriba, para que después de dos segundos tenga una velocidad de ?
1 Elegimos la dirección hacia arriba como positiva
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es negativa ya que elegimos al inicio que la dirección hacia arriba es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituyendo los valores conocidos, se obtiene
10Una pelota se lanza verticalmente hacia abajo desde la parte más alta de un edificio. Si recorre 50 metros en 3 segundos, ¿cuál es su velocidad inicial?
1 Elegimos la dirección hacia abajo como positiva
2 Indicamos los datos conocidos
3Al tratarse de caida libre, la aceleración es igual a la gravedad , la cual es positiva ya que elegimos al inicio que la dirección hacia abajo es positiva
4Como la aceleración es constante, se tiene un problema de MRUA, por ello empleamos la fórmula
5Sustituimos los datos conocidos
Si tienes dudas, aquí puedes consultar la teoría.
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Gaspar
Soy matemático y es un placer poder compartir mi gusto por las matemáticas.
Resumen
### ¿Qué es la cinemática y cuál es su importancia?
Teoría
### Entendiendo la Posición en Cinemática
### Analizando la Velocidad en el Movimiento
### Conceptos Fundamentales de la Aceleración
### Desglosando el Movimiento Rectilíneo
### Explorando el Movimiento en Dos y Tres Dimensiones
Fórmulas
### Lista de fórmulas clave de cinemática
Ejercicios interactivos
### Ejercicios Interactivos de Movimiento rectilíneo uniforme (MRU)
### Ejercicios Interactivos: Movimiento rectilíneo uniformemente acelerado (MRUA)
### Ejercicios Interactivos de Caída libre
### Ejercicios interactivos de Tiro parabólico
### Ejercicios interactivos de Movimiento circular uniforme (MCU)
### Ejercicios Interactivos de Movimiento circular uniformemente acelerado (MCUA)
### Ejercicios interactivos de aplicaciones cotidianas de la cinemática
Otros ejercicios
### Ejercicios Resueltos de Movimiento rectilíneo uniforme (MRU)
### Ejercicios de Movimiento rectilíneo uniformemente acelerado (MRUA)
### Ejercicios resueltos de Caída libre
### Ejercicios resueltos de Tiro parabólico
### Ejercicios de Movimiento circular uniforme (MCU)
### Ejercicios de Movimiento circular uniformemente acelerado (MCUA)
### Aplicaciones prácticas y cotidianas de la cinemática
Déjanos un comentario
Cancelar la respuesta
Martin Escutia
En el problema siete está mal la unidad de medida de la aceleración, está se mide en m/s^2 no en m^2
Antonio Tapia de Superprof
Hola, una disculpa por el error ya se corrigió.
William Toalombo
5.- Dos motocicletas están separadas una distancia de 2 km. La primera se mueve a
velocidad (constante) de 25 m·s-1. La segunda, parte de parado con una aceleración
constante de 3 m·s-2. Calcule:
a) Cuanto tiempo tardarán en encontrase, si una va en dirección opuesta a la otra.
b) Si las dos van en la misma dirección, ¿cuánto tiempo tardará la segunda
moticicleta en pillar a la primera?
c) En este último caso, ¿en qué punto la pillará?
María J. Rosales
Hola, disculpa por las molestias. Estoy estudiando para mí examen de Física con el tema MCU y estoy practicando con tus ejercicios, pero algo que no entiendo es en el ejercicio 7. En aquel ejercicio, no me pide Fuerza Centrípeta y si la calculo, la aceleración Centrípeta está mal y tmp me dicen acerca del peso de algún objeto. Espero mi comentario se entienda, sigo aprendiendo sobre este tema y no quiero tener dudas ni equivocaciones. Saludos y le mando buena suerte. Buen artículo!!
Antonio Tapia de Superprof
Hola tienes razón, era un error una disculpa ya se corrigió.
Al octavo ejercicio el creador le dio una solusion erronea al obviar que 1 rev/s es igual a 1 rad/s
Antonio Tapia de Superprof
Hola una disculpa por el error, ya se corrigió.
Hola pero en el ejercicio 7 dice que son 4 revoluciones por 1 segundo, cuando luego se cambia a 1 minuto, asi que da 4 revoluciones por minuto?
