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Kidney dialysis [ a ] is the process of removing excess water , solutes , and toxins from the blood in people whose kidneys can no longer perform these functions naturally. Along with kidney transplantation , it is a type of renal replacement therapy .
Dialysis may need to be initiated when there is a sudden rapid loss of kidney function, known as acute kidney injury (previously called acute renal failure), or when a gradual decline in kidney function, chronic kidney failure , reaches stage 5. Stage 5 chronic renal failure is reached when the glomerular filtration rate is less than 15% of the normal, creatinine clearance is less than 10 mL per minute, and uremia is present. [ 1 ]
Dialysis is used as a temporary measure in either acute kidney injury or in those awaiting kidney transplant and as a permanent measure in those for whom a transplant is not indicated or not possible. [ 2 ]
In West European countries, Australia, Canada, the United Kingdom, and the United States, dialysis is paid for by the government for those who are eligible. [ 3 ] [ 4 ] The first successful dialysis was performed in 1943.
The kidneys have an important role in maintaining health. When the person is healthy, the kidneys maintain the body's internal equilibrium of water and minerals (sodium, potassium, chloride, calcium, phosphorus, magnesium, sulphate). The acidic metabolism end-products that the body cannot get rid of via respiration are also excreted through the kidneys. The kidneys also function as a part of the endocrine system , producing erythropoietin , calcitriol and renin . Erythropoietin is involved in the production of red blood cells and calcitriol plays a role in bone formation. [ 5 ] Dialysis is an imperfect treatment to replace kidney function because it does not correct the compromised endocrine functions of the kidney. Dialysis treatments replace some of these functions through diffusion (waste removal) and ultrafiltration (fluid removal). [ 6 ] Dialysis uses highly purified (also known as "ultrapure") water. [ 7 ]
Dialysis works on the principles of the diffusion of solutes and ultrafiltration of fluid across a semipermeable membrane . Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration. [ 8 ] Blood flows by one side of a semipermeable membrane, and a dialysate, or special dialysis fluid, flows by the opposite side. A semipermeable membrane is a thin layer of material that contains holes of various sizes, or pores. Smaller solutes and fluid pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells and large proteins). This replicates the filtering process that takes place in the kidneys when the blood enters the kidneys and the larger substances are separated from the smaller ones in the glomerulus . [ 8 ]
The two main types of dialysis, hemodialysis and peritoneal dialysis , remove wastes and excess water from the blood in different ways. [ 2 ] Hemodialysis removes wastes and water by circulating blood outside the body through an external filter, called a dialyzer , that contains a semipermeable membrane . The blood flows in one direction and the dialysate flows in the opposite. The counter-current flow of the blood and dialysate maximizes the concentration gradient of solutes between the blood and dialysate, which helps to remove more urea and creatinine from the blood. The concentrations of solutes normally found in the urine (for example potassium , phosphorus and urea) are undesirably high in the blood, but low or absent in the dialysis solution, and constant replacement of the dialysate ensures that the concentration of undesired solutes is kept low on this side of the membrane. The dialysis solution has levels of minerals like potassium and calcium that are similar to their natural concentration in healthy blood. For another solute, bicarbonate , dialysis solution level is set at a slightly higher level than in normal blood, to encourage the diffusion of bicarbonate into the blood, to act as a pH buffer to neutralize the metabolic acidosis that is often present in these patients. The levels of the components of dialysate are typically prescribed by a nephrologist according to the needs of the individual patient.
In peritoneal dialysis , wastes and water are removed from the blood inside the body using the peritoneum as a natural semipermeable membrane. Waste and excess water move from the blood, across the visceral peritoneum due to its large surface area and into a special dialysis solution, called dialysate, in the peritoneal cavity within the abdomen.
There are three primary and two secondary types of dialysis: hemodialysis (primary), peritoneal dialysis (primary), hemofiltration (primary), hemodiafiltration (secondary) and intestinal dialysis (secondary).
In hemodialysis , the patient's blood is pumped through the blood compartment of a dialyzer, exposing it to a partially permeable membrane . The dialyzer is composed of thousands of tiny hollow synthetic fibers . The fiber wall acts as the semipermeable membrane. Blood flows through the fibers, dialysis solution flows around the outside of the fibers, and water and wastes move between these two solutions. [ 9 ] The cleansed blood is then returned via the circuit back to the body. Ultrafiltration occurs by increasing the hydrostatic pressure across the dialyzer membrane. This usually is done by applying a negative pressure to the dialysate compartment of the dialyzer. This pressure gradient causes water and dissolved solutes to move from blood to dialysate and allows the removal of several litres of excess fluid during a typical 4-hour treatment.
In the United States, hemodialysis treatments are typically given in a dialysis center three times per week (due in the United States to Medicare reimbursement rules); however, as of 2005 over 2,500 people in the United States are dialyzing at home more frequently for various treatment lengths. [ 10 ] Studies have demonstrated the clinical benefits of dialyzing 5 to 7 times a week, for 6 to 8 hours. This type of hemodialysis is usually called nocturnal daily hemodialysis and a study has shown it provides a significant improvement in both small and large molecular weight clearance and decreases the need for phosphate binders . [ 11 ] These frequent long treatments are often done at home while sleeping, but home dialysis is a flexible modality and schedules can be changed day to day, week to week. In general, studies show that both increased treatment length and frequency are clinically beneficial. [ 12 ]
Hemo-dialysis was one of the most common procedures performed in U.S. hospitals in 2011, occurring in 909,000 stays (a rate of 29 stays per 10,000 population). [ 13 ]
In peritoneal dialysis, a sterile solution containing glucose (called dialysate) is run through a tube into the peritoneal cavity , the abdominal body cavity around the intestine , where the peritoneal membrane acts as a partially permeable membrane.
This exchange is repeated 4–5 times per day; automatic systems can run more frequent exchange cycles overnight. Peritoneal dialysis is less efficient than hemodialysis, but because it is carried out for a longer period of time the net effect in terms of removal of waste products and of salt and water are similar to hemodialysis. Peritoneal dialysis is carried out at home by the patient, often without help. This frees patients from the routine of having to go to a dialysis clinic on a fixed schedule multiple times per week. Peritoneal dialysis can be performed with little to no specialized equipment (other than bags of fresh dialysate).
Hemofiltration is a similar treatment to hemodialysis, but it makes use of a different principle. The blood is pumped through a dialyzer or "hemofilter" as in dialysis, but no dialysate is used. A pressure gradient is applied; as a result, water moves across the very permeable membrane rapidly, "dragging" along with it many dissolved substances, including ones with large molecular weights, which are not cleared as well by hemodialysis. Salts and water lost from the blood during this process are replaced with a "substitution fluid" that is infused into the extracorporeal circuit during the treatment.
Hemodiafiltration is a combination between hemodialysis and hemofiltration, thus used to purify the blood from toxins when the kidney is not working normally and also used to treat acute kidney injury (AKI).
In intestinal dialysis, the diet is supplemented with soluble fibres such as acacia fibre , which is digested by bacteria in the colon. This bacterial growth increases the amount of nitrogen that is eliminated in fecal waste. [ 14 ] [ 15 ] An alternative approach utilizes the ingestion of 1 to 1.5 liters of non-absorbable solutions of polyethylene glycol or mannitol every fourth hour. [ 16 ]
The decision to initiate dialysis or hemofiltration in patients with kidney failure depends on several factors. These can be divided into acute or chronic indications.
Depression and kidney failure symptoms can be similar to each other. [ citation needed ] It is important that there is open communication between a dialysis team and the patient. Open communication will allow giving a better quality of life. Knowing the patients' needs will allow the dialysis team to provide more options like: changes in dialysis type like home dialysis for patients to be able to be more active or changes in eating habits to avoid unnecessary waste products.
Indications for dialysis in a patient with acute kidney injury are summarized with the vowel mnemonic of "AEIOU": [ 17 ]
Chronic dialysis may be indicated when a patient has symptomatic kidney failure and low glomerular filtration rate (GFR < 15 mL/min). [ 18 ] Between 1996 and 2008, there was a trend to initiate dialysis at progressively higher estimated GFR, eGFR.
A review of the evidence shows no benefit or potential harm with early dialysis initiation, which has been defined by start of dialysis at an estimated GFR of greater than 10 ml/min/1.73 2 . Observational data from large registries of dialysis patients suggests that early start of dialysis may be harmful. [ 19 ] The most recent published guidelines from Canada, for when to initiate dialysis, recommend an intent to defer dialysis until a patient has definite kidney failure symptoms, which may occur at an estimated GFR of 5–9 ml/min/1.73 2 . [ 20 ]
Even though it is not a cure for kidney failure, dialysis is a very effective treatment. [ 21 ] Survival rates of kidney failure are generally longer with dialysis than without (having only conservative kidney management). However, from the age of 80 and in elderly patients with comorbidities there is no difference in survival between the two groups. [ 22 ]
Dialysis is an intensive treatment that has a serious impact on those treated with it. Being on dialysis usually leads to a poor quality of life . However, there are strategies that can make it more tolerable. [ 23 ] Receiving dialysis at home might improve people's quality of life and autonomy. [ 23 ]
Dialysis is typically on a regular schedule of three times a week.
Given that dialysis patients have little or no capacity to filtrate solutes and regulate their fluid volume due to kidney dysfunction, [ 24 ] missing dialysis is potentially lethal. These patients can be hyperkalaemic leading to cardiac dysrhythmias and potential cardiac arrest , [ 25 ] as well as fluid in the alveoli of their lungs which can impair breathing. [ 26 ]
Some medications can be used in the short term to decrease serum potassium and stabilise the cardiac muscle so as to facilitate stabilisation of acute patients in the setting of missed dialysis. Salbutamol and insulin can decrease serum potassium by up to 1.0mmol/L each by shifting potassium from the extracellular space into the intracellular spaces within skeletal muscle cells , and calcium gluconate is used to stabilise the myocardium in hyperkalaemic patients, in an attempt to reduce the likelihood of lethal arrhythmias arising from a high serum potassium. [ 27 ]
People who decide against dialysis treatment when reaching end-stage chronic kidney disease could survive several years and experience improvements in their mental well-being in addition to sustained physical well-being and overall quality of life until late in their illness course. However, use of acute care services in these cases is common and intensity of end-of-life care is highly variable among people opting out of dialysis. [ 28 ] [ 29 ]
The average annual total cost per dialysis patient varies between countries, for example in South Korea 19,812 USD, in New Zealand 26,479 USD and in Netherlands 89,958 USD, according to an 2021 article. [ 30 ]
Over the past 20 years, children have benefited from major improvements in both technology and clinical management of dialysis. Morbidity during dialysis sessions has decreased with seizures being exceptional and hypotensive episodes rare. Pain and discomfort have been reduced with the use of chronic internal jugular venous catheters and anesthetic creams for fistula puncture. Non-invasive technologies to assess patient target dry weight and access flow can significantly reduce patient morbidity and health care costs. [ 31 ] Mortality in paediatric and young adult patients on chronic hemodialysis is associated with multifactorial markers of nutrition, inflammation , anaemia and dialysis dose, which highlights the importance of multimodal intervention strategies besides adequate hemodialysis treatment as determined by Kt/V alone. [ 32 ]
Biocompatible synthetic membranes , specific small size material dialyzers and new low extra-corporeal volume tubing have been developed for young infants. Arterial and venous tubing length is made of minimum length and diameter, a <80 ml to <110 ml volume tubing is designed for pediatric patients and a >130 to <224 ml tubing are for adult patients, regardless of blood pump segment size, which can be of 6.4 mm for normal dialysis or 8.0mm for high flux dialysis in all patients. All dialysis machine manufacturers design their machine to do the pediatric dialysis. In pediatric patients, the pump speed should be kept at low side, according to patient blood output capacity, and the clotting with heparin dose should be carefully monitored. The high flux dialysis (see below) is not recommended for pediatric patients. [ 33 ]
In children, hemodialysis must be individualized and viewed as an "integrated therapy" that considers their long-term exposure to chronic renal failure treatment. Dialysis is seen only as a temporary measure for children compared with renal transplantation because this enables the best chance of rehabilitation in terms of educational and psychosocial functioning. Long-term chronic dialysis, however, the highest standards should be applied to these children to preserve their future "cardiovascular life"—which might include more dialysis time and on-line hemodiafiltration online hdf with synthetic high flux membranes with the surface area of 0.2 m 2 to 0.8 m 2 and blood tubing lines with the low volume yet large blood pump segment of 6.4/8.0 mm, if we are able to improve on the rather restricted concept of small-solute urea dialysis clearance. [ 34 ]
Dialyzable substances—substances removable with dialysis—have these properties:
The National Health Service provides dialysis in the United Kingdom. In 2022, there were more than 30,000 people on dialysis in the UK. [ 23 ]
For people who need to travel to dialysis centres, patient transport services are generally provided without charge. Cornwall Clinical Commissioning Group proposed to restrict this provision to people who did not have specific medical or financial reasons in 2018 but changed their minds after a campaign led by Kidney Care UK and decided to fund transport for people requiring dialysis three times a week for a minimum or six times a month for a minimum of three months. [ 35 ]
UK clinical guidelines recommend offering people a choice regarding where they get their dialysis. [ 23 ] Research in the UK found that receiving dialysis at home can lead to better quality of life and is less costly than receiving dialysis in hospital. [ 36 ] [ 37 ] However, many people in the UK prefer to receive dialysis in hospital: In 2022, only 1 in 6 chose receiving it at home. [ 36 ] [ 38 ] [ 23 ]
There are various reasons why people do not choose home dialysis. Among these are preferring hospitals as a way of getting regular social contact, being concerned about necessary changes to their homes and their family members becoming carers. [ 36 ] [ 38 ] Other reasons include a lack of motivation, doubting abilities for self-managed treatment, and not having suitable housing or support at home. [ 23 ] Hospital dialysis is also often presented as the norm by healthcare professionals. [ 23 ]
Encouraging people to have dialysis at home could reduce the impact of dialysis on people's social and professional lives. [ 36 ] [ 38 ] Some ways to help are offering peer support from other people on home dialysis, better education materials, and professionals being more familiar with home dialysis and its impact. Choosing home dialysis is more likely at kidney centers which have better organisational culture, leadership and attitude. [ 23 ]
Since 1972, insurance companies in the United States have covered the cost of dialysis and transplants for all citizens. [ 39 ] By 2014, more than 460,000 Americans were undergoing treatment, the costs of which amount to six percent of the entire Medicare budget. Kidney disease is the ninth leading cause of death, and the U.S. has one of the highest mortality rates for dialysis care in the industrialized world. The rate of patients getting kidney transplants has been lower than expected. These outcomes have been blamed on a new for-profit dialysis industry responding to government payment policies. [ 40 ] [ 41 ] [ 42 ] A 1999 study concluded that "patients treated in for-profit dialysis facilities have higher mortality rates and are less likely to be placed on the waiting list for a renal transplant than are patients who are treated in not-for-profit facilities", possibly because transplantation removes a constant stream of revenue from the facility. [ 43 ] The insurance industry has complained about kickbacks and problematic relationships between charities and providers. [ 44 ]
The Government of China provides the funding for dialysis treatment. There is a challenge to reach everyone who needs dialysis treatment because of the unequal distribution of health care resources and dialysis centers. [ 45 ] There are 395,121 individuals who receive hemodialysis or peritoneal dialysis in China per year. The percentage of the Chinese population with Chronic Kidney Disease is 10.8%. [ 46 ] The Chinese Government is trying to increase the amount of peritoneal dialysis taking place to meet the needs of the nation's individuals with Chronic Kidney Disease. [ 47 ]
Dialysis is provided without cost to all patients through Medicare , with 75% of all dialysis being administered as haemodialysis to patients three times per week in a dialysis facility. [ 48 ] The Northern Territory has the highest incidence rate per population of haemodialysis, [ 49 ] with Indigenous Australians having higher rates of Chronic Kidney Disease and lower rates of functional kidney transplants than the broader population. [ 50 ] The remote Central Australian town of Alice Springs , despite having a population of approximately 25000, has the largest dialysis unit in the Southern Hemisphere . [ 51 ] Many people must move to Alice Springs from remote Indigenous communities to access health services such as haemodialysis, which results in housing shortages, overcrowding, and poor living conditions. [ 52 ]
In 1913, Leonard Rowntree and John Jacob Abel of Johns Hopkins Hospital developed the first dialysis system which they successfully tested in animals. [ 53 ] A Dutch doctor, Willem Johan Kolff , constructed the first working dialyzer in 1943 during the Nazi occupation of the Netherlands . [ 54 ] Due to the scarcity of available resources, Kolff had to improvise and build the initial machine using sausage casings , beverage cans , a washing machine and various other items that were available at the time. Over the following two years (1944–1945), Kolff used his machine to treat 16 patients with acute kidney failure , but the results were unsuccessful. Then, in 1945, a 67-year-old comatose woman regained consciousness following 11 hours of hemodialysis with the dialyzer and lived for another seven years before dying from an unrelated condition. She was the first-ever patient successfully treated with dialysis. [ 54 ] Gordon Murray of the University of Toronto independently developed a dialysis machine in 1945. Unlike Kolff's rotating drum, Murray's machine used fixed flat plates, more like modern designs. [ 55 ] Like Kolff, Murray's initial success was in patients with acute renal failure. [ 56 ] Nils Alwall of Lund University in Sweden modified a similar construction to the Kolff dialysis machine by enclosing it inside a stainless steel canister. This allowed the removal of fluids, by applying a negative pressure to the outside canister, thus making it the first truly practical device for hemodialysis. Alwall treated his first patient in acute kidney failure on 3 September 1946. [ 57 ] | https://en.wikipedia.org/wiki/Kidney_dialysis |
Kihachiro Ueda (上田毅八郎) was a Japanese painter. His primary subject matter is Japanese ships.
During the Second World War , Ueda was an anti-aircraft gunner assigned to several merchant ships. [ 1 ] He served on about 28 merchant ships, and 6 of those were attacked and destroyed while he was aboard. During his spare time, he would draw ships. On 13 November 1944, while assigned to Kinka Maru , he was severely injured during a U.S. air raid on Manila Bay. He lost the use of his right hand, but subsequently was able to draw and paint with his left hand. [ 1 ] [ 2 ]
His painting of the Ōryoku Maru (sunk 15 December 1944) is part of the Hell Ship Memorial Project 's display at the Subic Bay Historical Center , Philippines.
Ueda started working with the model company Tamiya Corporation in the early 1960s. Ueda illustrated most of the package artwork for Tamiya's Waterline Series (which consists of about 150 models). [ 2 ]
Ueda died on 26 June 2016. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/Kihachiro_Ueda |
Kiiti Morita ( 森田 紀一 , Morita Kiichi , February 11, 1915 – August 4, 1995) was a Japanese mathematician working in algebra and topology .
Morita was born in 1915 in Hamamatsu , Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo University of Science . [ 1 ] He received his Ph.D. from Osaka University in 1950, with a thesis in topology. [ 1 ] [ 2 ] After teaching at the Tokyo Higher Normal School, he became professor at the University of Tsukuba in 1951. He held this position until 1978, after which he taught at Sophia University . [ 1 ] [ 3 ] Morita died of heart failure in 1995 at the Sakakibara Heart Institute in Tokyo ; [ 3 ] he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson. [ 3 ] [ 4 ]
He introduced the concepts now known as Morita equivalence and Morita duality which were given wide circulation in the 1960s by Hyman Bass in a series of lectures. The Morita conjectures on normal topological spaces are also named after him.
This article about a Japanese scientist is a stub . You can help Wikipedia by expanding it .
This article about an Asian mathematician is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kiiti_Morita |
The Kiliani–Fischer synthesis , named for German chemists Heinrich Kiliani and Emil Fischer , is a method for synthesizing monosaccharides . It proceeds via synthesis and hydrolysis of a cyanohydrin , followed by reduction of the intermediate acid to the aldehyde, thus elongating the carbon chain of an aldose by one carbon atom while preserving stereochemistry on all the previously present chiral carbons. The new chiral carbon is produced with both stereochemistries, so the product of a Kiliani–Fischer synthesis is a mixture of two diastereomeric sugars, called epimers . For example, D - arabinose is converted to a mixture of D - glucose and D - mannose .
The original version of the Kiliani–Fischer synthesis proceeds through cyanohydrin and aldonic acid lactone intermediates. The first step is to react the starting sugar with aqueous cyanide (typically NaCN); the cyanide undergoes nucleophilic addition to the carbonyl group of the sugar (while sugars tend to exist mainly as cyclic hemiacetal , they are always in chemical equilibrium with their open-chain aldehyde or ketone forms, and in the case of these aldoses it is that aldehyde form that reacts in this synthesis). The cyanohydrin resulting from this addition is heated in water, which hydrolyzes the cyanide into a carboxylic acid group that quickly reacts with itself to form a more stable lactone . Now there are two diastereomeric lactones in the reaction mixture. They are separated (by chromatography , partitioned into different solvents, or any of the numerous other separation methods) and then the desired lactone is reduced with a sodium amalgam . As illustrated below, D - arabinose is converted to a mixture of D - glucononitrile and D - mannononitrile , which is then converted to D - gluconolactone and D - mannonolactone , separated, and reduced to D - glucose or D - mannose . The chemical yield by this method is estimated to be around 30%.
More recently, an improved reduction method has been developed that produces somewhat higher yields of the larger sugars. Instead of conversion of the cyanohydrin to a lactone, the cyanohydrin is reduced with hydrogen , using palladium on barium sulfate as the catalyst and water as the solvent , to form an imine . Due to the presence of water, the imine quickly hydrolyzes to form an aldehyde, thus the final sugars are produced in just two steps rather than three. The separation of the isomers is then performed at the stage of the sugar products themselves rather than at the lactone intermediates.
The special catalyst is needed to avoid further reduction of the aldehyde group to a hydroxyl group, which would yield an alditol . These catalysts that limit hydrogenation to one step are called poisoned catalysts ; Lindlar palladium is another example.
The reactions below illustrate this improved method for the conversion of L - threose to L - lyxose and L - xylose .
Both enantiomers of glyceraldehyde are commercially available, so one can access any stereoisomer of any chain-length aldose by an appropriate number of repeated applications of the Kiliani–Fischer synthesis. The triose D -glyceraldehyde ( 1 ) leads to the tetroses D -erythrose ( 2a ) and D -threose ( 2b ). Those lead to the pentoses D -ribose ( 3a ) and D -arabinose ( 3b ), and D -xylose ( 3c ) and D -lyxose ( 3d ), respectively. The next iteration leads to the hexoses D -allose ( 4a ) and D -altrose ( 4b ), D -glucose ( 4c ) and D -mannose ( 4d ), D -gulose ( 4e ) and D -idose ( 4f ), and D -galactose ( 4g ) and D -talose ( 4h ). The D - heptoses and beyond are available by continuing the sequence, and enantiomeric L series is available by starting the sequence with L -glyceraldehyde.
In practice, the Kiliani–Fischer synthesis is usually used for production of sugars that are difficult or impossible to obtain from natural sources. While it does provide access to every possible stereoisomer of any desired aldose, the process is limited by its low yield and use of toxic reagents. In addition, the process requires having a supply of the previous sugar in the series, which may itself require substantial synthetic work if it is not readily available. For example, if successive iterations of the Kiliani–Fischer synthesis are used, the overall yield drops approximately exponentially for each additional iteration.
The process only provides direct access to aldoses, whereas some sugars of interest may instead be ketoses. Some ketoses may be accessible from similar aldoses by isomerization via an enediol intermediate; for example, on standing in aqueous base, glucose , fructose , and mannose will slowly interconvert since they share an enediol form. (See Lobry de Bruyn–van Ekenstein transformation ). Some unusual sugars are also accessible via aldol addition . | https://en.wikipedia.org/wiki/Kiliani–Fischer_synthesis |
Kilju ( Finnish pronunciation: [ˈkilju] ) is the Finnish word for a mead -like homemade alcoholic beverage made from a source of carbohydrates (such as cane sugar or honey), yeast , and water, making it both affordable and cheap to produce. The ABV depends on the yeast that was used, and since it does not contain a sweet reserve it is completely dry . Crude product may be distilled into moonshine . Kilju intended for direct consumption is usually clarified and stabilized to avoid wine faults . It is a flax -colored alcoholic beverage with no discernible taste other than that of ethanol . It can be used as an ethanol base for drink mixers .
Kilju is commonly associated with the punk subculture. [ 1 ]
Kilju is a well-established part of the Finnish alcohol and counter-culture, as witnessed even in the leading engineer school's making-and-use-of video of yore. "Four thousand litres of gases are generated. They are led to the neighbours' delight." The drink tends to invite such black humour , of the deadpan kind.
The first commercially produced kilju was introduced in 2022. [ 2 ] [ 3 ]
The process is similar to that of homebrewing wine. If done slowly, it requires rigorous hygiene and filtering of the product. If brewed fast, specialized dried yeasts are available in amounts to drive the fermentation process through before bacterial infiltration can take place, in about three days. In Finnish, the latter are called pikahiiva (lit. quick-yeast), and they are sold in about a hundred gramme packs dry, as opposed to the live standard pack of brewer's yeast of 50 g wet.
Properly made kilju is a clear, colorless, or off-white liquid with no discernible taste other than that of ethanol . It can be produced by natural settling of the yeast over time, but nowadays various fining agents are used to hasten the process as well.
Kilju is often produced improperly by home brewers who allow contaminants to disrupt fermentation or do not adequately filter or rack the liquid, or do not use a fining agent . The latter mistakes result in yeast being suspended, causing the mixture to be cloudy rather than clear. The yeast is not harmful, but can yield an unpleasant taste and intestinal discomfort. [ citation needed ] It is also a common mistake to leave the carbon dioxide produced by fermentation into the suspension, so that the yeast provides it with nucleation sites, keeping the yeast up in the solution. Proper technique calls for airing the product after fermentation, stirring, and perhaps for fining agents such as microsilica or various semipolar proteinaceacous or carbohydrate agents.
An easy way to produce fermented water is to obtain turbo yeast kits (contains Saccharomyces cerevisiae yeast strain, enzymes, vitamins, and minerals) that instructs on the package the quantity of white sugar, and tap water needed.
Inverted sugar syrup for fermented water is usually home-made by fully dissolving sugar in cold tap water . Yeast requires oxygen rich water that do not exceed 25 degrees Celsius.
A common manual way to dissolve refined sugar is to mix with water in a container which is half filled, and then sealed and shaken. However, a mixer or blender may be used to automatically dissolve the sugar, in turns, if necessary.
Yeast, and yeast nutrition, is mixed in the syrup. One gram pure yeast consumes approximately 0.2 grams sugar.
Yeasts will usually die out once the alcohol level reaches about 15% due to the toxicity of alcohol on the yeast cells' physiology while the more alcohol tolerant Saccharomyces species take over. In addition to S. cerevisiae , Saccharomyces bayanus is a species of yeast that can tolerate alcohol levels of 17–20%. [ 7 ]
Since fermented water contains no flavors, water may be added to cut down the ABV if desired.
Fermented water contains a similar alcoholic content of wines as both beverages are fermented on yeast, however fermented water differs from wine and other fermented beverages in that it contains no fruit juice or residual sugar after manufacture.
Kilju can be produced by fermenting sugar, yeast, and water, but it was illegal in Finland before March 2018; [ 8 ] therefore, grain, potatoes, fruits [ 9 ] or berries were used during fermentation to avoid legal problems and to flavor the drink. Oranges and lemons are a popular choice for this purpose.
It often has additives such as citrus fruits, apples, berry juices, or artificial flavorings. Flavored kilju from fruits for example doesn't necessarily have to be sweet as long as all sugar is consumed by the yeast.
Kilju (15-17% ABV) contains 2.4-2.7 times more water than 40% distilled spirit. Since kilju contains approximately 85% water, it can be mixed with concentrates such as a drink mixer , fruit syrup, or squash concentrate.
Alternatively, it can be made as a carbonated soft drink by two methods.
When served before the fermentation process is complete . Kilju made this way is high in sugar and carbon dioxide (CO 2 ) content, and has little to no alcohol, being similar to a sweet lemon soda. It is a family tradition to many. The simple production process also makes it accessible to underage drinkers . Cf. sima , commonly seasoned with lemon and unpurified cane sugar, leading to a small beer or a light mead .
To make homemade alcopop (typically to 3–7%) water is added to kilju after the fermentation process is complete to dilute the ABV accordingly. The solution is then carbonated with a soda machine , and soft drink syrup (which will lower the ABV approximately 10%) is added. Alternatively, it can be made as a carbonated soft drink when served before the fermentation process is complete. Fermented water made this way is high in sugar and carbon dioxide (CO 2 ) content, and do not need to be diluted with water because it has little to no alcohol depending on how many days it has been fermented, being similar to a sweet lemon soda.
Kilju can be refined into moonshine by means of distillation to vodka or rectified spirit, but it is illegal in most countries . It is distinct from rum because it is typically made by molasses, a byproduct of the sugar refining process, or fresh sugar cane juice that has a discernible taste of its own.
Moonshine by country , often distilled from fermented water:
Winemaking is legal in most countries. However, kilju is fermented from pure carbohydrates like white sugar (a plant extract) instead of grapes.
The Finnish Alcoholic Beverages Act 1 March 2018 legalized the manufacture of fermented water and wine from fruits, berries and other carbohydrate sources, without the pretense of making proper wine. [ 10 ]
In Sweden, it is legal to produce if the final product is not distilled. [ 11 ]
Kilju is often mixed with juice or some other beverage to mask off tastes, of which there can be several.
Compared to wines, kilju most closely resembles Beaujolais nouveau , which is drunk after only a few weeks of fermentation. However, properly made kilju will not easily turn into vinegar, lacking the nutrients necessary for further fermentation. It is possible to drink kilju years after it was made if it has been properly stored. In fact as white wines, it ages well into 2-3a, especially when made from impure cane sugar, molasses included ( fariinisokeri ), or if brewed partially from oat malt and hops, as an extra strong beer.
Kilju is regarded as a low-quality drink that is primarily consumed for its alcohol content, mainly associated with binge drinking . [ 1 ] Due to its low cost, potential wine fault (when not clarified enough), and simple production process, kilju is mostly drunk by low-income people.
When homebrewing grew in popularity during the economic depression that followed the Finnish banking crisis of the early 1990s, yeast strains known as "turbo yeast" ("turbohiiva", "pikahiiva") were introduced to the market. These yeast strains enable a very rapid fermentation to full cask strength, in some cases in as little as three days (compared to several weeks required by traditional wine yeast strains). Such a short production time naturally does not allow the yeast to become lees . The introduction of turbo yeast reinforced the public's view of kilju as an easy method of procuring cheap alcohol. | https://en.wikipedia.org/wiki/Kilju |
In computing , a kill pill is a mechanism or a technology designed to render systems useless either by user command, or under a predefined set of circumstances. Kill pill technology is most commonly used to disable lost or stolen devices for security purposes, but can also be used for the enforcement of rules and contractual obligations.
Kill pill technology is used prominently in smartphones , especially in the disablement of lost or stolen devices. A notable example is Find My iPhone , a service that allows the user to password protect or wipe their iDevice(s) remotely, aiding in the protection of private data. [ 1 ] Similar applications exist for other smartphone operating systems , including Android , [ 2 ] BlackBerry , [ 3 ] and Windows Phone . [ 4 ]
Kill pill technology has been notably used as an anti-piracy measure. Windows Vista was released with the ability to severely limit its own functionality if it was determined that the copy was obtained through piracy . The feature was later dropped after complaints that false positives caused genuine copies of Vista to act as though they were pirated. [ 5 ]
The concept of a kill pill is also applied to the remote removal by a server of malicious files or applications from a client's system. Such technology is a standard component of most handheld computing devices, mainly due to their generally more limited operating systems and means of obtaining applications. Such functionality is also reportedly available to applications downloaded from the Windows Store on Windows 8 operating systems. [ 6 ]
Kill pill technology is used frequently in vehicles for a variety of reasons. Remote vehicle disablement can be used to prevent a vehicle from starting, to prevent it from moving, and to prevent the vehicle's continued operation. Non-remotely, vehicles can require driver recognition before starting or moving, such as asking for a password or some form of biometrics from the driver. [ 7 ]
Kill pill technology is often used by governments to prevent drunk driving by repeat offenders as a punishment and deterrent. The installation of an ignition interlock devices is a sentencing alternative for drunk drivers in almost all 50 of the United States . Such a device requires the driver to blow into a breathalyzer before starting the vehicle. If the driver is found to be over the legal blood alcohol content limit, the vehicle will not start [ 8 ]
Kill pill technology can also be implemented to contextually disable certain aspects of a smartphone's functionality. A patent obtained by Apple claims the ability to disable the antenna, screen, or camera of a smartphone in settings like theaters, schools, and areas of high security sensitivity. [ 9 ]
Kill pill technology has been criticized for allowing for the suppression of personal liberties. While a kill pill can be utilized in a school setting to prevent academic dishonesty, it has been suggested that governments may also use it to suppress their people, for example, by disabling a phone's camera or antenna in the area of a protest. [ 10 ]
The ability to remotely remove files and applications from a user's device has also come under fire. Apple's apparent ability to blacklist applications, rendering them unusable on any iDevice, has raised concerns about the user's rights when downloading from the App Store . [ 11 ] As of July 2014, no applications appear on Apple's blacklist website. | https://en.wikipedia.org/wiki/Kill_pill |
Killer Activation Receptors (KARs) are receptors expressed on the plasma membrane (cell membrane) of Natural Killer cells ( NK cells ). KARs work together with Killer I nhibitory Receptors (abbreviated as KIRs in the text), which inactivate KARs in order to regulate the NK cells functions on hosted or transformed cells . [ 1 ] These receptors have a broad binding specificity and are able to broadcast opposite signals. It is the balance between these competing signals that determines if the cytotoxic activity of the NK cell and apoptosis of distressed cell occurs. [ 2 ]
There is sometimes confusion regarding the KIR acronym . The KIR term has been started to be being used parallelly both for the Killer-cell immunoglobulin-like receptors (KIRs) and for the Killer Inhibitory Receptors. The Killer-cell immunoglobulin-like receptors involve both activation and inhibitory receptors. [ 3 ] Killer-cell inhibitory receptors involve both immunoglobulin-like receptors and C-type lectin-like receptors . [ 4 ]
KARs and KIRs have some morphological features in common, such as being transmembrane proteins . The similarities are specially found in the extracellular domains . [ 1 ]
The differences between KARs and KIRs tend to be in the intracellular domains . They can have a tyrosine containing activation or inhibitory motifs in the intracellular part of the receptor molecule (they are called ITAMs and ITIMs ).
At first, it was thought that there was only one KAR and one KIR receptor present on the NK cell, known as the two-receptor model. [ 2 ] In the last decade, many different KARs and KIRs, such as NKp46 or NKG2D , have been discovered creating the opposing-signals model. [ 1 ] NKG2D is activated by the cell-surface ligands MICA and ULBP2 . [ 5 ]
Even though KARs and KIRs are receptors with antagonistic effects on NK cells, they have some structural characteristics in common. Both receptors are usually transmembrane proteins . Also , the extracellular domains of these proteins tend to have similar molecular features and are responsible for ligand recognition.
The opposing functions of these receptors are due to differences in their intracellular domains. KARs proteins possess positively charged transmembrane residues and short cytoplasmic tails that contain few intracellular signaling domains. In contrast, KIRs proteins usually have long cytoplasmic tails.
As the chains from KARs are not able to mediate any signal transduction in isolation, a common feature of such receptors is the presence of noncovalently linked subunits that contain immunoreceptor tyrosine-based activation motifs (ITAMs) in their cytoplasmic tails. ITAMs are composed of a conserved sequence of amino acids , including two Tyr-x-x-Leu/Ile elements (where x is any amino acid) separated by six to eight amino acid residues. When the binding of an activation ligand to an activation receptor complex occurs, the tyrosine residues in the ITAMs in the associated chain are phosphorylated by kinases , and a signal that promotes natural cytotoxicity is conveyed to the interior of the NK cell. Therefore, ITAMs are involved in the facilitation of signal transduction. These subunits are moreover composed of an accessory signaling molecule such as CD3ζ , the γc chain, or one of two adaptor proteins called DAP10 and DAP12 . All of these molecules possess negatively charged transmembrane domains. [ 6 ]
A common feature of members of all KIR is the presence of immunoreceptor tyrosine-based inhibition motifs (ITIMs) in their cytoplasmic tails. ITIMs are composed of the sequence Ile/Val/Leu/Ser-x-Tyr-x-x-Leu/Val, where x denotes any amino acid. The latter are essential to the signaling functions of these molecules. When an inhibitory receptor is stimulated by the binding of MHC class I , kinases and phosphatases are recruited to the receptor complex. This is how ITIMs counteract the effect of kinases initiated by activating receptors and manage to inhibit the signal transduction within the NK cell.
Based on their structure there are three different groups of KARS. The first group of receptors is called Natural Cytotoxicity Receptors (NCR), which only includes activation receptors. The two other classes are: Natural Killer Group 2 ( NKG2 ), which includes activation and inhibition receptors, and some KIRs which do not have an inhibitor role. [ 7 ]
The three receptors that are included in the NCR class are NKp46, NKp44 and NKp30 . The crystal structure of NKp46, which is representative for all three NCR, has been determined. It has two C2-set immunoglobulin domains, and it’s probable that the binding site for its ligand is near the interdomain hinge. [ 8 ]
There are two NKG2-class receptors which are NKG2D and CD94/NKG2C. NKG2D, which doesn’t bind to CD94 , is a homodimeric lectin -like receptor. CD94/NKG2C consists in a complex formed by the CD94 protein, which is a C-type lectin molecule bound to the NKG2C protein. This molecule can bind to five classes of NKG2 (A, B, C, E and H), but the union can trigger an activation or an inhibition response, depending on the NKG2 molecule (CD94/NKG2A, for example, is an inhibitor complex). [ 8 ]
Most KIRs have an inhibitor function, however, a few KIRs that have an activator role also exist. One of these activating KIRs is KIR2DS1 , which has an Ig-like structure, like KIRs in general.
Finally, there is CD16 , a low affinity Fc receptor (FcγRIII) which contains N-glycosylation sites; therefore, it is a glycoprotein .
Killer Activation Receptors are associated with signaling intracellular chains. In fact, these intracellular domains determine the opposite functions of activation and inhibitory receptors. Activation receptors are associated with an accessory signaling molecule (for instance, CD3ζ) or with an adaptor protein, which can be either DAP10 or DAP12 . All of these signaling molecules contain immunoreceptor tyrosine-based activated motifs (ITAMs), which are phosphorylated and consequently facilitate signal transduction.
Each of these receptors has a specific ligand, although some receptors that belong to the same class, such as NCR, recognize similar molecules.
KARs can detect a specific type of molecules: MICA and MICB . These molecules are in MHC class I of human cells and they are related to cellular stress: this is why MICA and MICB appear in infected or transformed cells but they aren't very common in healthy cells. KARs recognize MICA and MICB when they are in a huge proportion and get engaged. This engagement activates the natural killer cell to attack the transformed or infected cells. This action can be done in different ways. NK can kill directly the hosted cell, it can do it by segregating cytokines , IFN-β and IFN-α , or by doing both things.
There are other less common ligands, like carbohydrate domains, which are recognized by a group of receptors: C-type lectins (so named because they have calcium -dependent carbohydrates recognition domains).
In addition to lectins, there are other molecules implicated in the activation of NK. These additional proteins are: CD2 and CD16. The CD16 works in antibody-mediated recognition.
Finally, there is a group of proteins which are related to the activation in an unknown way. These are NKp30, Nkp44 and Nkp46. [ 8 ]
These ligands activate the NK cell, however, before the activation, Killer Inhibition Receptors (KIRs) recognize certain molecules in the MHC class I of the hosted cell and get engaged with them. These molecules are typical of healthy cells but some of these molecules are repressed in infected or transformed cells. For this reason when the hosted cell is really infected the proportion of KARs engaged with ligands is bigger than the proportion of KIRs engaged with MHC I molecules. When this happens the NK is activated and the hosted cell is destroyed. On the other hand, if there are more KIRs engaged with MHC class I molecules than KARs engaged with ligands, the NK isn't activated and the suspicious hosted cell remains alive. [ 2 ]
One way by which NK cells are able to distinguish between normal and infected or transformed cells is by monitoring the amount of MHC class I molecules cells have on their surface. When it come to an infected and a tumor cell , the expression of MHC class I decreases. [ 2 ]
In cancers , a Killer Activation Receptor (KAR), located on the surface of the NK cell, binds to certain molecules which only appear on cells that are undergoing stress situations. In humans, this KAR is called NKG2D and the molecules it recognizes MICA and MICB. This binding provides a signal which induces the NK cell to kill the target cell. [ 9 ]
Then, Killer Inhibitory Receptors (KIRs) examine the surface of the tumor cell in order to determine the levels of MHC class I molecules it has. If KIRs bind sufficiently to MHC class I molecules, the “killing signal” is overridden to prevent the killing of the cell. However, if KIRs are not sufficiently engaged to MHC class I molecules, killing of the target cell proceeds. [ 2 ] | https://en.wikipedia.org/wiki/Killer_activation_receptor |
A killing jar or killing bottle is a device used by entomologists to kill captured insects quickly and with minimum damage. [ 1 ] The jar typically contains gypsum plaster (plaster of paris) on the bottom to absorb a killing fluid. The killing fluid evaporates into the air and gasses the insect. Typically only adult hard bodied insects are killed in a killing jar; other insects require different methods of killing.
The jar, typically glass, must be hermetically sealable . One design has a thin layer of hardened plaster of Paris on the bottom to absorb the killing agent. The killing agent will then slowly evaporate, allowing the jar to be used many times before it needs to be refreshed. The absorbent plaster of Paris layer also helps prevent the agent sticking to and damaging insects. Crumpled paper tissue is also placed in the jar for the same reason. A second method utilises a wad of cotton or other absorbent material placed in the bottom of the jar. Liquid killing agent is added until the absorbent material is nearly saturated. A piece of stiff paper or cardboard cut to fit the inside of the jar tightly is then pressed in. [ 2 ]
The most common killing agents are ether , chloroform and ethyl acetate . Ethyl acetate has many advantages and is very widely used. Its fumes are less toxic to humans than those of the other agents, and specimens will remain limp if they are left in an ethyl acetate killing jar for several days and the ethyl acetate is not allowed to entirely evaporate from the specimens. It also preserves the body colors of some insects, such as dragonflies, that would otherwise lose their color, especially if there is a liquid layer to saturate their body tissues. A disadvantage is that, although the insects are quickly stunned by ethyl acetate, it kills them slowly and specimens may revive if removed from the killing jar too soon. Isopropyl alcohol is an easy to find and use killing agent for amateurs. Potassium cyanide or other cyanide compounds, including calcium cyanide , are also used, but only by experts due to its extreme toxicity. It also has the disadvantages that it makes the specimens brittle when left in the jar for several hours and that it may also cause some discoloration of colored specimens. It does kill rapidly, and the cyanide charge will last a long time. A few drops of acetic acid will increase the cyanide gas production. If the jar is not used for long periods it may dry out and produce little gas; a few drops of water will help get the process going again. [ 2 ] The potassium cyanide slowly decomposes, releasing hydrogen cyanide . In former times, amateur entomologists commonly used the thick green leaves of the Cherry Laurel ( Prunus laurocerasus or Prunus caroliniana ) which, crushed or finely sliced, will similarly release hydrogen cyanide. [ 3 ] [ 4 ]
Gassing in a killing jar is typically only used on adult hard-bodied insects. Soft-bodied and hard-bodied immature insects, such as the larval stage of many insects, are generally fixed in ethanol at 70–80% concentration in a vial. Dropping them in the vial containing ethanol both kills and preserves them. Higher concentrations of ethanol are not recommend for use with soft bodied insects, as it can distort and harden the body. [ 5 ] Parasitic wasps are typically fixed in 95% alcohol to preserve the wing structure. [ 5 ] Butterflies and moths are generally killed manually by crushing the thorax, as they can often destroy their wings by beating them against the jar. [ 6 ] | https://en.wikipedia.org/wiki/Killing_jar |
A Killough platform is a three-wheel drive system that uses traditional wheels to achieve omni-directional movement without the use of omni-directional wheels (such as omni wheels / Mecanum wheels ). Designed by Stephen Killough, after which the platform is named, with help from Francois Pin, wanted to achieve omni-directional movement without using the complicated six motor arrangement required to achieve a controllable three caster wheel system (one motor to control wheel rotation and one motor to control pivoting of the wheel). He first looked into solutions by other inventors that used rollers on the rims larger wheels but considered them flawed in some critical way. This led to the Killough system:
Picture a round platform with three motors underneath, each governing the motion of two wheels that look like miniature balloon tires. The wheels in each pair are mounted in a cage at right angles to each other; the motor can rotate the cage so that one wheel or the other is touching the ground at any one time. By configuring the three pairs of wheels to allow the same type of motion found in three pivoting casters, and by changing the relative speeds of the motors, Killough can make his robotic platform rotate, follow a straight or curved path, and even rotate while moving forward.
With Francois Pin, who helped with the computer control and choreography aspects of the design, Killough and Pin readied a public demonstration in 1994. This led to a partnership with Cybertrax Innovative Technologies in 1996, which was developing a motorized wheelchair. [ 1 ]
By combining the motion of two wheels the vehicle can move in the direction of the third, perpendicular, wheel; or, by setting the cages so that no wheel is perpendicular and rotating all the wheels in the same direction, the vehicle can rotate in place. By using the resultant motion of the vector addition of the wheels a Killough platform is able to achieve omni-directional motion. [ 2 ]
This robotics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Killough_platform |
A kiln is a thermally insulated chamber, a type of oven , that produces temperatures sufficient to complete some process, such as hardening, drying, or chemical changes . Kilns have been used for millennia to turn objects made from clay into pottery , tiles and bricks . Various industries use rotary kilns for pyroprocessing (to calcinate ores, such as limestone to lime for cement ) and to transform many other materials.
According to the Oxford English Dictionary , kiln was derived from the words cyline, cylene, cyln(e) in Old English , in turn derived from Latin culina ('kitchen'). In Middle English , the word is attested as kulne, kyllne, kilne, kiln, kylle, kyll, kil, kill, keele, kiele. [ 1 ] [ 2 ] In Greek the word καίειν, kaiein , means 'to burn'.
The word 'kiln' was originally pronounced 'kil' with the 'n' silent, as is referenced in Webster's Dictionary of 1828 [ 3 ] and in English Words as Spoken and Written for Upper Grades by James A. Bowen 1900: "The digraph ln, n silent, occurs in kiln. A fall down the kiln can kill you." [ 4 ] Bowen was noting that "kill" and "kiln" are homophones . [ 5 ]
Pit fired pottery was produced for thousands of years before the earliest known kiln, which dates to around 6000 BCE , and was found at the Yarim Tepe site in modern Iraq . [ 6 ] Neolithic kilns were able to produce temperatures greater than 900 °C (1652 °F). [ 7 ] Uses include:
Kilns are an essential part of the manufacture of almost all types of ceramics . Ceramics require high temperatures so chemical and physical reactions will occur to permanently alter the unfired body. In the case of pottery, clay materials are shaped, dried and then fired in a kiln. The final characteristics are determined by the composition and preparation of the clay body and the temperature at which it is fired. After a first firing, glazes may be used and the ware is fired a second time to fuse the glaze into the body. A third firing at a lower temperature may be required to fix overglaze decoration. Modern kilns often have sophisticated electronic control systems, although pyrometric devices are often also used.
Clay consists of fine-grained particles that are relatively weak and porous. Clay is combined with other minerals to create a workable clay body. The firing process includes sintering . This heats the clay until the particles partially melt and flow together, creating a strong, single mass, composed of a glassy phase interspersed with pores and crystalline material. Through firing, the pores are reduced in size, causing the material to shrink slightly.
In the broadest terms, there are two types of kilns: intermittent and continuous, both being an insulated box with a controlled inner temperature and atmosphere.
A continuous kiln , sometimes called a tunnel kiln , is long with only the central portion directly heated. From the cool entrance, ware is slowly moved through the kiln, and its temperature is increased steadily as it approaches the central, hottest part of the kiln. As it continues through the kiln, the temperature is reduced until the ware exits the kiln nearly at room temperature. A continuous kiln is energy-efficient, because heat given off during cooling is recycled to pre-heat the incoming ware. In some designs, the ware is left in one place, while the heating zone moves across it. Kilns in this type include:
In the intermittent kiln , the ware is placed inside the kiln, the kiln is closed, and the internal temperature is increased according to a schedule. After the firing is completed, both the kiln and the ware are cooled. The ware is removed, the kiln is cleaned and the next cycle begins. Kilns in this type include: [ 9 ]
Kiln technology is very old. Kilns developed from a simple earthen trench filled with pots and fuel pit firing , to modern methods. One improvement was to build a firing chamber around pots with baffles and a stoking hole. This conserved heat. A chimney stack improved the air flow or draw of the kiln, thus burning the fuel more completely.
Chinese kiln technology has always been a key factor in the development of Chinese pottery , and until recent centuries was the most advanced in the world. The Chinese developed kilns capable of firing at around 1,000 °C before 2000 BCE . These were updraft kilns, often built below ground. Two main types of kiln were developed by about 200 AD and remained in use until modern times. These are the dragon kiln of hilly southern China , usually fuelled by wood, long and thin and running up a slope, and the horseshoe-shaped mantou kiln of the north Chinese plains, smaller and more compact. Both could reliably produce the temperatures of up to 1300 °C or more needed for porcelain . In the late Ming, the egg-shaped kiln or zhenyao was developed at Jingdezhen and mainly used there. This was something of a compromise between the other types, and offered locations in the firing chamber with a range of firing conditions. [ 10 ]
Both Ancient Roman pottery and medieval Chinese pottery could be fired in industrial quantities, with tens of thousands of pieces in a single firing. [ 11 ] Early examples of simpler kilns found in Britain include those that made roof-tiles during the Roman occupation. These kilns were built up the side of a slope, such that a fire could be lit at the bottom and the heat would rise up into the kiln.
Traditional kilns include:
With the industrial age , kilns were designed to use electricity and more refined fuels, including natural gas and propane . Many large industrial pottery kilns use natural gas, as it is generally clean, efficient and easy to control. Modern kilns can be fitted with computerized controls allowing for fine adjustments during the firing. A user may choose to control the rate of temperature climb or ramp , hold or soak the temperature at any given point, or control the rate of cooling. Both electric and gas kilns are common for smaller scale production in industry and craft, handmade and sculptural work.
Modern kilns include:
Green wood coming straight from the felled tree has far too high a moisture content to be commercially useful and will rot, warp and split. Both hardwoods and softwood must be left to dry out until the moisture content is between 18% and 8%. This can be a long process unless accelerated by use of a kiln. A variety of kiln technologies exist today: conventional, dehumidification, solar, vacuum and radio frequency.
Conventional wood dry kilns [ 13 ] are either package-type (side-loader) or track-type (tram) construction. Most hardwood lumber kilns are side-loader kilns in which fork trucks are used to load lumber packages into the kiln. Most softwood kilns are track types in which the timber is loaded on kiln/track cars for loading the kiln. Modern high-temperature, high-air-velocity conventional kilns can typically dry 25-millimetre-thick (1 in) green wood in 10 hours down to a moisture content of 18%. However, 25-mm-thick green red oak requires about 28 days to dry down to a moisture content of 8%. [ citation needed ]
Heat is typically introduced via steam running through fin/tube heat exchangers controlled by on/off pneumatic valves. Humidity is removed by a system of vents, the specific layout of which are usually particular to a given manufacturer. In general, cool dry air is introduced at one end of the kiln while warm moist air is expelled at the other. Hardwood conventional kilns also require the introduction of humidity via either steam spray or cold water misting systems to keep the relative humidity inside the kiln from dropping too low during the drying cycle. Fan directions are typically reversed periodically to ensure even drying of larger kiln charges. [ citation needed ]
Most softwood kilns operate below 115 °C (240 °F) temperature. Hardwood kiln drying schedules typically keep the dry bulb temperature below 80 °C (180 °F). Difficult-to-dry species might not exceed 60 °C (140 °F).
Dehumidification kilns are similar to other kilns in basic construction and drying times are usually comparable. Heat comes primarily from an integral dehumidification unit that also removes humidity. Auxiliary heat is often provided early in the schedule to supplement the dehumidifier.
Solar kilns are conventional kilns, typically built by hobbyists to keep initial investment costs low. Heat is provided via solar radiation, while internal air circulation is typically passive.
Vacuum and radio frequency kilns reduce the air pressure to attempt to speed up the drying process. A variety of these vacuum technologies exist, varying primarily in the method heat is introduced into the wood charge. Hot water platten vacuum kilns use aluminum heating plates with the water circulating within as the heat source, and typically operate at significantly reduced absolute pressure. Discontinuous and SSV (super-heated steam) use atmosphere pressure to introduce heat into the kiln charge. The entire kiln charge comes up to full atmospheric pressure, the air in the chamber is then heated and finally a vacuum is pulled as the charge cools. SSV run at partial-atmospheres, typically around 1/3 of full atmospheric pressure, in a hybrid of vacuum and conventional kiln technology (SSV kilns are significantly more popular in Europe where the locally harvested wood is easier to dry than the North American woods.) RF/V (radio frequency + vacuum) kilns use microwave radiation to heat the kiln charge, and typically have the highest operating cost due to the heat of vaporization being provided by electricity rather than local fossil fuel or waste wood sources. [ citation needed ]
The economics of different wood drying technologies are based on the total energy, capital, insurance/risk, environmental impacts, labor, maintenance, and product degradation costs. These costs, which can be a significant part of plant costs, involve the differential impact of the presence of drying equipment in a specific plant. Every piece of equipment from the green trimmer to the infeed system at the planer mill is part of the "drying system". The true costs of the drying system can only be determined when comparing the total plant costs and risks with and without drying. [ citation needed ]
Kiln dried firewood was pioneered during the 1980s, and was later adopted extensively in Europe due to the economic and practical benefits of selling wood with a lower moisture content (with optimal moisture levels of under 20% being much easier to achieve). [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ]
The total (harmful) air emissions produced by wood kilns, including their heat source, can be significant. Typically, the higher the temperature at which the kiln operates, the larger the quantity of emissions that are produced (per mass unit of water removed). This is especially true in the drying of thin veneers and high-temperature drying of softwoods. [ citation needed ] | https://en.wikipedia.org/wiki/Kiln |
Kiln furniture are devices and implements inside furnaces used during the heating of manufactured individual pieces, such as pottery or other ceramic or metal components. [ 1 ] Kiln furniture is made of refractory materials , i.e., materials that withstand high temperatures without deformation. [ 2 ] Kiln furniture can account for up to 80% of the mass of a kiln charge. [ 3 ]
Commonly used materials are cordierite (up to 1275 °C), mullite (up to 1750 °C), silicon carbide (up to 1500 °C), alumina (up to 1750 °C), zirconia (up to 1650 °C). The choice depends on cost, weight, and physical properties. [ 2 ]
Functions of kiln furniture include carrying the kiln / furnace load and protecting the load from various kind of damage: open file, smoke, debris, from deforming or sticking the components to each other. In addition to various carriers and plates, capsules with heating material may be used. [ 1 ]
Kiln furniture influences the heat distribution in the furnace and the interaction of the load with the atmosphere in the furnace. Since the furniture is being heated along with the load, this increases energy consumption hence the operating costs increase. An additional increase of costs comes from wear of the furniture due to thermomechanical and chemical stresses. To decrease heat capacity porous materials or thinner furniture components may be used. However this calls for a trade-off with load-bearing capacity and stress resistance. [ 1 ]
A saggar (also misspelled as sagger or segger) is a ceramic boxlike container used in the firing of pottery to enclose or protect ware being fired inside a kiln . [ 4 ] [ 5 ] [ 6 ] Saggars have been used to protect, or safeguard, ware from open flame, smoke, gases and kiln debris. [ 7 ] Traditionally, saggars were made primarily from fireclay . [ 8 ] [ 9 ] Modern saggars are made of alumina ceramic, cordierite ceramic, mullite ceramic silicon carbide [ 10 ] [ 11 ] and in special cases from zirconia . [ 12 ]
A pernette or stilt is a prop to support pottery in a kiln so that pottery does not touch each other or kiln's floor. [ 13 ] In archaeology, they may be upside-down fired clay tripods, leaving characteristic marks at the bottoms of the pottery/porcelain. [ 14 ] [ 15 ] They expose the bottom of the fired piece to the full heat. [ 16 ]
Other types of furniture and furniture systems include kiln cars, kiln shelves, batts, tiles, and plates; tubes and beams; props and fittings, profile setters, rollers, stools; T-cranks, Y-cranks, pin cranks. [ 17 ] The design of kiln furniture system depends on the wares manufactured: structural clay products, dinnerware, tiles, electronics ceramics, sanitaryware, electrical porcelain, etc. [ 2 ] | https://en.wikipedia.org/wiki/Kiln_furniture |
The kilobyte is a multiple of the unit byte for digital information .
The International System of Units (SI) defines the prefix kilo as a multiplication factor of 1000 (10 3 ); therefore, one kilobyte is 1000 bytes. [ 1 ] The internationally recommended unit symbol for the kilobyte is kB . [ 1 ]
In some areas of information technology , particularly in reference to random-access memory capacity, kilobyte instead often refers to 1024 (2 10 ) bytes. This arises from the prevalence of sizes that are powers of two in modern digital memory architectures, coupled with the coincidence that 2 10 differs from 10 3 by less than 2.5%.
The kibibyte is defined as 1024 bytes, avoiding the ambiguity issues of the kilobyte . [ 1 ]
In the International System of Units (SI) the metric prefix kilo means 1,000 (10 3 ); therefore, one kilobyte is 1000 bytes. The unit symbol is kB.
This is the definition standardised by the International Electrotechnical Commission (IEC). [ 2 ] This definition, and the related definitions of the prefixes mega ( 1,000,000 ), giga ( 1,000,000,000 ), etc., are most commonly used for data transfer rates in computer networks , internal bus, hard drive and flash media transfer speeds, and for the capacities of most storage media , particularly hard disk drives , [ 3 ] flash -based storage, [ 4 ] and DVDs . It is also consistent with the other uses of the metric prefixes in computing, such as CPU clock speeds or measures of performance .
The international standard IEC 80000-13 uses the term "byte" to mean eight bits (1 B = 8 bit). Therefore, 1 kB = 8000 bit. One thousand kilobytes (1000 kB) is equal to one megabyte (1 MB), where 1 MB is one million bytes.
The term 'kilobyte' has traditionally been used to refer to 1024 bytes (2 10 B). [ 5 ] [ 6 ] [ 7 ] The usage of the metric prefix kilo for binary multiples arose as a convenience, because 1024 is approximately 1000. [ 8 ]
The binary interpretation of metric prefixes is still prominently used by the Microsoft Windows operating system. [ 9 ] Binary interpretation is also used for random-access memory capacity, such as main memory and CPU cache size, due to the prevalent binary addressing of memory.
The binary meaning of the kilobyte for 1024 bytes typically uses the symbol KB, with an uppercase letter K . The B is sometimes omitted in informal use. For example, a processor with 65,536 bytes of cache memory might be said to have "64 K" of cache. In this convention, one thousand and twenty-four kilobytes (1024 KB) is equal to one megabyte (1 MB), where 1 MB is 1024 2 bytes.
In December 1998, the IEC addressed such multiple usages and definitions by creating prefixes such as kibi, mebi, gibi, etc., to unambiguously denote powers of 1024. [ 10 ] Thus the kibibyte, symbol KiB, represents 2 10 bytes = 1024 bytes. These prefixes are now part of IEC 80000-13. The IEC further specified that the kilobyte should only be used to refer to 1000 bytes. The International System of Units restricts the use of the SI prefixes strictly to powers of 10. [ 11 ] | https://en.wikipedia.org/wiki/Kilobyte |
The kilocalorie per mole is a unit to measure an amount of energy per number of molecules , atoms , or other similar particles . It is defined as one kilocalorie of energy (1000 thermochemical gram calories ) per one mole of substance. The unit symbol is written kcal/mol or kcal⋅mol −1 . As typically measured, one kcal/mol represents a temperature increase of one degree Celsius in one liter of water (with a mass of 1 kg) resulting from the reaction of one mole of reagents.
In SI units , one kilocalorie per mole is equal to 4.184 kilojoules per mole (kJ/mol), which comes to approximately 6.9477 × 10 −21 joules per molecule, or about 0.043 eV per molecule. At room temperature (25 °C, 77 °F, or 298.15 K), one kilocalorie per mole is approximately equal to 1.688 kT per molecule.
Even though it is not an SI unit, the kilocalorie per mole is still widely used in chemistry [ 1 ] and biology [ 2 ] for thermodynamical quantities such as thermodynamic free energy , heat of vaporization , heat of fusion and ionization energy . This is due to a variety of factors, including the ease with which it can be calculated based on the units of measure typically employed in quantifying a chemical reaction, especially in aqueous solution. In addition, for many important biological processes, thermodynamic changes are on a convenient order of magnitude when expressed in kcal/mol. For example, for the reaction of glucose with ATP to form glucose-6-phosphate and ADP , the free energy of reaction is −4.0 kcal/mol using the pH = 7 standard state . [ 2 ]
This thermodynamics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kilocalorie_per_mole |
The kilogram per cubic metre (symbol: kg·m −3 , or kg/m 3 ) is the unit of density in the International System of Units (SI). It is defined by dividing the SI unit of mass , the kilogram , by the SI unit of volume , the cubic metre . [ 1 ]
The density of water is about 1000 kg/m 3 or 1 g/cm 3 , because the size of the gram was originally based on the mass of a cubic centimetre of water.
In chemistry , g/cm 3 is more commonly used. | https://en.wikipedia.org/wiki/Kilogram_per_cubic_metre |
A kilonova (also called a macronova ) is a transient astronomical event that occurs in a compact binary system when two neutron stars (BNS) or a neutron star and a black hole collide. [ 1 ] The kilonova, visible over the weeks and months following the merger, is an isotropically expanding luminous afterglow of electromagnetic radiation emitted by the radioactive decay of r -process nuclei synthesized by—and then ejected from—the initial cataclysmic event. [ 2 ] [ 3 ]
The high sphericity of kilonovae through its early epochs was deduced from the blackbody nature of the spectrum observed for the most important recorded BNS merger , GW170817 / AT2017gfo . [ 4 ] [ 5 ]
The existence of thermal transient events from neutron star mergers was first introduced by Li & Paczyński in 1998. [ 1 ] The radioactive glow arising from the merger ejecta was originally called mini-supernova, as it is 1 ⁄ 10 to 1 ⁄ 100 the brightness of a typical supernova , the self-detonation of a massive star. [ 6 ] The term kilonova was later introduced by Metzger et al. in 2010 [ 7 ] to characterize the peak brightness, which they showed reaches 1000 times that of a classical nova .
The first candidate kilonova to be found was detected on June 3, 2013 as short gamma-ray burst GRB 130603B by instruments on board the Swift Gamma-Ray Burst Explorer and KONUS/WIND spacecraft, and then imaged by the Hubble Space Telescope 9 and 30 days later. [ 8 ]
On October 16, 2017, the LIGO and Virgo collaborations announced the detection of GW170817 , [ 9 ] the first gravitational wave (GW) shown to have originated from the binary merger of neutron stars . [ 10 ] From its kilonova, it would also become the first GW to be definitively pinpointed by its corresponding electromagnetic observation. The GW detection co-occurred with a short GRB ( GRB 170817A ) , and then after several hours, a longer lasting astronomical transient ( AT 2017gfo ), visible for weeks in the optical and near-infrared electromagnetic spectrum.
The kilonova observations allowed the event to be precisely located at just 140 million light-years away in the nearby galaxy NGC 4993 . [ 11 ] Observations of AT 2017gfo confirmed that it was the first conclusive observation of a kilonova. [ 12 ] Spectral modelling of AT2017gfo identified the r -process elements strontium and yttrium , which conclusively ties the formation of heavy elements to neutron-star mergers. [ 13 ] [ 14 ] Further modelling showed the ejected fireball of heavy elements was highly spherical in early epochs. [ 4 ] [ 15 ] Some researchers have suggested that "thanks to this work, astronomers could use kilonovae as a standard candle to measure cosmic expansion. Since kilonovae explosions are spherical, astronomers could compare the apparent size of a supernova explosion with its actual size as seen by the gas motion, and thus measure the rate of cosmic expansion at different distances." [ 16 ]
The inspiral and merging of two compact objects are a strong source of gravitational waves (GW). [ 7 ] The basic model for thermal transients from neutron star mergers was introduced by Li-Xin Li and Bohdan Paczyński in 1998. [ 1 ] In their work, they suggested that the radioactive ejecta from a neutron star merger is a source for powering thermal transient emission, later dubbed kilonova . [ 17 ]
A first observational suggestion of a kilonova came in 2008 following the gamma-ray burst GRB 080503, [ 19 ] where a faint object appeared in optical light after one day and rapidly faded. However, other factors such as the lack of a galaxy and the detection of X-rays were not in agreement with the hypothesis of a kilonova. Another kilonova was suggested in 2013, in association with the short-duration gamma-ray burst GRB 130603B, where the faint infrared emission from the distant kilonova was detected using the Hubble Space Telescope . [ 8 ]
In October 2017, astronomers reported that observations of AT 2017gfo showed that it was the first definitive case of a kilonova following a merger of two neutron stars . [ 12 ]
In October 2018, astronomers reported that GRB 150101B , a gamma-ray burst event detected in 2015, may be analogous to the historic GW170817 . The similarities between the two events, in terms of gamma ray , optical and x-ray emissions, as well as to the nature of the associated host galaxies , are considered "striking", and this remarkable resemblance suggests the two separate and independent events may both be the result of the merger of neutron stars, and both may be a hitherto-unknown class of kilonova transients. Kilonova events, therefore, may be more diverse and common in the universe than previously understood, according to the researchers. [ 20 ] [ 21 ] [ 22 ] [ 23 ] In retrospect, GRB 160821B, a gamma-ray burst detected in August 2016, is now believed to also have been due to a kilonova, by its resemblance of its data to AT2017gfo . [ 24 ]
A kilonova was also thought to have caused the long gamma-ray burst GRB 211211A , discovered in December 2021 by Swift ’s Burst Alert Telescope (BAT) and the Fermi Gamma-ray Burst Monitor (GBM). [ 25 ] [ 26 ] These discoveries challenge the formerly prevailing theory that long GRBs exclusively come from supernovae , the end-of-life explosions of massive stars. [ 27 ] GRB 211211A lasted 51s; [ 28 ] [ 29 ] GRB 191019A (2019) [ 30 ] and GRB 230307A (2023), [ 31 ] [ 32 ] with durations of around 64s and 35s respectively, have been also argued to belong to this class of long GBRs from neutron star mergers . [ 33 ] [ 34 ]
In 2023, GRB 230307A was observed and associated with tellurium and lanthanides . [ 35 ] | https://en.wikipedia.org/wiki/Kilonova |
Kim Guldstrand Larsen R (born 1957) is a Danish scientist and professor of computer science at Aalborg University , Denmark . His field of research includes modeling , validation and verification, performance analysis, and synthesing of real-time , embedded , and cyber-physical systems utilizing and contributing to concurrency theory and model checking . Within this domain, he has been instrumental in the invention and continuous development of one of the most widely used verification tools, and has received several awards and honors for his work.
Larsen has an MSc in mathematics from Aalborg University, 1982. [ 1 ] In 1986, he received his PhD in Computer Science from University of Edinburgh , advised by Robin Milner . [ 2 ]
Since 1993, Larsen has been a professor in Computer Science at Aalborg Universitet. [ 3 ] He has also been a visiting professor at several places around the world, including the National Institute for Research in Digital Science and Technology (INRIA) (as an international chair 2016-2020). [ 4 ]
Larsen heads the Center for Embedded Software Systems (CISS). [ 5 ] From 2007 to 2011, he was director of the university-industry consortium Danish Network of Embedded Systems (DaNES), and from 2011 to 2017, he was the Danish co-lead of the Danish-Chinese Center for IDEA4CPS: Foundations for Cyber-Physical Systems, established by the Danish National Research Foundation [ da ] and the Natural Science Foundation of China (NSFC) . [ 6 ] [ 7 ] [ 8 ]
In addition, he was director of the Danish ICT Innovation Network (InfinIT) from 2009 to 2020, director of the Center for Data-Intensive Cyber-Physical Systems (DiCyPS) funded by Innovation Fund Denmark [ da ] from 2015 to 2021, and head of project on the Learning, Analysis, Synthesis, and Optimization of Cyber-Physical Systems (LASSO) project from 2015 to 2020, funded by an ERC Advanced Grant . [ 9 ] [ 10 ] [ 11 ] [ 12 ]
Larsen is one of the key figures behind the award-winning tool UPPAAL , which is one of the most widely used tools for the verification of real-time models. [ 13 ] [ 14 ] "UPPAAL in a Nutshell," written by Larsen and colleagues, is one of the most cited papers in The Journal Software Tools for Technology Transfer, published by Springer (citation rank in the 99th percentile). [ 15 ] [ 16 ]
He is a member of Royal Danish Academy of Sciences and Letters and elected fellow and digital expert (vismand) in the Danish Academy of Technical Sciences [ da ] . He has served as the national expert for the Information and Communication Technology theme under the EU's 7th Framework Programme (FP7-ICT) , and currently he is a member of the Digital, Industry, and Space referencegroup that serves the Danish Ministry of Higher Education and Science in connection to the EU Horizon Europe program . [ 17 ] [ 18 ] [ 19 ] [ 20 ]
Larsen has published six books (monographs) and more than 400 peer-reviewed papers and he has been cited many times ( Google Scholar Citation Tracker ). Selected works: | https://en.wikipedia.org/wiki/Kim_Guldstrand_Larsen |
The Kim reformer is a type of syngas plant invented by Hyun Yong Kim. It is a high temperature furnace (as shown in figure 1), filled with steam and/or carbon dioxide gas and maintaining a thermal equilibrium at a temperature just above 1200 °C, in which the reforming reaction is at its thermodynamic equilibrium and carbonaceous substance is reformed with the highest efficiency.
In December 2000, Kim discovered that the reforming reaction (C + H 2 O ↔ CO + H 2 ) proceeds at a temperature just above 1200 °C, but not below it. This work was published in International Journal [1] and registered in KR patent, US patent, CN patent, and JP patent.
The reformer reforms all carbon atoms of carbonaceous feedstock to produce just syngas , no other hydrocarbons . The high temperature furnace is packed with castables to minimize heat loss in such a way as to maintain the inner temperature of a reduction reactor filled with steam and carbon dioxide (CO 2 ) gas at a temperature just above 1200 °C (aka Kim temperature, see figure 2), [ 1 ] and it reforms all carbonaceous substances most efficiently to produce syngas. The produced syngas exits from the reduction reactor at a temperature of 1200 °C. The reduction chamber is heated by super-hot gases (steam and CO 2 ) generated in the syngas burner with oxygen gas. The reduction chamber must be constructed to withstand, physically and chemically, the reforming reaction at 1200 °C.
Both steam reforming and dry reforming are carried out in this reformer; therefore, it is possible to configure the H 2 /CO ratio by adjusting the H 2 O/CO 2 ratio in the reduction chamber. The reforming reaction is a very specific elementary reaction; all carbon atoms on the left are reformed into carbon monoxide and all hydrogen atoms are reduced to hydrogen gas. The mixture of two product gases is called syngas. These reforming reactions are an endothermic reduction reaction. In contrast, the conventional gasification reaction is a combination of several reactions operating below 1200 °C and the product is a mixture of many gases.
The process for producing water gas (C + H 2 O → CO + H 2 ) has been known since the 19th century and it was later found that it is applicable to all carbonaceous substances. Reactions C + H 2 O ↔ CO + H 2 and (-CH 2 ) + H 2 O → CO + 2H 2 are called steam reforming and reactions C + CO 2 → 2CO and (-CH 2 ) + CO 2 → 2CO + H 2 , carbon dioxide or dry reforming. The oil industry has used the reforming reactions extensively for the cracking process and to generate hydrogen gas. | https://en.wikipedia.org/wiki/Kim_reformer |
Kimito Funatsu ( 船津 公人 , Funatsu Kimito , born December 3, 1955) [ 1 ] is a Japanese chemist specializing in chemoinformatics and data-driven chemistry, a professor emeritus at University of Tokyo , and the research director of the Data Science Center at Nara Institute of Science and Technology .
He graduated from Kagoshima Prefectural Konan High School in 1974 [ 2 ] and from Department of Chemistry, School of Science, Kyushu University in 1978. [ 3 ] He completed Department of Chemistry, Graduate School of Science, Kyushu University and obtained a doctorate in science in 1983. [ 3 ] After he served as an associate professor at Toyohashi University of Technology , he became a professor at Department of Chemical System Engineering, School of Engineering, University of Tokyo in 2004. [ 3 ] He concurrently holds the posts of a professor and the research director of the Data Science Center at Nara Institute of Science and Technology from 2017. [ 3 ] He was also invited as visiting professor at University of Strasbourg in France in 2011. [ 3 ]
The Division of Chemical Information of the American Chemical Society gave him the Herman Skolnik Award in 2019 for his contributions to structure elucidation, de novo structure generation and applications of cheminformatics methods to materials design and chemical process control. [ 4 ] [ 5 ] He also received the Chemical Society of Japan Award for Creative Work ( 日本化学会 学術賞 ) for 2020. [ 6 ] In 2021, he retired from University of Tokyo at mandatory age [ 7 ] and was given the title of professor emeritus. [ 8 ] | https://en.wikipedia.org/wiki/Kimito_Funatsu |
Kin recognition , also called kin detection , is an organism's ability to distinguish between close genetic kin and non-kin. In evolutionary biology and psychology , such an ability is presumed to have evolved for inbreeding avoidance , [ 1 ] though animals do not typically avoid inbreeding. [ 2 ]
An additional adaptive function sometimes posited for kin recognition is a role in kin selection . There is debate over this, since in strict theoretical terms kin recognition is not necessary for kin selection or the cooperation associated with it. Rather, social behaviour can emerge by kin selection in the demographic conditions of 'viscous populations' with organisms interacting in their natal context, without active kin discrimination, since social participants by default typically share recent common origin. Since kin selection theory emerged, much research has been produced investigating the possible role of kin recognition mechanisms in mediating altruism. Some researchers suggest that, taken as a whole, active powers of recognition play a negligible role in mediating social cooperation relative to less elaborate cue-based and context-based mechanisms, such as familiarity and imprinting , whereas other researchers argue that specialized kin recognition mechanisms, such as phenotype matching, are widespread in facilitating nepotism. [ 3 ]
Because cue-based 'recognition' predominates in social mammals, outcomes are non-deterministic in relation to actual genetic kinship, instead outcomes simply reliably correlate with genetic kinship in an organism's typical conditions. A well-known human example of an inbreeding avoidance mechanism is the Westermarck effect , in which unrelated individuals who happen to spend their childhood in the same household find each other sexually unattractive. Similarly, due to the cue-based mechanisms that mediate social bonding and cooperation, unrelated individuals who grow up together in this way are also likely to demonstrate strong social and emotional ties, and enduring altruism .
The English evolutionary biologist W. D. Hamilton 's theory of inclusive fitness , and the related theory of kin selection , were formalized in the 1960s and 1970s to explain the evolution of social behaviours. Hamilton's early papers, as well as giving a mathematical account of the selection pressure, discussed possible implications and behavioural manifestations. Hamilton considered potential roles of cue-based mechanisms mediating altruism versus 'positive powers' of kin discrimination:
The selective advantage which makes behaviour conditional in the right sense on the discrimination of factors which correlate with the relationship of the individual concerned is therefore obvious. It may be, for instance, that in respect of a certain social action performed towards neighbours indiscriminately, an individual is only just breaking even in terms of inclusive fitness. If he could learn to recognise those of his neighbours who really were close relatives and could devote his beneficial actions to them alone an advantage to inclusive fitness would at once appear. Thus, a mutation causing such discriminatory behaviour itself benefits inclusive fitness and would be selected. In fact, the individual may not need to perform any discrimination so sophisticated as we suggest here; a difference in the generosity of his behaviour according to whether the situations evoking it were encountered near to, or far from, his own home might occasion an advantage of a similar kind." (1996 [1964], 51) [ 4 ]
These two possibilities, altruism mediated via 'passive situation' or via 'sophisticated discrimination', stimulated a generation of researchers to look for evidence of any 'sophisticated' kin discrimination. However, Hamilton later (1987) developed his thinking to consider that "an innate kin recognition adaptation" was unlikely to play a role in mediating altruistic behaviours:
But once again, we do not expect anything describable as an innate kin recognition adaptation, used for social behaviour other than mating, for the reasons already given in the hypothetical case of the trees.(Hamilton 1987, 425) [ 5 ]
The implication that the inclusive fitness criterion can be met by mediating mechanisms of cooperative behaviour that are context and location-based has been clarified by recent work by West et al. :
In his original papers on inclusive fitness theory, Hamilton pointed out a sufficiently high relatedness to favour altruistic behaviours could accrue in two ways—kin discrimination or limited dispersal (Hamilton, 1964, 1971, 1972, 1975). There is a huge theoretical literature on the possible role of limited dispersal reviewed by Platt & Bever (2009) and West et al. (2002a), as well as experimental evolution tests of these models (Diggle et al., 2007; Griffin et al., 2004; Kümmerli et al., 2009 ). However, despite this, it is still sometimes claimed that kin selection requires kin discrimination (Oates & Wilson, 2001; Silk, 2002 ). Furthermore, a large number of authors appear to have implicitly or explicitly assumed that kin discrimination is the only mechanism by which altruistic behaviours can be directed towards relatives... [T]here is a huge industry of papers reinventing limited dispersal as an explanation for cooperation. The mistakes in these areas seem to stem from the incorrect assumption that kin selection or indirect fitness benefits require kin discrimination (misconception 5), despite the fact that Hamilton pointed out the potential role of limited dispersal in his earliest papers on inclusive fitness theory (Hamilton, 1964; Hamilton, 1971; Hamilton, 1972; Hamilton, 1975). (West et al. 2010, p. 243 and supplement) [ 6 ]
For a recent review of the debates around kin recognition and their role in the wider debates about how to interpret inclusive fitness theory, including its compatibility with ethnographic data on human kinship, see Holland (2012). [ 7 ]
Leading inclusive fitness theorists such as Alan Grafen have argued that the whole research program around kin recognition is somewhat misguided:
Do animals really recognise kin in a way that is different from the way they recognise mates, neighbours, and other organisms and objects? Certainly animals use recognition systems to recognise their offspring, their siblings and their parents. But to the extent that they do so in the same way that they recognise their mates and their neighbours, I feel it is unhelpful to say they have a kin recognition system."
(Grafen 1991, 1095) [ 8 ]
Others have cast similar doubts over the enterprise:
[T]he fact that animals benefit from engaging in spatially mediated behaviors is not evidence that these animals can recognize their kin, nor does it support the conclusion that spatially based differential behaviors represent a kin recognition mechanism (see also discussions by Blaustein, 1983; Waldman, 1987; Halpin 1991). In other words, from an evolutionary perspective it may well be advantageous for kin to aggregate and for individuals to behave preferentially towards nearby kin, whether or not this behaviour is the result of kin recognition per se" (Tang-Martinez 2001, 25) [ 9 ]
Kin recognition has been documented in many species of vertebrates and invertebrates. Recent studies have shown that kin recognition can result from a multitude of sensory input. Jill Mateo notes that there are three components prominent in kin recognition. First, "production of unique phenotypic cues or labels". Second, "perception of these labels and the degree of correspondence of these labels with a 'recognition template'", and finally the recognition of the phenotypes should lead to "action taken by the animal as a function of the perceived similarity between its template and an encountered phenotype". [ 10 ]
The three components allow for several possible mechanisms of kin recognition. Sensory information gathered from visual, olfactory and auditory stimuli are the most prevalent. The Belding's ground squirrel kin produce similar odors in comparison to non-kin. [ 11 ] Mateo notes that the squirrels spent longer investigating non-kin scents suggesting recognition of kin odor. It's also noted that Belding's ground squirrels produce at least two scents arising from dorsal and oral secretions, giving two opportunities for kin recognition. In addition, the Black Rock Skink is also able to use olfactory stimuli as a mechanism of kin recognition. Egernia saxatilis have been found to discriminate kin from non-kin based on scent. Egernia striolata also use some form of scent, most likely through skin secretions. [ 12 ] However, Black Rock Skinks discriminate based on familiarity rather than genotypic similarity. Juvenile E. saxatilis can recognize the difference between the scent of adults from their own family group and unrelated adults. Black Rock Skink recognize their family groups based on prior association and not how genetically related the other lizards are to themselves. [ 13 ] Auditory distinctions have been noted among avian species. Long-tailed tits ( Aegithalos caudatus ) are capable of discriminating kin and non-kin based on contact calls. Distinguishing calls are often learned from adults during the nestling period. [ 14 ] Studies suggest that the bald-faced hornet, Dolichovespula maculata , can recognize nest mates by their cuticular hydrocarbon profile, which produces a distinct smell. [ 15 ]
Some animals can recognize kin by "self-referencing:" comparing the phenotypes of others to themselves. For example, Belding's ground squirrels identify relatives by comparing their own odor and those of littermates with odors of squirrels they encounter. [ 16 ] The phenotypes that are used are odors from dorsal and anal glands, and each animal has its own repertoire of odors. [ 17 ] If another individual's odor phenotype matches itself closely enough, it is likely a relative. Laboratory tests [ 16 ] indicate that females can discriminate between kin and nonkin, close and distant relatives and, within-litters, between full-siblings and maternal half-siblings. Field observations [ 18 ] [ 19 ] confirm that females cooperate with their closest kin more than with distant kin, and behave aggressively toward nonrelatives. [ 16 ] Golden hamsters [ 20 ] and bluegill sunfish [ 21 ] also can use themselves as
referents to discriminate close relatives from distant kin and nonkin.
Kin recognition in some species may also be mediated by immunogenetic similarity of the major histocompatibility complex (MHC). [ 22 ] For a discussion of the interaction of these social and biological kin recognition factors see Lieberman, Tooby, and Cosmides (2007). [ 23 ] Some have suggested that, as applied to humans, this nature-nurture interactionist perspective allows a synthesis [ 7 ] between theories and evidence of social bonding and cooperation across the fields of evolutionary biology, psychology ( attachment theory ) and cultural anthropology ( nurture kinship ).
A study has shown that humans are about as genetically equivalent to their friends as they are their fourth cousins. [ 24 ]
Kin recognition is an adaptive behavior observed in living beings to prevent inbreeding, and increase fitness of populations, individuals and genes. Kin recognition is the key to successful reciprocal altruism , a behavior that increases reproductive success of both organisms involved. Reciprocal altruism as a product of kin recognition has been observed and studied in many animals, and more recently, plants . Due to the nature of plant reproduction and growth, plants are more likely than animals to live in close proximity to family members, and therefore stand to gain more from the ability to differentiate kin from strangers. [ 25 ]
In recent years, botanists have been conducting studies to determine which plant species can recognize kin, and discover the responses of plants to neighboring kin. Murphy and Dudley (2009) shows that Impatiens pallida has the ability to recognize individuals closely related to them and those not related to them. The physiological response to this recognition is increasingly interesting. I. pallida responds to kin by increasing branchiness and stem elongation, to prevent shading relatives, and responds to strangers by increasing leaf to root allocation, as a form of competition. [ 26 ]
Root allocation has been a very common trait shown through research in plants. Limited amounts of biomass can cause trade-offs among the construction of leaves, stems, and roots overall. But, in plants that recognize kin, the movement of resources in the plant has been shown to be affected by proximity to related individuals. [ 27 ] It is well documented that roots can emit volatile compounds in the soil and that interactions also occur below-ground between plant roots and soil organisms. This has mainly focused on organisms in the kingdom Animalia, however.
Regarding this, root systems are known to exchange carbon and defense related molecular signals via connected mycorrhizal networks. For instance, it has been demonstrated that tobacco plants can detect the volatile chemical ethylene in order to form a “shade-avoidance phenotype.” [ 28 ] Barley plants were also shown to allocate biomass to their roots when exposed to chemical signals from members of the same species, [ 28 ] showing that, if they can recognize those signals for competition, recognition of kin in the plant could be likely via a similar chemical response.
Similarly, Bhatt et al. (2010) show that Cakile edentula , the American sea rocket, has the ability to allocate more energy to root growth, and competition, in response to growing next to a stranger, and allocates less energy to root growth when planted next to a sibling. This reduces competition between siblings and increases fitness of relatives growing next to each other, while still allowing competition between non-relative plants. [ 29 ]
Little is known about the mechanisms involved in kin recognition. They most likely vary between species as well as within species. A study by Bierdrzycki et al. (2010) shows that root secretions are necessary for Arabidopsis thaliana to recognize kin vs. strangers, but not necessary to recognize self vs. non-self roots. This study was performed using secretion inhibitors, which disabled the mechanism responsible for kin recognition in this species, and showed similar growth patterns to Bhatt et al., (2010) and Murphy and Dudley (2009) in control groups. The most interesting result of this study was that inhibiting root secretions did not reduce the ability of Arabidopsis to recognize their own roots, which implicates a separate mechanism for self/non-self recognition than that for kin/stranger recognition. [ 30 ]
While this mechanism in the roots responds to exudates and involves competition over resources like nitrogen and phosphorus, another mechanism has been recently proposed, which involves competition over light, in which kin recognition takes place in leaves. In their 2014 study, Crepy and Casal conducted multiple experiments on different accessions of A. thaliana . These experiments showed that Arabidopsis accessions have distinct R:FR and blue light signatures, and that these signatures can be detected by photoreceptors, which allows the plant to recognize its neighbor as a relative or non-relative. Not much is known about the pathway that Arabidopsis uses to associate these light patterns with kin, however, researchers ascertained that photoreceptors phyB, cry 1, cry 2, phot1, and phot2 are involved in the process by performing a series of experiments with knock-out mutants. Researchers also concluded that the auxin-synthesis gene TAA1 is involved in the process, downstream of the photoreceptors, by performing a similar experiments using Sav3 knock-out mutants. This mechanism leads to altered leaf direction to prevent shading of related neighbors and to reduce competition for sunlight. [ 31 ]
When mice inbreed with close relatives in their natural habitat, there is a significant detrimental effect on progeny survival. [ 32 ] Since inbreeding can be detrimental, it tends to be avoided by many species. In the house mouse , the major urinary protein (MUP) gene cluster provides a highly polymorphic scent signal of genetic identity that appears to underlie kin recognition and inbreeding avoidance. Thus there are fewer matings between mice sharing MUP haplotypes than would be expected if there were random mating. [ 33 ] Another mechanism for avoiding inbreeding is evident when a female house mouse mates with multiple males. In such a case, there appears to be egg-driven sperm selection against sperm from related males. [ 34 ]
In toads , male advertisement vocalizations may serve as cues by which females recognize their kin and thus avoid inbreeding. [ 35 ]
In dioecious plants, the stigma may receive pollen from several different potential donors. As multiple pollen tubes from the different donors grow through the stigma to reach the ovary , the receiving maternal plant may carry out pollen selection favoring pollen from less related donor plants. [ 36 ] Thus, kin recognition at the level of the pollen tube apparently leads to post-pollination selection to avoid inbreeding depression . Also, seeds may be aborted selectively depending on donor–recipient relatedness. [ 36 ] | https://en.wikipedia.org/wiki/Kin_recognition |
The term Kinase tyrosine-based inhibitory motif (KTIM) was coined by a group of immunology researchers from McGill University , Montreal, Quebec , Canada [ 1 ] in 2008 to represent any immunoreceptor tyrosine-based inhibitory -like sequence motif , with the consensus sequence I/V/L/SxYxxL/V, found in a kinase and regulating its activity.
One major way in which cells respond to stimuli is through signal transduction pathways, whereby a ligand binds to a receptor, causing conformational changes that lead to a cascade of events in intracellular signalling molecules. This ultimately ends up in the translocation of transcription factors to the nucleus altering the expression of target genes, therefore affecting specific cellular functions. One way activation signals can be counteracted is through the triggering of different receptors bearing immunoreceptor tyrosine-based inhibitory motifs (ITIMs) in their cytoplasmic tails.
Several transmembrane receptors are negatively regulated through recruiting cytosolic SH2 domain -containing proteins such as SHP-1 to immunoreceptor tyrosine-based inhibitory motifs (ITIMs) that they possess. ITIM-like motifs (KTIMs) were shown to exist in non-receptor proteins and to play a key role in their regulation.
The main difference between ITIM and KTIM is that KTIM was shown to be found in a cytosolic protein (IL-1 receptor-associated Kinase 1 (IRAK-1)) and not in a transmembrane protein such as the case with ITIMs. [ 1 ]
KTIMs are speculated to play a regulatory role in the negative regulation of many kinases in addition to IRAK-1, and can thus represent a novel regulatory mechanism in a wide range of cellular kinases. [ 1 ] [ 2 ] The mode of action of KTIMs is similar to ITIMs and involves recruiting SH2 domain-containing proteins such as SHP-1. | https://en.wikipedia.org/wiki/Kinase_tyrosine-based_inhibitory_motif |
A physical quantity (or simply quantity ) [ 1 ] [ a ] is a property of a material or system that can be quantified by measurement . A physical quantity can be expressed as a value , which is the algebraic multiplication of a numerical value and a unit of measurement . For example, the physical quantity mass , symbol m , can be quantified as m = n kg, where n is the numerical value and kg is the unit symbol (for kilogram ). Quantities that are vectors have, besides numerical value and unit, direction or orientation in space.
Following ISO 80000-1 , [ 1 ] any value or magnitude of a physical quantity is expressed as a comparison to a unit of that quantity. The value of a physical quantity Z is expressed as the product of a numerical value { Z } (a pure number) and a unit [ Z ]:
For example, let Z {\displaystyle Z} be "2 metres"; then, { Z } = 2 {\displaystyle \{Z\}=2} is the numerical value and [ Z ] = m e t r e {\displaystyle [Z]=\mathrm {metre} } is the unit.
Conversely, the numerical value expressed in an arbitrary unit can be obtained as:
The multiplication sign is usually left out, just as it is left out between variables in the scientific notation of formulas. The convention used to express quantities is referred to as quantity calculus . In formulas, the unit [ Z ] can be treated as if it were a specific magnitude of a kind of physical dimension : see Dimensional analysis for more on this treatment.
International recommendations for the use of symbols for quantities are set out in ISO/IEC 80000 , the IUPAP red book and the IUPAC green book . For example, the recommended symbol for the physical quantity "mass" is m , and the recommended symbol for the quantity "electric charge" is Q .
Physical quantities are normally typeset in italics.
Purely numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in italics. Symbols for elementary functions (circular trigonometric, hyperbolic, logarithmic etc.), changes in a quantity like Δ in Δ y or operators like d in d x , are also recommended to be printed in roman type.
Examples:
A scalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the Latin or Greek alphabet , and are printed in italic type.
Vectors are physical quantities that possess both magnitude and direction and whose operations obey the axioms of a vector space . Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, if u is the speed of a particle, then the straightforward notations for its velocity are u , u , or u → {\displaystyle {\vec {u}}} .
Scalar and vector quantities are the simplest tensor quantities , which are tensors that can be used to describe more general physical properties. For example, the Cauchy stress tensor possesses magnitude, direction, and orientation qualities.
The notion of dimension of a physical quantity was introduced by Joseph Fourier in 1822. [ 2 ] By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension.
There is often a choice of unit, though SI units are usually used in scientific contexts due to their ease of use, international familiarity and prescription. For example, a quantity of mass might be represented by the symbol m , and could be expressed in the units kilograms (kg), pounds (lb), or daltons (Da).
Dimensional homogeneity is not necessarily sufficient for quantities to be comparable; [ 1 ] for example, both kinematic viscosity and thermal diffusivity have dimension of square length per time (in units of m 2 /s ).
Quantities of the same kind share extra commonalities beyond their dimension and units allowing their comparison;
for example, not all dimensionless quantities are of the same kind. [ 1 ]
A systems of quantities relates physical quantities, and due to this dependence, a limited number of quantities can serve as a basis in terms of which the dimensions of all the remaining quantities of the system can be defined. A set of mutually independent quantities may be chosen by convention to act as such a set, and are called base quantities. The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units and dimensions are listed in the following table. [ 3 ] : 136 Other conventions may have a different number of base units (e.g. the CGS and MKS systems of units).
The angular quantities, plane angle and solid angle , are defined as derived dimensionless quantities in the SI. For some relations, their units radian and steradian can be written explicitly to emphasize the fact that the quantity involves plane or solid angles. [ 3 ] : 137
Derived quantities are those whose definitions are based on other physical quantities (base quantities).
Important applied base units for space and time are below. Area and volume are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.
Important and convenient derived quantities such as densities, fluxes , flows , currents are associated with many quantities. Sometimes different terms such as current density and flux density , rate , frequency and current , are used interchangeably in the same context; sometimes they are used uniquely.
To clarify these effective template-derived quantities, we use q to stand for any quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions – where [ q ] denotes the dimension of q .
For time derivatives, specific, molar, and flux densities of quantities, there is no one symbol; nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we use q m , q n , and F respectively. No symbol is necessarily required for the gradient of a scalar field, since only the nabla/del operator ∇ or grad needs to be written. For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts.
For current density, t ^ {\displaystyle \mathbf {\hat {t}} } is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the dot product with the unit normal for a surface, since the amount of current passing through the surface is reduced when the current is not normal to the area. Only the current passing perpendicular to the surface contributes to the current passing through the surface, no current passes in the (tangential) plane of the surface.
The calculus notations below can be used synonymously.
If X is a n -variable function X ≡ X ( x 1 , x 2 ⋯ x n ) {\displaystyle X\equiv X\left(x_{1},x_{2}\cdots x_{n}\right)} , then
Differential The differential n -space volume element is d n x ≡ d V n ≡ d x 1 d x 2 ⋯ d x n {\displaystyle \mathrm {d} ^{n}x\equiv \mathrm {d} V_{n}\equiv \mathrm {d} x_{1}\mathrm {d} x_{2}\cdots \mathrm {d} x_{n}} ,
No common symbol for n -space density, here ρ n is used.
(length, area, volume or higher dimensions)
q = ∫ q λ d λ {\displaystyle q=\int q_{\lambda }\mathrm {d} \lambda } q = ∫ q ν d ν {\displaystyle q=\int q_{\nu }\mathrm {d} \nu }
[q]T ( q ν )
Transport mechanics , nuclear physics / particle physics : q = ∭ F d A d t {\displaystyle q=\iiint F\mathrm {d} A\mathrm {d} t}
Vector field : Φ F = ∬ S F ⋅ d A {\displaystyle \Phi _{F}=\iint _{S}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }
k -vector q : m = r ∧ q {\displaystyle \mathbf {m} =\mathbf {r} \wedge q} | https://en.wikipedia.org/wiki/Kind_of_quantity |
Kindara is a femtech company headquartered in Colorado that develops apps that help women identify their fertile window. The products are used for women trying to get pregnant, or women who want to track their menstrual cycle for overall health. [ 1 ] [ 2 ] Their latest product, Priya Fertility and Ovulation Monitor, maximizes a woman's chance of getting pregnancy by identifying her most fertile days.
Kindara was founded in 2011 by husband-and-wife team Will Sacks and Kati Bicknell. [ 3 ] The company launched its free mobile application in 2012. Kindara's mobile application allows women to track signs of fertility, such as basal body temperature , cervical fluid , and the position of the cervix to determine when ovulation is occurring. [ 1 ] [ 2 ] Kindara also sells a thermometer, Wink, which records basal body temperature and syncs automatically to the Kindara fertility application. [ 4 ]
In 2018, Kindara was acquired by the company Prima-Temp. [ 5 ]
This article related to medical technology is a stub . You can help Wikipedia by expanding it .
This human reproduction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kindara |
A kinemage (short for kinetic image ) is an interactive graphic scientific illustration. It often is used to visualize molecules , especially proteins although it can also represent other types of 3-dimensional data (such as geometric figures, social networks, [ 1 ] or tetrahedra of RNA base composition ). The kinemage system is designed to optimize ease of use, interactive performance, and the perception and communication of detailed 3D information. The kinemage information is stored in a text file, human- and machine-readable, that describes the hierarchy of display objects and their properties, and includes optional explanatory text. The kinemage format is a defined chemical MIME type of 'chemical/x-kinemage' with the file extension '.kin'.
Kinemages were first developed by David Richardson at Duke University School of Medicine , for the Protein Society's journal Protein Science that premiered in January 1992. [ 2 ] For its first 5 years (1992–1996), each issue of Protein Science included a supplement on floppy disk of interactive, kinemage 3D computer graphics to illustrate many of the articles, plus the Mage software ( cross-platform , free, open-source ) to display them; [ 3 ] kinemage supplementary material is still available on the journal web site. Mage and RasMol [ 4 ] were the first widely used macro molecular graphics programs to support interactive display on personal computers . Kinemages are used for teaching, [ 5 ] [ 6 ] and for textbook supplements, [ 7 ] [ 8 ] individual exploration, and analysis of macromolecular structures.
More recently, with the availability of a much wider variety of other molecular graphics tools, presentation use of kinemages has been overtaken by a wide variety of research uses, concomitant with new display features and with the development of software that produces kinemage-format output from other types of molecular calculations. All-atom contact analysis [ 9 ] adds and optimizes explicit hydrogen atoms, [ 10 ] and then uses patches of dot surface to display the hydrogen bond , van der Waals , and steric clash interactions between atoms. The results can be used visually (in kinemages) and quantitatively to analyze the detailed interactions between molecular surfaces, [ 11 ] [ 12 ] most extensively for the purpose of validating and improving the molecular models from experimental x-ray crystallography data. [ 13 ] [ 14 ] [ 15 ] [ 16 ] Both Mage and KiNG (see below) have been enhanced for kinemage display of data in higher than 3 dimensions (moving between views in various 3-D projections, coloring and selecting candidate clusters of datapoints, and switching to a parallel coordinates representation), used for instance for defining clusters of favorable RNA backbone conformations in the 7-dimensional space of backbone dihedral angles between one ribose and the next. [ 17 ]
KiNG is an open-source kinemage viewer, written in the programming language Java by Ian Davis and Vincent Chen, [ 18 ] that can work interactively either standalone on a user machine with no network connection, or as a web service in a web page . The interactive nature of kinemages is their primary purpose and attribute. To appreciate their nature, the demonstration KiNG in browser has two examples that can be moved around in 3D, plus instructions for how to embed a kinemage on a web page. [ 19 ] The figure below shows KiNG being used to remodel a lysine sidechain in a high-resolution crystal structure. KiNG is one of the viewers provided on each structure page at the Protein Data Bank site, [ 20 ] and displays validation results in 3D on the MolProbity site. [ 21 ] [ 22 ] [ 23 ] Kinemages can also be shown in immersive virtual reality systems, with the open-source KinImmerse software. [ 24 ] All of the kinemage display and all-atom contact software is available free and open-source on the kinemage web site . | https://en.wikipedia.org/wiki/Kinemage |
In mechanical engineering , a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system . [ 1 ] As the word chain suggests, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is the kinematic model for a typical robot manipulator . [ 2 ]
Mathematical models of the connections, or joints, between two links are termed kinematic pairs . Kinematic pairs model the hinged and sliding joints fundamental to robotics , often called lower pairs and the surface contact joints critical to cams and gearing , called higher pairs. These joints are generally modeled as holonomic constraints . A kinematic diagram is a schematic of the mechanical system that shows the kinematic chain.
The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in compliant mechanisms and micro-electro-mechanical systems , and cable compliance in cable robotic and tensegrity systems. [ 3 ] [ 4 ]
The degrees of freedom , or mobility, of a kinematic chain is the number of parameters that define the configuration of the chain. [ 2 ] [ 5 ] A system of n rigid bodies moving in space has 6 n degrees of freedom measured relative to a fixed frame. This frame is included in the count of bodies, so that mobility does not depend on link that forms the fixed frame. This means the degree-of-freedom of this system is M = 6( N − 1) , where N = n + 1 is the number of moving bodies plus the fixed body.
Joints that connect bodies impose constraints. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints c that a joint imposes in terms of the joint's freedom f , where c = 6 − f . In the case of a hinge or slider , which are one-degree-of-freedom joints, have f = 1 and therefore c = 6 − 1 = 5 .
The result is that the mobility of a kinematic chain formed from n moving links and j joints each with freedom f i , i = 1, 2, …, j , is given by
Recall that N includes the fixed link.
The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form algebraic equations that are solved to determine the configuration of the chain associated with specific values of input parameters, called degrees of freedom .
The constraint equations for a kinematic chain are obtained using rigid transformations [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the dimensions of each link. In the case of a serial open chain, the result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link. A chain of n links connected in series has the kinematic equations,
where [ T ] is the transformation locating the end-link—notice that the chain includes a "zeroth" link consisting of the ground frame to which it is attached. These equations are called the forward kinematics equations of the serial chain. [ 6 ]
Kinematic chains of a wide range of complexity are analyzed by equating the kinematics equations of serial chains that form loops within the kinematic chain. These equations are often called loop equations .
The complexity (in terms of calculating the forward and inverse kinematics ) of the chain is determined by the following factors:
Explanation
Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system. [ 5 ]
The constraint equations of a kinematic chain can be used in reverse to determine the dimensions of the links from a specification of the desired movement of the system. This is termed kinematic synthesis. [ 7 ]
Perhaps the most developed formulation of kinematic synthesis is for four-bar linkages , which is known as Burmester theory . [ 8 ] [ 9 ] [ 10 ]
Ferdinand Freudenstein is often called the father of modern kinematics for his contributions to the kinematic synthesis of linkages beginning in the 1950s. His use of the newly developed computer to solve Freudenstein's equation became the prototype of computer-aided design systems. [ 7 ]
This work has been generalized to the synthesis of spherical and spatial mechanisms. [ 2 ] | https://en.wikipedia.org/wiki/Kinematic_chain |
Kinematic determinacy is a term used in structural mechanics to describe a structure where material compatibility conditions alone can be used to calculate deflections . [ 1 ] A kinematically determinate structure can be defined as a structure where, if it is possible to find nodal displacements compatible with member extensions, those nodal displacements are unique. The structure has no possible mechanisms, i.e. nodal displacements, compatible with zero member extensions, at least to a first-order approximation. Mathematically, the mass matrix of the structure must have full rank. Kinematic determinacy can be loosely used to classify an arrangement of structural members as a structure (stable) instead of a mechanism (unstable). The principles of kinematic determinacy are used to design precision devices such as mirror mounts for optics, and precision linear motion bearings .