Antonio Tapia de Superprof
Hola que bueno que visitas nuestra pagina, en cuanto a tu duda, si te refieres al ejercicio de la rueda de un auto la frecuencia es 4 rev/s y el tiempo es un 1 minuto, para la velocidad angular solo se usa la frecuencia, en cuanto el tiempo se cambia a segundos, entonces no hay cambio a revoluciones por minuto. |
6593 | https://courses.grainger.illinois.edu/phys487/fa2018/discussion/Disc12.pdf | Phys 487 Discussion 12 – Time-Dependent PT with 2-State Systems Same formula summary as last week : if our Hamiltonian H consists of a time-independent part H0 with known eigenvalues En (0) and eigenstates n(0) plus and a much smaller time-dependent part H’, then we get the following results : Given ● H(t) = H (0) + ′ H (t) , ● En (0), n(0) { } = the eigen- of H (0) ● initial state ψ (t = 0) = i(0) then ψ (t) = cn(t) e−iωnt n(0) n ∑ with iℏ" cf (t) = ′ H fn eiω fntcn(t) n ∑ • ω fn ≡E f (0) −En (0) ( ) / ℏ • ′ H fn ≡f (0) ′ H n(0) & to 1st order in ′ H ≪H (0), cf (t) ≈δ fi + 1 iℏ ′ H fi( ′ t ) 0 t ∫ eiω fi ′ t d ′ t The first box is the solution form, the second box is the exact differential equation it must satisfy, and the third box is the 1st-order perturbative solution. Some names: ωfi is called the transition frequency for going from initial state i to final state f, while cf(t) is the transition amplitude for doing the same thing. The transition probability that we are usually trying to calculate is Pf →i = cf (t) 2 Problem 1 : Two-State hydrogen atom in electric field adapted from Griffiths 9.1 Systems with only two independent states are excellent sandboxes for playing around with time-dependent potentials, e.g. you can write down all the matrix elements of ′ H fn since there are only four of them. Let’s take a specific case. A hydrogen atom is placed in a time-dependent electric field !
E = E(t) ˆ z . (a) Calculate all four matrix elements ′ Hij of the perturbation ′ H = eE z between the ground state (n = 1) and the quadruply degenerate first excited states (n = 2). (b) Show that the diagonal elements ′ Hii of the perturbation are both zero for all five states. ▶ NOTE: There is only one integral to be done here, if you exploit oddness with respect to z; only one of the n = 2 states is “accessible” from the ground state by a perturbation of this form, and therefore the system functions as a two-state configuration — assuming transitions to higher excited states can be ignored. (c) It is very commonly the case that the diagonal elements ′ Hii of a time-dependent perturbation are zero. In this case, the exact differential equations iℏ" cf (t) = ′ H fn eiω fntcn(t) n ∑ for the two coefficients ca(t) and cb(t) (using Griffiths’ notation for two states) reduce to two coupled differential equations without summation signs. Show that the above reduces to these two in the case ′ Hii = 0 : !
ca = 1 iℏ ′ Hab e−iω0tcb and !
cb = 1 iℏ ′ Hbaeiω0t ca These are Griffiths’ equation 9.13, on which the first part of his chapter 9 are based. NOTATION CHANGE : We will henceforth stop placing a superscript (0) on the states and energies of the unperturbed system. Reason? We are ALWAYS referring to the states of the unperturbed system in time-dependent PT, so there is no reason to flag it with such notation. As we mentioned in class, the goal of time-dependent PT is quite different from that of time-independent PT. • t-indep. PT : calculate changes to the eigenenergies & eigenstates of H0 produced by the perturbation H ′ • t-dep. PT : calculate transition probabilities caused by H ′ between states of the unperturbed system H0 Problem 2 : Rabi flopping frequency adapted from Griffiths 9.7 A rare example of a system that can be solved exactly is the important case of a two-state problem with a sinusoidal oscillating potential. The system is often an atom or a molecule, with two states of particular interest or relevance ; the perturbation usually comes from an incident electromagnetic wave whose frequency ω is tuned to the transition frequency ωab = (Ea-Eb)/ℏ between the two states. Here is a simplified expression for the sinusoidal perturbation V(!
r) cos(ωt).after it is applied to a system with two states a and b in the case that the driving frequency ω is very close to the transition frequency ωab : 1 ′ Hba = Vba 2 e−iωt , ′ Hab = Vab 2 eiωt , ′ Haa = ′ Hbb = 0 with Vab ≡ψ a V ψ b .