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kinematic_determinacy |
In mechanical engineering , a kinematic diagram or kinematic scheme (also called a joint map or skeleton diagram ) illustrates the connectivity of links and joints of a mechanism or machine rather than the dimensions or shape of the parts. Often links are presented as geometric objects, such as lines, triangles or squares, that support schematic versions of the joints of the mechanism or machine. [ 1 ]
For example, the figures show the kinematic diagrams (i) of the slider-crank that forms a piston and crank-shaft in an engine, and (ii) of the first three joints for a PUMA manipulator .
A kinematic diagram can be formulated as a graph by representing the joints of the mechanism as vertices and the links as edges of the graph. This version of the kinematic diagram has proven effective in enumerating kinematic structures in the process of machine design. [ 2 ]
An important consideration in this design process is the degree of freedom of the system of links and joints, which is determined using the Chebychev–Grübler–Kutzbach criterion .
Elements of kinematics diagrams include the frame, which is the frame of reference for all the moving components, as well as links ( kinematic pairs ), and joints. Primary Joints include pins, sliders and other elements that allow pure rotation or pure linear motion. Higher order joints also exist that allow a combination of rotation or linear motion. Kinematic diagrams also include points of interest, and other important components. | https://en.wikipedia.org/wiki/Kinematic_diagram |
In mechanical engineering , kinematic synthesis (also known as mechanism synthesis ) determines the size and configuration of mechanisms that shape the flow of power through a mechanical system , or machine , to achieve a desired performance. [ 1 ] The word synthesis refers to combining parts to form a whole. [ 2 ] Hartenberg and Denavit describe kinematic synthesis as [ 3 ]
...it is design, the creation of something new. Kinematically, it is the conversion of a motion idea into hardware.
The earliest machines were designed to amplify human and animal effort, later gear trains and linkage systems captured wind and flowing water to rotate millstones and pumps . Now machines use chemical and electric power to manufacture, transport, and process items of all types. And kinematic synthesis is the collection of techniques for designing those elements of these machines that achieve required output forces and movement for a given input.
Applications of kinematic synthesis include determining:
Kinematic synthesis for a mechanical system is described as having three general phases, known as type synthesis, number synthesis and dimensional synthesis. [ 3 ] Type synthesis matches the general characteristics of a mechanical system to the task at hand, selecting from an array of devices such as a cam-follower mechanism, linkage, gear train, a fixture or a robotic system for use in a required task. Number synthesis considers the various ways a particular device can be constructed, generally focussed on the number and features of the parts. Finally, dimensional synthesis determines the geometry and assembly of the components that form the device.
A linkage is an assembly of links and joints that is designed to provide required force and movement. Number synthesis of linkages which considers the number of links and the configuration of the joints is often called type synthesis, because it identifies the type of linkage. [ 10 ] Generally, the number of bars, the joint types, and the configuration of the links and joints are determined before starting dimensional synthesis. [ 11 ] However, design strategies have been developed that combine type and dimensional synthesis. [ 12 ]
Dimensional synthesis of linkages begins with a task defined as the movement of an output link relative to a base reference frame. This task may consist of the trajectory of a moving point or the trajectory of a moving body. The kinematics equations , or loop equations, of the mechanism must be satisfied in all of the required positions of the moving point or body. The result is a system of equations that are solved to compute the dimensions of the linkage. [ 4 ]
There are three general tasks for dimensional synthesis, i) path generation , in which the trajectory of a point in the output link is required, ii) motion generation , in which the trajectory of the output link is required, and iii) function generation in which the movement of the output link relative to an input link is required. [ 3 ] The equations for function generation can be obtained from those for motion generation by considering the movement of the output link relative to an input link, rather than relative to the base frame.
The trajectory and motion requirements for dimensional synthesis are defined as sets of either instantaneous positions or finite positions . Instantaneous positions is a convenient way to describe requirements on the differential properties of the trajectory of a point or body, which are geometric versions of velocity, acceleration and rate of change of acceleration. The mathematical results that support instantaneous position synthesis are called curvature theory. [ 13 ]
Finite-position synthesis has a task defined as a set of positions of the moving body relative to a base frame, or relative to an input link. A crank that connects a moving pivot to a base pivot constrains the center of the moving pivot to follow a circle. This yields constraint equations that can be solved graphically using techniques developed by L. Burmester , [ 14 ] and called Burmester theory .
A cam and follower mechanism uses the shape of the cam to guide the movement of the follower by direct contact. Kinematic synthesis of a cam and follower mechanism consists of finding the shape of the cam that guides a particular follower through the required movement. [ 15 ]
A plate cam is connected to a base frame by hinged joint and the contour of the cam forms a surface that pushes on a follower. The connection of the follower to the base frame can be either a hinged or sliding joint to form a rotating and translating follower. The portion of the follower that contacts the cam can have any shape, such as a knife-edge, a roller, or flat-faced contact. As the cam rotates its contact with the follower face drives its output rotation or sliding movement.
The task for a cam and follower mechanism is provided by a displacement diagram , which defines the rotation angle or sliding distance of the follower as a function of the rotation of the cam. Once the contact shape of follower and its motion are defined, the cam can be constructed using graphical or numerical techniques. [ 15 ]
A pair of mating gears can be viewed as a cam and follower mechanism designed to use the rotary movement of an input shaft to drive the rotary movement of an output shaft. [ 15 ] This is achieved by providing a series of cam and followers, or gear teeth, distributed around the circumferences of two circles that form the mating gears. Early implementation of this rotary movement used cylindrical and rectangular teeth without concern for smooth transmission of movement, while the teeth were engaged---see the photo of the main drive gears for the windmill Doesburgermolen in Ede, Netherlands.
The geometric requirement that ensures smooth movement of contacting gear teeth is known as the fundamental law of gearing . This law states that for two bodies rotating about separate centers and in contact along their profiles, the relative angular velocity of the two will be constant as long as the line perpendicular to the point of contact of their two profiles, the profile normal, passes through the same point along the line between their centers throughout their movement. [ 15 ] A pair of tooth profiles that satisfy the fundamental law of gearing are said to be conjugate to each other. The involute profile that is used for most gear teeth today is self-conjugate, which means that if the teeth of two gears are the same size then they will mesh smoothly independent of the diameters of the mating gears.
The relative movement of gears with conjugate tooth profiles is defined by the distance from the center of each gear to the point at which the profile normal intersects the line of centers. This is known as the radius of the pitch circle for each gear. The calculation of the speed ratios for a gear train with conjugate gear teeth becomes a calculation using the ratios of the radii of the pitch circles that make up the gear train . [ 15 ]
Gear train design uses the desired speed ratio for a system of gears to select the number of gears, their configuration, and the size of their pitch circles. This is independent of the selection of the gear teeth as long as the tooth profiles are conjugate, with the exception that the circumferences of the pitch circles must provide for a whole number of teeth. | https://en.wikipedia.org/wiki/Kinematic_synthesis |
In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associated wave phenomena. [ 1 ] These waves are also applied to model the motion of highway traffic flows . [ 2 ] [ 3 ]
In these flows, mass and momentum equations can be combined to yield a kinematic wave equation. Depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave phase speed is a constant or a variable. Kinematic wave can be described by a simple partial differential equation with a single unknown field variable (e.g., the flow or wave height, h {\displaystyle h} ) in terms of the two independent variables, namely the time ( t {\displaystyle t} ) and the space ( x {\displaystyle x} ) with some parameters (coefficients) containing information about the physics and geometry of the flow. In general, the wave can be advecting and diffusing. However, in simple situations, the kinematic wave is mainly advecting.
Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients: ∂ h ∂ t + C ∂ h ∂ x = D ∂ 2 h ∂ x 2 , {\displaystyle {\frac {\partial h}{\partial t}}+C{\frac {\partial h}{\partial x}}=D{\frac {\partial ^{2}h}{\partial x^{2}}},} where h {\displaystyle h} is the debris flow height, t {\displaystyle t} is the time, x {\displaystyle x} is the downstream channel position, C {\displaystyle C} is the pressure gradient and the depth dependent nonlinear variable wave speed, and D {\displaystyle D} is a flow height and pressure gradient dependent variable diffusion term.
This equation can also be written in the conservative form :
where F {\displaystyle F} is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For F = h 2 / 2 {\displaystyle F=h^{2}/2} , this equation reduces to the Burgers' equation . | https://en.wikipedia.org/wiki/Kinematic_wave |
In physics , kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.
Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like cartesian , Curvilinear coordinates like polar coordinates or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used.
Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks.
Kinematics is a subfield of physics and mathematics , developed in classical mechanics , that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. [ 1 ] [ 2 ] [ 3 ] The study of how forces act on bodies falls within kinetics or dynamics (including analytical dynamics ), not kinematics.
Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. [ 4 ] [ 5 ] [ 6 ] A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined.
Another way to describe kinematics is as the specification of the possible states of a physical system. Dynamics then describes the evolution of a system through such states. Robert Spekkens argues that this division cannot be empirically tests and thus has no physical basis. [ 7 ]
Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering , robotics , and biomechanics , [ 8 ] kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine , a robotic arm or the human skeleton .
Geometric transformations , including called rigid transformations , are used to describe the movement of components in a mechanical system , simplifying the derivation of the equations of motion. They are also central to dynamic analysis .
Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. [ 9 ] In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.
Relativistic_kinematics applies the special theory of relativity to the geometry of object motion. The topics include time dilation , length contraction and the Lorentz transformation . [ 10 ] : 12.8 The kinematics of relativity operates in a spacetime geometry where spatial points are augmented with a time coordinate to form 4-vectors . [ 11 ] : 221
Werner Heisenberg reinterpreted classical kinetics for quantum systems in his 1925 paper "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships" . [ 12 ] Dirac noted the similarity in structure between Heisenberg's formulations and classical Poisson brackets . [ 13 ] : 143 In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measure, a result he called indeterminacy but which became known as the uncertainty principle . [ 14 ]
The term kinematic is the English version of A.M. Ampère 's cinématique , [ 15 ] which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move"). [ 16 ] [ 17 ]
Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek γρᾰ́φω grapho ("to write"). [ 18 ]
Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the x -axis and north is in the direction of the y -axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If the tower is 50 m high, and this height is measured along the z -axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m).
In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame.
The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector r {\displaystyle {\bf {r}}} can be expressed as r = ( x , y , z ) = x x ^ + y y ^ + z z ^ , {\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},} where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are the Cartesian coordinates and x ^ {\displaystyle {\hat {\mathbf {x} }}} , y ^ {\displaystyle {\hat {\mathbf {y} }}} and z ^ {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} coordinate axes, respectively. The magnitude of the position vector | r | {\displaystyle \left|\mathbf {r} \right|} gives the distance between the point r {\displaystyle \mathbf {r} } and the origin. | r | = x 2 + y 2 + z 2 . {\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.} The direction cosines of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector.
The trajectory of a particle is a vector function of time, r ( t ) {\displaystyle \mathbf {r} (t)} , which defines the curve traced by the moving particle, given by r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x ( t ) {\displaystyle x(t)} , y ( t ) {\displaystyle y(t)} , and z ( t ) {\displaystyle z(t)} describe each coordinate of the particle's position as a function of time.
The velocity of a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ( displacement ) by the time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ = Δ r Δ t = Δ x Δ t x ^ + Δ y Δ t y ^ + Δ z Δ t z ^ = v ¯ x x ^ + v ¯ y y ^ + v ¯ z z ^ {\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,} where Δ r {\displaystyle \Delta \mathbf {r} } is the displacement vector during the time interval Δ t {\displaystyle \Delta t} . In the limit that the time interval Δ t {\displaystyle \Delta t} approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^ + v y y ^ + v z z ^ . {\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.} Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
The speed of an object is the magnitude of its velocity. It is a scalar quantity: v = | v | = d s d t , {\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},} where s {\displaystyle s} is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s d t {\displaystyle {\frac {{\text{d}}s}{{\text{d}}t}}} is non-negative, which implies that speed is also non-negative.
The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio. a ¯ = Δ v ¯ Δ t = Δ v ¯ x Δ t x ^ + Δ v ¯ y Δ t y ^ + Δ v ¯ z Δ t z ^ = a ¯ x x ^ + a ¯ y y ^ + a ¯ z z ^ {\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,} where Δ v is the average velocity and Δ t is the time interval.
The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative, a = lim Δ t → 0 Δ v Δ t = d v d t = a x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.}
Alternatively, a = lim ( Δ t ) 2 → 0 Δ r ( Δ t ) 2 = d 2 r d t 2 = a x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.}
Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
The magnitude of the acceleration of an object is the magnitude | a | of its acceleration vector. It is a scalar quantity: | a | = | v ˙ | = d v d t . {\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.}
A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points.
The position of one point A relative to another point B is simply the difference between their positions
which is the difference between the components of their position vectors.
If point A has position components r A = ( x A , y A , z A ) {\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)}
and point B has position components r B = ( x B , y B , z B ) {\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)}
then the position of point A relative to point B is the difference between their components: r A / B = r A − r B = ( x A − x B , y A − y B , z A − z B ) {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)}
The velocity of one point relative to another is simply the difference between their velocities v A / B = v A − v B {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}} which is the difference between the components of their velocities.
If point A has velocity components v A = ( v A x , v A y , v A z ) {\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)} and point B has velocity components v B = ( v B x , v B y , v B z ) {\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)} then the velocity of point A relative to point B is the difference between their components: v A / B = v A − v B = ( v A x − v B x , v A y − v B y , v A z − v B z ) {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)}
Alternatively, this same result could be obtained by computing the time derivative of the relative position vector r B/A .
The acceleration of one point C relative to another point B is simply the difference between their accelerations. a C / B = a C − a B {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}} which is the difference between the components of their accelerations.
If point C has acceleration components a C = ( a C x , a C y , a C z ) {\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)} and point B has acceleration components a B = ( a B x , a B y , a B z ) {\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)} then the acceleration of point C relative to point B is the difference between their components: a C / B = a C − a B = ( a C x − a B x , a C y − a B y , a C z − a B z ) {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)}
Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration yields the velocity of the particle as a function of time. [ 19 ] v ( t ) = v 0 + ∫ 0 t a ( τ ) d τ {\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} (\tau )\,{\text{d}}\tau }
Additional relations between displacement, velocity, acceleration, and time can be derived. If the acceleration is constant, a = Δ v Δ t = v − v 0 t {\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}} can be substituted into the above equation to give: r ( t ) = r 0 + ( v + v 0 2 ) t . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\right)t.}
A relationship between velocity, position and acceleration without explicit time dependence can be obtained by solving the average acceleration for time and substituting and simplifying
t = v − v 0 a {\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}}
( r − r 0 ) ⋅ a = ( v − v 0 ) ⋅ v + v 0 2 , {\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,} where ⋅ {\displaystyle \cdot } denotes the dot product , which is appropriate as the products are scalars rather than vectors. 2 ( r − r 0 ) ⋅ a = | v | 2 − | v 0 | 2 . {\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.}
The dot product can be replaced by the cosine of the angle α between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case: 2 | r − r 0 | | a | cos α = | v | 2 − | v 0 | 2 . {\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.}
In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors ( α ) is 0, so cos 0 = 1 {\displaystyle \cos 0=1} , and | v | 2 = | v 0 | 2 + 2 | a | | r − r 0 | . {\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.} This can be simplified using the notation for the magnitudes of the vectors | a | = a , | v | = v , | r − r 0 | = Δ r {\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r} [ citation needed ] where Δ r {\displaystyle \Delta r} can be any curvaceous path taken as the constant tangential acceleration is applied along that path [ citation needed ] , so v 2 = v 0 2 + 2 a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.}
This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that Δ r = ∫ v d t {\textstyle \Delta r=\int v\,{\text{d}}t} or Δ r {\displaystyle \Delta r} is the area under a velocity–time graph. [ 20 ]
We can take Δ r {\displaystyle \Delta r} by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the A ⋅ B {\displaystyle A\cdot B} where A {\displaystyle A} is the width and B {\displaystyle B} is the height. In this case A = t {\displaystyle A=t} and B = v 0 {\displaystyle B=v_{0}} (the A {\displaystyle A} here is different from the acceleration a {\displaystyle a} ). This means that the bottom area is t v 0 {\displaystyle tv_{0}} . Now let's find the top area (a triangle). The area of a triangle is 1 2 B H {\textstyle {\frac {1}{2}}BH} where B {\displaystyle B} is the base and H {\displaystyle H} is the height. [ 21 ] In this case, B = t {\displaystyle B=t} and H = a t {\displaystyle H=at} or A = 1 2 B H = 1 2 a t t = 1 2 a t 2 = a t 2 2 {\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac {at^{2}}{2}}} . Adding v 0 t {\displaystyle v_{0}t} and a t 2 2 {\textstyle {\frac {at^{2}}{2}}} results in the equation Δ r {\displaystyle \Delta r} results in the equation Δ r = v 0 t + a t 2 2 {\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}} . [ 22 ] This equation is applicable when the final velocity v is unknown.
It is often convenient to formulate the trajectory of a particle r ( t ) = ( x ( t ), y ( t ), z ( t )) using polar coordinates in the X – Y plane. In this case, its velocity and acceleration take a convenient form.
Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F . As the particle moves, its coordinate vector r ( t ) traces its trajectory, which is a curve in space, given by: r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂ , ŷ , and ẑ are the unit vectors along the x , y and z axes of the reference frame F , respectively.
Consider a particle P that moves only on the surface of a circular cylinder r ( t ) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x – y plane can be used to define the trajectory as, r ( t ) = r cos ( θ ( t ) ) x ^ + r sin ( θ ( t ) ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where the constant distance from the center is denoted as r , and θ ( t ) is a function of time.
The cylindrical coordinates for r ( t ) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos ( θ ( t ) ) x ^ + sin ( θ ( t ) ) y ^ , θ ^ = − sin ( θ ( t ) ) x ^ + cos ( θ ( t ) ) y ^ . {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d ( ω θ ^ ) d t = α θ ^ − ω 2 r ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega ^{2}{\hat {\mathbf {r} }}.}
d θ ^ d t = − ω r ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\omega {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d ( − ω r ^ ) d t = − α r ^ − ω 2 θ ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\omega {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.}
Using this notation, r ( t ) takes the form, r ( t ) = r r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r ( t ) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r , θ , and z might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector v P is the time derivative of the trajectory r ( t ), which yields: v P = d d t ( r r ^ + z z ^ ) = v r ^ + r ω θ ^ + v z z ^ = v ( r ^ + θ ^ ) + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.}
Similarly, the acceleration a P , which is the time derivative of the velocity v P , is given by: a P = d d t ( v r ^ + v θ ^ + v z z ^ ) = ( a − v ω ) r ^ + ( a + v ω ) θ ^ + a z z ^ . {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\omega ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.}
The term − v ω r ^ {\displaystyle -v\omega {\hat {\mathbf {r} }}} acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration . The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta } }}} is called the Coriolis acceleration .
If the trajectory of the particle is constrained to lie on a cylinder, then the radius r is constant and the velocity and acceleration vectors simplify. The velocity of v P is the time derivative of the trajectory r ( t ), v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ + v z z ^ = v θ ^ + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.}
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the z axis: r ( t ) = r r ^ + z z ^ , {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},} where r and z 0 are constants. In this case, the velocity v P is given by: v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ = v θ ^ , {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},} where ω {\displaystyle \omega } is the angular velocity of the unit vector θ ^ around the z axis of the cylinder.
The acceleration a P of the particle P is now given by: a P = d ( v θ ^ ) d t = a θ ^ − v θ r ^ . {\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.}
The components a r = − v θ , a θ = a , {\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,} are called, respectively, the radial and tangential components of acceleration.
The notation for angular velocity and angular acceleration is often defined as ω = θ ˙ , α = θ ¨ , {\displaystyle \omega ={\dot {\theta }},\quad \alpha ={\ddot {\theta }},} so the radial and tangential acceleration components for circular trajectories are also written as a r = − r ω 2 , a θ = r α . {\displaystyle a_{r}=-r\omega ^{2},\quad a_{\theta }=r\alpha .}
The movement of components of a mechanical system are analyzed by attaching a reference frame to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts.
Geometry is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations. [ 24 ] These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.
The coordinates of points in a plane are two-dimensional vectors in R 2 (two dimensional space). Rigid transformations are those that preserve the distance between any two points. The set of rigid transformations in an n -dimensional space is called the special Euclidean group on R n , and denoted SE( n ) .
The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say M , on one that moves relative to a fixed frame, F, on the other. The rigid transformation, or displacement, of M relative to F defines the relative position of the two components. A displacement consists of the combination of a rotation and a translation .
The set of all displacements of M relative to F is called the configuration space of M. A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of M relative to F. The motion of a body consists of a continuous set of rotations and translations.
The combination of a rotation and translation in the plane R 2 can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 rotation matrix A ( φ ) and the 2×1 translation vector d = ( d x , d y ), as: [ T ( ϕ , d ) ] = [ A ( ϕ ) d 0 1 ] = [ cos ϕ − sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] . {\displaystyle [T(\phi ,\mathbf {d} )]={\begin{bmatrix}A(\phi )&\mathbf {d} \\\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}.} These homogeneous transforms perform rigid transformations on the points in the plane z = 1, that is, on points with coordinates r = ( x , y , 1).
In particular, let r define the coordinates of points in a reference frame M coincident with a fixed frame F . Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F , the new coordinates in F of points in M are given by: P = [ T ( ϕ , d ) ] r = [ cos ϕ − sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] [ x y 1 ] . {\displaystyle \mathbf {P} =[T(\phi ,\mathbf {d} )]\mathbf {r} ={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\1\end{bmatrix}}.}
Homogeneous transforms represent affine transformations . This formulation is necessary because a translation is not a linear transformation of R 2 . However, using projective geometry, so that R 2 is considered a subset of R 3 , translations become affine linear transformations. [ 25 ]
If a rigid body moves so that its reference frame M does not rotate ( θ = 0) relative to the fixed frame F , the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory d ( t ) of the origin of M, that is: r ( t ) = [ T ( 0 , d ( t ) ) ] p = d ( t ) + p . {\displaystyle \mathbf {r} (t)=[T(0,\mathbf {d} (t))]\mathbf {p} =\mathbf {d} (t)+\mathbf {p} .}
Thus, for bodies in pure translation, the velocity and acceleration of every point P in the body are given by: v P = r ˙ ( t ) = d ˙ ( t ) = v O , a P = r ¨ ( t ) = d ¨ ( t ) = a O , {\displaystyle \mathbf {v} _{P}={\dot {\mathbf {r} }}(t)={\dot {\mathbf {d} }}(t)=\mathbf {v} _{O},\quad \mathbf {a} _{P}={\ddot {\mathbf {r} }}(t)={\ddot {\mathbf {d} }}(t)=\mathbf {a} _{O},} where the dot denotes the derivative with respect to time and v O and a O are the velocity and acceleration, respectively, of the origin of the moving frame M . Recall the coordinate vector p in M is constant, so its derivative is zero.
Objects like a playground merry-go-round , ventilation fans, or hinged doors can be modeled as rigid bodies rotating about a single fixed axis. [ 26 ] : 37 The z -axis has been chosen by convention.
This allows the description of a rotation as the angular position of a planar reference frame M relative to a fixed F about this shared z -axis. Coordinates p = ( x , y ) in M are related to coordinates P = (X, Y) in F by the matrix equation: P ( t ) = [ A ( t ) ] p , {\displaystyle \mathbf {P} (t)=[A(t)]\mathbf {p} ,}
where [ A ( t ) ] = [ cos ( θ ( t ) ) − sin ( θ ( t ) ) sin ( θ ( t ) ) cos ( θ ( t ) ) ] , {\displaystyle [A(t)]={\begin{bmatrix}\cos(\theta (t))&-\sin(\theta (t))\\\sin(\theta (t))&\cos(\theta (t))\end{bmatrix}},} is the rotation matrix that defines the angular position of M relative to F as a function of time.
If the point p does not move in M , its velocity in F is given by v P = P ˙ = [ A ˙ ( t ) ] p . {\displaystyle \mathbf {v} _{P}={\dot {\mathbf {P} }}=[{\dot {A}}(t)]\mathbf {p} .} It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P ( t ), v P = [ A ˙ ( t ) ] [ A ( t ) − 1 ] P = [ Ω ] P , {\displaystyle \mathbf {v} _{P}=[{\dot {A}}(t)][A(t)^{-1}]\mathbf {P} =[\Omega ]\mathbf {P} ,} where the matrix [ Ω ] = [ 0 − ω ω 0 ] , {\displaystyle [\Omega ]={\begin{bmatrix}0&-\omega \\\omega &0\end{bmatrix}},} is known as the angular velocity matrix of M relative to F . The parameter ω is the time derivative of the angle θ , that is: ω = d θ d t . {\displaystyle \omega ={\frac {{\text{d}}\theta }{{\text{d}}t}}.}
The acceleration of P ( t ) in F is obtained as the time derivative of the velocity, A P = P ¨ ( t ) = [ Ω ˙ ] P + [ Ω ] P ˙ , {\displaystyle \mathbf {A} _{P}={\ddot {P}}(t)=[{\dot {\Omega }}]\mathbf {P} +[\Omega ]{\dot {\mathbf {P} }},} which becomes A P = [ Ω ˙ ] P + [ Ω ] [ Ω ] P , {\displaystyle \mathbf {A} _{P}=[{\dot {\Omega }}]\mathbf {P} +[\Omega ][\Omega ]\mathbf {P} ,} where [ Ω ˙ ] = [ 0 − α α 0 ] , {\displaystyle [{\dot {\Omega }}]={\begin{bmatrix}0&-\alpha \\\alpha &0\end{bmatrix}},} is the angular acceleration matrix of M on F , and α = d 2 θ d t 2 . {\displaystyle \alpha ={\frac {{\text{d}}^{2}\theta }{{\text{d}}t^{2}}}.}
The description of rotation then involves these three quantities:
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges: ω f = ω i + α t {\displaystyle \omega _{\mathrm {f} }=\omega _{\mathrm {i} }+\alpha t\!} θ f − θ i = ω i t + 1 2 α t 2 {\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }=\omega _{\mathrm {i} }t+{\tfrac {1}{2}}\alpha t^{2}} θ f − θ i = 1 2 ( ω f + ω i ) t {\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }={\tfrac {1}{2}}(\omega _{\mathrm {f} }+\omega _{\mathrm {i} })t} ω f 2 = ω i 2 + 2 α ( θ f − θ i ) . {\displaystyle \omega _{\mathrm {f} }^{2}=\omega _{\mathrm {i} }^{2}+2\alpha (\theta _{\mathrm {f} }-\theta _{\mathrm {i} }).}
Here θ i and θ f are, respectively, the initial and final angular positions, ω i and ω f are, respectively, the initial and final angular velocities, and α is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.
Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations .
In order to define these formulas, the movement of a component B of a mechanical system is defined by the set of rotations [A( t )] and translations d ( t ) assembled into the homogeneous transformation [T( t )]=[A( t ), d ( t )]. If p is the coordinates of a point P in B measured in the moving reference frame M , then the trajectory of this point traced in F is given by: P ( t ) = [ T ( t ) ] p = [ P 1 ] = [ A ( t ) d ( t ) 0 1 ] [ p 1 ] . {\displaystyle \mathbf {P} (t)=[T(t)]\mathbf {p} ={\begin{bmatrix}\mathbf {P} \\1\end{bmatrix}}={\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.} This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.
This equation for the trajectory of P can be inverted to compute the coordinate vector p in M as: p = [ T ( t ) ] − 1 P ( t ) = [ p 1 ] = [ A ( t ) T − A ( t ) T d ( t ) 0 1 ] [ P ( t ) 1 ] . {\displaystyle \mathbf {p} =[T(t)]^{-1}\mathbf {P} (t)={\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}A(t)^{\text{T}}&-A(t)^{\text{T}}\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}.} This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is: [ A ( t ) ] T [ A ( t ) ] = I . {\displaystyle [A(t)]^{\text{T}}[A(t)]=I.\!}
The velocity of the point P along its trajectory P ( t ) is obtained as the time derivative of this position vector, v P = [ T ˙ ( t ) ] p = [ v P 0 ] = ( d d t [ A ( t ) d ( t ) 0 1 ] ) [ p 1 ] = [ A ˙ ( t ) d ˙ ( t ) 0 0 ] [ p 1 ] . {\displaystyle \mathbf {v} _{P}=[{\dot {T}}(t)]\mathbf {p} ={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}=\left({\frac {d}{dt}}{\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}\right){\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}{\dot {A}}(t)&{\dot {\mathbf {d} }}(t)\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.} The dot denotes the derivative with respect to time; because p is constant, its derivative is zero.
This formula can be modified to obtain the velocity of P by operating on its trajectory P ( t ) measured in the fixed frame F . Substituting the inverse transform for p into the velocity equation yields: v P = [ T ˙ ( t ) ] [ T ( t ) ] − 1 P ( t ) = [ v P 0 ] = [ A ˙ d ˙ 0 0 ] [ A d 0 1 ] − 1 [ P ( t ) 1 ] = [ A ˙ d ˙ 0 0 ] A − 1 [ 1 − d 0 A ] [ P ( t ) 1 ] = [ A ˙ A − 1 − A ˙ A − 1 d + d ˙ 0 0 ] [ P ( t ) 1 ] = [ A ˙ A T − A ˙ A T d + d ˙ 0 0 ] [ P ( t ) 1 ] v P = [ S ] P . {\displaystyle {\begin{aligned}\mathbf {v} _{P}&=[{\dot {T}}(t)][T(t)]^{-1}\mathbf {P} (t)\\[4pt]&={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}={\begin{bmatrix}{\dot {A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}A&\mathbf {d} \\0&1\end{bmatrix}}^{-1}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}A^{-1}{\begin{bmatrix}1&-\mathbf {d} \\0&A\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}A^{-1}&-{\dot {A}}A^{-1}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[4pt]&={\begin{bmatrix}{\dot {A}}A^{\text{T}}&-{\dot {A}}A^{\text{T}}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\[6pt]\mathbf {v} _{P}&=[S]\mathbf {P} .\end{aligned}}} The matrix [ S ] is given by: [ S ] = [ Ω − Ω d + d ˙ 0 0 ] {\displaystyle [S]={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}} where [ Ω ] = A ˙ A T , {\displaystyle [\Omega ]={\dot {A}}A^{\text{T}},} is the angular velocity matrix.
Multiplying by the operator [ S ], the formula for the velocity v P takes the form: v P = [ Ω ] ( P − d ) + d ˙ = ω × R P / O + v O , {\displaystyle \mathbf {v} _{P}=[\Omega ](\mathbf {P} -\mathbf {d} )+{\dot {\mathbf {d} }}=\omega \times \mathbf {R} _{P/O}+\mathbf {v} _{O},} where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω]; the vector R P / O = P − d , {\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,} is the position of P relative to the origin O of the moving frame M ; and v O = d ˙ , {\displaystyle \mathbf {v} _{O}={\dot {\mathbf {d} }},} is the velocity of the origin O .
The acceleration of a point P in a moving body B is obtained as the time derivative of its velocity vector: A P = d d t v P = d d t ( [ S ] P ) = [ S ˙ ] P + [ S ] P ˙ = [ S ˙ ] P + [ S ] [ S ] P . {\displaystyle \mathbf {A} _{P}={\frac {d}{dt}}\mathbf {v} _{P}={\frac {d}{dt}}\left([S]\mathbf {P} \right)=[{\dot {S}}]\mathbf {P} +[S]{\dot {\mathbf {P} }}=[{\dot {S}}]\mathbf {P} +[S][S]\mathbf {P} .}
This equation can be expanded firstly by computing [ S ˙ ] = [ Ω ˙ − Ω ˙ d − Ω d ˙ + d ¨ 0 0 ] = [ Ω ˙ − Ω ˙ d − Ω v O + A O 0 0 ] {\displaystyle [{\dot {S}}]={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega {\dot {\mathbf {d} }}+{\ddot {\mathbf {d} }}\\0&0\end{bmatrix}}={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega \mathbf {v} _{O}+\mathbf {A} _{O}\\0&0\end{bmatrix}}} and [ S ] 2 = [ Ω − Ω d + v O 0 0 ] 2 = [ Ω 2 − Ω 2 d + Ω v O 0 0 ] . {\displaystyle [S]^{2}={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +\mathbf {v} _{O}\\0&0\end{bmatrix}}^{2}={\begin{bmatrix}\Omega ^{2}&-\Omega ^{2}\mathbf {d} +\Omega \mathbf {v} _{O}\\0&0\end{bmatrix}}.}
The formula for the acceleration A P can now be obtained as: A P = Ω ˙ ( P − d ) + A O + Ω 2 ( P − d ) , {\displaystyle \mathbf {A} _{P}={\dot {\Omega }}(\mathbf {P} -\mathbf {d} )+\mathbf {A} _{O}+\Omega ^{2}(\mathbf {P} -\mathbf {d} ),} or A P = α × R P / O + ω × ω × R P / O + A O , {\displaystyle \mathbf {A} _{P}=\alpha \times \mathbf {R} _{P/O}+\omega \times \omega \times \mathbf {R} _{P/O}+\mathbf {A} _{O},} where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; R P / O = P − d , {\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,} is the relative position vector (the position of P relative to the origin O of the moving frame M ); and A O = d ¨ {\displaystyle \mathbf {A} _{O}={\ddot {\mathbf {d} }}} is the acceleration of the origin of the moving frame M .
Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called holonomic constraints , and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called non-holonomic constraints . The following are some common examples.
A kinematic coupling exactly constrains all 6 degrees of freedom.
An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass: v G ( t ) = Ω × r G / O . {\displaystyle {\boldsymbol {v}}_{G}(t)={\boldsymbol {\Omega }}\times {\boldsymbol {r}}_{G/O}.}
For the case of an object that does not tip or turn, this reduces to v = r ω {\displaystyle v=r\omega } .
This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero. [ 27 ] [ 28 ] [ 29 ] A dynamic problem of this type is the pendulum . Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. [ 30 ] An equilibrium problem (i.e. not kinematic) of this type is the catenary . [ 31 ]
Reuleaux called the ideal connections between components that form a machine kinematic pairs . He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification. [ 32 ]
A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:
Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a cam joint . Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.
Rigid bodies ("links") connected by kinematic pairs ("joints") are known as kinematic chains . Mechanisms and robots are examples of kinematic chains. The degree of freedom of a kinematic chain is computed from the number of links and the number and type of joints using the mobility formula . This formula can also be used to enumerate the topologies of kinematic chains that have a given degree of freedom, which is known as type synthesis in machine design.
The planar one degree-of-freedom linkages assembled from N links and j hinges or sliding joints are:
For larger chains and their linkage topologies, see R. P. Sunkari and L. C. Schmidt , "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", Mechanism and Machine Theory #41, pp. 1021–1030 (2006). | https://en.wikipedia.org/wiki/Kinematics |
Kinesis , like a taxis or tropism , is a movement or activity of a cell or an organism in response to a stimulus (such as gas exposure , light intensity or ambient temperature ).
Unlike taxis, the response to the stimulus provided is non-directional. The animal does not move toward or away from the stimulus but moves at either a slow or fast rate depending on its " comfort zone ." In this case, a fast movement (non-random) means that the animal is searching for its comfort zone while a slow movement indicates that it has found it.
There are two main types of kineses, both resulting in aggregations. However, the stimulus does not act to attract or repel individuals.
Orthokinesis : in which the speed of movement of the individual is dependent upon the stimulus intensity. For example, the locomotion of the collembola , Orchesella cincta , in relation to water. With increased water saturation in the soil there is an increase in the direction of its movement towards the aimed place. [ 1 ]
Klinokinesis : in which the frequency or rate of turning is proportional to stimulus intensity. For example, the behaviour of the flatworm ( Dendrocoelum lacteum ) which turns more frequently in response to increasing light thus ensuring that it spends more time in dark areas. [ 2 ]
The kinesis strategy controlled by the locally and instantly evaluated well-being ( fitness ) can be described in simple words: Animals stay longer in good conditions and leave bad conditions more quickly. If the well-being is measured by the local reproduction coefficient then the minimal reaction-diffusion model of kinesis can be written as follows: [ 3 ]
For each population in the biological community,
∂ t u i ( x , t ) = D 0 i ∇ ( e − α i r i ( u 1 , … , u k , s ) ∇ u i ) + r i ( u 1 , … , u k , s ) u i , {\displaystyle \partial _{t}u_{i}(x,t)=D_{0i}\nabla \left(e^{-\alpha _{i}r_{i}(u_{1},\ldots ,u_{k},s)}\nabla u_{i}\right)+r_{i}(u_{1},\ldots ,u_{k},s)u_{i},}
where: u i {\displaystyle u_{i}} is the population density of i th species, s {\displaystyle s} represents the abiotic characteristics of the living conditions (can be multidimensional), r i {\displaystyle r_{i}} is the reproduction coefficient, which depends on all u i {\displaystyle u_{i}} and on s , D 0 i > 0 {\displaystyle D_{0i}>0} is the equilibrium diffusion coefficient (defined for equilibrium r i = 0 {\displaystyle r_{i}=0} ). The coefficient α i > 0 {\displaystyle \alpha _{i}>0} characterises dependence of the diffusion coefficient on the reproduction coefficient.
The models of kinesis were tested with typical situations. It was demonstrated that kinesis is beneficial for assimilation of both patches and fluctuations of food distribution. Kinesis may delay invasion and spreading of species with the Allee effect . | https://en.wikipedia.org/wiki/Kinesis_(biology) |
Kinetic-segregation is a model proposed for the mechanism of T-cell receptor (TCR) triggering. [ 1 ] [ 2 ] It offers an explanation for how TCR binding to its ligand triggers T-cell activation, based on size-sensitivity for the molecules involved. Simon J. Davis and Anton van der Merwe , University of Oxford , proposed this model in 1996.
According to the model, TCR signalling is initiated by segregation of phosphatases with large extracellular domains from the TCR complex when binding to its ligand, allowing small kinases to phosphorylate intracellular domains of the TCR without inhibition. Its might also be applicable to other receptors of the Non-catalytic tyrosine-phosphorylated receptors family such as CD28 .
On plasma membrane of a T cell there is the T-cell receptor (consists of α,β chains and multiple CD3 adaptor proteins), as well as molecules that induce signalling (tyrosine kinase Lck phosphorylates ITAMs in CD3 complex) and factors that inhibit signalling (tyrosine phosphatases CD45 and CD148 ). In the resting T-cell, all molecules are repeatedly colliding by means of diffusion .
The TCR/ CD3 complex is constantly being phosphorylated by Lck . Because of an abundance of CD45 and CD148 in the cell membrane, phosphorylations are readily removed before they can recruit downstream signalling molecules. Overall phosphorylation of the TCR is low and tonic TCR signalling is avoided. [ 3 ]
The TCR/peptide- MHC complex, formed when a T cell recognises its ligand on an antigen presenting cell (APC) and the T-cell-APC contact occurs, spans a short length. This results in the formation of close contact zones between the membranes of the T cell and antigen presenting cell (~15 nm apart) around the TCR/peptide-MHC complex. [ 3 ] Phosphatases CD45 and CD148 with much larger ectodomains than TCR are sterically excluded from the close contact zones, while the region is still accessible for the small kinase Lck. This perturbs the balance of kinase activity to phosphatase activity and ITAM phosphorylation is strongly favoured. [ 3 ] Such prolonged phosphorylation of ITAMs by Lck kinase allows time for ZAP-70 recruitment, its activation by phosphorylation and subsequent phosphorylation of adaptor proteins LAT and SLP-76 . Full T-cell activation is initiated by multiple triggering events described above. When T-cell and APC membranes separate, the close-contact zone vanishes and large-ectodomain tyrosine phosphatases are allowed to restore the ground state. [ 3 ]
During ligand binding, CD45 and CD148 are excluded from the TCR region. [ 4 ] [ 5 ] It was also shown that both the truncation of CD45 and CD148 (hence are able to enter the close contact zone) and the elongation of the MHC inhibit TCR triggering. [ 6 ] [ 7 ] [ 8 ] [ 9 ] Furthermore CAR cell function is affected by the size of the ligand it recognises. [ 9 ] [ 10 ] Finally, T cells can be activated by pMHC immobilised on a plate surface but not by soluble, monomeric pMHC, providing evidence that TCR triggering depends on restricting width between two membranes. [ 11 ] [ 12 ]
In the resting T-cell there is no net phosphorylation of CD28 (one of the molecules providing co-stimulatory signals required for T-cell activation). Kinetic-segregation model uses here the same explanation as it provides for low net phosphorylation of TCR in the resting T-cell described previously.
Binding of both conventional and superagonistic (mitogenic) antibodies in suspension does not constrict the dephosphorylation effect of phosphatases acting on CD28. However, when these antibodies are immobilized (either by secondary antibody bound to plastic or by Fc receptors on other cells ) considerable steric constraints emerge. It is of note, that the immobilized conventional antibody poses less prominent spatial constraints than the immobilized superagonistic antibody. CD45 phosphatase is not completely excluded from the close-contact zone and thus the signal generated in the case of a conventional antibody is weaker. Immobilized superagonistic antibodies bound to CD28 exclude CD45 phosphatases completely and the signal leading to T-cell activation is stronger.
The tyrosine kinase Lck functions either in conjunction with a co-receptor molecule ( CD4 or CD8 ) or as a free Lck kinase. The kinetic-segregation model might be applied to both co-receptor dependent and co-receptor independent signaling through TCR. | https://en.wikipedia.org/wiki/Kinetic-segregation_model_of_T_cell_activation |
The kinetic Monte Carlo (KMC) method is a Monte Carlo method computer simulation intended to simulate the time evolution of some processes occurring in nature. Typically these are processes that occur with known transition rates among states. These rates are inputs to the KMC algorithm; the method itself cannot predict them.
The KMC method is essentially the same as the dynamic Monte Carlo method and the Gillespie algorithm .
One possible classification of KMC algorithms is as rejection-KMC (rKMC) and rejection-free-KMC (rfKMC).
A rfKMC algorithm, often only called KMC, for simulating the time evolution of a system, where some processes can occur with known rates r, can be written for instance as follows: [ 1 ] : 13–14
(Note: because the average value of ln ( 1 / u ′ ) {\displaystyle \ln(1/u^{\prime })} is equal to unity, the same average time scale can be obtained by instead using Δ t = Q k − 1 {\displaystyle \Delta t=Q_{k}^{-1}} in step 9. In this case, however, the delay associated with transition i will not be drawn from the Poisson distribution described by the rate Q k {\displaystyle Q_{k}} , but will instead be the mean of that distribution. [ citation needed ] )
This algorithm is known in different sources variously as the residence-time algorithm or the n -fold way or the Bortz-Kalos-Lebowitz (BKL) algorithm. It is important to note that the timestep involved is a function of the probability that all events i , did not occur. [ 1 ] : 13–14
Rejection KMC has typically the advantage of an easier data handling, and faster computations for each attempted step, since the time consuming action of getting all r k i {\displaystyle r_{ki}} is not needed.
On the other hand, the time evolved at each step is smaller than for rfKMC. The relative weight of pros and cons varies with the case at hand, and with available resources.
An rKMC associated with the same transition rates as above can be written as follows:
(Note: r 0 {\displaystyle r_{0}} can change from one MC step to another.)
This algorithm is usually called a standard algorithm .
Theoretical [ 2 ] and numerical [ 1 ] [ 3 ] comparisons between the algorithms were provided.
If the rates r k i ( t ) {\displaystyle r_{ki}(t)} are time dependent, step 9 in the rfKMC must be modified by: [ 4 ]
The reaction (step 6) has to be chosen after this by
Another very similar algorithm is called the First Reaction Method (FRM). It consists of choosing the first-occurring reaction, meaning to choose the smallest time Δ t i {\displaystyle \Delta t_{i}} , and the corresponding reaction number i , from the formula
where the u i ∈ ( 0 , 1 ] {\displaystyle u_{i}\in (0,1]} are N random numbers.
The key property of the KMC algorithm (and of the FRM one) is that if the rates are correct, if the processes associated with the rates are of the Poisson process type, and if different processes are independent (i.e. not correlated) then the KMC algorithm gives the correct time scale for the evolution of the simulated system. There was some debate about the correctness of the time scale for rKMC algorithms, but this was also rigorously shown to be correct. [ 2 ]
If furthermore the transitions follow detailed balance , the KMC algorithm can be used to simulate thermodynamic equilibrium . However, KMC is widely used to simulate non-equilibrium processes, [ 5 ] in which case detailed balance need not be obeyed.
The rfKMC algorithm is efficient in the sense that every iteration is guaranteed to produce a transition. However, in the form presented above it requires N {\displaystyle N} operations for each transition, which is not too efficient. In many cases this can be much improved on by binning the same kinds of transitions into bins, and/or forming a tree data structure of the events. A constant-time scaling algorithm of this type has recently been developed and tested. [ 6 ]
The major disadvantage with rfKMC is that all possible rates r k i {\displaystyle r_{ki}} and reactions have to be known in advance. The method itself can do nothing about predicting them. The rates and reactions must be obtained from other methods, such as diffusion (or other) experiments, molecular dynamics or density-functional theory simulations.
KMC has been used in simulations of the following physical systems:
To give an idea what the "objects" and "events" may be in practice, here is one concrete simple example, corresponding to example 2 above.
Consider a system where individual atoms are deposited on a surface one at a time (typical of physical vapor deposition ), but also may migrate on the surface with some known jump rate w {\displaystyle w} . In this case the "objects" of the KMC algorithm are simply the individual atoms.
If two atoms come right next to each other, they become immobile. Then the flux of incoming atoms determines a rate r deposit , and the system can be simulated with KMC considering all deposited mobile atoms which have not (yet) met a counterpart and become immobile. This way there are the following events possible at each KMC step:
After an event has been selected and carried out with the KMC algorithm, one then needs to check whether the new or just jumped atom has become immediately adjacent to some other atom. If this has happened, the atom(s) which are now adjacent needs to be moved away from the list of mobile atoms, and correspondingly their jump events removed from the list of possible events.
Naturally in applying KMC to problems in physics and chemistry, one has to first consider whether the real system follows the assumptions underlying KMC well enough.
Real processes do not necessarily have well-defined rates, the
transition processes may be correlated, in case of atom or particle jumps
the jumps may not occur in random directions, and so on. When simulating
widely disparate time scales one also needs to consider whether
new processes may be present at longer time scales. If any of these
issues are valid, the time scale and system evolution predicted by KMC
may be skewed or even completely wrong.
The first publication which described the basic features of the KMC method (namely using a cumulative function to select an event and a time scale calculation of the form 1/ R ) was by Young and Elcock in 1966. [ 8 ] The residence-time algorithm was also published at about the same time. [ 10 ]
Apparently independent of the work of Young and Elcock, Bortz, Kalos and Lebowitz [ 1 ] developed a KMC algorithm for simulating the Ising model , which they called the n-fold way . The basics of their algorithm is the same as that of Young, [ 8 ] but they do provide much greater detail on the method.
The following year Dan Gillespie published what is now known as the Gillespie algorithm to describe chemical reactions. [ 11 ] The algorithm is similar and the time advancement scheme essentially the same as in KMC.
There is as of the writing of this (June 2006) no definitive treatise of the theory of KMC, but Fichthorn and Weinberg have discussed the theory for thermodynamic equilibrium KMC simulations in detail. [ 12 ] A good introduction is given also by Art Voter, [ 13 ] [1] and by A.P.J. Jansen, [ 14 ] [2] , and a recent review is (Chatterjee 2007) [ 15 ] or (Chotia 2008). [ 16 ] The justification of KMC as a coarse-graining of the Langevin dynamics using the quasi-stationary distribution approach has been developed by T. Lelièvre and collaborators. [ 17 ] [ 18 ]
In March 2006 the, probably, first commercial software using Kinetic Monte Carlo to simulate the diffusion and activation/deactivation of dopants in Silicon and Silicon like materials is released by Synopsys , reported by Martin-Bragado et al. [ 19 ]
The KMC method can be subdivided by how the objects are moving or reactions occurring. At least the following subdivisions are used: | https://en.wikipedia.org/wiki/Kinetic_Monte_Carlo |
The Kinetic PreProcessor (KPP) is an open-source software tool used in atmospheric chemistry . Taking a set of chemical reactions and their rate coefficients as input, KPP generates Fortran 90 , FORTRAN 77 , C , or Matlab code
of the resulting ordinary differential equations (ODEs) . Solving the ODEs allows the temporal integration of the kinetic system. Efficiency is obtained by exploiting the sparsity structures of the Jacobian and of the Hessian . A comprehensive suite of stiff numerical integrators is also provided. Moreover, KPP can be used to generate the tangent linear model, as well as the continuous and discrete adjoint models of the chemical system. | https://en.wikipedia.org/wiki/Kinetic_PreProcessor |
Kinetic architecture is a concept through which buildings are designed to allow parts of the structure to move, without reducing overall structural integrity.
A building's capability for motion can be used just to: enhance its aesthetic qualities; respond to environmental conditions; and/or, perform functions that would be impossible for a static structure.
The possibilities for practical implementations of kinetic architecture increased sharply in the late 20th century due to advances in mechanics, electronics, and robotics.
Rudimentary forms of kinetic architecture such as the drawbridge can be traced back to the Middle Ages or earlier. Yet it was only in the early 20th century that architects began to widely discuss the possibility for movement to be enabled for a significant portion of a buildings' superstructure . In the first third of the 20th century, interest in kinetic architect was one of the stands of thought emerging from the Futurism movement. Various papers and books included plans and drawings for moving buildings, a notable example being Chernikhov's 101 Architectural Fantasies (1933). For the first few decades of the 20th century kinetic architecture was almost entirely theoretical, but by the 1940s innovators such as Buckminster Fuller began experimenting with concrete implementations, though his early efforts in this direction are not regarded as totally successful. [ 1 ]
In 1970, engineer/architect William Zuk published the book Kinetic architecture , [ 2 ] which helped inspire a new generation of architects to design an increasingly wide range of actual working kinetic buildings. Assisted by new concepts such as Fuller's Tensegrity and by developments in robotics , kinetic buildings have become increasingly common worldwide since the 1980s. [ 1 ]
By the early 21st century three interrelated themes had emerged. The first is for functional buildings such as bridges which can elevate their midsections to allow tall ships to pass, or stadiums with retractable roofs such as the Veltins-Arena , Millennium Stadium in Cardiff, or Wembley Stadium . [ 3 ]
A second theme is for fantastic structures that can perform Transformer style changes of shape or which have a visually stunning appearance.
The bird-like Burke Brise soleil at the Milwaukee Art Museum is a well regarded example of this, though it also has a functional aspect in that its movement allows it to shade the crowds from the sun or protect them from storms. [ 1 ]
The third theme is for movement to occur on the surface of the building, creating what Buckminster Fuller called a "skin-like articulation" effect. A classic example of this is the Institut du Monde Arabe . [ 1 ]
Architects Sarah Bonnemaison and Christine Macy have suggested that movement can be an inspiring idea for architecture without the designs having to allow for actual movement – they can merely suggest it as was the case for some of the constructions of Gaudi or their own recent work. [ 4 ] The term Kinetic architecture can also refer to static buildings designed to accentuate human movement, such as the performing arts. [ 5 ] The phrase has been chosen as a title for performing groups including a dance company. [ 6 ] | https://en.wikipedia.org/wiki/Kinetic_architecture |
Kinetic capillary electrophoresis or KCE is capillary electrophoresis of molecules that interact during electrophoresis.