(a) Solve the two coupled equations you obtained in problem 1(c) using the initial conditions ca(0) = 1 and cb(0) = 0 (i.e. the system starts in state a at time t = 0). Express your results for ca(t) and cb(t) in terms of ωr ≡1 2 ω −ω ab ( ) 2 + Vab / ℏ ( ) 2 , called the Rabi flopping frequency. Answers in footnote2.
(b) Determine the transition probability P a→b(t) and show that it never exceeds 1. Confirm that ca(t) 2 + cb(t) 2 = 1 at all times. (What would it mean if that were not true? Ask if you’re not sure!) (c) Check that P a→b(t) reduces to the perturbation theory result Pa→b(t) = cb(t) 2 ≈Vab 2 sin2 ω ab −ω ( )t / 2 ⎡ ⎣ ⎤ ⎦ ω ab −ω ( ) 2 when the perturbation is “small”, and state precisely what small means in this context, as a constraint on V. (d) At what time does the system first return to its initial state? 1 Griffiths §9.1.3 shows how the approximation ω ≈ ω0≡ωab is applied. 2 Q2(a) results : cb(t) = − i 2ℏωr Vbaei(ωab−ω )t/2 sin(ωrt), ca(t) = e i ω−ωab ( )t/2 cos ωrt ( )+ i ωab −ω 2ωr ⎛ ⎝ ⎜ ⎞ ⎠ ⎟sin ωrt ( ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥. |
6594 | https://proofwiki.org/wiki/Definition:Excircle_of_Triangle/Excenter | Definition:Excircle of Triangle/Excenter - ProofWiki
Definition:Excircle of Triangle/Excenter
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<Definition:Excircle of Triangle
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1 Definition
2 Also known as
3 Also see
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Definition
The center of an excircle of a triangle is called an excenter of the triangle.
In the above diagram, I a I a is the excenter of △A B C△A B C with respect to a a.
Also known as
In British English the spelling is usually excentre.
Some (usually older) sources refer to this as an ex-center or ex-centre.
The term ecenter or ecentre can also be found, particularly in the U.S.
Also see
Results about excenters of triangles can be found here.
Sources
1998:David Nelson: The Penguin Dictionary of Mathematics(2nd ed.)... (previous)... (next): excentre
2008:David Nelson: The Penguin Dictionary of Mathematics(4th ed.)... (previous)... (next): excentre (ecentre)
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6595 | http://www.csgnetwork.com/barcorrecttcalc.html | Temperature Barometer Reading Correction Calculator
Home | Top Free Apps |
CALCULATORS and CONVERTERS | Back
Temperature Barometer Reading Correction Calculator =================================================== This calculator requires the use of Javascript enabled and capable browsers. This script determines the error in an observed barometer reading, based on temperature (only) expansion and contraction differences. The metal housing of the barometer and the mercury both expand as the temperature increases. However, they expand at different rates. The barometer is calibrated for 0 degrees C (32 degrees F). For different temperatures a correction must be applied to compensate for the different expansions of the housing and the mercury. For temperatures greater than 0 degrees C, the correction should be added to the observed height; for temperatures less than 0 degrees C, the very minor correction should be subtracted. Contraction differences are extremely small and almost negligible. The default values are typical for the general lower desert area of Southern California. To use the calculator, enter the temperature and the known and observed barometer reading, in any of the designated values, then click on Calculate. Results are returned in both inches of mercury and millibars. There is also a temperature correction chart below the calculator. The standard value for atmospheric pressure at sea level (ATM) is equal to: 1 atm = 29.92 in Hg (inches of mercury) 1 atm = 760 mm Hg (millimeters of mercury) 1 atm = 1013.20 millibars 1 atm = 14.7 psi (pounds force per square inch) 1 atm = 1013.20 hPa (hectopascals) To further complicate matters, there are actually 2 values for an atmosphere. The first is generally called "standard" and the other is termed as "scientific". These are based on standard atmosphere calculations. Other pressure correction calculators are for altitude correction only, for gravity (latitude) correction only, for temperature and altitude correction and for gravity, altitude, and temperature correction. Discover more calculators Calculator browsers Computer Calculators Web browser computer browser Unit Number Required Data Entry Air Temperature Degrees C F Observed Barometer Reading Inches Hg Millibars Calculated Results Actual Barometer Correction inches of mercury Rounded Barometer Correction inches of mercury Actual Barometer Corrected inches of mercury Rounded Barometer Corrected inches of mercury Actual Barometer Correction millibars Rounded Barometer Correction millibars Actual Barometer Corrected millibars Rounded Barometer Corrected millibars Correction Chart For Relative Expansion Of Mercury And Brass Scale To 32°F Standard ------------------------------------------------------------------------------------- Temp hg col °F Observed reading of the barometer, in inches 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30 30.5 31.0 Correction to be subtracted from observed reading 40.026.026.027.027.028.028.029.029.030.030.031.031.032 42.030.031.032.032.033.033.034.035.035.036.036.037.038 44.035.036.036.037.038.038.039.040.040.041.041.042.043 46.039.040.041.042.043.043.044.045.046.046.047.048.049 48.044.045.046.047.047.048.049.050.051.052.053.054.054 50.048.048.050.051.052.053.054.055.056.057.058.059.060 52.053.054.055.056.057.058.059.060.061.062.064.065.066 54.057.059.060.061.062.063.064.066.067.068.069.070.071 56.062.063.064.066.067.068.069.071.072.073.074.076.077 58.066.068.069.070.072.073.074.076.077.078.080.081.082 60.071.072.074.075.077.078.080.081.082.084.085.087.088 62.076.077.079.080.082.083.085.086.088.089.091.092.094 64.080.082.083.085.086.088.090.091.093.095.096.098.099 66.085.086.088.090.091.093.095.096.098.100.101.103.105 68.089.091.093.094.096.098.100.102.103.105.107.109.110 70.094.095.097.099.101.103.105.107.109.110.112.114.116 71.096.098.100.102.103.105.107.109.111.113.115.117.119 72.098.100.102.104.106.108.110.112.114.116.118.120.122 73.100.102.104.106.108.110.112.115.116.118.120.122.124 74.103.105.107.109.111.113.115.117.119.121.123.125.127 75.105.107.109.111.113.115.117.119.122.124.126.128.130 76.107.109.111.113.116.118.120.122.124.126.128.131.133 77.109.112.114.116.118.120.122.125.127.129.131.133.136 78.112.114.116.118.120.123.125.127.129.132.134.136.138 79.114.116.118.121.123.125.127.130.132.134.137.139.141 80.116.118.121.123.125.128.130.132.135.137.139.142.144 81.118.121.123.125.128.130.132.135.137.140.142.144.147 82.121.123.125.128.130.133.135.137.140.142.145.147.149 83.123.125.128.130.133.135.138.140.142.145.147.150.152 84.125.128.130.133.135.138.140.143.145.148.150.153.155 85.127.130.132.135.137.140.143.145.148.150.153.155.158 86.130.132.135.137.140.142.145.148.150.153.155.158.161 87.132.134.137.140.142.145.148.150.153.155.158.161.163 88.134.137.139.142.145.147.150.153.155.158.161.163.166 89.136.139.142.144.147.150.153.155.158.161.164.166.169 90.138.141.144.147.150.152.155.158.161.163.166.169.172 92.143.146.149.152.154.157.160.163.166.169.172.174.177 94.147.150.153.156.159.162.165.168.171.174.177.180.183 96.152.155.158.161.164.167.170.173.176 179.182.185.188 98.156.160.163.166.169.172.175.178.181.185.188.191.194 100.161.164.167.171.174.177.180.183.187.190.193.196.200 Updated 8.15.11 Leave us a question or comment on Facebook Search or Browse Our Site Discover more computer calculators Web browser Calculators browsers Computer Calculator browserAutomotive Aviation Business Communications Computer Construction Electronics Everyday Use Financial General Interest Geography Related HealthHome Repair Javascript Lottery Marine Math Help Math Scripts Mechanical Money Science Sports Time Weather Registered® Trademark™ and Copyright© 1973 - 2025 CSG, Computer Support Group, Inc. and CSGNetwork.Com All Rights Reserved Home | Advertising | Calculators and Converters | Contact Us | Javascript | Sitemap | Glossary | Top Free Apps |
6596 | https://www.gauthmath.com/solution/1835764050479201/What-is-the-percent-by-mass-of-oxygen-in-CaOH_2-formula-mass-74-1J-1-21-6-2-43-2 | Question
Solution
Risposta: 𝑍L = $$2,5$$2,5 + j$$\sqrt{96,8^2 - 2,5^2}$$96,82−2,52.