KCE was introduced and developed by Professor Sergey Krylov and his research group at York University , Toronto, Canada. [ 1 ] It serves as a conceptual platform for development of homogeneous chemical affinity methods for studies of molecular interactions (measurements of binding and rate constants) and affinity purification (purification of known molecules and search for unknown molecules). Different KCE methods are designed by varying initial and boundary conditions – the way interacting molecules enter and exit the capillary. Several KCE methods were described: non-equilibrium capillary electrophoresis of the equilibrium mixtures (NECEEM), [ 2 ] sweeping capillary electrophoresis (SweepCE), [ 3 ] and plug-plug KCE (ppKCE). [ 4 ] | https://en.wikipedia.org/wiki/Kinetic_capillary_electrophoresis |
In polymer chemistry , the kinetic chain length ( ν ) of a polymer is the average number of units called monomers added to a growing chain during chain-growth polymerization . During this process, a polymer chain is formed when monomers are bonded together to form long chains known as polymers. Kinetic chain length is defined as the average number of monomers that react with an active center such as a radical from initiation to termination . [ 1 ]
This definition is a special case of the concept of chain length in chemical kinetics . For any chemical chain reaction , the chain length is defined as the average number of times that the closed cycle of chain propagation steps is repeated. It is equal to the rate of the overall reaction divided by the rate of the initiation step in which the chain carriers are formed. [ 2 ] [ 3 ] For example, the decomposition of ozone in water is a chain reaction which has been described in terms of its chain length. [ 4 ]
In chain-growth polymerization the propagation step is the addition of a monomer to the growing chain. The word kinetic is added to chain length in order to distinguish the number of reaction steps in the kinetic chain from the number of monomers in the final macromolecule, a quantity named the degree of polymerization . In fact the kinetic chain length is one factor which influences the average degree of polymerization, but there are other factors as described below. The kinetic chain length and therefore the degree of polymerization can influence certain physical properties of the polymer, including chain mobility, glass-transition temperature , and modulus of elasticity .
For most chain-growth polymerizations , the propagation steps are much faster than the initiation steps, so that each growing chain is formed in a short time compared to the overall polymerization reaction. During the formation of a single chain, the reactant concentrations and therefore the propagation rate remain effectively constant. Under these conditions, the ratio of the number of propagation steps to the number of initiation steps is just the ratio of reaction rates :
ν = R p R i = R p R t {\displaystyle \nu ={\frac {R_{p}}{R_{i}}}={\frac {R_{p}}{R_{t}}}}
where R p is the rate of propagation , R i is the rate of initiation of polymerization, and R t is the rate of termination of the polymer chain. The second form of the equation is valid at steady-state polymerization, as the chains are being initiated at the same rate they are being terminated ( R i = R t ). [ 5 ]
An exception is the class of living polymerizations , in which propagation is much slower than initiation, and chain termination does not occur until a quenching agent is added. In such reactions the reactant monomer is slowly consumed and the propagation rate varies and is not used to obtain the kinetic chain length. Instead the length at a given time is usually written as:
ν = [ M ] 0 − [ M ] [ I ] 0 {\displaystyle \nu ={\frac {[{\ce {M}}]_{0}-[{\ce {M}}]}{[{\ce {I}}]_{0}}}}
where [M] 0 – [M] represents the number of monomer units consumed, and [I] 0 the number of radicals that initiate polymerization. When the reaction goes to completion, [M] = 0 , and then the kinetic chain length is equal to the number average degree of polymerization of the polymer.
In both cases kinetic chain length is an average quantity, as not all polymer chains in a given reaction are identical in length. The value of ν depends on the nature and concentration of both the monomer and initiator involved.
In chain-growth polymerization, the degree of polymerization depends not only on the kinetic chain length but also on the type of termination step and the possibility of chain transfer .
Termination by disproportionation occurs when an atom is transferred from one polymer free radical to another. The atom is usually hydrogen, and this results in two polymer chains.
With this type of termination and no chain transfer, the number average degree of polymerization (DP n ) is then equal to the average kinetic chain length:
Combination simply means that two radicals are joined together, destroying the radical character of each and forming one polymeric chain. With no chain transfer, the average degree of polymerization is then twice the average kinetic chain length
Some chain-growth polymerizations include chain transfer steps, in which another atom (often hydrogen) is transferred from a molecule in the system to the polymer radical. The original polymer chain is terminated and a new one is initiated. [ 6 ] The kinetic chain is not terminated if the new radical can add monomer. [ 1 ] However the degree of polymerization is reduced without affecting the rate of polymerization (which depends on kinetic chain length), since two (or more) macromolecules are formed instead of one. [ 7 ] For the case of termination by disproportionation, the degree of polymerization becomes:
where R tr is the rate of transfer. The greater R tr is, the shorter the final macromolecule.
The kinetic chain length is important in determining the degree of polymerization, which in turn influences many physical properties of the polymer. | https://en.wikipedia.org/wiki/Kinetic_chain_length |
In physics and engineering , a free body diagram ( FBD ; also called a force diagram ) [ 1 ] is a graphical illustration used to visualize the applied forces , moments , and resulting reactions on a free body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss ), or be a compact body (such as a beam ). A series of free bodies and other diagrams may be necessary to solve complex problems. Sometimes in order to calculate the resultant force graphically the applied forces are arranged as the edges of a polygon of forces [ 2 ] or force polygon (see § Polygon of forces ).
A body is said to be "free" when it is singled out from other bodies for the purposes of dynamic or static analysis. The object does not have to be "free" in the sense of being unforced, and it may or may not be in a state of equilibrium; rather, it is not fixed in place and is thus "free" to move in response to forces and torques it may experience.
Figure 1 shows, on the left, green, red, and blue widgets stacked on top of each other, and for some reason the red cylinder happens to be the body of interest. (It may be necessary to calculate the stress to which it is subjected, for example.) On the right, the red cylinder has become the free body. In figure 2, the interest has shifted to just the left half of the red cylinder and so now it is the free body on the right. The example illustrates the context sensitivity of the term "free body". A cylinder can be part of a free body, it can be a free body by itself, and, as it is composed of parts, any of those parts may be a free body in itself.
Figure 1 and 2 are not yet free body diagrams. In a completed free body diagram, the free body would be shown with forces acting on it. [ 3 ]
Free body diagrams are used to visualize forces and moments applied to a body and to calculate reactions in mechanics and design problems. These diagrams are frequently used both to determine the loading of individual structural components and to calculate internal forces within a structure. They are used by most engineering disciplines from biomechanics to Structural Engineering . [ 4 ] [ 5 ] In the educational environment , a free body diagram is an important step in understanding certain topics, such as statics , dynamics and other forms of classical mechanics .
A free body diagram is not a scaled drawing, it is a diagram . The symbols used in a free body diagram depends upon how a body is modeled. [ 6 ]
Free body diagrams consist of:
The number of forces and moments shown depends upon the specific problem and the assumptions made. Common assumptions are neglecting air resistance and friction and assuming rigid body action.
In statics all forces and moments must balance to zero; the physical interpretation is that if they do not, the body is accelerating and the principles of statics do not apply. In dynamics the resultant forces and moments can be non-zero.
Free body diagrams may not represent an entire physical body. Portions of a body can be selected for analysis. This technique allows calculation of internal forces, making them appear external, allowing analysis. This can be used multiple times to calculate internal forces at different locations within a physical body.
For example, a gymnast performing the iron cross : modeling the ropes and person allows calculation of overall forces (body weight, neglecting rope weight, breezes, buoyancy, electrostatics, relativity, rotation of the earth, etc.). Then remove the person and show only one rope; you get force direction. Then only looking at the person the forces on the hand can be calculated. Now only look at the arm to calculate the forces and moments at the shoulders, and so on until the component you need to analyze can be calculated.
A body may be modeled in three ways:
An FBD represents the body of interest and the external forces acting on it.
Often a provisional free body is drawn before everything is known. The purpose of the diagram is to help to determine magnitude, direction, and point of application of external loads. When a force is originally drawn, its length may not indicate the magnitude. Its line may not correspond to the exact line of action. Even its orientation may not be correct.
External forces known to have negligible effect on the analysis may be omitted after careful consideration (e.g. buoyancy forces of the air in the analysis of a chair, or atmospheric pressure on the analysis of a frying pan).
External forces acting on an object may include friction , gravity , normal force , drag , tension , or a human force due to pushing or pulling. When in a non-inertial reference frame (see coordinate system, below), fictitious forces , such as centrifugal pseudoforce are appropriate.
At least one coordinate system is always included, and chosen for convenience. Judicious selection of a coordinate system can make defining the vectors simpler when writing the equations of motion or statics. The x direction may be chosen to point down the ramp in an inclined plane problem, for example. In that case the friction force only has an x component, and the normal force only has a y component. The force of gravity would then have components in both the x and y directions: mg sin( θ ) in the x and mg cos( θ ) in the y , where θ is the angle between the ramp and the horizontal.
A free body diagram should not show:
In an analysis, a free body diagram is used by summing all forces and moments (often accomplished along or about each of the axes). When the sum of all forces and moments is zero, the body is at rest or moving and/or rotating at a constant velocity, by Newton's first law . If the sum is not zero, then the body is accelerating in a direction or about an axis according to Newton's second law .
Determining the sum of the forces and moments is straightforward if they are aligned with coordinate axes, but it is more complex if some are not. It is convenient to use the components of the forces, in which case the symbols ΣF x and ΣF y are used instead of ΣF (the variable M is used for moments).
Forces and moments that are at an angle to a coordinate axis can be rewritten as two vectors that are equivalent to the original (or three, for three dimensional problems)—each vector directed along one of the axes ( F x ) and ( F y ).
A simple free-body diagram, shown above, of a block on a ramp, illustrates this.
Some care is needed in interpreting the diagram.
In the case of two applied forces, their sum ( resultant force ) can be found graphically using a parallelogram of forces .
To graphically determine the resultant force of multiple forces, the acting forces can be arranged as edges of a polygon by attaching the beginning of one force vector to the end of another in an arbitrary order. Then the vector value of the resultant force would be determined by the missing edge of the polygon. [ 2 ] In the diagram, the forces P 1 to P 6 are applied to the point O. The polygon is constructed starting with P 1 and P 2 using the parallelogram of forces ( vertex a). The process is repeated (adding P 3 yields the vertex b, etc.). The remaining edge of the polygon O-e represents the resultant force R.
In dynamics a kinetic diagram is a pictorial device used in analyzing mechanics problems when there is determined to be a net force and/or moment acting on a body. They are related to and often used with free body diagrams, but depict only the net force and moment rather than all of the forces being considered.
Kinetic diagrams are not required to solve dynamics problems; their use in teaching dynamics is argued against by some [ 7 ] in favor of other methods that they view as simpler. They appear in some dynamics texts [ 8 ] but are absent in others. [ 9 ] | https://en.wikipedia.org/wiki/Kinetic_diagram |
Kinetic diameter is a measure applied to atoms and molecules that expresses the likelihood that a molecule in a gas will collide with another molecule. It is an indication of the size of the molecule as a target. The kinetic diameter is not the same as atomic diameter defined in terms of the size of the atom's electron shell , which is generally a lot smaller, depending on the exact definition used. Rather, it is the size of the sphere of influence that can lead to a scattering event. [ 1 ]
Kinetic diameter is related to the mean free path of molecules in a gas. Mean free path is the average distance that a particle will travel without collision. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by, [ 2 ]
However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. Here, the Maxwell–Boltzmann distribution of energies must be considered, which leads to the modified expression, [ 3 ]
The following table lists the kinetic diameters of some common molecules;
Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For such cases, the above formula for scattering cross section has to be modified.
The scattering cross section, σ, in a collision between two dissimilar particles or molecules is defined by the sum of the kinetic diameters of the two particles,
We define an intensive quantity , the scattering coefficient α, as the product of the gas number density and the scattering cross section,
The mean free path is the inverse of the scattering coefficient,
For similar particles, r 1 = r 2 and,
as before. [ 7 ] | https://en.wikipedia.org/wiki/Kinetic_diameter |
Kinetic energy metamorphosis ( KEM ) is a tribological process of gradual crystal re-orientation and foliation of component minerals in certain rocks. It is caused by very high, localized application of kinetic energy . The required energy may be provided by prolonged battery of fluvially propelled bed load of cobbles, by glacial abrasion , tectonic deformation, and even by human action. It can result in the formation of laminae on specific metamorphic rocks that, while being chemically similar to the protolith , differ significantly in appearance and in their resistance to weathering or deformation. These tectonite layers are of whitish color and tend to survive granular or mass exfoliation much longer than the surrounding protolith. [ 1 ]
The products of KEM were first identified in 2015 in cupules , a form of rock art consisting of spherical cap or dome-shaped depressions created by percussion with hammer-stones. KEM laminae, caused by solid state re-metamorphosis of metamorphic rock , have been observed in cupules on three rock types:
Replication has established that cupules produced on very hard rocks, such as quartzite, require many tens of thousands of blows with hammer-stones to make. [ 4 ] Therefore, the cumulative force applied to very small surface areas (<15 cm2) is in the order of tens of kN (kilo Newtons). In one extreme case, the KEM lamina has been developed to a thickness of c. 10 mm, but the most commonly observed thickness is about 1–2 mm. The tectonite layer is always thickest in the central part of the cupule, i.e. where the greatest amount of energy was applied.
These phenomena have since also been observed in geological contexts, generally of three types:
Kinetic energy metamorphosis products are tribological [ 5 ] [ 6 ] phenomena, caused by very focused, localized cumulative effect of kinetic energy on the syntaxial silica (and the voids it contains) that forms the cement of such rocks as sandstones and quartzites. The conversion to tectonite does not appear to be reversible, and the high resistance of that product to weathering processes protects the parent rock it conceals from both granular and mass exfoliation. Its susceptibility to dating techniques needs to be explored. | https://en.wikipedia.org/wiki/Kinetic_energy_metamorphosis |
Kinetic exchange models are multi-agent dynamic models inspired by the statistical physics of energy distribution , which try to explain the robust and universal features of income/wealth distributions.
Understanding the distributions of income and wealth in an economy has been a classic problem in economics for more than a hundred years. Today it is one of the main branches of econophysics .
In 1897, Vilfredo Pareto first found a universal feature in the distribution of wealth . After that, with some notable exceptions, this field had been dormant for many decades, although accurate data had been accumulated over this period. Considerable investigations with the real data during the last fifteen years (1995–2010) revealed [ 1 ] that the tail (typically 5 to 10 percent of agents in any country) of the income / wealth distribution indeed follows a power law . However, the majority of the population (i.e., the low-income population) follows a different distribution which is debated to be either Gibbs or log-normal .
Basic tools used in this type of modelling are probabilistic and statistical methods mostly taken from the kinetic theory of statistical physics . Monte Carlo simulations often come handy in solving these models.
Since the distributions of income/wealth are the results of the interaction among many heterogeneous agents , there is an analogy with statistical mechanics , where many particles interact. This similarity was noted by Meghnad Saha and B. N. Srivastava in 1931 [ 2 ] and thirty years later by Benoit Mandelbrot . [ 3 ] In 1986, an elementary version of the stochastic exchange model was first proposed by J. Angle. [ 4 ] for open online view only.
In the context of kinetic theory of gases, such an exchange model was first investigated by A. Dragulescu and V. Yakovenko. [ 5 ] [ 6 ] Later, scholars found that in 1988, Bennati had independently introduced the same kinetic exchange dynamics, thus leading to the nomenclature of this model as Bennati-Dragulescu-Yakovenko (BDY) game. [ 7 ] The main modelling efforts since then have been put to introduce the concepts of savings , [ 8 ] [ 9 ] and taxation [ 10 ] in the setting of an ideal gas -like system. Basically, it assumes that in the short-run, an economy remains conserved in terms of income/wealth; therefore law of conservation for income/wealth can be applied. Millions of such conservative transactions lead to a steady state distribution of money ( gamma function -like in the Chakraborti - Chakrabarti model with uniform savings, [ 8 ] and a gamma-like bulk distribution ending with a Pareto tail [ 11 ] in the Chatterjee-Chakrabarti-Manna model with distributed savings [ 9 ] ) and the distribution converges to it. The distributions derived thus have close resemblance with those found in empirical cases of income/wealth distributions.
Though this theory had been originally derived from the entropy maximization principle of statistical mechanics , it had been shown by A. S. Chakrabarti and B. K. Chakrabarti [ 12 ] that the same could be derived from the utility maximization principle as well, following a standard exchange-model with Cobb-Douglas utility function . Recently it has been shown [ 13 ] that an extension of the Cobb-Douglas utility function (in the above-mentioned Chakrabarti-Chakrabarti formulation) by adding a production savings factor leads to the desired feature of growth of the economy in conformity with some earlier phenomenologically established growth laws in the economics literature. The exact distributions produced by this class of kinetic models are known only in certain limits and extensive investigations have been made on the mathematical structures of this class of models. [ 14 ] [ 15 ] The general forms have not been derived so far. For a recent review (in 2024) on these developments, see the article by M. Greenberg (Dept. Economics, University of Massachusetts Amherst & Systems Engineering, Cornell University ) and H. Oliver Gao (Systems Engineering, Cornell University ) in the last twenty five years of research on kinetic exchange modeling of income or wealth dynamics and the resulting statistical properties. [ 7 ]
A very simple model, based on the same kinetic exchange framework, was introduced by Chakraborti in 2002, [ 16 ] now popularly called the "yard sale model", [ 17 ] because it had few features of a real one-on-one economic transactions which led to an oligarchy; this has been extensively studied and reviewed by Boghosian. [ 18 ] [ 19 ]
This class of models has attracted criticisms from many dimensions. [ 20 ] It has been debated for long whether the distributions derived from these models are representing the income distributions or wealth distributions. The law of conservation for income/wealth has also been a subject of criticism. | https://en.wikipedia.org/wiki/Kinetic_exchange_models_of_markets |
A kinetic exclusion assay ( KinExA ) is a type of bioassay in which a solution containing receptor , ligand , and receptor-ligand complex is briefly exposed to additional ligand immobilized on a solid phase. [ 1 ] [ 2 ]
During the assay, a fraction of the free receptor is captured by the solid phase ligand and subsequently labeled with a fluorescent secondary molecule (Figure 1). [ 1 ] [ 2 ] The short contact time with the solid phase does not allow significant dissociation of the pre-formed complexes in the solution. [ 3 ] Solution dissociation is thus “kinetically excluded” from contributing to the captured receptor and the resulting signal provides a measure of the free receptor in the solution.
Measuring the free receptor as a function of total ligand in a series of equilibrated solutions enables calculation of the equilibrium dissociation constant (K d ). [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] Measuring the free receptor with several points before equilibrium enables measurement of the association rate constant (k on ). The off rate (k off ) can also be directly measured, however it is usually calculated from the measured K d and measured k on , (k off = K d * k on ).
Kinetic exclusion assays have been used to measure K d ’s in the nanomolar to femtomolar range. [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 9 ] [ 10 ]
Because the fluorescent secondary molecule is applied after capture of the free receptor from solution (Figure 2) the binding constants measured using a kinetic exclusion assay are for unmodified molecules in solution and thus more accurately reflects endogenous binding interactions than methods requiring modification (typically labeling or immobilization) before measurement. [ 1 ] [ 2 ] Kinetic exclusion assays have been performed using unpurified molecules, [ 4 ] [ 5 ] in serum, [ 7 ] and have measured binding to cell membrane proteins on intact whole cell [ 8 ] [ 11 ] which brings the measured binding interactions closer to their endogenous state.
Molecules suited for measurement by KinExA are antibodies , [ 4 ] [ 7 ] [ 12 ] [ 13 ] [ 14 ] recombinant proteins , [ 15 ] [ 16 ] [ 17 ] small molecules, [ 6 ] [ 18 ] [ 19 ] [ 20 ] aptamers , [ 21 ] [ 22 ] lipids, [ 23 ] [ 24 ] nanobodies , [ 25 ] and toxins. [ 12 ] [ 26 ] [ 27 ]
Kinetic exclusion assay have also been applied for concentration immunoassay , where it has proven capable of providing the maximum theoretical, K d limited, sensitivity. [ 28 ] [ 29 ] An example of this technique has been employed for sensitive detection of environmental contaminants i n near real-time. [ 30 ]
A series of samples are prepared with all the same receptor (R) concentration but in which the ligand (L) concentration is titrated . After equilibrium is reached each sample is measured by flowing it through the column (Figure 2).
For 1:1 reversible binding Equilibrium Kd is defined as
(1) K d ≡k off /k on =R*L/RL
the binding is reversible so conservation of mass can be written as
(2) R T = R+RL
(3) L T = L +RL
Where:
K d = equilibrium dissociation constant
k on = forward rate constant
k off = reverse rate constant
R = free receptor site concentration at equilibrium
L = free ligand site concentration at equilibrium
RL = concentration of complex at equilibrium
R T = total concentration of receptors
L T = total concentration of ligand
A simple equation [ 1 ] [ 2 ] relating the free fraction of R (=R/R T ) to the K d and L T is then fit to the measured data to find the K d of the interaction.
To measure the rate constants , known concentrations of receptor and ligand are mixed in solution and the quantity of free receptor is repeatedly measured over time as the solution phase reaction occurs. The time course of the free receptor depletion is then fit with a standard bimolecular rate equation.
(4) d LR /d t = k on ∙R∙L - K d ∙k on ∙RL
where K d * k on has been substituted for k off . | https://en.wikipedia.org/wiki/Kinetic_exclusion_assay |
Kinetic fractionation is an isotopic fractionation process that separates stable isotopes from each other by their mass during unidirectional processes. Biological processes are generally unidirectional and are very good examples of "kinetic" isotope reactions. All organisms preferentially use lighter isotopes, because "energy costs" are lower, resulting in a significant fractionation between the substrate (heavier) and the biologically mediated product (lighter). For example, photosynthesis preferentially takes up the light isotope of carbon 12 C during assimilation of atmospheric CO 2 . This kinetic isotope fractionation explains why plant material (and thus fossil fuels, which are derived from plants) is typically depleted in 13 C by 25 per mil (2.5%) relative to most inorganic carbon on Earth. [ 1 ]
A naturally occurring example of non-biological kinetic fractionation occurs during the evaporation of seawater to form clouds under conditions in which some part of the transport is unidirectional, such as evaporation into very dry air. In this case, lighter water molecules (i.e., those with 16 O ) evaporate slightly more easily than heavier water molecules with 18 O ; this difference will be greater than it would be if the evaporation was taking place under equilibrium conditions (with bidirectional transport).
During this process the oxygen isotopes are fractionated : the clouds become enriched with 16 O, and the seawater becomes enriched in 18 O. Whereas equilibrium fractionation makes the vapor about 10 per mil (1%) depleted in 18 O relative to the liquid water, kinetic fractionation enhances this fractionation and often makes vapor that is about 15 per mil (1.5%) depleted. Condensation occurs almost exclusively by equilibrium processes, and so it enriches cloud droplets somewhat less than evaporation depletes the vapor. This explains part of the reason why rainwater is observed to be isotopically lighter than seawater.
The heavy isotope of hydrogen in water, deuterium ( 2 H), is much less sensitive to kinetic fractionation than oxygen isotopes, relative to the very large equilibrium fractionation of deuterium. Therefore kinetic fractionation does not deplete 2 H nearly as much, in a relative sense, as 18 O. This gives rise to an excess of deuterium in vapor and rainfall, relative to seawater. The value of this "deuterium excess", as it is called, is about +10 per mil (1%) in most meteoric waters and its non-zero value is a direct manifestation of kinetic isotope fractionation.
A generalized treatment of kinetic isotopic effects is via the GEBIK and GEBIF equations describing transient kinetic isotope effects . [ 2 ]
This isotope -related article is a stub . You can help Wikipedia by expanding it .
This geochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kinetic_fractionation |
Kinetic inductance is the manifestation of the inertial mass of mobile charge carriers in alternating electric fields as an equivalent series inductance . Kinetic inductance is observed in high carrier mobility conductors (e.g. superconductors ) and at very high frequencies.
A change in electromotive force (emf) will be opposed by the inertia of the charge carriers since, like all objects with mass, they prefer to be traveling at constant velocity and therefore it takes a finite time to accelerate the particle. This is similar to how a change in emf is opposed by the finite rate of change of magnetic flux in an inductor. The resulting phase lag in voltage is identical for both energy storage mechanisms, making them indistinguishable in a normal circuit.
Kinetic inductance ( L K {\displaystyle L_{K}} ) arises naturally in the Drude model of electrical conduction considering not only the DC conductivity but also the finite relaxation time (collision time) τ {\displaystyle \tau } of the mobile charge carriers when it is not tiny compared to the wave period 1/f. This model defines a complex conductance at radian frequency ω=2πf given by σ ( ω ) = σ 1 − i σ 2 {\displaystyle {\sigma (\omega )=\sigma _{1}-i\sigma _{2}}} . The imaginary part, -σ 2 , represents the kinetic inductance. The Drude complex conductivity can be expanded into its real and imaginary components:
σ = n e 2 τ m ( 1 + i ω τ ) = n e 2 τ m ( 1 1 + ω 2 τ 2 − i ω τ 1 + ω 2 τ 2 ) {\displaystyle \sigma ={\frac {ne^{2}\tau }{m(1+i\omega \tau )}}={\frac {ne^{2}\tau }{m}}\left({\frac {1}{1+\omega ^{2}\tau ^{2}}}-i{\frac {\omega \tau }{1+\omega ^{2}\tau ^{2}}}\right)}
where m {\displaystyle m} is the mass of the charge carrier (i.e. the effective electron mass in metallic conductors ) and n {\displaystyle n} is the carrier number density. In normal metals the collision time is typically ≈ 10 − 14 {\displaystyle \approx 10^{-14}} s, so for frequencies < 100 GHz ω τ {\displaystyle {\omega \tau }} is very small and can be ignored; then this equation reduces to the DC conductance σ 0 = n e 2 τ / m {\displaystyle \sigma _{0}=ne^{2}\tau /m} . Kinetic inductance is therefore only significant at optical frequencies, and in superconductors whose τ → ∞ {\displaystyle {\tau \rightarrow \infty }} .
For a superconducting wire of cross-sectional area A {\displaystyle A} , the kinetic inductance of a segment of length l {\displaystyle l} can be calculated by equating the total kinetic energy of the Cooper pairs in that region with an equivalent inductive energy due to the wire's current I {\displaystyle I} : [ 1 ]
1 2 ( 2 m e v 2 ) ( n s l A ) = 1 2 L K I 2 {\displaystyle {\frac {1}{2}}(2m_{e}v^{2})(n_{s}lA)={\frac {1}{2}}L_{K}I^{2}}
where m e {\displaystyle m_{e}} is the electron mass ( 2 m e {\displaystyle 2m_{e}} is the mass of a Cooper pair), v {\displaystyle v} is the average Cooper pair velocity, n s {\displaystyle n_{s}} is the density of Cooper pairs, l {\displaystyle l} is the length of the wire, A {\displaystyle A} is the wire cross-sectional area, and I {\displaystyle I} is the current. Using the fact that the current I = 2 e v n s A {\displaystyle I=2evn_{s}A} , where e {\displaystyle e} is the electron charge, this yields: [ 2 ]
L K = ( m e 2 n s e 2 ) ( l A ) {\displaystyle L_{K}=\left({\frac {m_{e}}{2n_{s}e^{2}}}\right)\left({\frac {l}{A}}\right)}
The same procedure can be used to calculate the kinetic inductance of a normal (i.e. non-superconducting) wire, except with 2 m e {\displaystyle 2m_{e}} replaced by m e {\displaystyle m_{e}} , 2 e {\displaystyle 2e} replaced by e {\displaystyle e} , and n s {\displaystyle n_{s}} replaced by the normal carrier density n {\displaystyle n} . This yields:
L K = ( m e n e 2 ) ( l A ) {\displaystyle L_{K}=\left({\frac {m_{e}}{ne^{2}}}\right)\left({\frac {l}{A}}\right)}
The kinetic inductance increases as the carrier density decreases. Physically, this is because a smaller number of carriers must have a proportionally greater velocity than a larger number of carriers in order to produce the same current, whereas their energy increases according to the square of velocity. The resistivity also increases as the carrier density n {\displaystyle n} decreases, thereby maintaining a constant ratio (and thus phase angle) between the (kinetic) inductive and resistive components of a wire's impedance for a given frequency. That ratio, ω τ {\displaystyle \omega \tau } , is tiny in normal metals up to terahertz frequencies.
Kinetic inductance is the principle of operation of the highly sensitive photodetectors known as kinetic inductance detectors (KIDs). The change in the Cooper pair density brought about by the absorption of a single photon in a strip of superconducting material produces a measurable change in its kinetic inductance.
Kinetic inductance is also used in a design parameter for superconducting flux qubits : β {\displaystyle \beta } is the ratio of the kinetic inductance of the Josephson junctions in the qubit to the geometrical inductance of the flux qubit. A design with a low beta behaves more like a simple inductive loop, while a design with a high beta is dominated by the Josephson junctions and has more hysteretic behavior. [ 3 ] | https://en.wikipedia.org/wiki/Kinetic_inductance |
In physical organic chemistry , a kinetic isotope effect ( KIE ) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes . [ 3 ] Formally, it is the ratio of rate constants for the reactions involving the light ( k L ) and the heavy ( k H ) isotopically substituted reactants ( isotopologues ): KIE = k L /k H .
This change in reaction rate is a quantum effect that occurs mainly because heavier isotopologues have lower vibrational frequencies than their lighter counterparts. In most cases, this implies a greater energy input needed for heavier isotopologues to reach the transition state (or, in rare cases, dissociation limit ), and therefore, a slower reaction rate. The study of KIEs can help elucidate reaction mechanisms , and is occasionally exploited in drug development to improve unfavorable pharmacokinetics by protecting metabolically vulnerable C-H bonds.
KIE is considered one of the most essential and sensitive tools for studying reaction mechanisms, the knowledge of which allows improvement of the desirable qualities of said reactions. For example, KIEs can be used to reveal whether a nucleophilic substitution reaction follows a unimolecular (S N 1) or bimolecular (S N 2) pathway.
In the reaction of methyl bromide and cyanide (shown in the introduction), the observed methyl carbon KIE is 1.082, a small effect which indicates an S N 2 mechanism in which the C-Br bond is formed as the C-CN bond is broken. For S N 1 reactions in which the leaving group leaves first to form a trivalent carbon transition state, the KIE is close to the maximum observed value for a secondary KIE (SKIE, see below) of 1.22. [ 1 ] Depending on the pathway, different strategies may be used to stabilize the transition state of the rate-determining step of the reaction and improve the reaction rate and selectivity, which are important for industrial applications.
Isotopic rate changes are most pronounced when the relative mass change is greatest, since the effect is related to vibrational frequencies of the affected bonds. Thus, replacing normal hydrogen ( 1 H) with its isotope deuterium (D or 2 H), doubles the mass; whereas in replacing carbon-12 with carbon-13 , the mass increases by only 8%. The rate of a reaction involving a C– 1 H bond is typically 6–10x faster than with a C– 2 H bond, whereas a 12 C reaction is only 4% faster than the corresponding 13 C reaction; [ 4 ] : 445 even though, in both cases, the isotope is one atomic mass unit (amu) ( dalton ) heavier.
Isotopic substitution can modify the reaction rate in a variety of ways. In many cases, the rate difference can be rationalized by noting that the mass of an atom affects the vibrational frequency of the chemical bond that it forms, even if the potential energy surface for the reaction is nearly identical. Heavier isotopes will ( classically ) lead to lower vibration frequencies, or, viewed quantum mechanically , have lower zero-point energy (ZPE). With a lower ZPE, more energy must be supplied to break the bond, resulting in a higher activation energy for bond cleavage, which in turn lowers the measured rate (see, for example, the Arrhenius equation ). [ 3 ] [ 4 ] : 427
A primary kinetic isotope effect (PKIE) may be found when a bond to the isotopically labeled atom is being formed or broken. [ 3 ] [ 4 ] : 427 Depending on the way a KIE is probed (parallel measurement of rates vs. intermolecular competition vs. intramolecular competition), the observation of a PKIE is indicative of breaking/forming a bond to the isotope at the rate-limiting step, or subsequent product-determining step(s). (The misconception that a PKIE must reflect bond cleavage/formation to the isotope at the rate-limiting step is often repeated in textbooks and the primary literature: see the section on experiments below. ) [ 5 ]
For the aforementioned nucleophilic substitution reactions, PKIEs have been investigated for both the leaving groups, the nucleophiles, and the α-carbon at which the substitution occurs. Interpretation of the leaving group KIEs was difficult at first due to significant contributions from temperature independent factors. KIEs at the α-carbon can be used to develop some understanding into the symmetry of the transition state in S N 2 reactions, though this KIE is less sensitive than what would be ideal, also due to contribution from non-vibrational factors. [ 1 ]
A secondary kinetic isotope effect (SKIE) is observed when no bond to the isotopically labeled atom in the reactant is broken or formed. [ 3 ] [ 4 ] : 427 SKIEs tend to be much smaller than PKIEs; however, secondary deuterium isotope effects can be as large as 1.4 per 2 H atom, and techniques have been developed to measure heavy-element isotope effects to very high precision, so SKIEs are still very useful for elucidating reaction mechanisms.
For the aforementioned nucleophilic substitution reactions, secondary hydrogen KIEs at the α-carbon provide a direct means to distinguish between S N 1 and S N 2 reactions. It has been found that S N 1 reactions typically lead to large SKIEs, approaching to their theoretical maximum at about 1.22, while S N 2 reactions typically yield SKIEs that are very close to or less than 1. KIEs greater than 1 are called normal kinetic isotope effects , while KIEs less than 1 are called inverse kinetic isotope effects (IKIE). In general, smaller force constants in the transition state are expected to yield a normal KIE, and larger force constants in the transition state are expected to yield an IKIE when stretching vibrational contributions dominate the KIE. [ 1 ]
The magnitudes of such SKIEs at the α-carbon atom are largely determined by the C α -H( 2 H) vibrations. For an S N 1 reaction, since the carbon atom is converted into an sp 2 hybridized carbenium ion during the transition state for the rate-determining step with an increase in C α -H( 2 H) bond order, an IKIE would be expected if only the stretching vibrations were important. The observed large normal KIEs are found to be caused by significant out-of-plane bending vibrational contributions when going from the reactants to the transition state of carbenium ion formation. For S N 2 reactions, bending vibrations still play an important role for the KIE, but stretching vibrational contributions are of more comparable magnitude, and the resulting KIE may be normal or inverse depending on the specific contributions of the respective vibrations. [ 1 ] [ 6 ] [ 7 ]
Theoretical treatment of isotope effects relies heavily on transition state theory , which assumes a single potential energy surface for the reaction, and a barrier between the reactants and the products on this surface, on top of which resides the transition state. [ 8 ] [ 9 ] The KIE arises largely from the changes to vibrational ground states produced by the isotopic perturbation along the minimum energy pathway of the potential energy surface, which may only be accounted for with quantum mechanical treatments of the system. Depending on the mass of the atom that moves along the reaction coordinate and nature (width and height) of the energy barrier, quantum tunneling may also make a large contribution to an observed KIE and may need to be separately considered, in addition to the "semi-classical" transition state theory model. [ 8 ]
The deuterium kinetic isotope effect ( 2 H KIE) is by far the most common, useful, and well-understood type of KIE. The accurate prediction of the numerical value of a 2 H KIE using density functional theory calculations is now fairly routine. Moreover, several qualitative and semi-quantitative models allow rough estimates of deuterium isotope effects to be made without calculations, often providing enough information to rationalize experimental data or even support or refute different mechanistic possibilities. Starting materials containing 2 H are often commercially available, making the synthesis of isotopically enriched starting materials relatively straightforward. Also, due to the large relative difference in the mass of 2 H and 1 H and the attendant differences in vibrational frequency, the isotope effect is larger than for any other pair of isotopes except 1 H and 3 H, [ 10 ] allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, secondary effects are generally very small for heavier elements and close in magnitude to the experimental uncertainty, which complicates their interpretation and limits their utility. In the context of isotope effects, hydrogen often means the light isotope, protium ( 1 H), specifically. In the rest of this article, reference to hydrogen and deuterium in parallel grammatical constructions or direct comparisons between them should be taken to mean 1 H and 2 H. [ a ]
The theory of KIEs was first formulated by Jacob Bigeleisen in 1949. [ 11 ] [ 4 ] : 427 Bigeleisen's general formula for 2 H KIEs (which is also applicable to heavier elements) is given below. It employs transition state theory and a statistical mechanical treatment of translational, rotational, and vibrational levels for the calculation of rate constants k H and k D . However, this formula is "semi-classical" in that it neglects the contribution from quantum tunneling, which is often introduced as a separate correction factor. Bigeleisen's formula also does not deal with differences in non-bonded repulsive interactions caused by the slightly shorter C– 2 H bond compared to a C–H bond. In the equation, subscript H or D refer to the species with 1 H or 2 H, respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively. [ 7 ] [ 12 ] (Strictly speaking, a κ H / κ D {\displaystyle \kappa _{\mathrm {H} }/\kappa _{\mathrm {D} }} term resulting from an isotopic difference in transmission coefficients should also be included. [ 13 ] )
where we define
Here, h = Planck constant ; k B = Boltzmann constant ; ν ~ i {\displaystyle {\tilde {\nu }}_{i}} = frequency of vibration, expressed in wavenumber ; c = speed of light ; N A = Avogadro constant ; and R = universal gas constant . The σ X (X = H or D) are the symmetry numbers for the reactants and transition states. The M X are the molecular masses of the corresponding species, and the I q X ( q = x , y , or z ) terms are the moments of inertia about the three principal axes. The u i X are directly proportional to the corresponding vibrational frequencies, ν i , and the vibrational zero-point energy (ZPE) (see below). The integers N and N ‡ are the number of atoms in the reactants and the transition states, respectively. [ 7 ] The complicated expression given above can be represented as the product of four separate factors: [ 7 ]
For the special case of 2 H isotope effects, we will argue that the first three terms can be treated as equal to or well approximated by unity. The first factor S (containing σ X ) is the ratio of the symmetry numbers for the various species. This will be a rational number (a ratio of integers) that depends on the number of molecular and bond rotations leading to the permutation of identical atoms or groups in the reactants and the transition state. [ 12 ] For systems of low symmetry, all σ X (reactant and transition state) will be unity; thus S can often be neglected. The MMI factor (containing the M X and I q X ) refers to the ratio of the molecular masses and the moments of inertia. Since hydrogen and deuterium tend to be much lighter than most reactants and transition states, there is little difference in the molecular masses and moments of inertia between H and D containing molecules, so the MMI factor is usually also approximated as unity. The EXC factor (containing the product of vibrational partition functions ) corrects for the KIE caused by the reactions of vibrationally excited molecules. The fraction of molecules with enough energy to have excited state A–H/D bond vibrations is generally small for reactions at or near room temperature (bonds to hydrogen usually vibrate at 1000 cm −1 or higher, so exp(- u i ) = exp(- hν i / k B T ) < 0.01 at 298 K, resulting in negligible contributions from the 1–exp(- u i ) factors). Hence, for hydrogen/deuterium KIEs, the observed values are typically dominated by the last factor, ZPE (an exponential function of vibrational ZPE differences), consisting of contributions from the ZPE differences for each of the vibrational modes of the reactants and transition state, which can be represented as follows: [ 7 ]
where we define
The sums in the exponent of the second expression can be interpreted as running over all vibrational modes of the reactant ground state and the transition state. Or, one may interpret them as running over those modes unique to the reactant or the transition state or whose vibrational frequencies change substantially upon advancing along the reaction coordinate. The remaining pairs of reactant and transition state vibrational modes have very similar Δ u i {\displaystyle \Delta u_{i}} and Δ u i ‡ {\displaystyle \Delta u_{i}^{\ddagger }} , and cancellations occur when the sums in the exponent are calculated. Thus, in practice, 2 H KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of k H / k D possible. [ 12 ]
As mentioned, especially for 1 H/ 2 H substitution, most KIEs arise from the difference in ZPE between the reactants and the transition state of the isotopologues; this difference can be understood qualitatively as follows: in the Born–Oppenheimer approximation , the potential energy surface is the same for both isotopic species. However, a quantum treatment of the energy introduces discrete vibrational levels onto this curve, and the lowest possible energy state of a molecule corresponds to the lowest vibrational energy level, which is slightly higher in energy than the minimum of the potential energy curve . This difference, known as the ZPE, is a manifestation of the uncertainty principle that necessitates an uncertainty in the C-H or C-D bond length. Since the heavier (in this case the deuterated) species behaves more "classically", its vibrational energy levels are closer to the classical potential energy curve, and it has a lower ZPE. The ZPE differences between the two isotopic species, at least in most cases, diminish in the transition state, since the bond force constant decreases during bond breaking. Hence, the lower ZPE of the deuterated species translates into a larger activation energy for its reaction, as shown in the following figure, leading to a normal KIE. [ 14 ] This effect should, in principle, be taken into account all 3 N− 6 vibrational modes for the starting material and 3 N ‡ − 7 vibrational modes at the transition state (one mode, the one corresponding to the reaction coordinate, is missing at the transition state, since a bond breaks and there is no restorative force against the motion). The harmonic oscillator is a good approximation for a vibrating bond, at least for low-energy vibrational states. Quantum mechanics gives the vibrational ZPE as ϵ i ( 0 ) = 1 2 h ν i {\displaystyle \epsilon _{i}^{(0)}={\frac {1}{2}}h\nu _{i}} . Thus, we can readily interpret the factor of 1 / 2 and the sums of u i = h ν i / k B T {\displaystyle u_{i}=h\nu _{i}/k_{\mathrm {B} }T} terms over ground state and transition state vibrational modes in the exponent of the simplified formula above. For a harmonic oscillator, vibrational frequency is inversely proportional to the square root of the reduced mass of the vibrating system:
where k f is the force constant . Moreover, the reduced mass is approximated by the mass of the light atom of the system, X = H or D. Because m D ≈ 2 m H ,
In the case of homolytic C–H/D bond dissociation, the transition state term disappears; and neglecting other vibrational modes, k H / k D = exp( 1 / 2 Δ u i ). Thus, a larger isotope effect is observed for a stiffer ("stronger") C–H/D bond. For most reactions of interest, a hydrogen atom is transferred between two atoms, with a transition-state [A···H···B] ‡ and vibrational modes at the transition state need to be accounted for. Nevertheless, it is still generally true that cleavage of a bond with a higher vibrational frequency will give a larger isotope effect.
To calculate the maximum possible value for a non-tunneling 2 H KIE, we consider the case where the ZPE difference between the stretching vibrations of a C- 1 H bond (3000 cm −1 ) and C- 2 H bond (2200 cm −1 ) disappears in the transition state (an energy difference of [3000 – 2200 cm −1 ]/2 = 400 cm −1 ≈ 1.15 kcal/mol), without any compensation from a ZPE difference at the transition state (e.g., from the symmetric A···H···B stretch, which is unique to the transition state). The simplified formula above, predicts a maximum for k H / k D as 6.9. If the complete disappearance of two bending vibrations is also included, k H / k D values as large as 15-20 can be predicted. Bending frequencies are very unlikely to vanish in the transition state, however, and there are only a few cases in which k H / k D values exceed 7-8 near room temperature. Furthermore, it is often found that tunneling is a major factor when they do exceed such values. A value of k H / k D ~ 10 is thought to be maximal for a semi-classical PKIE (no tunneling) for reactions at ≈298 K. (The formula for k H / k D has a temperature dependence, so larger isotope effects are possible at lower temperatures.) [ 15 ] Depending on the nature of the transition state of H-transfer (symmetric vs. "early" or "late" and linear vs. bent); the extent to which a primary 2 H isotope effect approaches this maximum, varies. A model developed by Westheimer predicted that symmetrical (thermoneutral, by Hammond's postulate ), linear transition states have the largest isotope effects, while transition states that are "early" or "late" (for exothermic or endothermic reactions, respectively), or nonlinear (e.g. cyclic) exhibit smaller effects. These predictions have since received extensive experimental support. [ 16 ]
For secondary 2 H isotope effects, Streitwieser proposed that weakening (or strengthening, in the case of an inverse isotope effect) of bending modes from the reactant ground state to the transition state are largely responsible for observed isotope effects. These changes are attributed to a change in steric environment when the carbon bound to the H/D undergoes rehybridization from sp 3 to sp 2 or vice versa (an α SKIE), or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away (a β SKIE). These isotope effects have a theoretical maximum of k H / k D = 2 0.5 ≈ 1.4. For a SKIE at the α position, rehybridization from sp 3 to sp 2 produces a normal isotope effect, while rehybridization from sp 2 to sp 3 results in an inverse isotope effect with a theoretical minimum of k H / k D = 2 -0.5 ≈ 0.7. In practice, k H / k D ~ 1.1-1.2 and k H /k D ~ 0.8-0.9 are typical for α SKIEs, while k H / k D ~ 1.15-1.3 are typical for β SKIE. For reactants containing several isotopically substituted β-hydrogens, the observed isotope effect is often the result of several H/D's at the β position acting in concert. In these cases, the effect of each isotopically labeled atom is multiplicative, and cases where k H / k D > 2 are not uncommon. [ 17 ]
The following simple expressions relating 2 H and 3 H KIEs, which are also known as the Swain equation (or the Swain-Schaad-Stivers equations), can be derived from the general expression given above using some simplifications: [ 8 ] [ 18 ]
i.e.,
In deriving these expressions, the reasonable approximation that reduced mass roughly equals the mass of the 1 H, 2 H, or 3 H, was used. Also, the vibrational motion was assumed to be approximated by a harmonic oscillator, so that u i X ∝ μ X − 1 / 2 ≅ m X − 1 / 2 {\displaystyle u_{i\mathrm {X} }\propto \mu _{\mathrm {X} }^{-1/2}\cong m_{\mathrm {X} }^{-1/2}} ; X = 1,2,3 H. The subscript " s " refers to these "semi-classical" KIEs, which disregard quantum tunneling. Tunneling contributions must be treated separately as a correction factor.
For isotope effects involving elements other than hydrogen, many of these simplifications are not valid, and the magnitude of the isotope effect may depend strongly on some or all of the neglected factors. Thus, KIEs for elements other than hydrogen are often much harder to rationalize or interpret. In many cases and especially for hydrogen-transfer reactions, contributions to KIEs from tunneling are significant (see below).
In some cases, a further rate enhancement is seen for the lighter isotope, possibly due to quantum tunneling . This is typically only observed for reactions involving bonds to hydrogen. Tunneling occurs when a molecule penetrates through a potential energy barrier rather than over it. [ 19 ] [ 20 ] Though not allowed by classical mechanics , particles can pass through classically forbidden regions of space in quantum mechanics based on wave–particle duality . [ 21 ]
Tunneling can be analyzed using Bell's modification of the Arrhenius equation , which includes the addition of a tunneling factor, Q:
where A is the Arrhenius parameter, E is the barrier height and
where α = E R T {\displaystyle \alpha ={\frac {E}{RT}}} and β = 2 a π 2 ( 2 m E ) 1 / 2 h {\displaystyle \beta ={\frac {2a\pi ^{2}(2mE)^{1/2}}{h}}}
Examination of the β term shows exponential dependence on the particle's mass. As a result, tunneling is much more likely for a lighter particle such as hydrogen. Simply doubling the mass of a tunneling proton by replacing it with a deuteron drastically reduces the rate of such reactions. As a result, very large KIEs are observed that can not be accounted for by differences in ZPEs.
Also, the β term depends linearly with barrier width, 2a. As with mass, tunneling is greatest for small barrier widths. Optimal tunneling distances of protons between donor and acceptor atom is 40 pm. [ 23 ]
Tunneling is a quantum effect tied to the laws of wave mechanics, not kinetics . Therefore, tunneling tends to become more important at low temperatures, where even the smallest kinetic energy barriers may not be overcome but can be tunneled through. [ 19 ]
Peter S. Zuev et al. reported rate constants for the ring expansion of 1-methylcyclobutylfluorocarbene to be 4.0 × 10 −6 /s in nitrogen and 4.0 × 10 −5 /s in argon at 8 kelvin. They calculated that at 8 kelvin, the reaction would proceed via a single quantum state of the reactant so that the reported rate constant is temperature independent and the tunneling contribution to the rate was 152 orders of magnitude greater than the contribution of passage over the transition state energy barrier. [ 24 ]
So even though conventional chemical reactions tend to slow down dramatically as the temperature is lowered, tunneling reactions rarely change at all. Particles that tunnel through an activation barrier are a direct result of the fact that the wave function of an intermediate species, reactant or product is not confined to the energy well of a particular trough along the energy surface of a reaction but can "leak out" into the next energy minimum. In light of this, tunneling should be temperature independent. [ 19 ] [ 3 ]
For the hydrogen abstraction from gaseous n-alkanes and cycloalkanes by hydrogen atoms over the temperature range 363–463 K, the H/D KIE data were characterized by small preexponential factor ratios A H / A D ranging from 0.43 to 0.54 and large activation energy differences from 9.0 to 9.7 kJ/mol. Basing their arguments on transition state theory , the small A factor ratios associated with the large activation energy differences (usually about 4.5 kJ/mol for C–H(D) bonds) provided strong evidence for tunneling. For the purpose of this discussion, it is important is that the A factor ratio for the various paraffins they used was roughly constant throughout the temperature range. [ 25 ]
To determine if tunneling is involved in KIE of a reaction with H or D, a few criteria are considered:
Also for reactions where isotopes include H, D and T, a criterion of tunneling is the Swain-Schaad relations which compare the rate constants ( k ) of the reactions where H, D or T are exchanged:
In organic reactions, this proton tunneling effect has been observed in such reactions as the deprotonation and iodination of nitropropane with hindered pyridine base [ 26 ] with a reported KIE of 25 at 25°C:
and in a 1,5-sigmatropic hydrogen shift , [ 27 ] though it is observed that it is hard to extrapolate experimental values obtained at high temperature to lower temperatures: [ 28 ] [ 29 ]
It has long been speculated that high efficiency of enzyme catalysis in proton or hydride ion transfer reactions could be due partly to the quantum mechanical tunneling effect. Environment at the active site of an enzyme positions the donor and acceptor atom close to the optimal tunneling distance, where the amino acid side chains can "force" the donor and acceptor atom closer together by electrostatic and noncovalent interactions. It is also possible that the enzyme and its unusual hydrophobic environment inside a reaction site provides tunneling-promoting vibration. [ 30 ] Studies on ketosteroid isomerase have provided experimental evidence that the enzyme actually enhances the coupled motion/hydrogen tunneling by comparing primary and secondary KIEs of the reaction under enzyme-catalyzed and non-enzyme-catalyzed conditions. [ 31 ]
Many examples exist for proton tunneling in enzyme-catalyzed reactions that were discovered by KIE. A well-studied example is methylamine dehydrogenase, where large primary KIEs of 5–55 have been observed for the proton transfer step. [ 32 ]
Another example of tunneling contribution to proton transfer in enzymatic reactions is the reaction carried out by alcohol dehydrogenase . Competitive KIEs for the hydrogen transfer step at 25°C resulted in 3.6 and 10.2 for primary and secondary KIEs, respectively. [ 33 ]
Isotopic effect expressed with the equations given above only refer to reactions that can be described with first-order kinetics . In all instances in which this is not possible, transient KIEs should be taken into account using the GEBIK and GEBIF equations. [ 34 ]
Simmons and Hartwig refer to the following three cases as the main types of KIE experiments involving C-H bond functionalization: [ 5 ]
In this experiment, the rate constants for the normal substrate and its isotopically labeled analogue are determined independently, and the KIE is obtained as a ratio of the two. The accuracy of the measured KIE is severely limited by the accuracy with which each of these rate constants can be measured. Furthermore, reproducing the exact conditions in the two parallel reactions can be very challenging. Nevertheless, a measurement of a large kinetic isotope effect through direct comparison of rate constants is indicative that C-H bond cleavage occurs at the rate-determining step. (A smaller value could indicate an isotope effect due to a pre-equilibrium, so that the C-H bond cleavage occurs somewhere before the rate-determining step.)
This type of experiment, uses the same substrates as used in Experiment A, but they are allowed in to react in the same container, instead of two separate containers. The KIE in this experiment is determined by the relative amount of products formed from C-H versus C-D functionalization (or it can be inferred from the relative amounts of unreacted starting materials). One must quench the reaction before it goes to completion to observe the KIE (see the Evaluation section below). Generally, the reaction is halted at low conversion (~5 to 10% conversion) or a large excess (> 5 equiv.) of the isotopic mixture is used. This experiment type ensures that both C-H and C-D bond functionalizations occur under exactly the same conditions, and the ratio of products from C-H and C-D bond functionalizations can be measured with much greater precision than the rate constants in Experiment A. Moreover, only a single measurement of product concentrations from a single sample is required. However, an observed kinetic isotope effect from this experiment is more difficult to interpret, since it may either mean that C-H bond cleavage occurs during the rate-determining step or at a product-determining step ensuing the rate-determining step. The absence of a KIE, at least according to Simmons and Hartwig, is nonetheless indicative of the C-H bond cleavage not occurring during the rate-determining step.
This type of experiment is analogous to Experiment B, except this time there is an intramolecular competition for the C-H or C-D bond functionalization. In most cases, the substrate possesses a directing group (DG) between the C-H and C-D bonds. Calculation of the KIE from this experiment and its interpretation follow the same considerations as that of Experiment B. However, the results of Experiments B and C will differ if the irreversible binding of the isotope-containing substrate takes place in Experiment B prior to the cleavage of the C-H or C-D bond. In such a scenario, an isotope effect may be observed in Experiment C (where choice of the isotope can take place even after substrate binding) but not in Experiment B (since the choice of whether C-H or C-D bond cleaves is already made as soon as the substrate binds irreversibly). In contrast to Experiment B, the reaction need not be halted at low consumption of isotopic starting material to obtain an accurate k H / k D , since the ratio of H and D in the starting material is 1:1, regardless of the extent of conversion.
One non-C-H activation example of different isotope effects being observed in the case of intermolecular (Experiment B) and intramolecular (Experiment C) competition is the photolysis of diphenyldiazomethane in the presence of t -butylamine. To explain this result, the formation of diphenylcarbene, followed by irreversible nucleophilic attack by t -butylamine was proposed. Because there is little isotopic difference in the rate of nucleophilic attack, the intermolecular experiment resulted in a KIE close to 1. In the intramolecular case, however, the product ratio is determined by the proton transfer that occurs after the nucleophilic attack, a process which has a substantial KIE of 2.6. [ 35 ]
Thus, Experiments A, B, and C will give results of differing levels of precision and require different experimental setup and ways of analyzing data. As a result, the feasibility of each type of experiment depends on the kinetic and stoichiometric profile of the reaction, as well as the physical characteristics of the reaction mixture (e.g. homogeneous vs. heterogeneous). Moreover, as noted in the paragraph above, the experiments provide KIE data for different steps of a multi-step reaction, depending on the relative locations of the rate-limiting step, product-determining steps, and/or C-H/D cleavage step.
The hypothetical examples below illustrate common scenarios. Consider the following reaction coordinate diagram. For a reaction with this profile, all three experiments (A, B, and C) will yield a significant primary KIE:
On the other hand, if a reaction follows the following energy profile, in which the C-H or C-D bond cleavage is irreversible but occurs after the rate-determining step (RDS), no significant KIE will be observed with Experiment A, since the overall rate is not affected by the isotopic substitution. Nevertheless, the irreversible C-H bond cleavage step will give a primary KIE with the other two experiments, since the second step would still affect the product distribution. Therefore, with Experiments B and C, it is possible to observe the KIE even if C-H or C-D bond cleavage occurs not in the rate-determining step, but in the product-determining step.
A large part of the KIE arises from vibrational ZPE differences between the reactant ground state and the transition state that vary between the reactant and its isotopically substituted analog. While one can carry out involved calculations of KIEs using computational chemistry, much of the work done is of simpler order that involves the investigation of whether particular isotopic substitutions produce a detectable KIE or not. Vibrational changes from isotopic substitution at atoms away from the site where the reaction occurs tend to cancel between the reactant and the transition state. Therefore, the presence of a KIE indicates that the isotopically labeled atom is at or very near the reaction site.
The absence of an isotope effect is more difficult to interpret: It may mean that the isotopically labeled atom is away from the reaction site, but it may also mean there are certain compensating effects that lead to the lack of an observable KIE. For example, the differences between the reactant and the transition state ZPEs may be identical between the normal reactant and its isotopically labeled version. Alternatively, it may mean that the isotopic substitution is at the reaction site, but vibrational changes associated with bonds to this atom occur after the rate-determining step. Such a case is illustrated in the following example, in which ABCD represents the atomic skeleton of a molecule.
Assuming steady state conditions for the intermediate ABC, the overall rate of reaction is the following:
If the first step is rate-determining, this equation reduces to:
Or if the second step is rate-determining, the equation reduces to:
In most cases, isotopic substitution at A, especially if it is a heavy atom, will not alter k 1 or k 2 , but it will most probably alter k 3 . Hence, if the first step is rate-determining, there will not be an observable kinetic isotope effect in the overall reaction with isotopic labeling of A, but there will be one if the second step is rate-determining. For intermediate cases where both steps have comparable rates, the magnitude of the kinetic isotope effect will depend on the ratio of k 3 and k 2 .
Isotopic substitution of D will alter k 1 and k 2 while not affecting k 3 . The KIE will always be observable with this substitution since k 1 appears in the simplified rate expression regardless of which step is rate-determining, but it will be less pronounced if the second step is rate-determining due to some cancellation between the isotope effects on k 1 and k 2 . This outcome is related to the fact that equilibrium isotope effects are usually smaller than KIEs.
Isotopic substitution of B will clearly alter k 3 , but it may also alter k 1 to a lesser extent if the B-C bond vibrations are affected in the transition state of the first step. There may thus be a small isotope effect even if the first step is rate-determining.
In competition reactions, KIE is calculated from isotopic product or remaining reactant ratios after the reaction, but these ratios depend strongly on the extent of completion of the reaction. Most often, the isotopic substrate consists of molecules labeled in a specific position and their unlabeled, ordinary counterparts. [ 8 ] One can also, in case of 13 C KIEs, as well as similar cases, simply rely on the natural abundance of the isotopic carbon for the KIE experiments, eliminating the need for isotopic labeling. [ 37 ] The two isotopic substrates will react through the same mechanism, but at different rates. The ratio between the amounts of the two species in the reactants and the products will thus change gradually over the course of the reaction, and this gradual change can be treated as follows: [ 8 ] Assume that two isotopic molecules, A 1 and A 2 , undergo irreversible competition reactions:
The KIE for this scenario is found to be:
Where F 1 and F 2 refer to the fraction of conversions for the isotopic species A 1 and A 2 , respectively.
In this treatment, all other reactants are assumed to be non-isotopic. Assuming further that the reaction is of first order with respect to the isotopic substrate A, the following general rate expression for both these reactions can be written:
Since f([B],[C],...) does not depend on the isotopic composition of A, it can be solved for in both rate expressions with A 1 and A 2 , and the two can be equated to derive the following relations:
Where [A 1 ] 0 and [A 2 ] 0 are the initial concentrations of A 1 and A 2 , respectively. This leads to the following KIE expression:
Which can also be expressed in terms of fraction amounts of conversion of the two reactions, F 1 and F 2 , where 1-F n =[A n ]/[A n ] 0 for n = 1 or 2, as follows:
As for finding the KIEs, mixtures of substrates containing stable isotopes may be analyzed with a mass spectrometer, which yields the ratios of the isotopic molecules in the initial substrate (defined here as [A 2 ] 0 /[A 1 ] 0 =R 0 ), in the substrate after some conversion ([A 2 ]/[A 1 ]=R), or in the product ([P 2 ]/[P 1 ]=R P ). When one of the species, e.g. 2, is a radioisotope, its mixture with the other species can also be analyzed by its radioactivity, which is measured in molar activities that are proportional to [A 2 ] 0 / ([A 1 ] 0 +[A 2 ] 0 ) ≈ [A 2 ] 0 /[A 1 ] 0 = R 0 in the initial substrate, [A 2 ] / ([A 1 ]+[A 2 ]) ≈ [A 2 ]/[A 1 ] = R in the substrate after some conversion, and [R 2 ] / ([R 1 ]+[R 2 ]) ≈ [R 2 ]/[R 1 ] = R P , so that the same ratios as in the other case can be measured as long as the radioisotope is present in tracer amounts. Such ratios may also be determined using NMR spectroscopy. [ 38 ]
When the substrate composition is followed, the following KIE expression in terms of R 0 and R can be derived:
Taking the ratio of R and R 0 using the previously derived expression for F 2 , one gets:
Isotopic enrichment of the starting material can be calculated from the dependence of R/R 0 on F 1 for various KIEs, yielding the following figure. Due to the exponential dependence, even very low KIEs lead to large changes in isotopic composition of the starting material at high conversions.
When the products are followed, the KIE can be calculated using the products ratio R P along with R 0 as follows:
KIE measurement at natural abundance is a simple general method for measuring KIEs for chemical reactions performed with materials of natural abundance . This technique for measuring KIEs overcomes many limitations of previous KIE measurement methods. KIE measurements from isotopically labeled materials require a new synthesis for each isotopically labeled material (a process often prohibitively difficult), a competition reaction, and an analysis. [ 5 ] The KIE measurement at natural abundance avoids these issues by taking advantage of high precision quantitative techniques ( nuclear magnetic resonance spectroscopy , isotope-ratio mass spectrometry ) to site selectively measure kinetic fractionation of isotopes , in either product or starting material for a given chemical reaction .
Quantitative single-pulse nuclear magnetic resonance spectroscopy (NMR) is a method amenable for measuring kinetic fractionation of isotopes for natural abundance KIE measurements. Pascal et al. were inspired by studies demonstrating dramatic variations of deuterium within identical compounds from different sources and hypothesized that NMR could be used to measure 2 H KIEs at natural abundance. [ 39 ] [ 40 ] Pascal and coworkers tested their hypothesis by studying the insertion reaction of dimethyl diazomalonate into cyclohexane . Pascal et al. measured a KIE of 2.2 using 2 H NMR for materials of natural abundance. [ 40 ]
Singleton and coworkers demonstrated the capacity of 13 C NMR based natural abundance KIE measurements for studying the mechanism of the [4 + 2] cycloaddition of isoprene with maleic anhydride . [ 37 ] Previous studies by Gajewski on isotopically enrich materials observed KIE results that suggested an asynchronous transition state, but were always consistent, within error, for a perfectly synchronous reaction mechanism . [ 41 ]
This work by Singleton et al. established the measurement of multiple 13 C KIEs within the design of a single experiment. These 2 H and 13 C KIE measurements determined at natural abundance found the "inside" hydrogens of the diene experience a more pronounced 2 H KIE than the "outside" hydrogens and the C1 and C4 experience a significant KIE. These key observations suggest an asynchronous reaction mechanism for the cycloaddition of isoprene with maleic anhydride .
The limitations for determining KIEs at natural abundance using NMR are that the recovered material must have a suitable amount and purity for NMR analysis (the signal of interest should be distinct from other signals), the reaction of interest must be irreversible, and the reaction mechanism must not change for the duration of the chemical reaction .
Experimental details for using quantitative single pulse NMR to measure KIE at natural abundance as follows: the experiment needs to be performed under quantitative conditions including a relaxation time of 5 T 1 , measured 90° flip angle, a digital resolution of at least 5 points across a peak, and a signal:noise greater than 250. The raw FID is zero-filled to at least 256K points before the Fourier transform. NMR spectra are phased and then treated with a zeroth order baseline correction without any tilt correction. Signal integrations are determined numerically with a minimal tolerance for each integrated signal. [ 37 ] [ clarification needed ]
Colletto et al. developed a regioselective β-arylation of benzo[b]thiophenes at room temperature with aryl iodides as coupling partners and sought to understand the mechanism of this reaction by performing natural abundance KIE measurements via single pulse NMR. [ 42 ]
The observation of a primary 13 C isotope effect at C3, an inverse 2 H isotope effect, a secondary 13 C isotope effect at C2, and the lack of a 2 H isotope effect at C2; led Colletto et al. to suggest a Heck-type reaction mechanism for the regioselective β -arylation of benzo[b]thiophenes at room temperature with aryl iodides as coupling partners. [ 42 ]
Frost et al. sought to understand the effects of Lewis acid additives on the mechanism of enantioselective palladium -catalyzed C-N bond activation using natural abundance KIE measurements via single pulse NMR. [ 43 ]
The primary 13 C KIE observed in the absence of BPh 3 suggests a reaction mechanism with rate limiting cis oxidation into the C–CN bond of the cyanoformamide . The addition of BPh 3 causes a relative decrease in the observed 13 C KIE which led Frost et al. to suggest a change in the rate limiting step from cis oxidation to coordination of palladium to the cyanoformamide. [ 43 ]
Though KIE measurements at natural abundance are a powerful tool for understanding reaction mechanisms, the amounts of material needed for analysis can make this technique inaccessible for reactions that use expensive reagents or unstable starting materials. To mitigate these limitations, Jacobsen and coworkers developed 1 H to 13 C polarization transfer as a means to reduce the time and material required for KIE measurements at natural abundance. The distortionless enhancement by polarization transfer (DEPT) takes advantage of the larger gyromagnetic ratio of 1 H over 13 C, to theoretically improve measurement sensitivity by a factor of 4 or decrease experiment time by a factor of 16. This method for natural abundance kinetic isotope measurement is favorable for analysis for reactions containing unstable starting materials, and catalysts or products that are relatively costly. [ 44 ]
Jacobsen and coworkers identified the thiourea -catalyzed glycosylation of galactose as a reaction that met both of the aforementioned criteria (expensive materials and unstable substrates) and was a reaction with a poorly understood mechanism. [ 45 ] Glycosylation is a special case of nucleophilic substitution that lacks clear definition between S N 1 and S N 2 mechanistic character. The presence of the oxygen adjacent to the site of displacement (i.e., C1) can stabilize positive charge. This charge stabilization can cause any potential concerted pathway to become asynchronous and approaches intermediates with oxocarbenium character of the S N 1 mechanism for glycosylation.
Jacobsen and coworkers observed small normal KIEs at C1, C2, and C5 which suggests significant oxocarbenium character in the transition state and an asynchronous reaction mechanism with a large degree of charge separation.
High precision isotope-ratio mass spectrometry (IRMS) is another method for measuring kinetic fractionation of isotopes for natural abundance KIE measurements. Widlanski and coworkers demonstrated 34 S KIE at natural abundance measurements for the hydrolysis of sulfate monoesters. Their observation of a large KIE suggests S-O bond cleavage is rate controlling and likely rules out an associate reaction mechanism . [ 46 ]
The major limitation for determining KIEs at natural abundance using IRMS is the required site selective degradation without isotopic fractionation into an analyzable small molecule, a non-trivial task. [ 37 ]
Primary hydrogen KIEs refer to cases in which a bond to the isotopically labeled hydrogen is formed or broken at a rate- and/or product-determining step of a reaction. [ 5 ] These are the most commonly measured KIEs, and much of the previously covered theory refers to primary KIEs. When there is adequate evidence that transfer of the labeled hydrogen occurs in the rate-determining step of a reaction, if a fairly large KIE is observed, e.g. k H /k D of at least 5-6 or k H /k T about 10–13 at room temperature, it is quite likely that the hydrogen transfer is linear and that the hydrogen is fairly symmetrically located in the transition state. It is usually not possible to make comments about tunneling contributions to the observed isotope effect unless the effect is very large. If the primary KIE is not as large, it is generally considered to be indicative of a significant contribution from heavy-atom motion to the reaction coordinate, though it may also mean that hydrogen transfer follows a nonlinear pathway. [ 8 ]
Secondary hydrogen isotope effects or secondary KIE (SKIE) arise in cases where the isotopic substitution is remote from the bond being broken. The remote atom nonetheless influences the internal vibrations of the system, which via changes in zero-point energy (ZPE) affect the rates of chemical reactions. [ 47 ] Such effects are expressed as ratios of rate for the light isotope to that of the heavy isotope and can be "normal" (ratio ≥ 1) or "inverse" (ratio < 1) effects. [ 48 ] SKIEs are defined as α,β (etc.) secondary isotope effects where such prefixes refer to the position of the isotopic substitution relative to the reaction center (see alpha and beta carbon ). [ 49 ] The prefix α refers to the isotope associated with the reaction center and the prefix β refers to the isotope associated with an atom neighboring the reaction center and so on.
In physical organic chemistry, SKIE is discussed in terms of electronic effects such as induction, bond hybridization, or hyperconjugation . [ 50 ] These properties are determined by electron distribution, and depend upon vibrationally averaged bond length and angles that are not greatly affected by isotopic substitution. Thus, the use of the term "electronic isotope effect" while legitimate is discouraged from use as it can be misinterpreted to suggest that the isotope effect is electronic in nature rather than vibrational. [ 49 ]
SKIEs can be explained in terms of changes in orbital hybridization. When the hybridization of a carbon atom changes from sp 3 to sp 2 , a number of vibrational modes (stretches, in-plane and out-of-plane bending) are affected. The in-plane and out-of-plane bending in an sp 3 hybridized carbon are similar in frequency due to the symmetry of an sp 3 hybridized carbon. In an sp 2 hybridized carbon the in-plane bend is much stiffer than the out-of-plane bending resulting in a large difference in the frequency, the ZPE and thus the SKIE (which exists when there is a difference in the ZPE of the reactant and transition state). [ 19 ] The theoretical maximum change caused by the bending frequency difference has been calculated as 1.4. [ 19 ]
When carbon undergoes a reaction that changes its hybridization from sp 3 to sp 2 , the out-of-plane bending force constant at the transition state is weaker as it is developing sp 2 character and a "normal" SKIE is observed with typical values of 1.1 to 1.2. [ 19 ] Conversely, when carbon's hybridization changes from sp 2 to sp 3 , the out of plane bending force constants at the transition state increase and an inverse SKIE is observed with typical values of 0.8 to 0.9. [ 19 ]
More generally the SKIE for reversible reactions can be "normal" one way and "inverse" the other if bonding in the transition state is midway in stiffness between substrate and product, or they can be "normal" both ways if bonding is weaker in the transition state, or "inverse" both ways if bonding is stronger in the transition state than in either reactant. [ 48 ]
An example of an "inverse" α SKIE can be seen in the work of Fitzpatrick and Kurtz who used such an effect to distinguish between two proposed pathways for the reaction of d-amino acid oxidase with nitroalkane anions. [ 51 ] Path A involved a nucleophilic attack on the coenzyme flavin adenine dinucleotide (FAD), while path B involves a free-radical intermediate. As path A results in the intermediate carbon changing hybridization from sp 2 to sp 3 an "inverse" SKIE is expected. If path B occurs then no SKIE should be observed as the free radical intermediate does not change hybridization. An SKIE of 0.84 was observed and Path A verified as shown in the scheme below.
Another example of SKIE is oxidation of benzyl alcohols by dimethyldioxirane, where three transition states for different mechanisms were proposed. Again, by considering how and if the hydrogen atoms were involved in each, researchers predicted whether or not they would expect an effect of isotopic substitution of them. Then, analysis of the experimental data for the reaction allowed them to choose which pathway was most likely based on the observed isotope effect. [ 52 ]
Secondary hydrogen isotope effects from the methylene hydrogens were also used to show that Cope rearrangement in 1,5-hexadiene follow a concerted bond rearrangement pathway, and not one of the alternatively proposed allyl radical or 1,4-diyl pathways, all of which are presented in the following scheme. [ 53 ]
Alternative mechanisms for the Cope rearrangement of 1,5-hexadiene: (from top to bottom), allyl radical, synchronous concerted, and 1,4-dyil pathways. The predominant pathway is found to be the middle one, which has six delocalized π electrons corresponding to an aromatic intermediate. [ 53 ]
The steric isotope effect (SIE) is a SKIE that does not involve bond breaking or formation. This effect is attributed to the different vibrational amplitudes of isotopologues . [ 54 ] An example of such an effect is the racemization of 9,10-dihydro-4,5-dimethylphenanthrene. [ 55 ] The smaller amplitude of vibration for 2 H than for 1 H in C– 1 H, C– 2 H bonds, results in a smaller van der Waals radius or effective size in addition to a difference in the ZPE between the two. When there is a greater effective bulk of molecules containing one over the other this may be manifested by a steric effect on the rate constant. For the example above, 2 H racemizes faster than 1 H resulting in a SIE. A model for the SIE was developed by Bartell. [ 56 ] A SIE is usually small, unless the transformations passes through a transition state with severe steric encumbrance, as in the racemization process shown above.
Another example of the SIE is in the deslipping reaction of rotaxanes. 2 H, due to its smaller effective size, allows easier passage of the stoppers through the macrocycle, resulting in faster deslipping for the deuterated rotaxanes . [ 57 ]
Reactions are known where the deuterated species reacts faster than the undeuterated one, and these cases are said to exhibit inverse KIEs (IKIE). IKIEs are often observed in the reductive elimination of alkyl metal hydrides, e.g. ( (Me 2 NCH 2 ) 2 )PtMe(H). [ b ] In such cases the C-D bond in the transition state, an agostic species, is highly stabilized relative to the C–H bond. [ 58 ]
An inverse effect can also occur in a multistep reaction if the overall rate constant depends on a pre-equilibrium prior to the rate-determining step which has an inverse equilibrium isotope effect . For example, the rates of acid-catalyzed reactions are usually 2-3 times greater for reactions in D 2 O catalyzed by D 3 O + than for the analogous reactions in H 2 O catalyzed by H 3 O + [ 4 ] : 433 This can be explained for a mechanism of specific hydrogen-ion catalysis of a reactant R by H 3 O + (or D 3 O + ).
The rate of formation of products is then d[P]/dt = k 2 [RH + ] = k 2 K 1 [H 3 O + ][R] = k obs [H 3 O + ][R]. In the first step, H 3 O + is usually a stronger acid than RH + . Deuteration shifts the equilibrium toward the more strongly bound acid species RD + in which the effect of deuteration on zero-point vibrational energy is greater, so that the deuterated equilibrium constant K 1D is greater than K 1H . This equilibrium isotope effect in the first step usually outweighs the kinetic isotope effect in the second step, so that there is an apparent inverse isotope effect and the observed overall rate constant k obs = k 2 K 1 decreases. [ 4 ] : 433
For the solvent isotope effects to be measurable, a fraction of the solvent must have a different isotopic composition than the rest. Therefore, large amounts of the less common isotopic species must be available, limiting observable solvent isotope effects to isotopic substitutions involving hydrogen. Detectable KIEs occur only when solutes exchange hydrogen with the solvent or when there is a specific solute-solvent interaction near the reaction site. Both such phenomena are common for protic solvents, in which the hydrogen is exchangeable, and they may form dipole-dipole interactions or hydrogen bonds with polar molecules. [ 8 ]
Most organic reactions involve breaking and making bonds to carbon; thus, it is reasonable to expect detectable carbon isotope effects. When 13 C is used as the label, the change in mass of the isotope is only ~8%, though, which limits the observable KIEs to much smaller values than the ones observable with hydrogen isotope effects.
Often, the largest source of error in a study that depends on the natural abundance of carbon is the slight variation in natural 13 C abundance itself. Such variations arise; because the starting materials in the reaction, are themselves products of other reactions that have KIEs and thus isotopically enrich the products. To compensate for this error when NMR spectroscopy is used to determine the KIE, the following guidelines have been proposed: [ 38 ]
If these as well as some other precautions listed by Jankowski are followed, KIEs with precisions of three decimal places can be achieved. [ 38 ]
Interpretation of carbon isotope effects is usually complicated by simultaneously forming and breaking bonds to carbon. Even reactions that involve only bond cleavage from the carbon, such as S N 1 reactions, involve strengthening of the remaining bonds to carbon. In many such reactions, leaving group isotope effects tend to be easier to interpret. For example, substitution and elimination reactions in which chlorine acts as a leaving group are convenient to interpret, especially since chlorine acts as a monatomic species with no internal bonding to complicate the reaction coordinate, and it has two stable isotopes, 35 Cl and 37 Cl, both with high abundance. The major challenge to the interpretation of such isotope affects is the solvation of the leaving group. [ 8 ]
Owing to experimental uncertainties, measurement of isotope effect may entail significant uncertainty. Often isotope effects are determined through complementary studies on a series of isotopomers. Accordingly, it is quite useful to combine hydrogen isotope effects with heavy-atom isotope effects. For instance, determining nitrogen isotope effect along with hydrogen isotope effect was used to show that the reaction of 2-phenylethyltrimethylammonium ion with ethoxide in ethanol at 40°C follows an E2 mechanism, as opposed to alternative non-concerted mechanisms. This conclusion was reached upon showing that this reaction yields a nitrogen isotope effect, k 14 / k 15 , of 1.0133±0.0002 along with a hydrogen KIE of 3.2 at the leaving hydrogen. [ 8 ]
Similarly, combining nitrogen and hydrogen isotope effects was used to show that syn eliminations of simple ammonium salts also follow a concerted mechanism, which was a question of debate before. In the following two reactions of 2-phenylcyclopentyltrimethylammonium ion with ethoxide, both of which yield 1-phenylcyclopentene, both isomers exhibited a nitrogen isotope effect k 14 / k 15 at 60°C. Though the reaction of the trans isomer, which follows syn elimination, has a smaller nitrogen KIE (1.0064) than the cis isomer which undergoes anti elimination (1.0108); both results are large enough to be indicative of weakening of the C-N bond in the transition state that would occur in a concerted process. [ c ]
Since KIEs arise from differences in isotopic mass, the largest observable KIEs are associated with substitution of 1 H with 2 H (2× increase in mass) or 3 H (3× increase in mass). KIEs from isotopic mass ratios can be as large as 36.4 using muons. They have produced the lightest "hydrogen" atom, 0.11 H (0.113 amu), in which an electron orbits a positive muon (μ + ) "nucleus" that has a mass of 206 electrons. They have also prepared the heaviest "hydrogen" atom by replacing one electron in helium with a negative muon μ − to form Heμ (mass 4.116 amu). Since μ − is much heavier than an electron, it orbits much closer to the nucleus, effectively shielding one proton, making Heμ behave as 4.1 H. With these exotic atoms , the reaction of H with 1 H 2 was investigated. Rate constants from reacting the lightest and the heaviest hydrogen analogs with 1 H 2 were then used to calculate k 0.11 / k 4.1 , in which there is a 36.4× difference in isotopic mass. For this reaction, isotopic substitution happens to produce an IKIE, and the authors report a KIE as low as 1.74×10 −4 , the smallest KIE ever reported. [ 59 ]
The KIE leads to a specific distribution of 2 H in natural products, depending on the route they were synthesized in nature. By NMR spectroscopy, it is therefore easy to detect whether the alcohol in wine was fermented from glucose , or from illicitly added saccharose .
Another reaction mechanism that was elucidated using the KIE is halogenation of toluene : [ 60 ]
In this particular "intramolecular KIE" study, a benzylic hydrogen undergoes radical substitution by bromine using N -bromosuccinimide as the brominating agent. It was found that PhCH 3 brominates 4.86x faster than PhC 2 H 3 (PhCD 3 ). A large KIE of 5.56 is associated with the reaction of ketones with bromine and sodium hydroxide . [ 61 ]
In this reaction the rate-limiting step is formation of the enolate by deprotonation of the ketone. In this study the KIE is calculated from the reaction rate constants for regular 2,4-dimethyl-3-pentanone and its deuterated isomer by optical density measurements.
In asymmetric catalysis, there are rare cases where a KIE manifests as a significant difference in the enantioselectivity observed for a deuterated substrate compared to a non-deuterated one. One example was reported by Toste and coworkers, in which a deuterated substrate produced an enantioselectivity of 83% ee, compared to 93% ee for the undeuterated substrate. The effect was taken to corroborate additional inter- and intramolecular competition KIE data that suggested cleavage of the C-H/D bond in the enantiodetermining step. [ 62 ] | https://en.wikipedia.org/wiki/Kinetic_isotope_effect |
The kinetic isotope effect (KIE) of ribulose-1,5-bisphosphate carboxylase oxygenase ( RuBisCO ) is the isotopic fractionation associated solely with the step in the Calvin-Benson cycle where a molecule of carbon dioxide (CO 2 ) is attached to the 5-carbon sugar ribulose-1,5-bisphosphate (RuBP) to produce two 3-carbon sugars called 3-phosphoglycerate (3 PGA). This chemical reaction is catalyzed by the enzyme RuBisCO, and this enzyme-catalyzed reaction creates the primary kinetic isotope effect of photosynthesis . [ 1 ] It is also largely responsible for the isotopic compositions of photosynthetic organisms and the heterotrophs that eat them. [ 2 ] [ 3 ] Understanding the intrinsic KIE of RuBisCO is of interest to earth scientists , botanists , and ecologists because this isotopic biosignature can be used to reconstruct the evolution of photosynthesis and the rise of oxygen in the geologic record , reconstruct past evolutionary relationships and environmental conditions, and infer plant relationships and productivity in modern environments. [ 4 ] [ 5 ] [ 6 ]
The fixation of CO 2 by RuBisCO is a multi-step process. First, a CO 2 molecule (that is not the CO 2 molecule that is eventually fixed) attaches to the uncharged ε-amino group of lysine 201 in the active site to form a carbamate. [ 7 ] This carbamate then binds to the magnesium ion (Mg 2+ ) in RuBisCO's active site . A molecule of RuBP then binds to the Mg 2+ ion. The bound RuBP then loses a proton to form a reactive, enodiolate species. [ 7 ] The rate-limiting step of the Calvin-Benson cycle is the addition of CO 2 to this 2,3-enediol form of RuBP. [ 8 ] [ 9 ] [ 10 ] This is the stage where the intrinsic KIE of Rubisco occurs because a new C-C bond is formed. The newly formed 2-carboxy-3-keto-D-arabinitol 1,5-bisphosphate molecule is then hydrated and cleaved to form two molecules of 3-phosphoglycerate (3 PGA). 3 PGA is then converted into hexoses to be used in the photosynthetic organism's central metabolism. [ 7 ]
The isotopic substitutions that can occur in this reaction are for carbon, oxygen, and/or hydrogen, though currently only a significant isotope effect is seen for carbon isotope substitution. [ 11 ] Isotopes are atoms that have the same number of protons but varying numbers of neutrons. "Lighter" isotopes (like the stable carbon-12 isotope) have a smaller overall mass, and "heavier" isotopes (like the stable carbon-13 isotope or radioactive carbon-14 isotope) have a larger overall mass. Stable isotope geochemistry is concerned with how varying chemical and physical processes preferentially enrich or deplete stable isotopes. Enzymes like RuBisCO cause isotopic fractionation because molecules containing lighter isotopes have higher zero-point energies (ZPE), the lowest possible quantum energy state for a given molecular arrangement. [ 12 ] For this reaction, 13 CO 2 has a lower ZPE than 12 CO 2 and sits lower in the potential energy well of the reactants. When enzymes catalyze chemical reactions, the lighter isotope is preferentially selected because it has a lower activation energy and is thus more energetically favorable to overcome the high potential-energy transition state and proceed through the reaction. Here, 12 CO 2 has a lower activation energy so more 12 CO 2 than 13 CO 2 goes through the reaction, resulting in the product (3 PGA) being lighter.
The observed intrinsic KIEs of RuBisCO have been correlated with two aspects of its enzyme kinetics : 1) Its "specificity" for CO 2 over O 2 , and 2) Its rate of carboxylation.
The reactive enodiolate species is also sensitive to oxygen (O 2 ), which results in the dual carboxylase / oxygenase activity of RuBisCO. [ 13 ] This reaction is considered wasteful as it produces products (3-phosphoglycerate and 2-phosphoglycolate) that must be catabolized through photorespiration . [ 14 ] This process requires energy and is a missed-opportunity for CO 2 fixation, which results in the net loss of carbon fixation efficiency for the organism. [ 13 ] The dual carboxylase / oxygenase activity of RuBisCO is exacerbated by the fact that O 2 and CO 2 are small, relatively indistinguishable molecules that can bind only weakly, if at all, in Michaelis-Menten complexes. [ 15 ] [ 16 ] There are four forms of RuBisCO (Form I, II, III, and IV), with Form I being the most abundantly used form. Form I is used extensively by higher plants , eukaryotic algae , cyanobacteria , and Pseudomonadota (formerly proteobacteria). [ 13 ] Form II is also used but much less widespread, and can be found in some species of Pseudomonadota and in dinoflagellates . [ 13 ] RuBisCOs from different photosynthetic organisms display varying abilities to distinguish between CO 2 and O 2 . This property can be quantified and is termed "specificity" (S c/o ). A higher value of S c/o means that a RuBisCO's carboxylase activity is greater than its oxygenase activity.
The rate of carboxylation (V C ) is the rate that RuBisCO fixes CO 2 to RuBP under substrate saturated conditions. [ 14 ] A higher value of V C corresponds to a higher rate of carboxylation. This rate of carboxylation can also be represented through its Michaelis-Menten constant K C , with a higher value of K C corresponding to a higher rate of carboxylation. V C is represented by V max , and K C is represented as K M in the generalized Michaelis-Menten curve. Although the rate of carboxylation varies among RuBisCO types, RuBisCO on average fixes only three molecules of CO 2 per second. [ 17 ] This is remarkably slow compared to typical enzyme catalytic rates, which usually catalyze reactions at the rate of thousands of molecules per second. [ 17 ]
It has been observed among natural RuBisCOs that an increased ability to distinguish between CO 2 and O 2 (larger values of S c/o ) corresponds with a decreased rate of carboxylation (lower values of V C and K C ). [ 18 ] The variation and trade-off between S c/o and K C has been observed across all photosynthetic organisms, from photosynthetic bacteria and algae to higher plants. [ 18 ] Organisms using RuBisCOs with high values of V C / K C , and low values of S c/o have localized RuBisCO to areas within the cell with artificially high local CO 2 concentrations. In cyanobacteria, concentrations of CO 2 are increased using a carboxysome , an icosahedral protein compartment about 100 nm in diameter that selectively uptakes bicarbonate and converts it to CO 2 in the presence of RuBisCO. [ 19 ] Organisms without a CCM, like certain plants, instead utilize RuBisCOs with high values of S c/o and low values of V C and K C . [ 18 ] It has been theorized that groups with a CCM have been able to maximize K C at the expense of decreasing S c/o , because artificially enhancing the concentration of CO 2 would decrease the concentration of O 2 and remove the need for high CO 2 specificity. However, the opposite is true for organisms without a CCM, who must optimize S c/o at the expense of K C because O 2 is readily present in the atmosphere.
This trade-off between S c/o and V C or K C observed in extant organisms suggest that RuBisCO has evolved through geologic time to be maximally optimized in its current, modern environment. [ 11 ] [ 18 ] RuBisCO evolved over 2.5 billion years ago when atmospheric CO 2 concentrations were 300 to 600 times higher than present day concentrations, and oxygen concentrations were only 5-18% of present-day levels. [ 14 ] Therefore, because CO 2 was abundant and O 2 rare, there was no need for the ancestral RuBisCO enzyme to have high specificity. This is supported by the biochemical characterization of an ancestral RuBisCO enzyme, which has intermediate values of V C and S C/O between the extreme end-members. [ 14 ]
It has been theorized that this ecological trade-off is due to the form that 2-carboxy-3-keto-D-arabinitol 1,5-bisphophate in its transient transition state before cleaving into two 3PGA molecules. [ 11 ] The more closely the Mg 2+ -bound CO 2 moiety resembles the carboxylate group in 2-carboxy-3-keto-D-arabinitol 1,5-bisphophate, the greater the structural difference between the transition states of carboxylation and oxygenation. [ 11 ] The larger structural difference allows RuBisCO to better distinguish between CO 2 and O 2 , resulting in larger values of S c/o . [ 11 ] However, this increasing structural similarity between the transition state and the product state requires strong binding at the carboxyketone group, and this binding is so strong that the rate of cleavage into two product 3PGA molecules is slowed. [ 11 ] Therefore, an increased specificity for CO 2 over O 2 necessitates a lower overall rate of carboxylation. This theory implies that there is a physical chemistry limitation at the heart of Rubisco's active site, and may preclude any efforts to engineer a simultaneously more selective and faster Rubisco. [ 11 ] [ 18 ]
S c/o has been positively correlated with the magnitude of carbon isotope fractionation (represented by Δ 13 C), with larger values of S c/o corresponding with a larger values of Δ 13 C. [ 11 ] It has been theorized that because increasing S c/o means the transition state is more like the product, the O 2 C---C-2 bond will be shorter, resulting in a higher overall potential energy & vibrational energy. [ 11 ] This creates a higher energy transition state, which makes it even harder for 13 CO 2 (lower in the potential energy well than 12 CO 2 ) to overcome the required activation energy. [ 11 ] The RuBisCOs used by varying photosynthetic organisms vary slightly in their enzyme structure, and this enzyme structure results in varying transition states. This diversity in enzyme structure is reflected in the resulting Δ 13 C values measured from different photosynthetic organisms. However, overlap exists between the Δ 13 C values of different groups because the carbon isotope values measured are generally of the entire organism, and not just its RuBisCO enzyme. Many other factors, including growth rate and the isotopic composition of the starting substrate, can affect the carbon isotope values of whole organism and cause the spread seen in C isotope measurements. [ 20 ] [ 21 ] | https://en.wikipedia.org/wiki/Kinetic_isotope_effects_of_RuBisCO |
Kinetic proofreading (or kinetic amplification ) is a mechanism for error correction in biochemical reactions , proposed independently by John Hopfield (1974) and Jacques Ninio (1975). Kinetic proofreading allows enzymes to discriminate between two possible reaction pathways leading to correct or incorrect products with an accuracy higher than what one would predict based on the difference in the activation energy between these two pathways. [ 1 ] [ 2 ]
Increased specificity is obtained by introducing an irreversible step exiting the pathway, with reaction intermediates leading to incorrect products more likely to prematurely exit the pathway than reaction intermediates leading to the correct product. If the exit step is fast relative to the next step in the pathway, the specificity can be increased by a factor of up to the ratio between the two exit rate constants. (If the next step is fast relative to the exit step, specificity will not be increased because there will not be enough time for exit to occur.) This can be repeated more than once to increase specificity further.
As an analogy, if we have a medicine assembly line sometimes produces empty boxes, and we are unable to upgrade the assembly line, then we can increase the ratio of full boxes over empty boxes (specificity) by placing a giant fan at the end. Empty boxes are more likely to be blown off the line (a higher exit rate) than full boxes, even though both kinds' production rates are lowered. By lengthening the final section and adding more giant fans (multistep proofreading), the specificity can be increased arbitrarily, at the cost of decreasing production rate.
In protein synthesis , the error rate is on the order of 10 − 4 = e − 9.2 {\displaystyle 10^{-4}=e^{-9.2}} . This means that when a ribosome is matching anticodons of tRNA to the codons of mRNA , it matches complementary sequences correctly nearly all the time. Hopfield noted that because of how similar the substrates are (the difference between a wrong codon and a right codon can be as small as a difference in a single base), an error rate that small is unachievable with a one-step mechanism. Both wrong and right tRNA can bind to the ribosome, and if the ribosome can only discriminate between them by complementary matching of the anticodon, it must rely on the small free energy difference between binding three matched complementary bases or only two.
A one-shot machine which tests whether the codons match or not by examining whether the codon and anticodon are bound will not be able to tell the difference between wrong and right codon with an error rate less than 10 − 4 = e − 9.2 {\displaystyle 10^{-4}=e^{-9.2}} unless the free energy difference is at least 9.2 kT , which is much larger than the free energy difference for single codon binding. This is a thermodynamic bound, so it cannot be evaded by building a different machine. However, this can be overcome by kinetic proofreading, which introduces an irreversible step through the input of energy. [ 3 ]
Another molecular recognition mechanism, which does not require expenditure of free energy is that of conformational proofreading . The incorrect product may also be formed but hydrolyzed at a greater rate than the correct product, giving the possibility of theoretically infinite specificity the longer you let this reaction run, but at the cost of large amounts of the correct product as well. (Thus there is a tradeoff between product production and its efficiency.) The hydrolytic activity may be on the same enzyme, as in DNA polymerases with editing functions, or on different enzymes.
Hopfield suggested a simple way to achieve smaller error rates using a molecular ratchet which takes many irreversible steps, each testing to see if the sequences match. At each step, energy is expended and specificity (the ratio of correct substrate to incorrect substrate at that point in the pathway) increases.
The requirement for energy in each step of the ratchet is due to the need for the steps to be irreversible; for specificity to increase, entry of substrate and analogue must occur largely through the entry pathway, and exit largely through the exit pathway. If entry were an equilibrium, the earlier steps would form a pre-equilibrium and the specificity benefits of entry into the pathway (less likely for the substrate analogue) would be lost; if the exit step were an equilibrium, then the substrate analogue would be able to re-enter the pathway through the exit step, bypassing the specificity of earlier steps altogether.
Although one test will fail to discriminate between mismatched and matched sequences a fraction p = e − Δ F / k T {\displaystyle p=e^{-\Delta F/kT}} of the time, two tests will both fail only p 2 {\displaystyle p^{2}} of the time, and N tests will fail p N {\displaystyle p^{N}} of the time. In terms of free energy, the discrimination power of N successive tests for two states with a free energy Δ F {\displaystyle \Delta F} is the same as one test between two states with a free energy N Δ F {\displaystyle N\Delta F} .
To achieve an error rate of e − 10 {\displaystyle e^{-10}} requires several comparison steps. Hopfield predicted on the basis of this theory that there is a multistage ratchet in the ribosome which tests the match several times before incorporating the next amino acid into the protein.
Biochemical processes that use kinetic proofreading to improve specificity implement the delay-inducing multistep ratchet by a variety of distinct biochemical networks. Nonetheless, many such networks result in the times to completion of the molecular assembly and the proofreading steps (also known as the first passage time ) that approach a near-universal, exponential shape for high proofreading rates and large network sizes. [ 10 ] Since exponential completion times are characteristic of a two-state Markov process , this observation makes kinetic proofreading one of only a few examples of biochemical processes where structural complexity results in a much simpler large-scale, phenomenological dynamics.
The increase in specificity, or the overall amplification factor of a kinetic proofreading network that may include multiple pathways and especially loops is intimately related to the topology of the network: the specificity grows exponentially with the number of loops in the network. [ 11 ] [ 12 ] An example is homologous recombination in which the number of loops scales like the square of DNA length. [ 5 ] [ 6 ] The universal completion time emerges precisely in this regime of large number of loops and high amplification. [ 11 ] | https://en.wikipedia.org/wiki/Kinetic_proofreading |
In organic chemistry , kinetic resolution is a means of differentiating two enantiomers in a racemic mixture . In kinetic resolution, two enantiomers react with different reaction rates in a chemical reaction with a chiral catalyst or reagent, resulting in an enantioenriched sample of the less reactive enantiomer. [ 1 ] As opposed to chiral resolution , kinetic resolution does not rely on different physical properties of diastereomeric products, but rather on the different chemical properties of the racemic starting materials. The enantiomeric excess (ee) of the unreacted starting material continually rises as more product is formed, reaching 100% just before full completion of the reaction. Kinetic resolution relies upon differences in reactivity between enantiomers or enantiomeric complexes.
Kinetic resolution can be used for the preparation of chiral molecules in organic synthesis . Kinetic resolution reactions utilizing purely synthetic reagents and catalysts are much less common than the use of enzymatic kinetic resolution in application towards organic synthesis, although a number of useful synthetic techniques have been developed in the past 30 years. [ 2 ]
The first reported kinetic resolution was achieved by Louis Pasteur . After reacting aqueous racemic ammonium tartrate with a mold from Penicillium glaucum, he reisolated the remaining tartrate and found it was levorotatory . [ 3 ] The chiral microorganisms present in the mold catalyzed the metabolization of ( R , R )-tartrate selectively, leaving an excess of ( S , S )-tartrate.
Kinetic resolution by synthetic means was first reported by Marckwald and McKenzie in 1899 in the esterification of racemic mandelic acid with optically active (−)- menthol . With an excess of the racemic acid present, they observed the formation of the ester derived from (+)- mandelic acid to be quicker than the formation of the ester from (−)-mandelic acid. The unreacted acid was observed to have a slight excess of (−)-mandelic acid, and the ester was later shown to yield (+)-mandelic acid upon saponification. The importance of this observation was that, in theory, if a half equivalent of (−)-menthol had been used, a highly enantioenriched sample of (−)-mandelic acid could have been prepared. This observation led to the successful kinetic resolution of other chiral acids, the beginning of the use of kinetic resolution in organic chemistry. [ 4 ] [ 5 ]
Kinetic resolution is a possible method for irreversibly differentiating a pair of enantiomers due to (potentially) different activation energies. While both enantiomers are at the same Gibbs free energy level by definition, and the products of the reaction with both enantiomers are also at equal levels, the Δ G ‡ {\displaystyle \Delta G^{\ddagger }} , or transition state energy, can differ. In the image below, the R enantiomer has a lower Δ G ‡ {\displaystyle \Delta G^{\ddagger }} and would thus react faster than the S enantiomer.
The ideal kinetic resolution is that in which only one enantiomer reacts, i.e. k R >>k S . The selectivity (s) of a kinetic resolution is related to the rate constants of the reaction of the R and S enantiomers, k R and k S respectively, by s=k R /k S , for k R >k S . This selectivity can also be referred to as the relative rates of reaction . This can be written in terms of the free energy difference between the high- and low-energy transitions states, Δ Δ G ‡ {\displaystyle \Delta \Delta G^{\ddagger }} . [ 6 ]
The selectivity can also be expressed in terms of ee of the recovered starting material and conversion (c), if first-order kinetics (in substrate) are assumed.
If it is assumed that the S enantiomer of the starting material racemate will be recovered in excess, it is possible to express the concentrations (mole fractions) of the S and R enantiomers as
where ee is the ee of the starting material. Note that for c=0, which signifies the beginning of the reaction, S 0 = R 0 = 1 2 {\displaystyle S_{0}=R_{0}={\frac {1}{2}}} , where these signify the initial concentrations of the enantiomers. Then, for stoichiometric chiral resolving agent B*,
Note that, if the resolving agent is stoichiometric and achiral, with a chiral catalyst, the [B*] term does not appear. Regardless, with a similar expression for R, we can express s as
If we wish to express this in terms of the enantiomeric excess of the product, ee", we must make use of the fact that, for products R' and S' from R and S, respectively
From here, we see that
which gives us
which, when we plug into our expression for s derived above, yield
The conversion ( c ) and selectivity factor ( s ) can be expressed in terms of starting material and product enantiomeric excesses (ee and ee'', respectively) only:
Additionally, the expressions for c and ee can be parametrized to give explicit expressions for C and ee in terms of t. First, solving explicitly for [S] and [R] as functions of t yields
which, plugged into expressions for ee and c, gives
Without loss of generality, we can allow k S =1, which gives k R =s, simplifying the above expressions. Similarly, an expression for ee″ as a function of t can be derived
Thus, plots of ee and ee″ vs. c can be generated with t as the parameter and different values of s generating different curves, as shown below.
As can be seen, high enantiomeric excesses are much more readily attainable for the unreacted starting material. There is however a tradeoff between ee and conversion, with higher ee (of the recovered substrate) obtained at higher conversion, and therefore lower isolated yield. For example, with a selectivity factor of just 10, 99% ee is possible with approximately 70% conversion, resulting in a yield of about 30%. In contrast, in order to get good ee's and yield of the product, very high selectivity factors are necessary. For example, with a selectivity factor of 10, ee″ above approximately 80% is unattainable, and significantly lower ee″ values are obtained for more realistic conversions. A selectivity in excess of 50 is required for highly enantioenriched product, in reasonable yield.
This is a simplified version of the true kinetics of kinetic resolution. The assumption that the reaction is first order in substrate is limiting, and it is possible that the dependence on substrate may depend on conversion, resulting in a much more complicated picture. As a result, a common approach is to measure and report only yields and ee's, as the formula for k rel only applies to an idealized kinetic resolution. It is simple to consider an initial substrate-catalyst complex forming, which could negate the first-order kinetics. However, the general conclusions drawn are still helpful to understand the effect of selectivity and conversion on ee.
With the advent of asymmetric catalysis , it is necessary to consider the practicality of utilizing kinetic resolution for the preparation of enantiopure products. Even for a product which can be attained through an asymmetric catalytic or auxiliary-based route, the racemate may be significantly less expensive than the enantiopure material, resulting in heightened cost-effectiveness even with the inherent "loss" of 50% of the material. The following have been proposed as necessary conditions for a practical kinetic resolution: [ 6 ]
To date, a number of catalysts for kinetic resolution have been developed that satisfy most, if not all of the above criteria, making them highly practical for use in organic synthesis. The following sections will discuss a number of key examples.
Gregory Fu and colleagues have developed a methodology utilizing a chiral DMAP analogue to achieve excellent kinetic resolution of secondary alcohols. [ 7 ] Initial studies utilizing ether as a solvent, low catalyst loadings (2 mol %), acetic anhydride as the acylating agent, and triethylamine at room temperature gave selectivities ranging from 14-52, corresponding to ee's of the recovered alcohol product as high as 99.2%. [ 8 ] However, solvent screening proved that the use of tert-amyl alcohol increased both the reactivity and selectivity. [ 9 ]
With the benchmark substrate 1-phenylethanol, this corresponded to 99% ee of the unreacted alcohol at 55% conversion when run at 0 °C. This system proved to be adept at resolution of a number of arylalkylcarbinols, with selectivities as high as 95 and low catalyst loadings of 1%, as shown below utilizing the (-)-enantiomer of the catalyst. This resulted in highly enantioenriched alcohols at very low conversions, giving excellent yields as well. In addition, the high selectivities result in highly enantioenriched acylated products, with a 90% ee sample of acylated alcohol for o-tolylmethylcarbinol, with s=71.
In addition, Fu reported the first highly selective acylation of racemic diols (as well as desymmetrization of meso diols). With low catalyst loading of 1%, enantioenriched diol was recovered in 98% ee and 43% yield, with the diacetate in 39% yield and 99% ee. The remainder of the material was recovered as a mixture of monoacetate.
The planar-chiral DMAP catalyst was also shown to be effective at kinetically resolving propargylic alcohols. [ 10 ] In this case, though, selectivities were found to be highest without any base present. When run with 1 mol% of the catalyst at 0 °C, selectivities as high as 20 could be attained. The limitations of this method include the requirement of an unsaturated functionality, such as carbonyl or alkenes, at the remote alkynyl position. Alcohols resolved using the (+)-enantiomer of the DMAP catalyst are shown below.