Passo 1: Calcolare il modulo dell'impedenza del carico: |𝑍L| = 𝑅𝑂𝑆 = $$2,5$$2,5.
Passo 2: Calcolare il valore massimo dell'impedenza del carico: |𝑍L| = $$2,5$$2,5.
Passo 3: Trovare il valore massimo della parte reale dell'impedenza del carico: Re(𝑍L) = $$2,5$$2,5.
Passo 4: Trovare il valore massimo della parte immaginaria dell'impedenza del carico: Im(𝑍L) = $$\sqrt{96,8^2 - 2,5^2}$$96,82−2,52.
Passo 5: Calcolare l'impedenza del carico massima: 𝑍L = Re(𝑍L) + jIm(𝑍L).
Passo 6: Sostituire i valori calcolati: 𝑍L = $$2,5$$2,5 + j$$\sqrt{96,8^2 - 2,5^2}$$96,82−2,52.
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6597 | https://rtullydo.github.io/cryptography-notes/section-fermat.html | Fermat's little theorem and computing inverses
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\numberwithin e q u a t i o n s e c t i o n\numberwithin e q u a t i o n s e c t i o n
Mathematical Cryptography
Ryan Tully-Doyle
Contents
PrevUpNext
ContentsPrevUpNext
Front Matter
1 Preliminaries
The course
Why “mathematical” cryptography?
A simple example: the shift cipher
2 Number theory
Integers and divisibility
Common divisors and the Euclidean algorithm
Modular arithmetic and the ring of integers modulo n n
The fast powering algorithm
Finite fields
Fermat's little theorem and computing inverses
Conclusion
3 Group based methods
A naïve cryptosystem and some related questions
Discrete logarithms
Diffie-Hellman key exchange
The Elgamal cryptosystem
Groups
Attacking the discrete log problem
The Chinese Remainder theorem and Pohlig-Hellman
Homomorphisms and the lunchtime attack
4 Elliptic curves and applications
Introduction to elliptic curves
Elliptic addition on E.E.
Elliptic curves over finite fields
Discrete logs and the ECDLP
Elliptic curve Diffie-Hellman key exchange and El Gamal
5 Digital signatures
Introduction to digital signatures
Elgamal digital signatures
Subgroups and cyclic subgroups
DSA and Elliptic Curve DSA
Bitcoin - an application of ECDSA
6 Probability and non-deterministic methods
Counting
Permutations and combinations
Primality testing
Pollard's ρ ρ method
Back Matter
A Code repository
7 Exercises
Set 1
Set 2
Set 3
Set 4
Authored in PreTeXt
🔗
Section 2.6 Fermat's little theorem and computing inverses
🔗
Exponents in F∗p F p∗ have several useful and noteworthy properties. First, we have Fermat's little theorem, which essentially states that multiplying some element g g in F∗p F p∗ by itself p p times will always return to g.g.
🔗
Theorem 2.6.1.(Fermat's little theorem).
p p a a p∤a,p∤a,
a p−1≡1 mod p.a p−1≡1 mod p.
🔗
###### Proof.
To be added, need to choose one that fits with the course.
🔗
Fermat's little theorem gives us the ability to perform a relatively fast computation of inverses with the help of the fast powering algorithm.
🔗
Algorithm 2.6.2.Fast inverse mod p p.
a∈F∗p.a∈F p∗.
1. Since a a then must be smaller than p,p,p∤a,p∤a, and so a p−1=1 a p−1=1 in F∗p.F p∗.
2. But then a p−2≡a−1.a p−2≡a−1.
3. Apply the fast powering algorithm to compute a p−2.a p−2. The result is a−1.a−1.
🔗
Our final observation to close out this first unit of the course is to note that it is possible for an element to have a power c c smaller than p−1 p−1 such that a c=1 a c=1 in F∗p.F p∗.
🔗
Definition 2.6.3.
The order of a a in F∗p F p∗ is the smallest positive integer k k so that a k=1.a k=1.
🔗
The following proposition follows directly from Fermat's little theorem (you should try to prove it!)
🔗
Proposition 2.6.4.
p p a a F∗p.F p∗.a k=1 a k=1 F∗p.F p∗.k∣(p−1).k∣(p−1).
🔗
Elements with order p−1 p−1 in F∗p F p∗ are special, and of fundamental importance in cryptography.