Fu also showed his chiral DMAP catalyst's ability to resolve allylic alcohols. [ 11 ] Effective selectivity was dependent upon the presence of either a geminal or cis substituent to the alcohol-bearing group, with a notable exception of a trans-phenyl alcohol which exhibited the highest selectivity. Using 1-2.5 mol% of the (+)-enantiomer of the DMAP catalyst, the alcohols shown below were resolved in the presence of triethylamine.
While Fu's DMAP analogue catalyst worked exceptionally well to kinetically resolve racemic alcohols, it was not successful in use for the kinetic resolution of amines. A similar catalyst, PPY*, was developed that, in use with a novel acylating agent, allowed for the successful kinetic resolution acylation of amines. With 10 mol% (-)-PPY* in chloroform at –50 °C, good to very good selectivities were observed in the acylation of amines, shown below. [ 12 ] A similar protocol was developed for the kinetic resolution of indolines. [ 13 ]
The Sharpless epoxidation , developed by K. Barry Sharpless in 1980, [ 14 ] has been utilized for the kinetic resolution of a racemic mixture of allylic alcohols. [ 15 ] [ 16 ] While extremely effective at resolving a number of allylic alcohols, this method has a number of drawbacks. Reaction times can run as long as 6 days, and the catalyst is not recyclable. However, the Sharpless asymmetric epoxidation kinetic resolution remains one of the most effective synthetic kinetic resolutions to date. A number of different tartrates can be used for the catalyst; a representative scheme is shown below utilizing diisopropyl tartrate . This method has seen general use on a number of secondary allylic alcohols. [ 17 ]
Sharpless asymmetric dihydroxylation has also seen use as a method for kinetic resolution. [ 18 ] [ 19 ] This method is not widely used, however, since the same resolution can be accomplished in different manners that are more economical. Additionally, the Shi epoxidation has been shown to affect kinetic resolution of a limited selection of olefins. [ 20 ] This method is also not widely used, but is of mechanistic interest.
While enantioselective epoxidations have been successfully achieved utilizing Sharpless epoxidation, Shi epoxidation , and Jacobsen epoxidation , none of these methods allows for the efficient asymmetric synthesis of terminal epoxides, which are key chiral building blocks. Due to the inexpensiveness of most racemic terminal epoxides and their inability to generally be subjected to classical resolution, an effective kinetic resolution of terminal epoxides would serve as a highly important synthetic methodology. In 1996, Jacobsen and coworkers developed a methodology for the kinetic resolution of epoxides via nucleophilic ring-opening with attack by an azide anion. The (R,R) catalyst is shown. [ 21 ] The catalyst could effectively, with loadings as low as 0.5 mol%, open the epoxide at the terminal position enantioselectively, yielding enantioenriched epoxide starting material and 1,2-azido alcohols. Yields are nearly quantitative and ee's were excellent (≥95% in nearly all cases). The 1,2-azido alcohols can be hydrogenated to give 1,2-amino alcohols, as shown below.
In 1997, Jacobsen's group published a methodology which improved upon their earlier work, allowing for the use of water as the nucleophile in the epoxide opening. Utilizing a nearly identical catalyst, ee's in excess of 98% for both the recovered starting material epoxide and 1,2-diol product were observed. In the example below, hydrolytic kinetic resolution (HKR) was carried out on a 58 gram scale, resulting in 26 g (44%) of the enantioriched epoxide in >99% ee and 38 g (50%) of the diol in 98% ee. [ 22 ]
A multitude of other substrates were examined, with yields of the recovered epoxide ranging from 36-48% for >99% ee. Jacobsen hydrolytic kinetic resolution can be used in tandem with Jacobsen epoxidation to yield enantiopure epoxides from certain olefins, as shown below. The first epoxidation yields a slightly enantioenriched epoxide, and subsequent kinetic resolution yields essentially a single enantiomer. The advantage of this approach is the ability to reduce the amount of hydrolytic cleavage necessary to achieve high enantioselectivity, allowing for overall yields up to approximately 90%, based on the olefin. [ 23 ]
Ultimately, the Jacobsen epoxide opening kinetic resolutions produce high enantiomeric purity in the epoxide and product, in solvent-free or low-solvent conditions, and have been applicable on a large scale. The Jacobsen methodology for HKR in particular is extremely attractive since it can be carried out on a multiton scale and utilizes water as the nucleophile, resulting in extremely cost-effective industrial processes.
Despite impressive achievements, HKR has generally been applied to the resolution of simple terminal epoxides with one stereocentre. Quite recently, D. A. Devalankar et al. reported an elegant protocol involving a two-stereocentered Co-catalyzed HKR of racemic terminal epoxides bearing adjacent C–C binding substituents. [ 24 ]
Ryōji Noyori and colleagues have developed a methodology for the kinetic resolution of benzylic and allylic secondary alcohols via transfer hydrogenation. The ruthenium complex catalyzes oxidation of the more reactive enantiomer from acetone , yielding an unreacted enantiopure alcohol, an oxidized ketone, and isopropanol. In the example illustrated below, exposure of 1-phenylethanol to the (S,S) enantiomer of the catalyst in the presence of acetone results in a 51% yield of 94% ee (R)-1-phenylethanol, along with 49% acetophenone and isopropanol as a byproduct. [ 25 ]
This methodology is essentially the reverse of Noyori's asymmetric transfer hydrogenation of ketones, [ 26 ] which yield enantioenriched alcohols via reduction. This limits the attractiveness of the kinetic resolution method, since there is a similar method to achieve the same products without the loss of half the material. Thus, the kinetic resolution would only be carried out in an instance for which the racemic alcohol was at least one half the price of the ketone or significantly easier to access.
In addition, Uemura and Hidai have developed a ruthenium catalyst for the kinetic resolution oxidation of benzylic alcohols, yielding highly enantioenriched alcohols in good yields. [ 27 ] The complex can, like Noyori's catalyst, affect transfer hydrogenation between a ketone and isopropanol to give an enantioenriched alcohol as well as affect kinetic resolution of a racemic alcohol, giving enantiopure alcohol (>99% ee) and oxidized ketone, with acetone as the byproduct. It is highly effective at reducing ketones enantioselectively, giving most benzylic alcohols in >99% ee and can resolve a number of racemic benzylic alcohols to give high yields (up to 49%) of single enantiomers, as shown below. This method has the same disadvantages as the Noyori kinetic resolution, namely that the alcohols can also be accessed via reduction of the ketones enantioselectively. Additionally, only one enantiomer of the catalyst has been reported.
Noyori has also demonstrated the kinetic resolution of allylic alcohols by asymmetric hydrogenation of the olefin. [ 28 ] Utilizing the Ru[BINAP] complex, selective hydrogenation can give high ee's of the unsaturated alcohol in addition to the hydrogenated alcohol, as shown below. Thus, a second hydrogenation of the enantioenriched allylic alcohol remaining will give enantiomerically pure samples of both enantiomers of the saturated alcohol. Noyori has resolved a number of allylic alcohols with good to excellent yields and good to excellent ee's (up to >99%).
Hoveyda and Schrock have developed a catalyst for ring-closing metathesis kinetic resolution of dienyl allylic alcohols. [ 29 ] The molybdenum alkylidene catalyst selectively catalyzes one enantiomer to perform ring closing metathesis, resulting in an enantiopure alcohol, and an enantiopure closed ring, as shown below. The catalyst is most effective at resolving 1,6-dienes. However, slight structural changes in the substrate, such as increasing the inter-alkene distance to 1,7, can sometimes necessitate the use of a different catalyst, reducing the efficacy of this method.
As with synthetic kinetic resolution procedures, enzymatic acylation kinetic resolutions have seen the broadest application in a synthetic context. Especially important has been the use of enzymatic kinetic resolution to efficiently and cheaply prepare amino acids. On a commercial scale, Degussa's methodology employing acylases is capable of resolving numerous natural and unnatural amino acids. The racemic mixtures can be prepared via Strecker synthesis , and the use of either porcine kidney acylase (for straight chain substrates) or an enzyme from the mold Aspergillus oryzae (for branched side chain substrates) can effectively yield enantioenriched amino acids in high (85-90%) yields. The unreacted starting material can be racemized in situ, thus making this a dynamic kinetic resolution. [ 30 ]
In addition, lipases are used extensively for kinetic resolution in both academic and industrial settings. [ 31 ] [ 32 ] Lipases have been used to resolve primary alcohols, secondary alcohols, a limited number of tertiary alcohols, carboxylic acids, diols, and even chiral allenes. Lipase from Pseudomonas cepacia (PSL) is the most widely used in the resolution of primary alcohols and has been used with vinyl acetate as an acylating agent to kinetically resolve the primary alcohols shown below.
For the resolution of secondary alcohols, pseudomonas cepecia lipase (PSL-C) has been employed effectively to generate excellent ee's of the ( R )-enantiomer of the alcohol. [ 33 ] The use of isopropenyl acetate as the acylating agent results in acetone as the byproduct, which is effectively removed from the reaction using molecular sieves .
Baker's yeast (BY) has been utilized for the kinetic resolution of α-stereogenic carbonyl compounds. [ 34 ] [ 35 ] The enzyme selectively reduces one enantiomer, yielding a highly enantioenriched alcohol and ketone, as shown below.
Baker's yeast has also been used in the kinetic resolution of secondary benzylic alcohols by oxidation. [ 36 ] While excellent ee's of the recovered alcohol have been reported, they typically require >60% conversion, resulting in diminished yields. Baker's yeast has also been used in the kinetic resolution via reduction of β-ketoesters. [ 37 ] However, given the success of Noyori's resolution of the same substrates, detailed later in this article, this has not seen much use.
Dynamic kinetic resolution (DKR) occurs when the starting material racemate is able to epimerize easily, resulting in an essentially racemic starting material mix at all points during the reaction. Then, the enantiomer with the lower barrier to activation can form in, theoretically, up to 100% yield. This is in contrast to standard kinetic resolution, which necessarily has a maximum yield of 50%. For this reason, dynamic kinetic resolution has extremely practical applications to organic synthesis. The observed dynamics are based on the Curtin-Hammett principle . The barrier to reaction of either enantiomer is necessarily higher than the barrier to epimerization, resulting in a kinetic well containing the racemate. This is equivalent to writing, for k R >k S ,
A number of excellent reviews have been published, most recently in 2008, detailing the theory and practical applications of DKR. [ 38 ] [ 39 ] [ 40 ]
The Noyori asymmetric hydrogenation of ketones is an excellent example of dynamic kinetic resolution at work. The enantiomeric β-ketoesters can undergo epimerization , and the choice of chiral catalyst, typically of the form Ru[(R)-BINAP]X 2 , where X is a halogen , leads to one of the enantiomers reacting preferentially faster. The relative free energy for a representative reaction is shown below. [ 41 ] [ 42 ] As can be seen, the epimerization intermediate is lower in free energy than the transition states for hydrogenation, resulting in rapid racemization and high yields of a single enantiomer of the product.
The enantiomers interconvert through their common enol , which is the energetic minimum located between the enantiomers. The shown reaction yields a 93% ee sample of the anti product shown above. Solvent choice appears to have a major influence on the diastereoselectivity, as dichloromethane and methanol both show effectiveness for certain substrates. Noyori and others have also developed newer catalysts which have improved on both ee and diastereomeric ratio (dr).
Genêt and coworkers developed SYNPHOS , a BINAP analogue which forms ruthenium complexes, which perform highly selective asymmetric hydrogenations. [ 43 ] Enantiopure Ru[SYNPHOS]Br 2 was shown to selectively hydrogenate racemic α-amino-β-ketoesters to enantiopure aminoalcohols, as shown below utilizing (R)-SYNPHOS. [ 44 ] 1,2- syn amino alcohols were prepared from benzoyl protected amino compounds, whereas anti products were prepared from hydrochloride salts of the amine.
Recently, Gregory Fu and colleagues reported a modification of their earlier kinetic resolution work to produce an effective dynamic kinetic resolution. [ 45 ] Using the ruthenium racemization catalyst shown to the right, and his planar chiral DMAP catalyst, Fu has demonstrated the dynamic kinetic resolution of secondary alcohols yielding up to 99% and 93% ee, as shown below. Work is ongoing to further develop the applications of the widely used DMAP catalyst to dynamic kinetic resolution.
A number of enzymatic dynamic kinetic resolutions have been reported. [ 46 ] A prime example using PSL effectively resolves racemic acyloins in the presence of triethylamine and vinyl acetate as the acylating agent. [ 47 ] As shown below, the product was isolated in 75% yield and 97% ee. Without the presence of the base, regular kinetic resolution occurred, resulting in 45% yield of >99% ee acylated product and 53% of the starting material in 92% ee.
Another excellent, though not high-yielding, example is the kinetic resolution of (±)-8-amino-5,6,7,8-tetrahydroquinoline. When exposed to Candida antarctica lipase B (CALB) in toluene and ethyl acetate for 3–24 hours, normal kinetic resolution occurs, resulting in 45% yield of 97% ee of starting material and 45% yield of >97% ee acylated amine product. However, when the reaction is allowed to stir for 40–48 hours, racemic starting material and >60% of >95% ee acylated product are recovered. [ 48 ]
Here, the unreacted starting material racemizes in situ via a dimeric enamine, resulting in a recovery of greater than 50% yield of the enantiopure acylated amine product.
There have been a number of reported procedures which take advantage of a chemical reagent/catalyst to perform racemization of the starting material and an enzyme to selectively react with one enantiomer, called chemoenzymatic dynamic kinetic resolutions. [ 49 ] PSL-C was utilized along with a ruthenium catalyst (for racemization) to produce enantiopure (>95% ee) δ-hydroxylactones. [ 50 ]
More recently, secondary alcohols have been resolved by Bäckvall with yields up to 99% and ee's up to >99% utilizing CALB and a ruthenium racemization complex. [ 51 ]
A second type of chemoenzymatic dynamic kinetic resolution involves a π-allyl complex from an allylic acetate with palladium . Here, racemization occurs with loss of the acetate, forming a cationic complex with the transition metal center, as shown below. [ 52 ] Palladium has been shown to facilitate this reaction, while ruthenium has been shown to affect a similar reaction, also shown below. [ 53 ]
In parallel kinetic resolution (PKR), a racemic mixture reacts to form two non-enantiomeric products, often through completely different reaction pathways. With PKR, there is no tradeoff between conversion and ee, as the formed products are not enantiomers. [ 54 ] [ 55 ] One strategy for PKR is to remove the less reactive enantiomer (towards the desired chiral catalyst) from the reaction mixture by subjecting it to a second set of reaction conditions that preferentially react with it, ideally with an approximately equal reaction rate. Thus, both enantiomers are consumed in different pathways at equal rates. PKR experiments can be stereodivergent, regiodivergent, or structurally divergent. [ 56 ] One of the most highly efficient PKR's reported to date was accomplished by Yoshito Kishi in 1998; CBS reduction of a racemic steroidal ketone resulted in stereoselective reduction, producing two diastereomers of >99% ee, as shown below. [ 57 ]
PKR have also been accomplished with the use of enzyme catalysts. Using the fungus Mortierella isabellina NRRL 1757, reduction of racemic β-ketonitriles affords two diastereomers, which can be separated and re-oxidized to give highly enantiopure β-ketonitriles. [ 58 ] Highly synthetically useful parallel kinetic resolutions have truly yet to be discovered, however. A number of procedures have been discovered that give acceptable ee's and yields, but there are very few examples which give highly selective parallel kinetic resolution and not simply somewhat selective reactions. For example, Fu's parallel kinetic resolution of 4-alkynals yields very enantioenriched cyclobutanone in low yield and slightly enantioenriched cyclopentenone, as shown below. [ 59 ]
In theory, parallel kinetic resolution can give the highest ee's of products, since only one enantiomer gives each desired product. For example, for two complementary reactions both with s=49, 100% conversion would give products in 50% yield and 96% ee. These same values would require s=200 for a simple kinetic resolution. As such, the promise of PKR continues to attract much attention. The Kishi CBS reduction remains one of the few examples to fulfill this promise. | https://en.wikipedia.org/wiki/Kinetic_resolution |
In physics , chemistry and related fields, a kinetic scheme is a network of states and connections between them representing a dynamical process. Usually a kinetic scheme represents a Markovian process , while for non-Markovian processes generalized kinetic schemes are used. Figure 1 illustrates a kinetic scheme.
A kinetic scheme is a network (a directed graph ) of distinct states (although repetition of states may occur and this depends on the system), where each pair of states i and j are associated with directional rates, A i j {\displaystyle A_{ij}} (and A j i {\displaystyle A_{ji}} ). It is described with a master equation : a first-order differential equation for the probability P → {\displaystyle {\vec {P}}} of a system to occupy each one its states at time t (element i represents state i ). Written in a matrix form, this states: d P → d t = A P → {\displaystyle {\frac {d{\vec {P}}}{dt}}=\mathbf {A} {\vec {P}}} , where A {\displaystyle \mathbf {A} } is the matrix of connections (rates) A i j {\displaystyle A_{ij}} .
In a Markovian kinetic scheme the connections are constant with respect to time (and any jumping time probability density function for state i is an exponential, with a rate equal the value of all the exiting connections).
When detailed balance exists in a system, the relation A j i P i ( t → ∞ ) = A i j P j ( t → ∞ ) {\displaystyle A_{ji}P_{i}(t\rightarrow \infty )=A_{ij}P_{j}(t\rightarrow \infty )} holds for every connected states i and j . The result represents the fact that any closed loop in a Markovian network in equilibrium does not have a net flow.
Matrix A {\displaystyle \mathbf {A} } can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, where then, the process is not in equilibrium. These terms are different than a birth–death process , where there is simply a linear kinetic scheme .
An example for such a process is a reduced dimensions form . | https://en.wikipedia.org/wiki/Kinetic_scheme |
The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases . Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion. These particles are now known to be the atoms or molecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain the relationship between the macroscopic properties of gases, such as volume , pressure , and temperature , as well as transport properties such as viscosity , thermal conductivity and mass diffusivity .
The basic version of the model describes an ideal gas . It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.
Due to the time reversibility of microscopic dynamics ( microscopic reversibility ), the kinetic theory is also connected to the principle of detailed balance , in terms of the fluctuation-dissipation theorem (for Brownian motion ) and the Onsager reciprocal relations .
The theory was historically significant as the first explicit exercise of the ideas of statistical mechanics .
In about 50 BCE , the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. [ 1 ] This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant. [ citation needed ]
One of the first and boldest statements on the relationship between motion of particles and heat was by the English philosopher Francis Bacon in 1620. "It must not be thought that heat generates motion, or motion heat (though in some respects this be true), but that the very essence of heat ... is motion and nothing else." [ 2 ] "not a ... motion of the whole, but of the small particles of the body." [ 3 ] In 1623, in The Assayer , Galileo Galilei , in turn, argued that heat, pressure, smell and other phenomena perceived by our senses are apparent properties only, caused by the movement of particles, which is a real phenomenon. [ 4 ] [ 5 ]
In 1665, in Micrographia , the English polymath Robert Hooke repeated Bacon's assertion, [ 6 ] [ 7 ] and in 1675, his colleague, Anglo-Irish scientist Robert Boyle noted that a hammer's "impulse" is transformed into the motion of a nail's constituent particles, and that this type of motion is what heat consists of. [ 8 ] Boyle also believed that all macroscopic properties, including color, taste and elasticity, are caused by and ultimately consist of nothing but the arrangement and motion of indivisible particles of matter. [ 9 ] In a lecture of 1681, Hooke asserted a direct relationship between the temperature of an object and the speed of its internal particles. "Heat ... is nothing but the internal Motion of the Particles of [a] Body; and the hotter a Body is, the more violently are the Particles moved." [ 10 ] In a manuscript published 1720, the English philosopher John Locke made a very similar statement: "What in our sensation is heat , in the object is nothing but motion ." [ 11 ] [ 12 ] Locke too talked about the motion of the internal particles of the object, which he referred to as its "insensible parts".
In his 1744 paper Meditations on the Cause of Heat and Cold , Russian polymath Mikhail Lomonosov made a relatable appeal to everyday experience to gain acceptance of the microscopic and kinetic nature of matter and heat: [ 13 ]
Movement should not be denied based on the fact it is not seen. Who would deny that the leaves of trees move when rustled by a wind, despite it being unobservable from large distances? Just as in this case motion remains hidden due to perspective, it remains hidden in warm bodies due to the extremely small sizes of the moving particles. In both cases, the viewing angle is so small that neither the object nor their movement can be seen.
Lomonosov also insisted that movement of particles is necessary for the processes of dissolution , extraction and diffusion , providing as examples the dissolution and diffusion of salts by the action of water particles on the of the “molecules of salt”, the dissolution of metals in mercury, and the extraction of plant pigments by alcohol. [ 14 ]
Also the transfer of heat was explained by the motion of particles. Around 1760, Scottish physicist and chemist Joseph Black wrote: "Many have supposed that heat is a tremulous ... motion of the particles of matter, which ... motion they imagined to be communicated from one body to another." [ 15 ]
In 1738 Daniel Bernoulli published Hydrodynamica , which laid the basis for the kinetic theory of gases . In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average kinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic. [ 16 ] : 36–37
Pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), [ 17 ] Georges-Louis Le Sage (ca. 1780, published 1818), [ 18 ] John Herapath (1816) [ 19 ] and John James Waterston (1843), [ 20 ] which connected their research with the development of mechanical explanations of gravitation .
In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles. [ 21 ] In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. [ 22 ] In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. [ 23 ] This was the first-ever statistical law in physics. [ 24 ] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. [ 25 ] In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases." [ 26 ] In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution . The logarithmic connection between entropy and probability was also first stated by Boltzmann.
At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein 's (1905) [ 27 ] and Marian Smoluchowski 's (1906) [ 28 ] papers on Brownian motion , which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
Following the development of the Boltzmann equation , a framework for its use in developing transport equations was developed independently by David Enskog and Sydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as Chapman–Enskog theory . The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
The application of kinetic theory to ideal gases makes the following assumptions:
Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.
As a simplifying assumption, the particles are usually assumed to have the same mass as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with Dalton's law of partial pressures . Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below). [ 29 ]
More modern developments, such as the revised Enskog theory and the extended Bhatnagar–Gross–Krook model, [ 30 ] relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with internal degrees of freedom , because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation. [ 31 ] While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results. [ 32 ]
In the kinetic theory of gases, the pressure is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.
Consider a gas particle traveling at velocity, v i {\textstyle v_{i}} , along the i ^ {\displaystyle {\hat {i}}} -direction in an enclosed volume with characteristic length , L i {\displaystyle L_{i}} , cross-sectional area, A i {\displaystyle A_{i}} , and volume, V = A i L i {\displaystyle V=A_{i}L_{i}} . The gas particle encounters a boundary after characteristic time t = L i / v i . {\displaystyle t=L_{i}/v_{i}.}
The momentum of the gas particle can then be described as p i = m v i = m L i / t . {\displaystyle p_{i}=mv_{i}=mL_{i}/t.}
We combine the above with Newton's second law , which states that the force experienced by a particle is related to the time rate of change of its momentum, such that F i = d p i d t = m L i t 2 = m v i 2 L i . {\displaystyle F_{i}={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}={\frac {mL_{i}}{t^{2}}}={\frac {mv_{i}^{2}}{L_{i}}}.}
Now consider a large number, N {\displaystyle N} , of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed, v {\textstyle v} , in every direction is identical v x 2 = v y 2 = v z 2 . {\displaystyle v_{x}^{2}=v_{y}^{2}=v_{z}^{2}.}
Further, assume that the volume is symmetrical about its three dimensions, i ^ , j ^ , k ^ {\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}} , such that V = V i = V j = V k , F = F i = F j = F k , A i = A j = A k . {\displaystyle {\begin{aligned}V={}&V_{i}=V_{j}=V_{k},\\F={}&F_{i}=F_{j}=F_{k},\\&A_{i}=A_{j}=A_{k}.\end{aligned}}} The total surface area on which the gas particles act is therefore A = 3 A i . {\displaystyle A=3A_{i}.}
The pressure exerted by the collisions of the N {\displaystyle N} gas particles with the surface can then be found by adding the force contribution of every particle and dividing by the interior surface area of the volume, P = N F ¯ A = N L F V {\displaystyle P={\frac {N{\overline {F}}}{A}}={\frac {NLF}{V}}} ⇒ P V = N L F = N 3 m v 2 . {\displaystyle \Rightarrow PV=NLF={\frac {N}{3}}mv^{2}.}
The total translational kinetic energy K t {\displaystyle K_{\text{t}}} of the gas is defined as K t = N 2 m v 2 , {\displaystyle K_{\text{t}}={\frac {N}{2}}mv^{2},} providing the result P V = 2 3 K t . {\displaystyle PV={\frac {2}{3}}K_{\text{t}}.}
This is an important, non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the translational kinetic energy of the molecules, which is a microscopic property.
The mass density of a gas ρ {\displaystyle \rho } is expressed through the total mass of gas particles and through volume of this gas: ρ = N m V {\displaystyle \rho ={\frac {Nm}{V}}} . Taking this into account, the pressure is equal to P = ρ v 2 3 . {\displaystyle P={\frac {\rho v^{2}}{3}}.}
Relativistic expression for this formula is [ 33 ]
P = 2 ρ c 2 3 ( ( 1 − v 2 ¯ / c 2 ) − 1 / 2 − 1 ) , {\displaystyle P={\frac {2\rho c^{2}}{3}}\left({\left(1-{\overline {v^{2}}}/c^{2}\right)}^{-1/2}-1\right),} where c {\displaystyle c} is speed of light . In the limit of small speeds, the expression becomes P ≈ ρ v 2 ¯ / 3 {\displaystyle P\approx \rho {\overline {v^{2}}}/3} .
Rewriting the above result for the pressure as P V = 1 3 N m v 2 {\textstyle PV={\frac {1}{3}}Nmv^{2}} , we may combine it with the ideal gas law
where k B {\displaystyle k_{\mathrm {B} }} is the Boltzmann constant and T {\displaystyle T} is the absolute temperature defined by the ideal gas law, to obtain k B T = 1 3 m v 2 , {\displaystyle k_{\mathrm {B} }T={\frac {1}{3}}mv^{2},} which leads to a simplified expression of the average translational kinetic energy per molecule, [ 34 ] 1 2 m v 2 = 3 2 k B T . {\displaystyle {\frac {1}{2}}mv^{2}={\frac {3}{2}}k_{\mathrm {B} }T.} The translational kinetic energy of the system is N {\displaystyle N} times that of a molecule, namely K t = 1 2 N m v 2 {\textstyle K_{\text{t}}={\frac {1}{2}}Nmv^{2}} . The temperature, T {\displaystyle T} is related to the translational kinetic energy by the description above, resulting in
which becomes
Equation ( 3 ) is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature .
From equations ( 1 ) and ( 3 ), we have
Thus, the product of pressure and volume per mole is proportional to the average
translational molecular kinetic energy.
Equations ( 1 ) and ( 4 ) are called the "classical results", which could also be derived from statistical mechanics ;
for more details, see: [ 35 ]
The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom , D . A monatomic gas is axially symmetric about each spatial axis, so that D = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that D = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water , is not radially symmetric about any axis, resulting in D = 6, comprising 3 translational and 3 rotational degrees of freedom.
Because the equipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is K = D K t = D 2 N m v 2 . {\displaystyle K=DK_{\text{t}}={\frac {D}{2}}Nmv^{2}.}
Thus, the energy added to the system per gas particle kinetic degree of freedom is K N D = 1 2 k B T . {\displaystyle {\frac {K}{ND}}={\frac {1}{2}}k_{\text{B}}T.}
Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas ( D = 3) is K = D 2 k B N A = 3 2 R , {\displaystyle K={\frac {D}{2}}k_{\text{B}}N_{\text{A}}={\frac {3}{2}}R,} where N A {\displaystyle N_{\text{A}}} is the Avogadro constant , and R is the ideal gas constant .
Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas can be calculated easily:
At standard temperature (273.15 K), the kinetic energy can also be obtained:
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on D and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions. [ 36 ]
For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated [ 37 ] based on naive kinetic theory, and the results can be used for analyzing effusive flow rates , which is useful in applications such as the gaseous diffusion method for isotope separation .
Assume that in the container, the number density (number per unit volume) is n = N / V {\displaystyle n=N/V} and that the particles obey Maxwell's velocity distribution : f Maxwell ( v x , v y , v z ) d v x d v y d v z = ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T d v x d v y d v z {\displaystyle f_{\text{Maxwell}}(v_{x},v_{y},v_{z})\,dv_{x}\,dv_{y}\,dv_{z}=\left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\,dv_{x}\,dv_{y}\,dv_{z}}
Then for a small area d A {\displaystyle dA} on the container wall, a particle with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal of the area d A {\displaystyle dA} , will collide with the area within time interval d t {\displaystyle dt} , if it is within the distance v d t {\displaystyle v\,dt} from the area d A {\displaystyle dA} . Therefore, all the particles with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal that can reach area d A {\displaystyle dA} within time interval d t {\displaystyle dt} are contained in the tilted pipe with a height of v cos ( θ ) d t {\displaystyle v\cos(\theta )dt} and a volume of v cos ( θ ) d A d t {\displaystyle v\cos(\theta )\,dA\,dt} .
The total number of particles that reach area d A {\displaystyle dA} within time interval d t {\displaystyle dt} also depends on the velocity distribution; All in all, it calculates to be: n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) . {\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).}
Integrating this over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time: J collision = ∫ 0 π / 2 cos ( θ ) sin ( θ ) d θ ∫ 0 π sin ( θ ) d θ × n v ¯ = 1 4 n v ¯ = n 4 8 k B T π m . {\displaystyle J_{\text{collision}}={\frac {\displaystyle \int _{0}^{\pi /2}\cos(\theta )\sin(\theta )\,d\theta }{\displaystyle \int _{0}^{\pi }\sin(\theta )\,d\theta }}\times n{\bar {v}}={\frac {1}{4}}n{\bar {v}}={\frac {n}{4}}{\sqrt {\frac {8k_{\mathrm {B} }T}{\pi m}}}.}
This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed v ¯ {\displaystyle {\bar {v}}} of the Maxwell's velocity distribution, one has to integrate over v > 0 {\displaystyle v>0} , 0 < θ < π {\displaystyle 0<\theta <\pi } , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } .
The momentum transfer to the container wall from particles hitting the area d A {\displaystyle dA} with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal, in time interval d t {\displaystyle dt} is: [ 2 m v cos ( θ ) ] × n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) . {\displaystyle [2mv\cos(\theta )]\times nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}\left(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi \right).} Integrating this over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the pressure (consistent with Ideal gas law ): P = 2 ∫ 0 π / 2 cos 2 ( θ ) sin ( θ ) d θ ∫ 0 π sin ( θ ) d θ × n m v rms 2 = 1 3 n m v rms 2 = 2 3 n ⟨ E kin ⟩ = n k B T {\displaystyle P={\frac {\displaystyle 2\int _{0}^{\pi /2}\cos ^{2}(\theta )\sin(\theta )\,d\theta }{\displaystyle \int _{0}^{\pi }\sin(\theta )\,d\theta }}\times nmv_{\text{rms}}^{2}={\frac {1}{3}}nmv_{\text{rms}}^{2}={\frac {2}{3}}n\langle E_{\text{kin}}\rangle =nk_{\mathrm {B} }T} If this small area A {\displaystyle A} is punched to become a small hole, the effusive flow rate will be: Φ effusion = J collision A = n A k B T 2 π m . {\displaystyle \Phi _{\text{effusion}}=J_{\text{collision}}A=nA{\sqrt {\frac {k_{\mathrm {B} }T}{2\pi m}}}.}
Combined with the ideal gas law , this yields Φ effusion = P A 2 π m k B T . {\displaystyle \Phi _{\text{effusion}}={\frac {PA}{\sqrt {2\pi mk_{\mathrm {B} }T}}}.}
The above expression is consistent with Graham's law .
To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with ( v , θ , ϕ ) {\displaystyle (v,\theta ,\phi )} that hit the area d A {\displaystyle dA} within the time interval d t {\displaystyle dt} are contained in the tilted pipe with a height of v cos ( θ ) d t {\displaystyle v\cos(\theta )\,dt} and a volume of v cos ( θ ) d A d t {\displaystyle v\cos(\theta )\,dA\,dt} ; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of v cos θ {\displaystyle v\cos \theta } : f ( v , θ , ϕ ) d v d θ d ϕ = λ v cos θ ( m 2 π k T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin θ d v d θ d ϕ ) {\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &=\lambda v\cos {\theta }\left({\frac {m}{2\pi kT}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )\end{aligned}}} with the constraint v > 0 {\textstyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } . The constant λ {\displaystyle \lambda } can be determined by the normalization condition ∫ f ( v , θ , ϕ ) d v d θ d ϕ = 1 {\textstyle \int f(v,\theta ,\phi )\,dv\,d\theta \,d\phi =1} to be 4 / v ¯ {\textstyle 4/{\bar {v}}} , and overall: f ( v , θ , ϕ ) d v d θ d ϕ = 1 2 π ( m k B T ) 2 e − m v 2 2 k B T ( v 3 sin θ cos θ d v d θ d ϕ ) ; v > 0 , 0 < θ < π 2 , 0 < ϕ < 2 π {\displaystyle {\begin{aligned}f(v,\theta ,\phi )\,dv\,d\theta \,d\phi &={\frac {1}{2\pi }}\left({\frac {m}{k_{\mathrm {B} }T}}\right)^{2}e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{3}\sin {\theta }\cos {\theta }\,dv\,d\theta \,d\phi )\\\end{aligned}};\quad v>0,\,0<\theta <{\frac {\pi }{2}},\,0<\phi <2\pi }
From the kinetic energy formula it can be shown that v p = 2 ⋅ k B T m , {\displaystyle v_{\text{p}}={\sqrt {2\cdot {\frac {k_{\mathrm {B} }T}{m}}}},} v ¯ = 2 π v p = 8 π ⋅ k B T m , {\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},} v rms = 3 2 v p = 3 ⋅ k B T m , {\displaystyle v_{\text{rms}}={\sqrt {\frac {3}{2}}}v_{p}={\sqrt {{3}\cdot {\frac {k_{\mathrm {B} }T}{m}}}},} where v is in m/s, T is in kelvin, and m is the mass of one molecule of gas in kg. The most probable (or mode) speed v p {\displaystyle v_{\text{p}}} is 81.6% of the root-mean-square speed v rms {\displaystyle v_{\text{rms}}} , and the mean (arithmetic mean, or average) speed v ¯ {\displaystyle {\bar {v}}} is 92.1% of the rms speed ( isotropic distribution of speeds ).
See:
In kinetic theory of gases, the mean free path is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let σ {\displaystyle \sigma } be the collision cross section of one molecule colliding with another. As in the previous section, the number density n {\displaystyle n} is defined as the number of molecules per (extensive) volume, or n = N / V {\displaystyle n=N/V} . The collision cross section per volume or collision cross section density is n σ {\displaystyle n\sigma } , and it is related to the mean free path ℓ {\displaystyle \ell } by ℓ = 1 n σ 2 {\displaystyle \ell ={\frac {1}{n\sigma {\sqrt {2}}}}}
Notice that the unit of the collision cross section per volume n σ {\displaystyle n\sigma } is reciprocal of length.
The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as viscosity , thermal conductivity , mass diffusivity and thermal diffusion .
In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures, Revised Enskog Theory , was developed in 1983-1987 by E. G. D. Cohen , J. M. Kincaid and M. Lòpez de Haro , [ 38 ] [ 39 ] [ 40 ] [ 41 ] building on work by H. van Beijeren and M. H. Ernst . [ 42 ]
In books on elementary kinetic theory [ 43 ] one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force F . The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component u {\displaystyle u} which increase uniformly with distance y {\displaystyle y} above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Inside a dilute gas in a Couette flow setup, let u 0 {\displaystyle u_{0}} be the forward velocity of the gas at a horizontal flat layer (labeled as y = 0 {\displaystyle y=0} ); u 0 {\displaystyle u_{0}} is along the horizontal direction. The number of molecules arriving at the area d A {\displaystyle dA} on one side of the gas layer, with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal, in time interval d t {\displaystyle dt} is n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin θ d v d θ d ϕ ) {\displaystyle nv\cos({\theta })\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2k_{\mathrm {B} }T}}}(v^{2}\sin {\theta }\,dv\,d\theta \,d\phi )}
These molecules made their last collision at y = ± ℓ cos θ {\displaystyle y=\pm \ell \cos \theta } , where ℓ {\displaystyle \ell } is the mean free path . Each molecule will contribute a forward momentum of p x ± = m ( u 0 ± ℓ cos θ d u d y ) , {\displaystyle p_{x}^{\pm }=m\left(u_{0}\pm \ell \cos \theta {\frac {du}{dy}}\right),} where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient d u / d y {\displaystyle du/dy} can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the forward momentum transfer per unit time per unit area (also known as shear stress ): τ ± = 1 4 v ¯ n ⋅ m ( u 0 ± 2 3 ℓ d u d y ) {\displaystyle \tau ^{\pm }={\frac {1}{4}}{\bar {v}}n\cdot m\left(u_{0}\pm {\frac {2}{3}}\ell {\frac {du}{dy}}\right)}
The net rate of momentum per unit area that is transported across the imaginary surface is thus τ = τ + − τ − = 1 3 v ¯ n m ⋅ ℓ d u d y {\displaystyle \tau =\tau ^{+}-\tau ^{-}={\frac {1}{3}}{\bar {v}}nm\cdot \ell {\frac {du}{dy}}}
Combining the above kinetic equation with Newton's law of viscosity τ = η d u d y {\displaystyle \tau =\eta {\frac {du}{dy}}} gives the equation for shear viscosity, which is usually denoted η 0 {\displaystyle \eta _{0}} when it is a dilute gas: η 0 = 1 3 v ¯ n m ℓ {\displaystyle \eta _{0}={\frac {1}{3}}{\bar {v}}nm\ell }
Combining this equation with the equation for mean free path gives η 0 = 1 3 2 m v ¯ σ {\displaystyle \eta _{0}={\frac {1}{3{\sqrt {2}}}}{\frac {m{\bar {v}}}{\sigma }}}
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as v ¯ = 2 π v p = 2 2 π k B T m {\displaystyle {\bar {v}}={\frac {2}{\sqrt {\pi }}}v_{p}=2{\sqrt {{\frac {2}{\pi }}{\frac {k_{\mathrm {B} }T}{m}}}}} where v p {\displaystyle v_{p}} is the most probable speed. We note that k B N A = R and M = m N A {\displaystyle k_{\text{B}}N_{\text{A}}=R\quad {\text{and}}\quad M=mN_{\text{A}}}
and insert the velocity in the viscosity equation above. This gives the well known equation [ 44 ] (with σ {\displaystyle \sigma } subsequently estimated below) for shear viscosity for dilute gases : η 0 = 2 3 π ⋅ m k B T σ = 2 3 π ⋅ M R T σ N A {\displaystyle \eta _{0}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {mk_{\mathrm {B} }T}}{\sigma }}={\frac {2}{3{\sqrt {\pi }}}}\cdot {\frac {\sqrt {MRT}}{\sigma N_{\text{A}}}}} and M {\displaystyle M} is the molar mass . The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in Revised Enskog theory , which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by σ = π ( 2 r ) 2 = π d 2 {\displaystyle \sigma =\pi \left(2r\right)^{2}=\pi d^{2}}
The radius r {\displaystyle r} is called collision cross section radius or kinetic radius, and the diameter d {\displaystyle d} is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero Lennard-Jones potential may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required collision integrals .
The expression for viscosity obtained from Revised Enskog Theory reduces to the above expression in the limit of infinite dilution, and can be written as η = ( 1 + α η ) η 0 + η c {\displaystyle \eta =(1+\alpha _{\eta })\eta _{0}+\eta _{c}} where α η {\displaystyle \alpha _{\eta }} is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and η c {\displaystyle \eta _{c}} is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.
Following a similar logic as above, one can derive the kinetic model for thermal conductivity [ 43 ] of a dilute gas:
Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as thermal reservoirs . The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy ε {\displaystyle \varepsilon } which increases uniformly with distance y {\displaystyle y} above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let ε 0 {\displaystyle \varepsilon _{0}} be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area d A {\displaystyle dA} on one side of the gas layer, with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal, in time interval d t {\displaystyle dt} is n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) {\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi )}
These molecules made their last collision at a distance ℓ cos θ {\displaystyle \ell \cos \theta } above and below the gas layer, and each will contribute a molecular kinetic energy of ε ± = ( ε 0 ± m c v ℓ cos θ d T d y ) , {\displaystyle \varepsilon ^{\pm }=\left(\varepsilon _{0}\pm mc_{v}\ell \cos \theta \,{\frac {dT}{dy}}\right),} where c v {\displaystyle c_{v}} is the specific heat capacity . Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient d T / d y {\displaystyle dT/dy} can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the energy transfer per unit time per unit area (also known as heat flux ): q y ± = − 1 4 v ¯ n ⋅ ( ε 0 ± 2 3 m c v ℓ d T d y ) {\displaystyle q_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}n\cdot \left(\varepsilon _{0}\pm {\frac {2}{3}}mc_{v}\ell {\frac {dT}{dy}}\right)}
Note that the energy transfer from above is in the − y {\displaystyle -y} direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus q = q y + − q y − = − 1 3 v ¯ n m c v ℓ d T d y {\displaystyle q=q_{y}^{+}-q_{y}^{-}=-{\frac {1}{3}}{\bar {v}}nmc_{v}\ell \,{\frac {dT}{dy}}}
Combining the above kinetic equation with Fourier's law q = − κ d T d y {\displaystyle q=-\kappa \,{\frac {dT}{dy}}} gives the equation for thermal conductivity, which is usually denoted κ 0 {\displaystyle \kappa _{0}} when it is a dilute gas: κ 0 = 1 3 v ¯ n m c v ℓ {\displaystyle \kappa _{0}={\frac {1}{3}}{\bar {v}}nmc_{v}\ell }
Similarly to viscosity, Revised Enskog theory yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as κ = α κ κ 0 + κ c {\displaystyle \kappa =\alpha _{\kappa }\kappa _{0}+\kappa _{c}} where α κ {\displaystyle \alpha _{\kappa }} is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and κ c {\displaystyle \kappa _{c}} is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.
Following a similar logic as above, one can derive the kinetic model for mass diffusivity [ 43 ] of a dilute gas:
Consider a steady diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities , but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density n {\displaystyle n} in the layer increases uniformly with distance y {\displaystyle y} above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell–Boltzmann equilibrium distribution of molecular motions.
Let n 0 {\displaystyle n_{0}} be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area d A {\displaystyle dA} on one side of the gas layer, with speed v {\displaystyle v} at angle θ {\displaystyle \theta } from the normal, in time interval d t {\displaystyle dt} is n v cos ( θ ) d A d t × ( m 2 π k B T ) 3 / 2 e − m v 2 2 k B T ( v 2 sin ( θ ) d v d θ d ϕ ) {\displaystyle nv\cos(\theta )\,dA\,dt\times \left({\frac {m}{2\pi k_{\mathrm {B} }T}}\right)^{3/2}e^{-{\frac {mv^{2}}{2k_{\text{B}}T}}}(v^{2}\sin(\theta )\,dv\,d\theta \,d\phi )}
These molecules made their last collision at a distance ℓ cos θ {\displaystyle \ell \cos \theta } above and below the gas layer, where the local number density is n ± = ( n 0 ± ℓ cos θ d n d y ) {\displaystyle n^{\pm }=\left(n_{0}\pm \ell \cos \theta \,{\frac {dn}{dy}}\right)}
Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient d n / d y {\displaystyle dn/dy} can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint v > 0 {\displaystyle v>0} , 0 < θ < π 2 {\textstyle 0<\theta <{\frac {\pi }{2}}} , 0 < ϕ < 2 π {\displaystyle 0<\phi <2\pi } yields the molecular transfer per unit time per unit area (also known as diffusion flux ): J y ± = − 1 4 v ¯ ⋅ ( n 0 ± 2 3 ℓ d n d y ) {\displaystyle J_{y}^{\pm }=-{\frac {1}{4}}{\bar {v}}\cdot \left(n_{0}\pm {\frac {2}{3}}\ell \,{\frac {dn}{dy}}\right)}
Note that the molecular transfer from above is in the − y {\displaystyle -y} direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus J = J y + − J y − = − 1 3 v ¯ ℓ d n d y {\displaystyle J=J_{y}^{+}-J_{y}^{-}=-{\frac {1}{3}}{\bar {v}}\ell {\frac {dn}{dy}}}
Combining the above kinetic equation with Fick's first law of diffusion J = − D d n d y {\displaystyle J=-D{\frac {dn}{dy}}} gives the equation for mass diffusivity, which is usually denoted D 0 {\displaystyle D_{0}} when it is a dilute gas: D 0 = 1 3 v ¯ ℓ {\displaystyle D_{0}={\frac {1}{3}}{\bar {v}}\ell }
The corresponding expression obtained from Revised Enskog Theory may be written as D = α D D 0 {\displaystyle D=\alpha _{D}D_{0}} where α D {\displaystyle \alpha _{D}} is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation chemical potentials with density.
The kinetic theory of gases entails that due to the microscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle of detailed balance . Specifically, the fluctuation-dissipation theorem applies to the Brownian motion (or diffusion ) and the drag force , which leads to the Einstein–Smoluchowski equation : [ 45 ] D = μ k B T , {\displaystyle D=\mu \,k_{\text{B}}T,} where
Note that the mobility μ = v d / F can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.
The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the Onsager reciprocal relations (i.e. the detailed balance of the reversible dynamics of the particles), when applied to the convection (matter flow due to temperature gradient, and heat flow due to pressure gradient) and advection (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas. | https://en.wikipedia.org/wiki/Kinetic_theory_of_gases |
A kinetochore ( / k ɪ ˈ n ɛ t ə k ɔːr / , /- ˈ n iː t ə k ɔːr / ) is a flared oblique-shaped protein structure associated with duplicated chromatids in eukaryotic cells where the spindle fibers , which can be thought of as the ropes pulling chromosomes apart, attach during cell division to pull sister chromatids apart. [ 1 ] The kinetochore assembles on the centromere and links the chromosome to microtubule polymers from the mitotic spindle during mitosis and meiosis . The term kinetochore was first used in a footnote in a 1934 Cytology book by Lester W. Sharp [ 2 ] and commonly accepted in 1936. [ 3 ] Sharp's footnote reads: "The convenient term kinetochore (= movement place) has been suggested to the author by J. A. Moore", likely referring to John Alexander Moore who had joined Columbia University as a freshman in 1932. [ 4 ]
Monocentric organisms, including vertebrates, fungi, and most plants, have a single centromeric region on each chromosome which assembles a single, localized kinetochore. Holocentric organisms , such as nematodes and some plants, assemble a kinetochore along the entire length of a chromosome. [ 5 ]
Kinetochores start, control, and supervise the striking movements of chromosomes during cell division. During mitosis, which occurs after the amount of DNA is doubled in each chromosome (while maintaining the same number of chromosomes) in S phase , two sister chromatids are held together by a centromere. Each chromatid has its own kinetochore, which face in opposite directions and attach to opposite poles of the mitotic spindle apparatus. Following the transition from metaphase to anaphase , the sister chromatids separate from each other, and the individual kinetochores on each chromatid drive their movement to the spindle poles that will define the two new daughter cells. The kinetochore is therefore essential for the chromosome segregation that is classically associated with mitosis and meiosis.
The kinetochore contains two regions:
Even the simplest kinetochores consist of more than 19 different proteins. Many of these proteins are conserved between eukaryotic species, including a specialized histone H3 variant (called CENP-A or CenH3) which helps the kinetochore associate with DNA. Other proteins in the kinetochore adhere it to the microtubules (MTs) of the mitotic spindle . There are also motor proteins , including both dynein and kinesin , which generate forces that move chromosomes during mitosis. Other proteins, such as Mad2 , monitor the microtubule attachment as well as the tension between sister kinetochores and activate the spindle checkpoint to arrest the cell cycle when either of these is absent. [ 6 ] The actual set of genes essential for kinetochore function varies from one species to another. [ 7 ] [ 8 ]
Kinetochore functions include anchoring of chromosomes to MTs in the spindle, verification of anchoring, activation of the spindle checkpoint and participation in the generation of force to propel chromosome movement during cell division. [ 9 ] On the other hand, microtubules are metastable polymers made of α- and β- tubulin , alternating between growing and shrinking phases, a phenomenon known as dynamic instability . [ 10 ] MTs are highly dynamic structures, whose behavior is integrated with kinetochore function to control chromosome movement and segregation. It has also been reported that the kinetochore organization differs between mitosis and meiosis and the integrity of meiotic kinetochore is essential for meiosis specific events such as pairing of homologous chromosomes, sister kinetochore monoorientation, protection of centromeric cohesin and spindle-pole body cohesion and duplication. [ 11 ] [ 12 ]
The kinetochore is composed of several layers, observed initially by conventional fixation and staining methods of electron microscopy , [ 13 ] [ 14 ] (reviewed by C. Rieder in 1982 [ 15 ] ) and more recently by rapid freezing and substitution. [ 16 ]
The deepest layer in the kinetochore is the inner plate , which is organized on a chromatin structure containing nucleosomes presenting a specialized histone (named CENP-A , which substitutes histone H3 in this region), auxiliary proteins, and DNA. DNA organization in the centromere ( satellite DNA ) is one of the least understood aspects of vertebrate kinetochores. The inner plate appears like a discrete heterochromatin domain throughout the cell cycle .
External to the inner plate is the outer plate , which is composed mostly of proteins. This structure is assembled on the surface of the chromosomes only after the nuclear envelope breaks down. [ 13 ] The outer plate in vertebrate kinetochores contains about 20 anchoring sites for MTs (+) ends (named kMTs, after kinetochore MTs ), whereas a kinetochore's outer plate in yeast ( Saccharomyces cerevisiae ) contains only one anchoring site.
The outermost domain in the kinetochore forms a fibrous corona, which can be visualized by conventional microscopy , yet only in the absence of MTs. This corona is formed by a dynamic network of resident and temporary proteins implicated in the spindle checkpoint , in microtubule anchoring, and in the regulation of chromosome behavior.
During mitosis, each sister chromatid forming the complete chromosome has its own kinetochore. Distinct sister kinetochores can be observed at first at the end of G2 phase in cultured mammalian cells. [ 17 ] These early kinetochores show a mature laminar structure before the nuclear envelope breaks down. [ 18 ] The molecular pathway for kinetochore assembly in higher eukaryotes has been studied using gene knockouts in mice and in cultured chicken cells, as well as using RNA interference (RNAi) in C. elegans , Drosophila and human cells, yet no simple linear route can describe the data obtained so far. [ citation needed ]
The first protein to be assembled on the kinetochore is CENP-A ( Cse4 in Saccharomyces cerevisiae ). This protein is a specialized isoform of histone H3. [ 19 ] CENP-A is required for incorporation of the inner kinetochore proteins CENP-C , CENP-H and CENP-I/MIS6 . [ 20 ] [ 21 ] [ 22 ] [ 23 ] [ 24 ] The relation of these proteins in the CENP-A-dependent pathway is not completely defined. For instance, CENP-C localization requires CENP-H in chicken cells, but it is independent of CENP-I/MIS6 in human cells. In C. elegans and metazoa, the incorporation of many proteins in the outer kinetochore depends ultimately on CENP-A.
Kinetochore proteins can be grouped according to their concentration at kinetochores during mitosis: some proteins remain bound throughout cell division, whereas some others change in concentration. Furthermore, they can be recycled in their binding site on kinetochores either slowly (they are rather stable) or rapidly (dynamic).
A 2010 study used a complex method (termed "multiclassifier combinatorial proteomics" or MCCP) to analyze the proteomic composition of vertebrate chromosomes, including kinetochores. [ 38 ] Although this study does not include a biochemical enrichment for kinetochores, obtained data include all the centromeric subcomplexes, with peptides from all 125 known centromeric proteins. According to this study, there are still about one hundred unknown kinetochore proteins, doubling the known structure during mitosis, which confirms the kinetochore as one of the most complex cellular substructures. Consistently, a comprehensive literature survey indicated that there had been at least 196 human proteins already experimentally shown to be localized at kinetochores. [ 39 ]
The number of microtubules attached to one kinetochore is variable: in Saccharomyces cerevisiae only one MT binds each kinetochore, whereas in mammals there can be 15–35 MTs bound to each kinetochore. [ 40 ] However, not all the MTs in the spindle attach to one kinetochore. There are MTs that extend from one centrosome to the other (and they are responsible for spindle length) and some shorter ones are interdigitated between the long MTs. Professor B. Nicklas (Duke University), showed that, if one breaks down the MT-kinetochore attachment using a laser beam , chromatids can no longer move, leading to an abnormal chromosome distribution. [ 41 ] These experiments also showed that kinetochores have polarity, and that kinetochore attachment to MTs emanating from one or the other centrosome will depend on its orientation. This specificity guarantees that only one chromatid will move to each spindle side, thus ensuring the correct distribution of the genetic material. Thus, one of the basic functions of the kinetochore is the MT attachment to the spindle, which is essential to correctly segregate sister chromatids. If anchoring is incorrect, errors may ensue, generating aneuploidy , with catastrophic consequences for the cell. To prevent this from happening, there are mechanisms of error detection and correction (as the spindle assembly checkpoint ), whose components reside also on the kinetochores. The movement of one chromatid towards the centrosome is produced primarily by MT depolymerization in the binding site with the kinetochore. These movements require also force generation, involving molecular motors likewise located on the kinetochores.
During the synthesis phase (S phase) in the cell cycle , the centrosome starts to duplicate. Just at the beginning of mitosis, both centrioles in each centrosome reach their maximal length, centrosomes recruit additional material and their nucleation capacity for microtubules increases. As mitosis progresses, both centrosomes separate to establish the mitotic spindle. [ 42 ] In this way, the spindle in a mitotic cell has two poles emanating microtubules. Microtubules are long proteic filaments with asymmetric extremes, a "minus"(-) end relatively stable next to the centrosome, and a "plus"(+) end enduring alternate phases of growing-shrinking, exploring the center of the cell. During this searching process, a microtubule may encounter and capture a chromosome through the kinetochore. [ 43 ] [ 44 ] Microtubules that find and attach a kinetochore become stabilized, whereas those microtubules remaining free are rapidly depolymerized. [ 45 ] As chromosomes have two kinetochores associated back-to-back (one on each sister chromatid), when one of them becomes attached to the microtubules generated by one of the cellular poles, the kinetochore on the sister chromatid becomes exposed to the opposed pole; for this reason, most of the times the second kinetochore becomes attached to the microtubules emanating from the opposing pole, [ 46 ] in such a way that chromosomes are now bi-oriented , one fundamental configuration (also termed amphitelic ) to ensure the correct segregation of both chromatids when the cell will divide. [ 47 ] [ 48 ]
When just one microtubule is anchored to one kinetochore, it starts a rapid movement of the associated chromosome towards the pole generating that microtubule. This movement is probably mediated by the motor activity towards the "minus" (-) of the motor protein cytoplasmic dynein, [ 49 ] [ 50 ] which is very concentrated in the kinetochores not anchored to MTs. [ 51 ] The movement towards the pole is slowed down as far as kinetochores acquire kMTs (MTs anchored to kinetochores) and the movement becomes directed by changes in kMTs length. Dynein is released from kinetochores as they acquire kMTs [ 30 ] and, in cultured mammalian cells, it is required for the spindle checkpoint inactivation, but not for chromosome congression in the spindle equator, kMTs acquisition or anaphase A during chromosome segregation. [ 52 ] In higher plants or in yeast there is no evidence of dynein, but other kinesins towards the (-) end might compensate for the lack of dynein.
Another motor protein implicated in the initial capture of MTs is CENP-E; this is a high molecular weight kinesin associated with the fibrous corona at mammalian kinetochores from prometaphase until anaphase. [ 53 ] In cells with low levels of CENP-E, chromosomes lack this protein at their kinetochores, which quite often are defective in their ability to congress at the metaphase plate. In this case, some chromosomes may remain chronically mono-oriented (anchored to only one pole), although most chromosomes may congress correctly at the metaphase plate. [ 54 ]
It is widely accepted that the kMTs fiber (the bundle of microtubules bound to the kinetochore) is originated by the capture of MTs polymerized at the centrosomes and spindle poles in mammalian cultured cells. [ 43 ] However, MTs directly polymerized at kinetochores might also contribute significantly. [ 55 ] The manner in which the centromeric region or kinetochore initiates the formation of kMTs and the frequency at which this happens are important questions, [ according to whom? ] because this mechanism may contribute not only to the initial formation of kMTs, but also to the way in which kinetochores correct defective anchoring of MTs and regulate the movement along kMTs.
MTs associated to kinetochores present special features: compared to free MTs, kMTs are much more resistant to cold-induced depolymerization, high hydrostatic pressures or calcium exposure. [ 56 ] Furthermore, kMTs are recycled much more slowly than astral MTs and spindle MTs with free (+) ends, and if kMTs are released from kinetochores using a laser beam, they rapidly depolymerize. [ 41 ]
When it was clear that neither dynein nor CENP-E is essential for kMTs formation, other molecules should be responsible for kMTs stabilization. Pioneer genetic work in yeast revealed the relevance of the Ndc80 complex in kMTs anchoring. [ 25 ] [ 57 ] [ 58 ] [ 59 ] In Saccharomyces cerevisiae , the Ndc80 complex has four components: Ndc80p , Nuf2p , Spc24p and Spc25p . Mutants lacking any of the components of this complex show loss of the kinetochore-microtubule connection, although kinetochore structure is not completely lost. [ 25 ] [ 57 ] Yet mutants in which kinetochore structure is lost (for instance Ndc10 mutants in yeast [ 60 ] ) are deficient both in the connection to microtubules and in the ability to activate the spindle checkpoint , probably because kinetochores work as a platform in which the components of the response are assembled.
The Ndc80 complex is highly conserved and it has been identified in S. pombe , C. elegans , Xenopus , chicken and humans. [ 25 ] [ 26 ] [ 57 ] [ 61 ] [ 62 ] [ 63 ] [ 64 ] Studies on Hec1 ( highly expressed in cancer cells 1 ), the human homolog of Ndc80p, show that it is important for correct chromosome congression and mitotic progression, and that it interacts with components of the cohesin and condensin complexes. [ 65 ]
Different laboratories have shown that the Ndc80 complex is essential for stabilization of the kinetochore-microtubule anchoring, required to support the centromeric tension implicated in the establishment of the correct chromosome congression in high eukaryotes . [ 26 ] [ 62 ] [ 63 ] [ 64 ] Cells with impaired function of Ndc80 (using RNAi , gene knockout , or antibody microinjection) have abnormally long spindles, lack of tension between sister kinetochores, chromosomes unable to congregate at the metaphase plate and few or any associated kMTs.
There is a variety of strong support for the ability of the Ndc80 complex to directly associate with microtubules and form the core conserved component of the kinetochore-microtubule interface. [ 66 ] However, formation of robust kinetochore-microtubule interactions may also require the function of additional proteins. In yeast, this connection requires the presence of the complex Dam1 -DASH-DDD. Some members of this complex bind directly to MTs, whereas some others bind to the Ndc80 complex. [ 58 ] [ 59 ] [ 67 ] This means that the complex Dam1-DASH-DDD might be an essential adapter between kinetochores and microtubules. However, in animals an equivalent complex has not been identified, and this question remains under intense investigation.
During S-Phase , the cell duplicates all the genetic information stored in the chromosomes, in the process termed DNA replication . At the end of this process, each chromosome includes two sister chromatids , which are two complete and identical DNA molecules. Both chromatids remain associated by cohesin complexes until anaphase, when chromosome segregation occurs. If chromosome segregation happens correctly, each daughter cell receives a complete set of chromatids, and for this to happen each sister chromatid has to anchor (through the corresponding kinetochore) to MTs generated in opposed poles of the mitotic spindle. This configuration is termed amphitelic or bi-orientation .
However, during the anchoring process some incorrect configurations may also appear: [ 68 ]
Both the monotelic and the syntelic configurations fail to generate centromeric tension and are detected by the spindle checkpoint. In contrast, the merotelic configuration is not detected by this control mechanism. However, most of these errors are detected and corrected before the cell enters in anaphase. [ 68 ] A key factor in the correction of these anchoring errors is the chromosomal passenger complex, which includes the kinase protein Aurora B, its target and activating subunit INCENP and two other subunits, Survivin and Borealin/Dasra B (reviewed by Adams and collaborators in 2001 [ 69 ] ). Cells in which the function of this complex has been abolished by dominant negative mutants, RNAi , antibody microinjection or using selective drugs, accumulate errors in chromosome anchoring. Many studies have shown that Aurora B is required to destabilize incorrect anchoring kinetochore-MT, favoring the generation of amphitelic connections. Aurora B homolog in yeast (Ipl1p) phosphorilates some kinetochore proteins, such as the constitutive protein Ndc10p and members of the Ndc80 and Dam1-DASH-DDD complexes. [ 70 ] Phosphorylation of Ndc80 complex components produces destabilization of kMTs anchoring. It has been proposed that Aurora B localization is important for its function: as it is located in the inner region of the kinetochore (in the centromeric heterochromatin), when the centromeric tension is established sister kinetochores separate, and Aurora B cannot reach its substrates, so that kMTs are stabilized. Aurora B is frequently overexpressed in several cancer types, and it is currently a target for the development of anticancer drugs. [ 71 ]
The spindle checkpoint, or SAC (for spindle assembly checkpoint ), also known as the mitotic checkpoint , is a cellular mechanism responsible for detection of:
When just one chromosome (for any reason) remains lagging during congression, the spindle checkpoint machinery generates a delay in cell cycle progression: the cell is arrested, allowing time for repair mechanisms to solve the detected problem. After some time, if the problem has not been solved, the cell will be targeted for apoptosis (programmed cell death), a safety mechanism to avoid the generation of aneuploidy , a situation which generally has dramatic consequences for the organism.
Whereas structural centromeric proteins (such as CENP-B ), remain stably localized throughout mitosis (including during telophase ), the spindle checkpoint components are assembled on the kinetochore in high concentrations in the absence of microtubules, and their concentrations decrease as the number of microtubules attached to the kinetochore increases. [ 30 ]
At metaphase, CENP-E , Bub3 and Bub1 levels decreases 3 to 4 fold as compared to the levels at unattached kinetochores, whereas the levels of dynein/dynactin , Mad1 , Mad2 and BubR1 decrease >10-100 fold. [ 30 ] [ 31 ] [ 32 ] [ 33 ] Thus at metaphase, when all chromosomes are aligned at the metaphase plate, all checkpoint proteins are released from the kinetochore. The disappearance of the checkpoint proteins out of the kinetochores indicates the moment when the chromosome has reached the metaphase plate and is under bipolar tension. At this moment, the checkpoint proteins that bind to and inhibit Cdc20 (Mad1-Mad2 and BubR1), release Cdc20, which binds and activates APC/C Cdc20 , and this complex triggers sister chromatids separation and consequently anaphase entry.
Several studies indicate that the Ndc80 complex participates in the regulation of the stable association of Mad1-Mad2 and dynein with kinetochores. [ 26 ] [ 63 ] [ 64 ] Yet the kinetochore associated proteins CENP-A, CENP-C, CENP-E, CENP-H and BubR1 are independent of Ndc80/Hec1. The prolonged arrest in prometaphase observed in cells with low levels of Ndc80/Hec1 depends on Mad2, although these cells show low levels of Mad1, Mad2 and dynein on kinetochores (<10-15% in relation to unattached kinetochores). However, if both Ndc80/Hec1 and Nuf2 levels are reduced, Mad1 and Mad2 completely disappear from the kinetochores and the spindle checkpoint is inactivated. [ 72 ]
Shugoshin (Sgo1, MEI-S332 in Drosophila melanogaster [ 73 ] ) are centromeric proteins which are essential to maintain cohesin bound to centromeres until anaphase. The human homolog, hsSgo1, associates with centromeres during prophase and disappears when anaphase starts. [ 74 ] When Shugoshin levels are reduced by RNAi in HeLa cells, cohesin cannot remain on the centromeres during mitosis, and consequently sister chromatids separate synchronically before anaphase initiates, which triggers a long mitotic arrest.
On the other hand, Dasso and collaborators have found that proteins involved in the Ran cycle can be detected on kinetochores during mitosis: RanGAP1 (a GTPase activating protein which stimulates the conversion of Ran-GTP in Ran-GDP) and the Ran binding protein called RanBP2/Nup358 . [ 75 ] During interphase, these proteins are located at the nuclear pores and participate in the nucleo-cytoplasmic transport. Kinetochore localization of these proteins seem to be functionally significant, because some treatments that increase the levels of Ran-GTP inhibit kinetochore release of Bub1, Bub3, Mad2 and CENP-E. [ 76 ]
Orc2 (a protein that belongs to the origin recognition complex -ORC- implicated in DNA replication initiation during S phase ) is also localized at kinetochores during mitosis in human cells; [ 77 ] in agreement with this localization, some studies indicate that Orc2 in yeast is implicated in sister chromatids cohesion, and when it is eliminated from the cell, spindle checkpoint activation ensues. [ 78 ] Some other ORC components (such orc5 in S. pombe ) have been also found to participate in cohesion. [ 79 ] However, ORC proteins seem to participate in a molecular pathway which is additive to cohesin pathway, and it is mostly unknown.
Most chromosome movements in relation to spindle poles are associated to lengthening and shortening of kMTs. One of the features of kinetochores is their capacity to modify the state of their associated kMTs (around 20) from a depolymerization state at their (+) end to polymerization state. This allows the kinetochores from cells at prometaphase to show "directional instability", [ 80 ] changing between persistent phases of movement towards the pole ( poleward ) or inversed ( anti-poleward ), which are coupled with alternating states of kMTs depolymerization and polymerization, respectively. This kinetochore bi-stability seem to be part of a mechanism to align the chromosomes at the equator of the spindle without losing the mechanic connection between kinetochores and spindle poles. It is thought that kinetochore bi-stability is based upon the dynamic instability of the kMTs (+) end, and it is partially controlled by the tension present at the kinetochore. In mammalian cultured cells, a low tension at kinetochores promotes change towards kMTs depolymerization, and high tension promotes change towards kMTs polymerization. [ 81 ] [ 82 ]
Kinetochore proteins and proteins binding to MTs (+) end (collectively called +TIPs) regulate kinetochore movement through the kMTs (+) end dynamics regulation. [ 83 ] However, the kinetochore-microtubule interface is highly dynamic, and some of these proteins seem to be bona fide components of both structures. Two groups of proteins seem to be particularly important: kinesins which work like depolymerases, such as KinI kinesins; and proteins bound to MT (+) ends, +TIPs, promoting polymerization, perhaps antagonizing the depolymerases effect. [ 84 ] | https://en.wikipedia.org/wiki/Kinetochore |
The King Abdullah Canal is the largest irrigation canal system in Jordan and runs parallel to the east bank of the Jordan River . It was previously known as the East Ghor Main Canal and renamed in 1987 after Abdullah I of Jordan . [ 1 ]
The main water source for the King Abdullah Canal (KAC) is the Yarmouk River and the Al-Mukhaibeh wells within the Yarmouk valley: farther south, additional water flows from Wadi el-Arab and from the Zarqa River , and its reservoir behind King Talal Dam . As a result of the 1994 Israel–Jordan peace treaty , some Yarmouk River water is also stored seasonally in Lake Tiberias , being conveyed through a pipe. [ 2 ] The canal's design capacity is 20 m 3 /second at the northern entrance of the Canal and 2.3 m 3 /second at its southern end. Water flows by gravity along its 110 km length, ranging in elevation from about 230 meters below sea level to almost 400 meters below. The Canal supplies water for irrigation and 90 million cubic meters/year of drinking water for Greater Amman through the Deir Allah-Amman carrier, which has been constructed in two phases in the mid-80s and in the early 2000s. The Zarqa River contains a mixture of treated wastewater and natural water flow, which influences the water quality downstream of the Zarqa River intake into the KAC.
The canal was designed in 1957 and was built in phases. Construction began in 1959, and the first section was completed in 1961. By 1966, the upstream portion to Wadi Zarqa was completed. The canal was then 70 km in length, and was subsequently extended three times between 1969 and 1987. The United States , through United States Agency for International Development (USAID) provided financing for the initial phase of project, after obtaining explicit assurances from the Jordanian government that Jordan would not withdraw more water from the Yarmouk than the amount allocated to it according to the Johnston Plan . [ 3 ] [ 4 ] It was also involved in later phases.
The original canal was part of a larger project - the Greater Yarmouk project - which envisioned two storage dams on the Yarmouk, and a future West Ghor Canal, on the West Bank of the Jordan. This other canal was never built, because Israel captured the West Bank from Jordan during the 1967 Six-Day War . After the Six-Day War, the Palestine Liberation Organization (PLO) operated from bases within Jordan, and launched several attacks on Israeli settlements in the Jordan Valley, including attacks on water facilities. Israel responded with raids in Jordan, in an attempt to force king Hussein to rein in the PLO. The canal was the target of at least four of these raids, and was virtually knocked out of commission. The United States intervened to resolve the conflict, and the canal was repaired after Hussein undertook to stop PLO activity in the area. [ 5 ] | https://en.wikipedia.org/wiki/King_Abdullah_Canal |
King Pharmaceuticals, Inc. is a pharmaceutical company, a wholly owned subsidiary of Pfizer based in Bristol, Tennessee . Before being acquired by Pfizer, it was the world's 39th largest pharmaceutical company . [ 1 ] [ 2 ] On October 12, 2010, King was acquired by Pfizer for $14.25 per share. [ 3 ] King produced a wide range of pharmaceuticals , including Altace for heart attack prevention, Levoxyl for hypothyroidism , Sonata , a sleeping aid , and Skelaxin , a muscle relaxant . King Pharmaceuticals operated manufacturing facilities in Bristol, Tennessee ; Rochester, Michigan ; St. Louis, Missouri ; St. Petersburg, Florida ; and Middleton, Wisconsin . They employed approximately 2,700 people including a sales force of over 1,000 individuals.
King Pharmaceuticals, Inc. was incorporated in the State of Tennessee in 1993. According to the King Pharmaceutcals, Inc. Form 10-K for the year ended December 31, 2007 filed with the U.S. Securities and Exchange Commission, the wholly owned subsidiaries of King Pharmaceuticals, Inc. are Monarch Pharmaceuticals, Inc.; King Pharmaceuticals Research and Development, Inc.; Meridian Medical Technologies, Inc.; Parkedale Pharmaceuticals, Inc.; King Pharmaceuticals Canada Inc.; and Monarch Pharmaceuticals Ireland Limited. [ 4 ]
King Pharmaceuticals was founded in 1993 John M. Gregory , Randal J. Kirk , Joseph "Joe" R. Gregory, Jefferson "Jeff" J. Gregory, and James E. Gregory. [ 5 ] [ 6 ] In January 1994, King acquired a former King College campus plant in Bristol, Tennessee. The 500,000-square-foot (46,000 m 2 ) facility was purchased for $1.18 million from RSR Pharmaceutical, who had been using it after Beecham merged with SmithKline . King initially manufactured drugs for other pharmaceutical companies, but soon established a strategy of acquiring branded prescription drugs, which have a much higher gross margin than contract manufactured drugs.
In February 1998, King acquired 15 branded pharmaceuticals, a sterile products manufacturing facility located in Rochester, Michigan that it called the "Parkedale Facility") and some contract manufacturing contracts. [ 7 ] By December 1998 King had placed its sterile products business into a subsidiary it named Parkedale Pharmaceuticals. [ 8 ]
King Pharmaceuticals obtained about twenty smaller branded drugs from the start up of the company until it went public in June 1998. The King Pharmaceuticals subsidiary Monarch Pharmaceuticals acquired one of its most profitable branded drugs, Altace, later the same year on December 18, 1998 from Hoechst Marion Roussel .
Hoechst merged with Marion Merrill Dow of Kansas City, Missouri in 1995, forming the Hoechst U.S. pharmaceutical subsidiary Hoechst Marion Roussel (HMR). Altace was bringing in under $90 million in U.S. revenues for HMR and Hoechst had stopped promoting Altace within the United States., [ 9 ] and King Pharmaceuticals President Jefferson "Jeff" Gregory also began negotiations in 1995 with Hoechst to acquire U.S. distribution rights to Altace. [ 9 ]
The King Pharmaceuticals wholly owned subsidiary Monarch Pharmaceuticals, Inc. (another brother of John Gregory - Joseph Gregory - was then the president of Monarch Pharmaceuticals) acquired ownership of the U.S. distribution and marketing rights to Altace and other Hoechst products from Hoechst AG subsidiary Hoechst Marion Roussel of Kansas City, Missouri on December 18, 1998, and [ 10 ] following a January 1999 merger a few weeks later with Rhône-Poulenc , Hoechst assumed the new corporate identity of Aventis ).
In 2001, Forbes magazine ranked John Gregory among the 400 richest Americans. The bulk of Gregory's personal fortune was due in large part due to the ability of King Pharmaceuticals, Inc. to reintroduce the Hoechst branded prescription drug Altace back into the U.S. market under the King Pharmaceuticals, Inc. subsidiary Monarch Pharmaceuticals brand following the 1998 U.S. marketing and distribution agreement between King Pharmaceuticals/Monarch Pharmaceuticals and Hoechst AG/HMR.
In late December 1998, King Pharmaceuticals (d.b.a. Monarch Pharmaceuticals, Inc.) purchased the U.S. marketing and distribution rights of the company's most successful drug, Altace, for $362.5 million from the U.S. subsidiary of Hoechst AG , Hoechst Marion Roussel of Kansas City. [ 11 ] As a result of increasing the number of sales representatives and the findings of the Heart Outcomes Prevention Evaluation (HOPE), [ 12 ] Altace sales increased. Using profits from Altace, King continued to add product lines, the most significant purchases being Levoxyl , Thrombin , and Cytomel in 2000. [ citation needed ] Also in 2000, seeing fewer opportunities to obtain branded drugs, the company acquired an R&D company based in North Carolina. [ citation needed ] In 2002, King and Meridian Medical Technologies agreed that King would purchase Merdian for $247.8 million in cash; Meridian was a manufacturer of autoinjectors, including EpiPen, an epinephrine autoinjector . [ 13 ] The deal was completed in January 2003. [ 14 ]
In 2002, John Gregory stepped down as CEO, and his brother Jefferson Gregory took over. Then in 2004, Jeff Gregory stepped down as well after the SEC began investigations into King's Medicaid billing practices. [ 15 ] The board named Brian Markison to replace him. Soon after, in July 2004, a deal was made for Mylan Laboratories to acquire King for $4 billion. [ 16 ] Investors, most notably Carl Icahn , were critical of the merger, saying that Mylan was overpaying for King. The next year the deal was called off.
In 2008 King Pharmaceuticals acquired Alpharma Pharmaceuticals to expand into the pain treatment market. From the acquisition, King gained the patents on the pain management drugs, Flector and Embeda. [ 17 ] They also gained a completely separate animal health division, which focuses on the many agricultural and animal health needs of livestock animals.
On October 12, 2010, Pfizer Inc. ( NYSE : PFE ) announced it would acquire Bristol-based King Pharmaceuticals, Inc. for a total cost to Pfizer of $3.6 billion in cash or $14.25 per share. The acquisition was expected to expand Pfizer's product line of pain relief and management medication by adding King products such as Embeda, Avinza, and the Flector Patch. [ 18 ] Other product lines that made King attractive to Pfizer included the EpiPen for emergency delivery of medications and the Alpharma animal health line. [ 19 ] | https://en.wikipedia.org/wiki/King_Pharmaceuticals |
A king post (or king-post or kingpost ) is a central vertical post used in architectural or bridge designs, working in tension to support a beam below from a truss apex above (whereas a crown post , though visually similar, supports items above from the beam below).
In aircraft design a strut called a king post acts in compression, similarly to an architectural crown post. Usage in mechanical plant and marine engineering differs again, as noted below.
A king post extends vertically from a crossbeam (the tie beam ) to the apex of a triangular truss . [ 1 ] The king post, itself in tension , connects the apex of the truss with its base, holding up the tie beam (also in tension) at the base of the truss. The post can be replaced with an iron rod called a king rod (or king bolt) and thus a king rod truss. [ 2 ] The king post truss is also called a "Latin truss". [ 3 ]
In traditional timber framing, a crown post looks similar to a king post, but it is very different structurally: whereas the king post is in tension, usually supporting the tie beam as a truss, the crown post is supported by the tie beam and is in compression . The crown post rises to a crown plate immediately below collar beams which it supports; it does not rise to the apex like a king post. Historically a crown post was called a king post in England but this usage is obsolete. [ 4 ]
An alternative truss construction uses two queen posts (or queen-posts). These vertical posts, positioned along the base of the truss, are supported by the sloping sides of the truss, rather than reaching its apex. A development adds a collar beam above the queen posts, which are then termed queen struts. A section of the tie beam between the queen posts may be removed to create a hammerbeam roof .
The king post truss is used for simple roof trusses and short-span bridges. It is the simplest form of truss in that it is constructed of the fewest truss members (individual lengths of wood or metal). The truss consists of two diagonal members that meet at the apex of the truss, one horizontal beam that serves to tie the bottom end of the diagonals together, and the king post which connects the apex to the horizontal beam below. For a roof truss, the diagonal members are called rafters , and the horizontal member may serve as a ceiling joist . A bridge would require two king post trusses with the driving surface between them. A roof usually uses many side-by-side trusses depending on the size of the structure. [ 5 ]
Pont-y-Cafnau , the world's first iron railway bridge, is of the king post type.
King posts were used in timber-framed roof construction in Roman buildings, [ 6 ] and in medieval architecture in buildings such as parish churches and tithe barns . The oldest surviving roof truss in the world is a king post truss in Saint Catherine's Monastery , Egypt, [ 7 ] built between 548 and 565. [ 8 ]
King posts also appear in Gothic Revival architecture , Queen Anne style architecture and occasionally in modern construction. King post trusses are also used as a structural element in wood and metal bridges.
A painting by Karl Blechen circa 1833 illustrating construction of the second Devil's Bridge ( Teufelsbrücke ) in the Schöllenen Gorge shows multiple king posts suspended from the apex of the falsework upon which the masonry arch has been laid. In this example, beams in compression are supported by each king post several feet below the apex, and the bottom of the king posts can clearly be seen to be unsupported.
Architectural historians in the French colonial cities St Louis , Missouri and New Orleans , Louisiana use the term "Norman roof" to refer to a steeply pitched roof; it is supported by what they call a "Norman truss" which is similar to a king post truss. This is a through-purlin truss consisting of a tie beam and paired truss blades, with a central king post to support the roof ridge. The name derives from a belief that this system of construction was introduced to North America by settlers from Normandy in northern France, but it is really a misnomer as the system was more widely used than that. [ 9 ] The difference between a Norman truss and a king post truss is the tie beam in a Norman truss is technically a collar beam (a beam between the rafters above the rafter feet) where the king post truss the rafters land on top of a tie beam.
King posts are also used in the construction of some wire-braced aircraft, [ 10 ] where a king post supports the top cables or "ground wires" supporting the wing. Only on the ground are these wires from the kingpost in tension, while in the air under positive g flight they are unloaded.
The very robust hinge connecting the boom to the chassis in a backhoe , similar in function and appearance to a large automotive kingpin , is called a king post.
On a cargo ship or oiler a king post is an upright with cargo-handling or fueling rig devices attached to it. On a cargo vessel king posts are designed for handling cargo, and so are located at the forward or after end of a hatch. For an oiler they are located over the fuel transfer lines. [ 11 ]
Notes
Bibliography | https://en.wikipedia.org/wiki/King_post |
In biology , a kingdom is the second highest taxonomic rank , just below domain . Kingdoms are divided into smaller groups called phyla (singular phylum).
Traditionally, textbooks from Canada and the United States have used a system of six kingdoms ( Animalia , Plantae , Fungi , Protista , Archaea /Archaebacteria, and Bacteria or Eubacteria), while textbooks in other parts of the world, such as Bangladesh, Brazil, Greece, India, Pakistan, Spain, and the United Kingdom have used five kingdoms (Animalia, Plantae, Fungi, Protista and Monera ).
Some recent classifications based on modern cladistics have explicitly abandoned the term kingdom , noting that some traditional kingdoms are not monophyletic , meaning that they do not consist of all the descendants of a common ancestor . The terms flora (for plants), fauna (for animals), and, in the 21st century, funga (for fungi) are also used for life present in a particular region or time. [ 1 ] [ 2 ]
When Carl Linnaeus introduced the rank-based system of nomenclature into biology in 1735, the highest rank was given the name "kingdom" and was followed by four other main or principal ranks: class , order , genus and species . [ 3 ] Later two further main ranks were introduced, making the sequence kingdom, phylum or division , class , order , family , genus and species . [ 4 ] In 1990, the rank of domain was introduced above kingdom. [ 5 ]
Prefixes can be added so subkingdom ( subregnum ) and infrakingdom (also known as infraregnum ) are the two ranks immediately below kingdom. Superkingdom may be considered as an equivalent of domain or empire or as an independent rank between kingdom and domain or subdomain. In some classification systems the additional rank branch (Latin: ramus ) can be inserted between subkingdom and infrakingdom, e.g., Protostomia and Deuterostomia in the classification of Cavalier-Smith. [ 6 ]
The classification of living things into animals and plants is an ancient one. Aristotle (384–322 BC) classified animal species in his History of Animals , while his pupil Theophrastus ( c. 371 – c. 287 BC ) wrote a parallel work, the Historia Plantarum , on plants. [ 7 ]
Carl Linnaeus (1707–1778) laid the foundations for modern biological nomenclature , now regulated by the Nomenclature Codes , in 1735. He distinguished two kingdoms of living things: Regnum Animale (' animal kingdom') and Regnum Vegetabile ('vegetable kingdom', for plants ). Linnaeus also included minerals in his classification system , placing them in a third kingdom, Regnum Lapideum .
Regnum Animale (animals)
Regnum Vegetabile ('vegetables'/plants)
Regnum Lapideum (minerals)
In 1674, Antonie van Leeuwenhoek , often called the "father of microscopy", sent the Royal Society of London a copy of his first observations of microscopic single-celled organisms. Until then, the existence of such microscopic organisms was entirely unknown. Despite this, Linnaeus did not include any microscopic creatures in his original taxonomy.
At first, microscopic organisms were classified within the animal and plant kingdoms. However, by the mid–19th century, it had become clear to many that "the existing dichotomy of the plant and animal kingdoms [had become] rapidly blurred at its boundaries and outmoded". [ 8 ]
In 1860 John Hogg proposed the Protoctista , a third kingdom of life composed of "all the lower creatures, or the primary organic beings"; he retained Regnum Lapideum as a fourth kingdom of minerals. [ 8 ] In 1866, Ernst Haeckel also proposed a third kingdom of life, the Protista , for "neutral organisms" or "the kingdom of primitive forms", which were neither animal nor plant; he did not include the Regnum Lapideum in his scheme. [ 8 ] Haeckel revised the content of this kingdom a number of times before settling on a division based on whether organisms were unicellular (Protista) or multicellular (animals and plants). [ 8 ]
Kingdom Protista or Protoctista
Kingdom Plantae
Kingdom Animalia
Regnum Lapideum (minerals)
The development of microscopy revealed important distinctions between those organisms whose cells do not have a distinct nucleus ( prokaryotes ) and organisms whose cells do have a distinct nucleus ( eukaryotes ). In 1937 Édouard Chatton introduced the terms "prokaryote" and "eukaryote" to differentiate these organisms. [ 9 ]
In 1938, Herbert F. Copeland proposed a four-kingdom classification by creating the novel Kingdom Monera of prokaryotic organisms; as a revised phylum Monera of the Protista, it included organisms now classified as Bacteria and Archaea . Ernst Haeckel, in his 1904 book The Wonders of Life , had placed the blue-green algae (or Phycochromacea) in Monera; this would gradually gain acceptance, and the blue-green algae would become classified as bacteria in the phylum Cyanobacteria . [ 8 ] [ 9 ]
In the 1960s, Roger Stanier and C. B. van Niel promoted and popularized Édouard Chatton's earlier work, particularly in their paper of 1962, "The Concept of a Bacterium"; this created, for the first time, a rank above kingdom—a superkingdom or empire —with the two-empire system of prokaryotes and eukaryotes. [ 9 ] The two-empire system would later be expanded to the three-domain system of Archaea, Bacteria, and Eukaryota. [ 10 ]
Kingdom Monera
Kingdom Protista or Protoctista
Kingdom Plantae
Kingdom Animalia
The differences between fungi and other organisms regarded as plants had long been recognised by some; Haeckel had moved the fungi out of Plantae into Protista after his original classification, [ 8 ] but was largely ignored in this separation by scientists of his time. Robert Whittaker recognized an additional kingdom for the Fungi . [ 11 ] The resulting five-kingdom system, proposed in 1969 by Whittaker, has become a popular standard and with some refinement is still used in many works and forms the basis for new multi-kingdom systems. It is based mainly upon differences in nutrition ; his Plantae were mostly multicellular autotrophs , his Animalia multicellular heterotrophs , and his Fungi multicellular saprotrophs .
The remaining two kingdoms, Protista and Monera, included unicellular and simple cellular colonies. [ 11 ] The five kingdom system may be combined with the two empire system. In the Whittaker system, Plantae included some algae. In other systems, such as Lynn Margulis 's system of five kingdoms, the plants included just the land plants ( Embryophyta ), and Protoctista has a broader definition. [ 12 ]
Following publication of Whittaker's system, the five-kingdom model began to be commonly used in high school biology textbooks. [ 13 ] But despite the development from two kingdoms to five among most scientists, some authors as late as 1975 continued to employ a traditional two-kingdom system of animals and plants, dividing the plant kingdom into subkingdoms Prokaryota (bacteria and cyanobacteria), Mycota (fungi and supposed relatives), and Chlorota (algae and land plants). [ 14 ]
Kingdom Monera
Kingdom Protista or Protoctista
Kingdom Plantae
Kingdom Fungi
Kingdom Animalia
Kingdom Monera
Kingdom Protista
Kingdom Plantae
Kingdom Fungi
Kingdom Animalia
In 1977, Carl Woese and colleagues proposed the fundamental subdivision of the prokaryotes into the Eubacteria (later called the Bacteria) and Archaebacteria (later called the Archaea), based on ribosomal RNA structure; [ 15 ] this would later lead to the proposal of three "domains" of life , of Bacteria, Archaea, and Eukaryota. [ 5 ] Combined with the five-kingdom model, this created a six-kingdom model, where the kingdom Monera is replaced by the kingdoms Bacteria and Archaea. [ 16 ] This six-kingdom model is commonly used in recent US high school biology textbooks, but has received criticism for compromising the current scientific consensus. [ 13 ] But the division of prokaryotes into two kingdoms remains in use with the recent seven kingdoms scheme of Thomas Cavalier-Smith, although it primarily differs in that Protista is replaced by Protozoa and Chromista . [ 17 ]
Kingdom Eubacteria (Bacteria)
Kingdom Archaebacteria (Archaea)
Kingdom Protista or Protoctista
Kingdom Plantae
Kingdom Fungi
Kingdom Animalia
Thomas Cavalier-Smith supported the consensus at that time, that the difference between Eubacteria and Archaebacteria was so great (particularly considering the genetic distance of ribosomal genes) that the prokaryotes needed to be separated into two different kingdoms. He then divided Eubacteria into two subkingdoms: Negibacteria ( Gram-negative bacteria ) and Posibacteria ( Gram-positive bacteria ). Technological advances in electron microscopy allowed the separation of the Chromista from the Plantae kingdom. Indeed, the chloroplast of the chromists is located in the lumen of the endoplasmic reticulum instead of in the cytosol . Moreover, only chromists contain chlorophyll c . Since then, many non-photosynthetic phyla of protists, thought to have secondarily lost their chloroplasts, were integrated into the kingdom Chromista.
Finally, some protists lacking mitochondria were discovered. [ 18 ] As mitochondria were known to be the result of the endosymbiosis of a proteobacterium , it was thought that these amitochondriate eukaryotes were primitively so, marking an important step in eukaryogenesis . As a result, these amitochondriate protists were separated from the protist kingdom, giving rise to the, at the same time, superkingdom and kingdom Archezoa . This superkingdom was opposed to the Metakaryota superkingdom, grouping together the five other eukaryotic kingdoms ( Animalia , Protozoa , Fungi , Plantae and Chromista ). This was known as the Archezoa hypothesis , which has since been abandoned; [ 19 ] later schemes did not include the Archezoa–Metakaryota divide. [ 6 ] [ 17 ]
Kingdom Eubacteria
Kingdom Archaebacteria
Kingdom Archezoa ‡
Kingdom Protozoa
Kingdom Chromista
Kingdom Plantae
Kingdom Fungi
Kingdom Animalia
‡ No longer recognized by taxonomists .
In 1998, Cavalier-Smith published a six-kingdom model, [ 6 ] which has been revised in subsequent papers. The version published in 2009 is shown below. [ 20 ] [ a ] [ 21 ] Cavalier-Smith no longer accepted the importance of the fundamental Eubacteria–Archaebacteria divide put forward by Woese and others and supported by recent research. [ 22 ] The kingdom Bacteria (sole kingdom of empire Prokaryota ) was subdivided into two sub-kingdoms according to their membrane topologies: Unibacteria and Negibacteria . Unibacteria was divided into phyla Archaebacteria and Posibacteria ; the bimembranous-unimembranous transition was thought to be far more fundamental than the long branch of genetic distance of Archaebacteria, viewed as having no particular biological significance.
Cavalier-Smith does not accept the requirement for taxa to be monophyletic ("holophyletic" in his terminology) to be valid. He defines Prokaryota, Bacteria, Negibacteria, Unibacteria, and Posibacteria as valid paraphyla (therefore "monophyletic" in the sense he uses this term) taxa, marking important innovations of biological significance (in regard of the concept of biological niche ).
In the same way, his paraphyletic kingdom Protozoa includes the ancestors of Animalia, Fungi, Plantae, and Chromista. The advances of phylogenetic studies allowed Cavalier-Smith to realize that all the phyla thought to be archezoans (i.e. primitively amitochondriate eukaryotes) had in fact secondarily lost their mitochondria, typically by transforming them into new organelles: Hydrogenosomes . This means that all living eukaryotes are in fact metakaryotes , according to the significance of the term given by Cavalier-Smith. Some of the members of the defunct kingdom Archezoa , like the phylum Microsporidia , were reclassified into kingdom Fungi . Others were reclassified in kingdom Protozoa , like Metamonada which is now part of infrakingdom Excavata .
Because Cavalier-Smith allows paraphyly , the diagram below is an "organization chart", not an "ancestor chart", and does not represent an evolutionary tree.
Kingdom Bacteria — includes Archaebacteria as part of a subkingdom
Kingdom Protozoa — e.g. Amoebozoa , Choanozoa , Excavata
Kingdom Chromista — e.g. Alveolata , cryptophytes , Heterokonta ( brown algae , diatoms etc.), Haptophyta , Rhizaria
Kingdom Plantae — e.g. glaucophytes , red and green algae , land plants
Kingdom Fungi
Kingdom Animalia
Cavalier-Smith and his collaborators revised their classification in 2015. In this scheme they introduced two superkingdoms of Prokaryota and Eukaryota and seven kingdoms. Prokaryota have two kingdoms: Bacteria and Archaea . (This was based on the consensus in the Taxonomic Outline of Bacteria and Archaea , and the Catalogue of Life ). The Eukaryota have five kingdoms: Protozoa, Chromista, Plantae, Fungi, and Animalia. In this classification a protist is any of the eukaryotic unicellular organisms . [ 17 ]
Kingdom Bacteria
Kingdom Archaea
Kingdom Protozoa — e.g. Amoebozoa , Choanozoa , Excavata
Kingdom Chromista — e.g. Alveolata , cryptophytes , Heterokonta ( Brown Algae , Diatoms etc.), Haptophyta , Rhizaria
Kingdom Plantae — e.g. glaucophytes , red and green algae , land plants
Kingdom Fungi
Kingdom Animalia
The kingdom-level classification of life is still widely employed as a useful way of grouping organisms, notwithstanding some problems with this approach:
While the concept of kingdoms continues to be used by some taxonomists, there has been a movement away from traditional kingdoms, as they are no longer seen as providing a cladistic classification, where there is emphasis in arranging organisms into natural groups . [ 42 ]
Based on RNA studies, Carl Woese thought life could be divided into three large divisions and referred to them as the "three primary kingdom" model or "urkingdom" model. [ 15 ]
In 1990, the name "domain" was proposed for the highest rank. [ 5 ] This term represents a synonym for the category of dominion (lat. dominium), introduced by Moore in 1974. [ 43 ] Unlike Moore, Woese et al. (1990) did not suggest a Latin term for this category, which represents a further argument supporting the accurately introduced term dominion. [ 44 ]
Woese divided the prokaryotes (previously classified as the Kingdom Monera) into two groups, called Eubacteria and Archaebacteria , stressing that there was as much genetic difference between these two groups as between either of them and all eukaryotes.
Domain Bacteria ( Eubacteria )
Domain Archaea ( Archaebacteria )
Domain Eukarya ( Eukaryota )
According to genetic data, although eukaryote groups such as plants, fungi, and animals may look different, they are more closely related to each other than they are to either the Eubacteria or Archaea. It was also found that the eukaryotes are more closely related to the Archaea than they are to the Eubacteria. Although the primacy of the Eubacteria-Archaea divide has been questioned, it has been upheld by subsequent research. [ 22 ] There is no consensus on how many kingdoms exist in the classification scheme proposed by Woese.
In 2004, a review article by Simpson and Roger noted that the Protista were "a grab-bag for all eukaryotes that are not animals, plants or fungi". They held that only monophyletic groups should be accepted as formal ranks in a classification and that – while this approach had been impractical previously (necessitating "literally dozens of eukaryotic 'kingdoms ' ") – it had now become possible to divide the eukaryotes into "just a few major groups that are probably all monophyletic". [ 42 ]
On this basis, the diagram opposite (redrawn from their article) showed the real "kingdoms" (their quotation marks) of the eukaryotes. [ 42 ] A classification which followed this approach was produced in 2005 for the International Society of Protistologists, by a committee which "worked in collaboration with specialists from many societies". It divided the eukaryotes into the same six "supergroups". [ 45 ] The published classification deliberately did not use formal taxonomic ranks, including that of "kingdom".
prokaryotic Bacteria
prokaryotic Archaeans
various flagellate protozoa
most lobose amoeboids and slime moulds
animals , fungi , choanoflagellates , etc.
Foraminifera , Radiolaria , and various other amoeboid protozoa
Stramenopiles ( Brown Algae , Diatoms , etc. ), Haptophyta , Cryptophyta (or cryptomonads), and Alveolata
Land plants , green algae , red algae , and glaucophytes
In this system the multicellular animals ( Metazoa ) are descended from the same ancestor as both the unicellular choanoflagellates and the fungi which form the Opisthokonta . [ 45 ] Plants are thought to be more distantly related to animals and fungi.
However, in the same year as the International Society of Protistologists' classification was published (2005), doubts were being expressed as to whether some of these supergroups were monophyletic, particularly the Chromalveolata, [ 46 ] and a review in 2006 noted the lack of evidence for several of the six proposed supergroups. [ 47 ]
As of 2019 [update] , there is widespread agreement that the Rhizaria belong with the Stramenopiles and the Alveolata, in a clade dubbed the SAR supergroup , [ 48 ] so that Rhizaria is not one of the main eukaryote groups. [ 20 ] [ 49 ] [ 50 ] [ 51 ] [ 52 ]
The Prokaryotic Code treats Bacteria and Archaea each as a domain. Since 2024, each domain each contains 4 kingdoms, for a total of 8. [ 53 ]
Some authors have added non-cellular life to their classifications. This can create a "superdomain" called "Acytota", also called "Aphanobionta", of non-cellular life; with the other superdomain being " cytota " or cellular life. [ 54 ] [ 55 ] (see section below for further discussion)
The eocyte hypothesis proposes that the eukaryotes emerged from a phylum within the archaea called the Thermoproteota (formerly known as eocytes or Crenarchaeota). [ 56 ] [ 57 ]
The International Committee on Taxonomy of Viruses uses the taxonomic rank "kingdom" in the classification of viruses (with the suffix -virae ); but this is beneath the top level classifications of realm and subrealm. [ 59 ]
There is ongoing debate as to whether viruses can be included in the tree of life. The arguments against include the fact that they are obligate intracellular parasites that lack metabolism and are not capable of replication outside of a host cell. [ 60 ] [ 61 ] Another argument is that their placement in the tree would be problematic, since it is suspected that viruses have various evolutionary origins, [ 60 ] and they have a penchant for harvesting nucleotide sequences from their hosts. | https://en.wikipedia.org/wiki/Kingdom_(biology) |
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems . It can be seen as a generalization of Birkhoff's ergodic theorem . [ 1 ] Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic). [ 2 ] As a result, it can be rephrased in the language of probability, e.g. using a sequence of random variables and expected values . The theorem is named after John Kingman .
Let T {\displaystyle T} be a measure-preserving transformation on the probability space ( Ω , Σ , μ ) {\displaystyle (\Omega ,\Sigma ,\mu )} , and let { g n } n ∈ N {\displaystyle \{g_{n}\}_{n\in \mathbb {N} }} be a sequence of L 1 {\displaystyle L^{1}} functions such that g n + m ( x ) ≤ g n ( x ) + g m ( T n x ) {\displaystyle g_{n+m}(x)\leq g_{n}(x)+g_{m}(T^{n}x)} (subadditivity relation). Then
for μ {\displaystyle \mu } -a.e. x , where g ( x ) is T -invariant.
In particular, if T is ergodic , then g ( x ) is a constant.
Given a family of real random variables X ( m , n ) {\textstyle X(m,n)} , with 0 ≤ m < n ∈ N {\textstyle 0\leq m<n\in \mathbb {N} } , such that they are subadditive in the sense that X ( m + 1 , n + 1 ) = X ( m , n ) ∘ T X ( 0 , n ) ≤ X ( 0 , m ) + X ( m , n ) {\displaystyle {\begin{aligned}&X(m+1,n+1)=X(m,n)\circ T\\&X(0,n)\leq X(0,m)+X(m,n)\end{aligned}}} Then there exists a random variable Y {\textstyle Y} such that Y ∈ [ − ∞ , + ∞ ) {\textstyle Y\in [-\infty ,+\infty )} , Y {\textstyle Y} is invariant with respect to T {\textstyle T} , and lim n 1 n X ( 0 , n ) = Y {\textstyle \lim _{n}{\frac {1}{n}}X(0,n)=Y} a.s..
They are equivalent by setting
Proof due to ( J. Michael Steele , 1989). [ 3 ]
Fix some n ≥ 1 {\textstyle n\geq 1} . By subadditivity, for any l ∈ 1 : n − 1 {\textstyle l\in 1:n-1} g n ≤ g n − l + g l ∘ T n − l {\displaystyle g_{n}\leq g_{n-l}+g_{l}\circ T^{n-l}}
We can picture this as starting with the set 0 : n − 1 {\textstyle 0:n-1} , and then removing its length l {\textstyle l} tail.
Repeating this construction until the set 0 : n − 1 {\textstyle 0:n-1} is all gone, we have a one-to-one correspondence between upper bounds of g n {\textstyle g_{n}} and partitions of 1 : n − 1 {\textstyle 1:n-1} .
Specifically, let { k i : ( k i + l i − 1 ) } i {\textstyle \{k_{i}:(k_{i}+l_{i}-1)\}_{i}} be a partition of 0 : n − 1 {\textstyle 0:n-1} , then we have g n ≤ ∑ i g l i ∘ T k i {\displaystyle g_{n}\leq \sum _{i}g_{l_{i}}\circ T^{k_{i}}}
Let g := lim inf g n / n {\textstyle g:=\liminf g_{n}/n} , then it is T {\textstyle T} -invariant.
By subadditivity, g n + 1 n + 1 ≤ g 1 + g n ∘ T n + 1 {\displaystyle {\frac {g_{n+1}}{n+1}}\leq {\frac {g_{1}+g_{n}\circ T}{n+1}}}
Taking the n → ∞ {\textstyle n\to \infty } limit, we have g ≤ g ∘ T {\displaystyle g\leq g\circ T} We can visualize T {\textstyle T} as hill-climbing on the graph of g {\textstyle g} . If T {\textstyle T} actually causes a nontrivial amount of hill-climbing, then we would get a spatial contraction, and so T {\textstyle T} does not preserve measure. Therefore g = g ∘ T {\textstyle g=g\circ T} a.e.
Let c ∈ R {\textstyle c\in \mathbb {R} } , then { g ≥ c } ⊂ { g ∘ T ≥ c } = T − 1 ( { g ≥ c } ) {\displaystyle \{g\geq c\}\subset \{g\circ T\geq c\}=T^{-1}(\{g\geq c\})} and since both sides have the same measure, by squeezing, they are equal a.e..
That is, g ( x ) ≥ c ⟺ g ( T x ) ≥ c {\textstyle g(x)\geq c\iff g(Tx)\geq c} , a.e..
Now apply this for all rational c {\textstyle c} .
By subadditivity, using the partition of 0 : n − 1 {\textstyle 0:n-1} into singletons. g 1 ≤ g 1 g 2 ≤ g 1 + g 1 ∘ T g 3 ≤ g 1 + g 1 ∘ T + g 1 ∘ T 2 ⋯ {\displaystyle {\begin{aligned}g_{1}&\leq g_{1}\\g_{2}&\leq g_{1}+g_{1}\circ T\\g_{3}&\leq g_{1}+g_{1}\circ T+g_{1}\circ T^{2}\\&\cdots \end{aligned}}} Now, construct the sequence f 1 = g 1 − g 1 f 2 = g 2 − ( g 1 + g 1 ∘ T ) f 3 = g 3 − ( g 1 + g 1 ∘ T + g 1 ∘ T 2 ) ⋯ {\displaystyle {\begin{aligned}f_{1}&=g_{1}-g_{1}\\f_{2}&=g_{2}-(g_{1}+g_{1}\circ T)\\f_{3}&=g_{3}-(g_{1}+g_{1}\circ T+g_{1}\circ T^{2})\\&\cdots \end{aligned}}} which satisfies f n ≤ 0 {\textstyle f_{n}\leq 0} for all n {\textstyle n} .
By the special case, f n / n {\textstyle f_{n}/n} converges a.e. to a T {\textstyle T} -invariant function.
By Birkhoff's pointwise ergodic theorem, the running average 1 n ( g 1 + g 1 ∘ T + g 1 ∘ T 2 + ⋯ ) {\displaystyle {\frac {1}{n}}(g_{1}+g_{1}\circ T+g_{1}\circ T^{2}+\cdots )} converges a.e. to a T {\textstyle T} -invariant function. Therefore, their sum does as well.