🔗
Theorem 2.6.5.(Primitive root theorem).
p p g∈F∗p g∈F p∗F∗p.F p∗.
F∗p={1,g,g 2,…,g p−2}.F p∗={1,g,g 2,…,g p−2}.
g g generator primitive root F∗p.F p∗.
🔗
For sufficiently large p,p,F∗p F p∗ has lots of generators. (In fact equal to the totient function of p−1.p−1.) Be careful thinking about successive powers of g:g: while the symbols have increasing exponents, the numbers themselves will not in general be increasing, as they are subject to the mixing effect of repeated multiplication mod p.p.
🔗
Exercises Exercises
🔗
1.
Proposition 2.6.4
🔗
2. |
6598 | https://mathtechconnections.com/mtc_lesson/4-2-rounding-to-the-nearest-hundred/ | 4.2 Rounding to the Nearest Hundred
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Home>Elementary Math Lessons>3rd Grade: Place Value> 4.2 Rounding to the Nearest Hundred
Lesson
4.2 Rounding to the Nearest Hundred
Objective:Students locate numbers on number lines between consecutive hundreds, identify endpoints and midpoints, round numbers to the nearest hundred, and compare the accuracy of rounding to the nearest 10 versus 100.
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Warm Up QuestionsMath CentersStandards
What are some situations in real life that we use estimates for?
What does accuracy mean?
Describe what digits to use when rounding to the nearest hundred.
Do numbers in the ones place of 3 digit numbers influence rounding to the nearest hundred?
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6599 | https://www.unitconverters.net/area/square-mile-to-square-foot.htm | Convert Square Mile to Square Foot
Home / Area Conversion / Convert Square Mile to Square Foot
Convert Square Mile to Square Foot
Please provide values below to convert square mile [mi^2] to square foot [ft^2], or vice versa.
From:square mile
To:square foot
Square mile
Definition: The square mile (symbol: sq mi or mi 2) is a unit of area in the imperial and United States customary (UCS) systems of units. It is defined as the area of a square with side lengths of one statute mile (5,280 feet). A square mile is a large area that is equal to 27,878,400 square feet, 3,097,600 square yards, and 640 acres.
History/origin: Area is commonly represented in the form of a square with sides of a specific unit of length, written as "square" followed by the chosen unit of length. The square mile is one of many such units and is derived from the area of a square with sides of one mile.
Current use: Square miles are used when the measurement involves large scales such as country sizes. It can be used when referencing population studies, such as a measurement of total population per square mile. As an imperial and USC unit, it is most commonly used in countries that use these systems of measurement. In countries that have adopted the International System of Units, the square meter (and its subsequent multiples and submultiples) is the preferred system of measurement.
Square foot
Definition: A square foot (symbol: sq ft) is a unit of area used in the imperial and US customary systems (UCS). It is defined as the area of a square with one-foot sides. One square foot equals 144 square inches.
History/origin: The origin of the square foot can be seen in the term itself. It is a measurement derived from the area of a square with a side length measured in feet.
Current use: The square foot is mainly used in the United States but is also used to some extent in countries such as the United Kingdom, Canada, Malaysia, Singapore, Pakistan, Bangladesh, India, and Hong Kong. In these countries, area in real estate, architecture, and of interior spaces is often specified in terms of square footage.
Square Mile to Square Foot Conversion Table
| Square Mile [mi^2] | Square Foot [ft^2] |
--- |
| 0.01 mi^2 | 278784 ft^2 |
| 0.1 mi^2 | 2787840 ft^2 |
| 1 mi^2 | 27878400 ft^2 |
| 2 mi^2 | 55756800 ft^2 |
| 3 mi^2 | 83635200 ft^2 |
| 5 mi^2 | 139392000 ft^2 |
| 10 mi^2 | 278784000 ft^2 |
| 20 mi^2 | 557568000 ft^2 |
| 50 mi^2 | 1393920000 ft^2 |
| 100 mi^2 | 2787840000 ft^2 |
| 1000 mi^2 | 27878400000 ft^2 |
How to Convert Square Mile to Square Foot
1 mi^2 = 27878400 ft^2
1 ft^2 = 3.5870064279155E-8 mi^2
Example: convert 15 mi^2 to ft^2:
15 mi^2 = 15 × 27878400 ft^2 = 418176000 ft^2
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