Fix arbitrary ϵ , M > 0 {\textstyle \epsilon ,M>0} , and construct the truncated function, still T {\textstyle T} -invariant: g ′ := max ( g , − M ) {\displaystyle g':=\max(g,-M)} With these, it suffices to prove an a.e. upper bound lim sup g n / n ≤ g ′ + ϵ {\displaystyle \limsup g_{n}/n\leq g'+\epsilon } since it would allow us to take the limit ϵ = 1 / 1 , 1 / 2 , 1 / 3 , … {\textstyle \epsilon =1/1,1/2,1/3,\dots } , then the limit M = 1 , 2 , 3 , … {\textstyle M=1,2,3,\dots } , giving us a.e.
lim sup g n / n ≤ lim inf g n / n =: g {\displaystyle \limsup g_{n}/n\leq \liminf g_{n}/n=:g} And by squeezing, we have g n / n {\textstyle g_{n}/n} converging a.e. to g {\textstyle g} .
Define two families of sets, one shrinking to the empty set, and one growing to the full set. For each "length" L = 1 , 2 , 3 , … {\displaystyle L=1,2,3,\dots } , define B L := { x : g l / l > g ′ + ϵ , ∀ l ∈ 1 , 2 , … , L } {\displaystyle B_{L}:=\{x:g_{l}/l>g'+\epsilon ,\forall l\in 1,2,\dots ,L\}} A L := B L c = { x : g l / l ≤ g ′ + ϵ , ∃ l ∈ 1 , 2 , … , L } {\displaystyle A_{L}:=B_{L}^{c}=\{x:g_{l}/l\leq g'+\epsilon ,\exists l\in 1,2,\dots ,L\}} Since g ′ ≥ lim inf g n / n {\textstyle g'\geq \liminf g_{n}/n} , the B {\textstyle B} family shrinks to the empty set.
Fix x ∈ X {\textstyle x\in X} . Fix L ∈ N {\textstyle L\in \mathbb {N} } . Fix n > L {\textstyle n>L} . The ordering of these qualifiers is vitally important, because we will be removing the qualifiers one by one in the reverse order.
To prove the a.e. upper bound, we must use the subadditivity, which means we must construct a partition of the set 0 : n − 1 {\textstyle 0:n-1} . We do this inductively:
Take the smallest k {\textstyle k} not already in a partition.
If T k x ∈ A N {\textstyle T^{k}x\in A_{N}} , then g l ( T k x ) / l ≤ g ′ ( x ) + ϵ {\textstyle g_{l}(T^{k}x)/l\leq g'(x)+\epsilon } for some l ∈ 1 , 2 , … L {\textstyle l\in 1,2,\dots L} . Take one such l {\textstyle l} – the choice does not matter.
If k + l − 1 ≤ n − 1 {\textstyle k+l-1\leq n-1} , then we cut out { k , … , k + l − 1 } {\textstyle \{k,\dots ,k+l-1\}} . Call these partitions “type 1”. Else, we cut out { k } {\textstyle \{k\}} . Call these partitions “type 2”.
Else, we cut out { k } {\textstyle \{k\}} . Call these partitions “type 3”.
Now convert this partition into an inequality: g n ( x ) ≤ ∑ i g l i ( T k i x ) {\displaystyle g_{n}(x)\leq \sum _{i}g_{l_{i}}(T^{k_{i}}x)} where k i {\textstyle k_{i}} are the heads of the partitions, and l i {\textstyle l_{i}} are the lengths.
Since all g n ≤ 0 {\textstyle g_{n}\leq 0} , we can remove the other kinds of partitions: g n ( x ) ≤ ∑ i : type 1 g l i ( T k i x ) {\displaystyle g_{n}(x)\leq \sum _{i:{\text{type 1}}}g_{l_{i}}(T^{k_{i}}x)} By construction, each g l i ( T k i x ) ≤ l i ( g ′ ( x ) + ϵ ) {\textstyle g_{l_{i}}(T^{k_{i}}x)\leq l_{i}(g'(x)+\epsilon )} , thus 1 n g n ( x ) ≤ g ′ ( x ) 1 n ∑ i : type 1 l i + ϵ {\displaystyle {\frac {1}{n}}g_{n}(x)\leq g'(x){\frac {1}{n}}\sum _{i:{\text{type 1}}}l_{i}+\epsilon } Now it would be tempting to continue with g ′ ( x ) 1 n ∑ i : type 1 l i ≤ g ′ ( x ) {\textstyle g'(x){\frac {1}{n}}\sum _{i:{\text{type 1}}}l_{i}\leq g'(x)} , but unfortunately g ′ ≤ 0 {\textstyle g'\leq 0} , so the direction is the exact opposite. We must lower bound the sum ∑ i : type 1 l i {\textstyle \sum _{i:{\text{type 1}}}l_{i}} .
The number of type 3 elements is equal to ∑ k ∈ 0 : n − 1 1 B L ( T k x ) {\displaystyle \sum _{k\in 0:n-1}1_{B_{L}}(T^{k}x)} If a number k {\textstyle k} is of type 2, then it must be inside the last L − 1 {\textstyle L-1} elements of 0 : n − 1 {\textstyle 0:n-1} . Thus the number of type 2 elements is at most L − 1 {\textstyle L-1} .
Together, we have the lower bound: 1 n ∑ i : type 1 l i ≥ 1 − L − 1 n − 1 n ∑ k ∈ 0 : n − 1 1 B L ( T k x ) {\displaystyle {\frac {1}{n}}\sum _{i:{\text{type 1}}}l_{i}\geq 1-{\frac {L-1}{n}}-{\frac {1}{n}}\sum _{k\in 0:n-1}1_{B_{L}}(T^{k}x)}
Remove the n > L {\textstyle n>L} qualifier by taking the n → ∞ {\textstyle n\to \infty } limit.
By Birkhoff's pointwise ergodic theorem, there exists an a.e. pointwise limit lim n 1 n ∑ k ∈ 0 : n − 1 1 B L ( T k x ) → 1 ¯ B L ( x ) {\displaystyle \lim _{n}{\frac {1}{n}}\sum _{k\in 0:n-1}1_{B_{L}}(T^{k}x)\to {\bar {1}}_{B_{L}}(x)} satisfying ∫ 1 ¯ B L = μ ( B L ) ; 1 ¯ B L ( x ) ∈ [ 0 , 1 ] {\displaystyle \int {\bar {1}}_{B_{L}}=\mu (B_{L});\quad {\bar {1}}_{B_{L}}(x)\in [0,1]} At the limit, we find that for a.e. x ∈ X , L ∈ N {\textstyle x\in X,L\in \mathbb {N} } , lim sup n g n ( x ) n ≤ g ′ ( x ) ( 1 − 1 ¯ B L ( x ) ) + ϵ {\displaystyle \limsup _{n}{\frac {g_{n}(x)}{n}}\leq g'(x)(1-{\bar {1}}_{B_{L}}(x))+\epsilon }
Remove the L ∈ N {\textstyle L\in \mathbb {N} } qualifier by taking the L → ∞ {\textstyle L\to \infty } limit.
Since we have ∫ 1 ¯ B L = μ ( B L ) → 0 {\displaystyle \int {\bar {1}}_{B_{L}}=\mu (B_{L})\to 0} and 1 ¯ B L ≥ 1 ¯ B L + 1 ≥ ⋯ {\displaystyle {\bar {1}}_{B_{L}}\geq {\bar {1}}_{B_{L+1}}\geq \cdots } as 1 B L ≥ 1 B L + 1 ≥ ⋯ {\displaystyle 1_{B_{L}}\geq 1_{B_{L+1}}\geq \cdots } , we can apply the same argument used for proving Markov's inequality , to obtain lim sup n g n ( x ) n ≤ g ′ ( x ) + ϵ {\displaystyle \limsup _{n}{\frac {g_{n}(x)}{n}}\leq g'(x)+\epsilon } for a.e. x ∈ X {\textstyle x\in X} .
In detail, the argument is as follows: since 1 ¯ B L ≥ 1 ¯ B L + 1 ≥ ⋯ ≥ 0 {\displaystyle {\bar {1}}_{B_{L}}\geq {\bar {1}}_{B_{L+1}}\geq \cdots \geq 0} , and ∫ 1 ¯ B L → 0 {\displaystyle \int {\bar {1}}_{B_{L}}\to 0} , we know that for any small δ , δ ′ > 0 {\displaystyle \delta ,\delta '>0} , all large enough L {\displaystyle L} satisfies 1 ¯ B L ( x ) < δ {\displaystyle {\bar {1}}_{B_{L}}(x)<\delta } everywhere except on a set of size ≥ δ ′ {\displaystyle \geq \delta '} . Thus, lim sup n g n ( x ) n ≤ g ′ ( x ) ( 1 − δ ) + ϵ {\displaystyle \limsup _{n}{\frac {g_{n}(x)}{n}}\leq g'(x)(1-\delta )+\epsilon } with probability ≥ 1 − δ ′ {\displaystyle \geq 1-\delta '} . Now take both δ , δ ′ → 0 {\displaystyle \delta ,\delta '\to 0} .
Taking g n ( x ) := ∑ j = 0 n − 1 f ( T j x ) {\displaystyle g_{n}(x):=\sum _{j=0}^{n-1}f(T^{j}x)} recovers Birkhoff's pointwise ergodic theorem.
Taking all g n {\displaystyle g_{n}} constant functions, we recover the Fekete's subadditive lemma.
Kingman's subadditive ergodic theorem can be used to prove statements about Lyapunov exponents . It also has applications to percolations and longest increasing subsequence . [ 4 ]
To study the longest increasing subsequence of a random permutation π {\displaystyle \pi } , we generate it in an equivalent way. A random permutation on 1 : n {\displaystyle 1:n} is equivalently generated by uniformly sampling n {\displaystyle n} points in a square, then find the longest increasing subsequence of that.
Now, define the Poisson point process with density 1 on [ 0 , ∞ ) 2 {\displaystyle [0,\infty )^{2}} , and define the random variables M k ∗ {\displaystyle M_{k}^{*}} to be the length of the longest increasing subsequence in the square [ 0 , k ) 2 {\displaystyle [0,k)^{2}} . Define the measure-preserving transform T {\displaystyle T} by shifting the plane by ( − 1 , − 1 ) {\displaystyle (-1,-1)} , then chopping off the parts that have fallen out of [ 0 , ∞ ) 2 {\displaystyle [0,\infty )^{2}} .
The process is subadditive, that is, M k + m ∗ ≥ M k ∗ + M m ∗ ∘ T k {\displaystyle M_{k+m}^{*}\geq M_{k}^{*}+M_{m}^{*}\circ T^{k}} . To see this, notice that the right side constructs an increasing subsequence first in the square [ 0 , k ) 2 {\displaystyle [0,k)^{2}} , then in the square [ k , k + m ) 2 {\displaystyle [k,k+m)^{2}} , and finally concatenate them together. This produces an increasing subsequence in [ 0 , k + m ) 2 {\displaystyle [0,k+m)^{2}} , but not necessarily the longest one.
Also, T {\displaystyle T} is ergodic, so by Kingman's theorem, M k ∗ / k {\displaystyle M_{k}^{*}/k} converges to a constant almost surely. Since at the limit, there are n = k 2 {\displaystyle n=k^{2}} points in the square, we have L n ∗ / n {\displaystyle L_{n}^{*}/{\sqrt {n}}} converging to a constant almost surely. | https://en.wikipedia.org/wiki/Kingman's_subadditive_ergodic_theorem |
Kinks are deviations of a dislocation defect along its glide plane. In edge dislocations, the constant glide plane allows short regions of the dislocation to turn, converting into screw dislocations and producing kinks. Screw dislocations have rotatable glide planes, thus kinks that are generated along screw dislocations act as an anchor for the glide plane. [ 1 ] Kinks differ from jogs in that kinks are strictly parallel to the glide plane, while jogs shift away from the glide plane.
Pure-edge and screw dislocations are conceptually straight in order to minimize its length, and through it, the strain energy of the system. Low-angle mixed dislocations, on the other hand, can be thought of as primarily edge dislocation with screw kinks in a stair-case structure (or vice versa), switching between straight pure-edge and pure-screw dislocation segments. In reality, kinks are not sharp transitions. Both the total length of the dislocation and the kink angle are dependent on the free energy of the system. The primary dislocation regions lie in Peierls-Nabarro potential minima, while the kink requires addition energy in the form of an energy peak. To minimize free energy, the kink equilibrates at a certain length and angle. Large energy peaks create short but sharp kinks in order to minimize dislocation length within the high energy region, while small energy peaks create long and drawn-out kinks in order to minimize total dislocation length. [ 2 ]
Kinks facilitate the movement of dislocations along its glide plane under shear stress , and is directly responsible for plastic deformation of crystals. When a crystal undergoes shear force, e.g. cut with scissors, the applied shear force causes dislocations to move through the material, displacing atoms and deforming the material. The entire dislocation does not move at once – rather, the dislocation produces a pair of kinks, which then propagates in opposite directions down the length of the dislocation, eventually shifting the entire dislocation by a Burgers vector . The velocity of dislocations through kink propagation also clearly limited on the nucleation frequency of kinks, as a lack of kinks compromises the mechanism by which dislocations move.
As shear force approaches infinity, the velocity at which dislocations migrate is limited by the physical properties of the material, maximizing at the material's sound velocity. At lower shear stresses, the velocity of dislocations end up relating exponentially with the applied shear force:
where
The above equation gives the upper limit on dislocation velocity. The interactions of dislocation movement on its environment, particularly other defects such as jogs and precipitates , results in drag and slows down the dislocation: [ 3 ]
where
Kink movement is strongly dependent on temperature as well. Higher thermal energy assists in the generation of kinks, as well as increasing atomic vibrations and promoting dislocation motion.
Kinks may also form under compressive stress due to the buckling of crystal planes into a cavity. At high compressive forces, masses of dislocations move at once. Kinks align with each other, forming walls of kinks that propagate all at once. [ 4 ] At sufficient forces, the tensile force produced by the dislocation core exceeds the fracture stress of the material, combining kink boundaries into sharp kinks and de-laminating the basal planes of the crystal. | https://en.wikipedia.org/wiki/Kink_(materials_science) |
In organophosphorus chemistry , the Kinnear–Perren reaction (sometimes the Clay-Kinnear-Perren reaction) is used to prepare alkylphosphonyl dichlorides (RP(O)Cl 2 ) and alkyl phosphonate esters (RP(O)(OR') 2 ). [ 1 ] The reactants are alkyl chloride, phosphorus trichloride , and aluminium trichloride as catalyst. [ 2 ] The reaction proceeds via the alkyltrichloro phosphonium salt:
Reduction of this trichlorophosphonium intermediate with aluminium powder gives alkyldichlorophosphines (RPCl 2 ).
Partial hydrolysis of the same intermediate gives the alkylphosphonyl dichloride:
The reaction was first reported by Clay [ 3 ] and expanded upon by Kinnear and Perren, who demonstrated that the four chlorinated methanes (CH 4−x Cl x ) give the corresponding CH 3 -, CH 2 Cl-, CHCl 2 -, and CCl 3 -substituted derivatives. They also demonstrated workup with hydrogen sulfide to give the alkylthiophosphoryl dichlorides. [ 4 ] | https://en.wikipedia.org/wiki/Kinnear–Perren_reaction |
The Kinyoun method or Kinyoun stain (cold method), developed by Joseph J. Kinyoun , is a procedure used to stain acid-fast species of the bacterial genus Mycobacterium . [ 1 ] It is a variation of a method developed by Robert Koch in 1882. Certain species of bacteria have a waxy lipid called mycolic acid, in their cell walls which allow them to be stained with Acid-Fast better than a Gram-Stain. The unique ability of mycobacteria to resist decolorization by acid-alcohol is why they are termed acid-fast. [ 2 ] It involves the application of a primary stain ( basic fuchsin ), a decolorizer (acid-alcohol), and a counterstain ( methylene blue ). [ 3 ] Unlike the Ziehl–Neelsen stain (Z-N stain), the Kinyoun method of staining does not require heating. [ 4 ] [ 5 ] In the Ziehl–Neelsen stain, heat acts as a physical mordant while phenol ( carbol of carbol fuchsin ) acts as the chemical mordant.
The Kinyoun method can be modified as a weak acid fast stain, which uses 0.5–1.0% sulfuric acid instead of hydrochloric acid . The weak acid fast stain, in addition to staining Mycobacteria , will also stain organisms that are not able to maintain the carbol fuchsin after decolorizing with HCl, such as Nocardia species and Cryptosporidium .
This microbiology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kinyoun_stain |
Kipp's apparatus , also called a Kipp generator , is an apparatus designed for preparation of small volumes of gases . It was invented around 1844 by the Dutch pharmacist Petrus Jacobus Kipp and widely used in chemical laboratories and for demonstrations in schools into the second half of the 20th century.
It later fell out of use, at least in laboratories, because most gases then became available in small gas cylinders . These industrial gases are much purer and drier than those initially obtained from a Kipp apparatus without further processing.
The apparatus is usually made of glass , or sometimes of polyethylene , and consists of three vertically stacked chambers, roughly resembling a snowman. The upper chamber extends downward as a tube that passes through the middle chamber into the lower chamber. There is no direct path between the middle and upper chambers, but the middle chamber is separated from the lower chamber by a retention plate, such as a conical piece of glass with small holes, which permits the passage of liquid and gas. The solid material (e.g., iron sulfide) is placed into the middle chamber in lumps sufficiently large to avoid falling through the retention plate. The liquid, such as an acid, is poured into the top chamber. Although the acid is free to flow down through the tube into the bottom chamber, it is prevented from rising there by the pressure of the gas contained above it, which is able to leave the apparatus only by a stopcock near the top of the middle chamber. This stopcock may be opened, initially to permit the air to leave the apparatus, allowing the liquid in the bottom chamber to rise through the retention plate into the middle chamber and react with the solid material. Gas is evolved from this reaction, which may be drawn off through the stopcock as desired. When the stopcock is closed, the pressure of the evolved gas in the middle chamber rises and pushes the acid back down into the bottom chamber, until it is not in contact with the solid material anymore. At that point the chemical reaction comes to a stop, until the stopcock is opened again and more gas is drawn off.
Kipp generators only work properly in the described manner if the solid material is insoluble in the acid, as otherwise the dissolved material would continue to evolve gas even after the level dropped. The produced gas often requires further purification and/or drying, due to content of water vapor and possibly mist if the reaction is vigorous.
For successful use in a Kipp's apparatus, the solid material has to be available in lumps large enough to stay on the retention plate without falling through its holes.
Generally, weak acidic gases can be released from their metal salts by dilute acids, and sometimes just with water: [ 1 ]
A version of the apparatus can be used for reaction between two liquid precursors. A mercury trap has to be added as a check valve, and the middle bulb is filled with an inert porous material, e.g. pumice , onto which one of the precursors is dropped. [ 3 ]
The prepared gas is usually impure, contaminated with fine aerosol of the reagents and water vapor. The gases may need to be filtered, washed and dried before further use.
Hydrogen can be washed from sulfane, arsine and oxygen with subsequent bubbling through solutions of lead acetate , silver nitrate , and alkaline pyrogallic acid . [ 4 ]
Acidic gases (e.g. hydrogen sulfide, hydrogen chloride, sulfur dioxide) can be dried with concentrated sulfuric acid, or with phosphorus pentoxide . Basic gases (e.g. ammonia) can be dried with calcium oxide , sodium hydroxide or soda lime .
Disposal of the gases can be done by burning the flammable ones (carbon monoxide, hydrogen, hydrocarbons), absorbing them in water (ammonia, hydrogen sulfide, sulfur dioxide, chlorine), or reacting them with a suitable reagent. [ 2 ]
Many variants of the gas production apparatus exist. Some are suitable for production of larger amounts of gases (Gay-Lussac and Verkhovsky), some for smaller amounts (Kiryushkin, U-tube).
A Döbereiner's lamp is a small modified Kipp's apparatus for production of hydrogen. The hydrogen is led over a platinum sponge catalyst , where it reacts with air oxygen, heats the catalyst and ignites from it, producing a gentle flame. It was commercialized for lighting fires and pipes. It's said that in 1820s over a million of the "tinderboxes" ("Feuerzeug") was sold. [ 5 ] | https://en.wikipedia.org/wiki/Kipp's_apparatus |
The Kirbee Kiln Site is a 19th-century kiln ruin located in Montgomery County, Texas , where stoneware was manufactured by the Kirbee family. It is one of the largest groundhog kilns ever recorded in the American South. The exact location of the site is restricted. It was listed on the National Register of Historic Places in 1973.
The Kirbee Kiln was founded and operated by James Kirbee, who was originally from Edgefield, South Carolina , and had relatives and acquaintances who were also potters. One of his acquaintances might have been David Drake , a potter who was enslaved by Kirbee's associate Rev. John Landrum. By 1830, Kirbee and his family had relocated to Georgia; and by 1840, they had migrated to Montgomery County, Texas. [ 3 ] The kiln itself was likely built around 1849, as it appeared in the 1850 Schedule of Industry and Manufacture. James was likely assisted by his sons M.J. and Louis. [ 4 ] The annual value of the stoneware produced did not exceed $500, much lower than other local kilns. [ 3 ] The kiln likely ceased operations in the 1860s. [ 4 ]
The site was one of several kilns surveyed by the Texas Historical Commission between 1973 and 1974. [ 5 ] It was listed on the National Register of Historic Places on August 28, 1973. [ 1 ] It was the first site from in the area to be added to the NRHP. [ 6 ]
At the time of the archaeological surveys in the 1970s, the Texas Historical Commission named the Kirbee Kiln Site as the largest groundhog kiln that had then been excavated in Texas, and it remains one of the largest ever recorded in the American South. [ 3 ] It measured 39 feet (12 m) across and 8 to 10 inches wide and was constructed of brick. The kiln was rectangular in shape, consisting of an opening at the very front for loading and firing, a depressed firebox , the loading shelf in the middle, and a fireplace-shaped chimney at the very back. A unique feature of this kiln was the presence of a second firing box located midway along the loading shelf; a side door would have provided access. The chimney is believed by the excavators to have decreased in width towards its top. The buttresses of the Kirbee Kiln were large and angled but also included several smaller ones, a rare feature that could have functioned to support its size, offer resistance against the sloped ground, and double as a retaining wall . The entire floor of the kiln was sandy soil. [ 5 ]
Kirbee's stoneware had similarities to techniques observed elsewhere in Georgia and South Carolina, particularly the alkaline glaze that was characteristic of contemporary Edgefield stoneware; and the vessels were also comparable in features such as their handles and shape. This style of pottery is very similar to Catawba Valley Pottery , which was developed in nearby North Carolina. The trademarks on the Kirbee stoneware were a round stamp resembling the letter "O". [ 3 ] | https://en.wikipedia.org/wiki/Kirbee_Kiln_Site |
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits . They were first described in 1845 by German physicist Gustav Kirchhoff . [ 1 ] This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell . Widely used in electrical engineering , they are also called Kirchhoff's rules or simply Kirchhoff's laws . These laws can be applied in time and frequency domains and form the basis for network analysis .
Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.
This law, also called Kirchhoff's first law , or Kirchhoff's junction rule , states that, for any node (junction) in an electrical circuit , the sum of currents flowing into that node is equal to the sum of currents flowing out of that node; or equivalently:
The algebraic sum of currents in a network of conductors meeting at a point is zero.
Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as: ∑ i = 1 n I i = 0 {\displaystyle \sum _{i=1}^{n}I_{i}=0} where n is the total number of branches with currents flowing towards or away from the node.
Kirchhoff's circuit laws were originally obtained from experimental results. However, the current law can be viewed as an extension of the conservation of charge , since charge is the product of current and the time the current has been flowing. If the net charge in a region is constant, the current law will hold on the boundaries of the region. [ 2 ] [ 3 ] This means that the current law relies on the fact that the net charge in the wires and components is constant.
A matrix version of Kirchhoff's current law is the basis of most circuit simulation software , such as SPICE . The current law is used with Ohm's law to perform nodal analysis .
The current law is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear.
This law, also called Kirchhoff's second law , or Kirchhoff's loop rule , states the following:
The directed sum of the potential differences (voltages) around any closed loop is zero.
Similarly to Kirchhoff's current law, the voltage law can be stated as: ∑ i = 1 n V i = 0 {\displaystyle \sum _{i=1}^{n}V_{i}=0}
Here, n is the total number of voltages measured.
A similar derivation can be found in The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits . [ 3 ]
Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation reveals that ∇ × E = − ∂ B ∂ t = 0 {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} } in the exterior region. If each of the components has a finite volume, then the exterior region is simply connected , and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that ∑ i V i = − ∑ i ∫ P i E ⋅ d l = ∮ E ⋅ d l = 0 {\displaystyle \sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0} where P i {\textstyle {\mathcal {P}}_{i}} are paths around the exterior of each of the components, from one terminal to another.
Note that this derivation uses the following definition for the voltage rise from a {\displaystyle a} to b {\displaystyle b} : V a → b = − ∫ P a → b E ⋅ d l {\displaystyle V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }
However, the electric potential (and thus voltage) can be defined in other ways, such as via the Helmholtz decomposition .
In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of Faraday's law of induction (which is one of Maxwell's equations ).
This has practical application in situations involving " static electricity ".
Kirchhoff's circuit laws are the result of the lumped-element model and both depend on the model being applicable to the circuit in question. When the model is not applicable, the laws do not apply.
The current law is dependent on the assumption that the net charge in any wire, junction or lumped component is constant. Whenever the electric field between parts of the circuit is non-negligible, such as when two wires are capacitively coupled , this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. [ 4 ] For example, in a transmission line , the charge density in the conductor may be constantly changing.
On the other hand, the voltage law relies on the fact that the actions of time-varying magnetic fields are confined to individual components, such as inductors. In reality, the induced electric field produced by an inductor is not confined, but the leaked fields are often negligible.
The lumped element approximation for a circuit is accurate at low frequencies. At higher frequencies, leaked fluxes and varying charge densities in conductors become significant. To an extent, it is possible to still model such circuits using parasitic components . If frequencies are too high, it may be more appropriate to simulate the fields directly using finite element modelling or other techniques .
To model circuits so that both laws can still be used, it is important to understand the distinction between physical circuit elements and the ideal lumped elements. For example, a wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively couple to each other (and to themselves), and have a finite propagation delay. Real conductors can be modeled in terms of lumped elements by considering parasitic capacitances distributed between the conductors to model capacitive coupling, or parasitic (mutual) inductances to model inductive coupling. [ 4 ] Wires also have some self-inductance.
Assume an electric network consisting of two voltage sources and three resistors.
According to the first law: i 1 − i 2 − i 3 = 0 {\displaystyle i_{1}-i_{2}-i_{3}=0} Applying the second law to the closed circuit s 1 , and substituting for voltage using Ohm's law gives: − R 2 i 2 + E 1 − R 1 i 1 = 0 {\displaystyle -R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0} The second law, again combined with Ohm's law, applied to the closed circuit s 2 gives: − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle -R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0}
This yields a system of linear equations in i 1 , i 2 , i 3 : { i 1 − i 2 − i 3 = 0 − R 2 i 2 + E 1 − R 1 i 1 = 0 − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle {\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}} which is equivalent to { i 1 + ( − i 2 ) + ( − i 3 ) = 0 R 1 i 1 + R 2 i 2 + 0 i 3 = E 1 0 i 1 + R 2 i 2 − R 3 i 3 = E 1 + E 2 {\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}} Assuming R 1 = 100 Ω , R 2 = 200 Ω , R 3 = 300 Ω , E 1 = 3 V , E 2 = 4 V {\displaystyle {\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}} the solution is { i 1 = 1 1100 A i 2 = 4 275 A i 3 = − 3 220 A {\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}}
The current i 3 has a negative sign which means the assumed direction of i 3 was incorrect and i 3 is actually flowing in the direction opposite to the red arrow labeled i 3 . The current in R 3 flows from left to right. | https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws |
In the mathematical field of graph theory , Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph , showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix ; specifically, the number is equal to any cofactor of the Laplacian matrix. Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph .
Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph, which is equal to the difference between the graph's degree matrix (the diagonal matrix of vertex degrees ) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise).
For a given connected graph G with n labeled vertices , let λ 1 , λ 2 , ..., λ n −1 be the non-zero eigenvalues of its Laplacian matrix. Then the number of spanning trees of G is
An English translation of Kirchhoff's original 1847 paper was made by J. B. O'Toole and published in 1958. [ 1 ]
First, construct the Laplacian matrix Q for the example diamond graph G (see image on the right):
Next, construct a matrix Q * by deleting any row and any column from Q . For example, deleting row 1 and column 1 yields
Finally, take the determinant of Q * to obtain t ( G ), which is 8 for the diamond graph. (Notice t ( G ) is the (1,1)-cofactor of Q in this example.)
(The proof below is based on the Cauchy–Binet formula . An elementary induction argument for Kirchhoff's theorem can be found on page 654 of Moore (2011). [ 2 ] )
First notice that the Laplacian matrix has the property that the sum of its entries across any row and any column is 0. Thus we can transform any minor into any other minor by adding rows and columns, switching them, and multiplying a row or a column by −1. Thus the cofactors are the same up to sign, and it can be verified that, in fact, they have the same sign.
We proceed to show that the determinant of the minor M 11 is the number of spanning trees. Let n be the number of vertices of the graph, and m the number of its edges. The incidence matrix E is an n -by- m matrix, which may be defined as follows: suppose that ( i , j ) is the k th edge of the graph, and that i < j . Then E ik = 1, E jk = −1, and all other entries in column k are 0 (see oriented incidence matrix for understanding this modified incidence matrix E ). For the preceding example (with n = 4 and m = 5):
Recall that the Laplacian L can be factored into the product of the incidence matrix and its transpose , i.e., L = EE T . Furthermore, let F be the matrix E with its first row deleted, so that FF T = M 11 .
Now the Cauchy–Binet formula allows us to write
where S ranges across subsets of [ m ] of size n − 1, and F S denotes the ( n − 1)-by-( n − 1) matrix whose columns are those of F with index in S . Then every S specifies n − 1 edges of the original graph, and it can be shown that those edges induce a spanning tree if and only if the determinant of F S is +1 or −1, and that they do not induce a spanning tree if and only if the determinant is 0. This completes the proof.
Cayley's formula follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, −1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n . These vectors together span a space of dimension n − 1, so there are no other non-zero eigenvalues.
Alternatively, note that as Cayley's formula gives the number of distinct labeled trees of a complete graph K n we need to compute any cofactor of the Laplacian matrix of K n . The Laplacian matrix in this case is
Any cofactor of the above matrix is n n −2 , which is Cayley's formula.
Kirchhoff's theorem holds for multigraphs as well; the matrix Q is modified as follows:
Cayley's formula for a complete multigraph is m n −1 ( n n −1 −( n −1) n n −2 ) by same methods produced above, since a simple graph is a multigraph with m = 1.
Kirchhoff's theorem can be strengthened by altering the definition of the Laplacian matrix. Rather than merely counting edges emanating from each vertex or connecting a pair of vertices, label each edge with an indeterminate and let the ( i , j )-th entry of the modified Laplacian matrix be the sum over the indeterminates corresponding to edges between the i -th and j -th vertices when i does not equal j , and the negative sum over all indeterminates corresponding to edges emanating from the i -th vertex when i equals j .
The determinant of the modified Laplacian matrix by deleting any row and column (similar to finding the number of spanning trees from the original Laplacian matrix), above is then a homogeneous polynomial (the Kirchhoff polynomial) in the indeterminates corresponding to the edges of the graph. After collecting terms and performing all possible cancellations, each monomial in the resulting expression represents a spanning tree consisting of the edges corresponding to the indeterminates appearing in that monomial. In this way, one can obtain explicit enumeration of all the spanning trees of the graph simply by computing the determinant.
For a proof of this version of the theorem see Bollobás (1998). [ 3 ]
The spanning trees of a graph form the bases of a graphic matroid , so Kirchhoff's theorem provides a formula for the number of bases in a graphic matroid. The same method may also be used to determine the number of bases in regular matroids , a generalization of the graphic matroids ( Maurer 1976 ).
Kirchhoff's theorem can be modified to give the number of oriented spanning trees in directed multigraphs.
The matrix Q is constructed as follows:
The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the i th row and column of Q
Kirchhoff's theorem can be generalized to count k -component spanning forests in an unweighted graph. [ 4 ] A k -component spanning forest is a subgraph with k connected components that contains all vertices and is cycle-free, i.e., there is at most one path between each pair of vertices. Given such a forest F with connected components F 1 , … , F k {\textstyle F_{1},\dots ,F_{k}} , define its weight w ( F ) = | V ( F 1 ) | ⋅ ⋯ ⋅ | V ( F k ) | {\textstyle w(F)=|V(F_{1})|\cdot \dots \cdot |V(F_{k})|} to be the product of the number of vertices in each component. Then
where the sum is over all k -component spanning forests and q k {\textstyle q_{k}} is the coefficient of x k {\textstyle x^{k}} of the polynomial
The last factor in the polynomial is due to the zero eigenvalue λ n = 0 {\textstyle \lambda _{n}=0} . More explicitly, the number q k {\textstyle q_{k}} can be computed as
where the sum is over all n − k -element subsets of { 1 , … , n } {\textstyle \{1,\dots ,n\}} . For example
q n − 1 = λ 1 + ⋯ + λ n − 1 = t r Q = 2 | E | q n − 2 = λ 1 λ 2 + λ 1 λ 3 + ⋯ + λ n − 2 λ n − 1 q 2 = λ 1 … λ n − 2 + λ 1 … λ n − 3 λ n − 1 + ⋯ + λ 2 … λ n − 1 q 1 = λ 1 … λ n − 1 {\displaystyle {\begin{aligned}q_{n-1}&=\lambda _{1}+\dots +\lambda _{n-1}=\mathrm {tr} Q=2|E|\\q_{n-2}&=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\dots +\lambda _{n-2}\lambda _{n-1}\\q_{2}&=\lambda _{1}\dots \lambda _{n-2}+\lambda _{1}\dots \lambda _{n-3}\lambda _{n-1}+\dots +\lambda _{2}\dots \lambda _{n-1}\\q_{1}&=\lambda _{1}\dots \lambda _{n-1}\\\end{aligned}}}
Since a spanning forest with n −1 components corresponds to a single edge, the k = n −1 case states that the sum of the eigenvalues of Q is twice the number of edges. The k = 1 case corresponds to the original Kirchhoff theorem since the weight of every spanning tree is n .
The proof can be done analogously to the proof of Kirchhoff's theorem. An invertible ( n − k ) × ( n − k ) {\displaystyle (n-k)\times (n-k)} submatrix of the incidence matrix corresponds bijectively to a k -component spanning forest with a choice of vertex for each component.
The coefficients q k {\textstyle q_{k}} are up to sign the coefficients of the characteristic polynomial of Q . | https://en.wikipedia.org/wiki/Kirchhoff's_theorem |
In fluid dynamics , the Kirchhoff equations , named after Gustav Kirchhoff , describe the motion of a rigid body in an ideal fluid .
d d t ∂ T ∂ ω = ∂ T ∂ ω × ω + ∂ T ∂ v × v + Q h + Q , d d t ∂ T ∂ v = ∂ T ∂ v × ω + F h + F , T = 1 2 ( ω T I ~ ω + m v 2 ) Q h = − ∫ p x × n ^ d σ , F h = − ∫ p n ^ d σ {\displaystyle {\begin{aligned}{\mathrm {d} \over \mathrm {d} t}{{\partial T} \over {\partial {\boldsymbol {\omega }}}}&={{\partial T} \over {\partial {\boldsymbol {\omega }}}}\times {\boldsymbol {\omega }}+{{\partial T} \over {\partial \mathbf {v} }}\times \mathbf {v} +\mathbf {Q} _{h}+\mathbf {Q} ,\\[10pt]{\mathrm {d} \over \mathrm {d} t}{{\partial T} \over {\partial \mathbf {v} }}&={{\partial T} \over {\partial \mathbf {v} }}\times {\boldsymbol {\omega }}+\mathbf {F} _{h}+\mathbf {F} ,\\[10pt]T&={1 \over 2}\left({\boldsymbol {\omega }}^{T}{\tilde {I}}{\boldsymbol {\omega }}+mv^{2}\right)\\[10pt]\mathbf {Q} _{h}&=-\int p\mathbf {x} \times {\hat {\mathbf {n} }}\,d\sigma ,\\[10pt]\mathbf {F} _{h}&=-\int p{\hat {\mathbf {n} }}\,d\sigma \end{aligned}}}
where ω {\displaystyle {\boldsymbol {\omega }}} and v {\displaystyle \mathbf {v} } are the angular and linear velocity vectors at the point x {\displaystyle \mathbf {x} } , respectively; I ~ {\displaystyle {\tilde {I}}} is the moment of inertia tensor, m {\displaystyle m} is the body's mass; n ^ {\displaystyle {\hat {\mathbf {n} }}} is
a unit normal vector to the surface of the body at the point x {\displaystyle \mathbf {x} } ; p {\displaystyle p} is a pressure at this point; Q h {\displaystyle \mathbf {Q} _{h}} and F h {\displaystyle \mathbf {F} _{h}} are the hydrodynamic
torque and force acting on the body, respectively; Q {\displaystyle \mathbf {Q} } and F {\displaystyle \mathbf {F} } likewise denote all other torques and forces acting on the
body. The integration is performed over the fluid-exposed portion of the
body's surface.
If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors Q h {\displaystyle \mathbf {Q} _{h}} and F h {\displaystyle \mathbf {F} _{h}} can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations:
d d t ∂ L ∂ ω = ∂ L ∂ ω × ω + ∂ L ∂ v × v , d d t ∂ L ∂ v = ∂ L ∂ v × ω , {\displaystyle {\mathrm {d} \over \mathrm {d} t}{{\partial L} \over {\partial {\boldsymbol {\omega }}}}={{\partial L} \over {\partial {\boldsymbol {\omega }}}}\times {\boldsymbol {\omega }}+{{\partial L} \over {\partial \mathbf {v} }}\times \mathbf {v} ,\quad {\mathrm {d} \over \mathrm {d} t}{{\partial L} \over {\partial \mathbf {v} }}={{\partial L} \over {\partial \mathbf {v} }}\times {\boldsymbol {\omega }},}
L ( ω , v ) = 1 2 ( A ω , ω ) + ( B ω , v ) + 1 2 ( C v , v ) + ( k , ω ) + ( l , v ) . {\displaystyle L({\boldsymbol {\omega }},\mathbf {v} )={1 \over 2}(A{\boldsymbol {\omega }},{\boldsymbol {\omega }})+(B{\boldsymbol {\omega }},\mathbf {v} )+{1 \over 2}(C\mathbf {v} ,\mathbf {v} )+(\mathbf {k} ,{\boldsymbol {\omega }})+(\mathbf {l} ,\mathbf {v} ).}
Their first integrals read J 0 = ( ∂ L ∂ ω , ω ) + ( ∂ L ∂ v , v ) − L , J 1 = ( ∂ L ∂ ω , ∂ L ∂ v ) , J 2 = ( ∂ L ∂ v , ∂ L ∂ v ) . {\displaystyle J_{0}=\left({{\partial L} \over {\partial {\boldsymbol {\omega }}}},{\boldsymbol {\omega }}\right)+\left({{\partial L} \over {\partial \mathbf {v} }},\mathbf {v} \right)-L,\quad J_{1}=\left({{\partial L} \over {\partial {\boldsymbol {\omega }}}},{{\partial L} \over {\partial \mathbf {v} }}\right),\quad J_{2}=\left({{\partial L} \over {\partial \mathbf {v} }},{{\partial L} \over {\partial \mathbf {v} }}\right).}
Further integration produces explicit expressions for position and velocities.
This fluid dynamics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kirchhoff_equations |
The Kiritimatiellota are a phylum of bacteria . [ 3 ]
This bacteria -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kiritimatiellota |
The Kirkendall effect is the motion of the interface between two metals that occurs due to the difference in diffusion rates of the metal atoms. The effect can be observed, for example, by placing insoluble markers at the interface between a pure metal and an alloy containing that metal, and heating to a temperature where atomic diffusion is reasonable for the given timescale; the boundary will move relative to the markers.
This process was named after Ernest Kirkendall (1914–2005), assistant professor of chemical engineering at Wayne State University from 1941 to 1946. The paper describing the discovery of the effect was published in 1947. [ 1 ]
The Kirkendall effect has important practical consequences. One of these is the prevention or suppression of voids formed at the boundary interface in various kinds of alloy-to-metal bonding. These are referred to as Kirkendall voids .
The Kirkendall effect was discovered by Ernest Kirkendall and Alice Smigelskas in 1947, in the course of Kirkendall's ongoing research into diffusion in brass . [ 2 ] The paper in which he discovered the famous effect was the third in his series of papers on brass diffusion, the first being his thesis. His second paper revealed that zinc diffused more quickly than copper in alpha-brass , which led to the research producing his revolutionary theory. Until this point, substitutional and ring methods were the dominant ideas for diffusional motion. Kirkendall's experiment produced evidence of a vacancy diffusion mechanism, which is the accepted mechanism to this day. At the time it was submitted, the paper and Kirkendall's ideas were rejected from publication by Robert Franklin Mehl , director of the Metals Research Laboratory at Carnegie Institute of Technology (now Carnegie Mellon University ). Mehl refused to accept Kirkendall's evidence of this new diffusion mechanism and denied publication for over six months, only relenting after a conference was held and several other researchers confirmed Kirkendall's results. [ 2 ]
A bar of brass (70% Cu, 30% Zn) was used as a core, with molybdenum wires stretched along its length, and then coated in a layer of pure copper. Molybdenum was chosen as the marker material due to it being very insoluble in brass, eliminating any error due to the markers diffusing themselves. Diffusion was allowed to take place at 785 °C over the course of 56 days, with cross-sections being taken at six times throughout the span of the experiment. Over time, it was observed that the wire markers moved closer together as the zinc diffused out of the brass and into the copper. A difference in location of the interface was visible in cross sections of different times. Compositional change of the material from diffusion was confirmed by x-ray diffraction . [ 1 ]
Early diffusion models postulated that atomic motion in substitutional alloys occurs via a direct exchange mechanism, in which atoms migrate by switching positions with atoms on adjacent lattice sites. [ 3 ] Such a mechanism implies that the atomic fluxes of two different materials across an interface must be equal, as each atom moving across the interface causes another atom to move across in the other direction. [ citation needed ]
Another possible diffusion mechanism involves lattice vacancies . An atom can move into a vacant lattice site, effectively causing the atom and the vacancy to switch places. If large-scale diffusion takes place in a material, there will be a flux of atoms in one direction and a flux of vacancies in the other.
[ citation needed ]
The Kirkendall effect arises when two distinct materials are placed next to each other and diffusion is allowed to take place between them. In general, the diffusion coefficients of the two materials in each other are not the same. This is only possible if diffusion occurs by a vacancy mechanism; if the atoms instead diffused by an exchange mechanism, they would cross the interface in pairs, so the diffusion rates would be identical, contrary to observation. By Fick's 1st law of diffusion , the flux of atoms from the material with the higher diffusion coefficient will be larger, so there will be a net flux of atoms from the material with the higher diffusion coefficient into the material with the lower diffusion coefficient. To balance this flux of atoms, there will be a flux of vacancies in the opposite direction—from the material with the lower diffusion coefficient into the material with the higher diffusion coefficient—resulting in an overall translation of the lattice relative to the environment in the direction of the material with the lower diffusion constant. [ 3 ]
Macroscopic evidence for the Kirkendall effect can be gathered by placing inert markers at the initial interface between the two materials, such as molybdenum markers at an interface between copper and brass. The diffusion coefficient of zinc is higher than the diffusion coefficient of copper in this case. Since zinc atoms leave the brass at a higher rate than copper atoms enter, the size of the brass region decreases as diffusion progresses. Relative to the molybdenum markers, the copper–brass interface moves toward the brass at an experimentally measurable rate. [ 1 ]
Shortly after the publication of Kirkendall's paper, L. S. Darken published an analysis of diffusion in binary systems much like the one studied by Smigelskas and Kirkendall. By separating the actual diffusive flux of the materials from the movement of the interface relative to the markers, Darken found the marker velocity v {\displaystyle v} to be [ 4 ] v = ( D 1 − D 2 ) d N 1 d x , {\displaystyle v=(D_{1}-D_{2}){\frac {dN_{1}}{dx}},} where D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} are the diffusion coefficients of the two materials, and N 1 {\displaystyle N_{1}} is an atomic fraction.
One consequence of this equation is that the movement of an interface varies linearly with the square root of time, which is exactly the experimental relationship discovered by Smigelskas and Kirkendall. [ 1 ]
Darken also developed a second equation that defines a combined chemical diffusion coefficient D {\displaystyle D} in terms of the diffusion coefficients of the two interfacing materials: [ 4 ] D = N 1 D 2 + N 2 D 1 . {\displaystyle D=N_{1}D_{2}+N_{2}D_{1}.} This chemical diffusion coefficient can be used to mathematically analyze Kirkendall effect diffusion via the Boltzmann–Matano method .
One important consideration deriving from Kirkendall's work is the presence of pores formed during diffusion. These voids act as sinks for vacancies, and when enough accumulate, they can become substantial and expand in an attempt to restore equilibrium. Porosity occurs due to the difference in diffusion rate of the two species. [ 5 ]
Pores in metals have ramifications for mechanical, thermal, and electrical properties, and thus control over their formation is often desired. The equation [ 6 ] X K = ( a 1 Δ C 1 ∘ + a 2 Δ C 2 ∘ + ⋯ + a n − 1 Δ C n − 1 ∘ ) t , {\displaystyle X^{K}=(a_{1}\Delta C_{1}^{\circ }+a_{2}\Delta C_{2}^{\circ }+\dots +a_{n-1}\Delta C_{n-1}^{\circ }){\sqrt {t}},} where X K {\displaystyle X^{K}} is the distance moved by a marker, a {\displaystyle a} is a coefficient determined by intrinsic diffusivities of the materials, and Δ C ∘ {\displaystyle \Delta C^{\circ }} is a concentration difference between components, has proven to be an effective model for mitigating Kirkendall porosity. Controlling annealing temperature is another method of reducing or eliminating porosity. Kirkendall porosity typically occurs at a set temperature in a system, so annealing can be performed at lower temperatures for longer times to avoid formation of pores. [ 7 ]
In 1972, C. W. Horsting of the RCA Corporation published a paper which reported test results on the reliability of semiconductor devices in which the connections were made using aluminium wires bonded ultrasonically to gold -plated posts. His paper demonstrated the importance of the Kirkendall effect in wire bonding technology, but also showed the significant contribution of any impurities present to the rate at which precipitation occurred at the wire bonds. Two of the important contaminants that have this effect, known as Horsting effect ( Horsting voids ) are fluorine and chlorine . Both Kirkendall voids and Horsting voids are known causes of wire-bond fractures, though historically this cause is often confused with the purple-colored appearance of one of the five different gold–aluminium intermetallics , commonly referred to as "purple plague" and less often "white plague". [ 8 ] | https://en.wikipedia.org/wiki/Kirkendall_effect |
The Kirkhill Astronomical Pillar was constructed in 1776 by David Stewart Erskine, 11th Earl of Buchan [ a ] and erected in the grounds of his estate at Kirkhill House, near Broxburn , Scotland . The pillar fell into disrepair and eventually collapsed in the 1970s but fortunately the stones were preserved and the pillar was reconstructed (1988) in Almondell Country Park on land once owned by the Erskine family. [ b ] The pillar records the details of an adjacent scale model of the Solar System constructed by Erskine following the measurements of the size of the Solar System deduced from the observations of the Transits of Venus in 1761 and 1769. The model, centred on a Sun of stone six feet in diameter with planets at distances and sizes to scale, has long since disappeared; only the pillar remains.
As a young child Erskine was taught at home by his parents, both of whom had studied (and met each other) in the classes of the famous mathematician Colin Maclaurin at Edinburgh University. [ 1 ] They also employed a private tutor, James Buchanan, a graduate of Glasgow university, well versed in mathematics and languages. [ 2 ] Under the guidance of this trio he developed a life interest in mathematics and astronomy.
At the age of 13, Erskine entered St. Andrews University (1755–1759) and then continued to Edinburgh University (1760–1762) and finally Glasgow University (1762–63). [ 3 ] Although Erskine's later intellectual activities were dominated by his investigation of Scottish antiquities, he remained interested in science and mathematics. He was honoured by election to the Royal Society of London in 1765. [ c ] At that time he was living in London and at meetings of the society he would have heard much of the following topical astronomical problem.
By the beginning of the eighteenth century the Copernican model of a heliocentric Solar System was well established and astronomers such as Tycho Brahe and Johannes Kepler were able to describe the motions of the planets with ever greater precision. [ 4 ] However, no one knew the absolute size in miles (or any other units) of the Solar System although the solar distances of the planets could all be expressed as definite ratios of the Earth-Sun distance by using Kepler's laws . This fundamental distance is termed the Astronomical Unit (AU). [ d ]
The breakthrough came in 1639 when Jeremiah Horrocks made the first scientific observation of a transit of Venus and used his results to estimate an approximation for the AU. [ 5 ] [ 6 ] A second method, proposed in 1663 by the Scottish mathematician James Gregory , [ 7 ] was promoted by Edmond Halley in a paper published in 1691 (revised 1716). [ 8 ] He demonstrated how the AU could be measured very accurately by comparing the duration of the Venus transit across the face of the Sun as measured by two observers spaced at latitudes a few thousand kilometres apart. [ e ]
The next opportunities of observing such a transit were in 1761 and 1769 but Halley had died in 1742 and it was left to others to organise observations in the first ever major international scientific collaboration. The event of 1761 produced sparse results because travel overseas was greatly hindered by the Seven Years' War but in 1769 many observers were again despatched all over the world, amongst them being Captain James Cook on behalf of the Royal Society of London . Various pairs of observation results were input into Halley's calculations giving many slightly different values and a mean value of the AU was published shortly afterwards in the Philosophical Transactions of the Royal Society. [ 9 ] The result was 93,726,900 miles, within one per cent of the presently accepted value is 92,955,807 miles.
In Scotland, both transits were observed by Erskine's friend and neighbour, Reverend Alexander Bryce, [ f ] [ 10 ] [ 11 ] minister of the church at Kirknewton , only 3 miles from Kirkhill. Bryce was a competent mathematician and he calculated [ g ] the AU and the other distance parameters of the Solar System: it is these values that Erskine used to create his scale model of the Solar System.
In his 'Account of the Parish of Uphall', Erskine writes: [ 12 ]
"In the year 1776, I caused a representation to be made of the solar system, on a scale of 12,283 miles and 28/100 to an inch; the table of which epitome [ h ] is engraved on a belfray which stands in the middle of my garden, and of which I shall insert a transcript below."
The scale appears unusual but it followed simply from Bryce's calculation of the diameter of the Sun as 884,396 miles and Erskine's arbitrary choice of a representation of the Sun by a freestone spheroid 6 feet, or 72 inches, in diameter. Dividing 884,396 by 72 gives 12,283.28 miles to one inch, or 778,268,621:1. Of the six planets known in the eighteenth century Jupiter and Saturn were modelled in stone, the latter having an iron band, and the smaller planets were made of bronze: all were mounted on plinths or pillars in the grounds of the Kirkhill estate at the correct scaled distance from the Sun. Primrose, writing in 1898, says that only a few of the plinths remained in his day. [ 13 ]
The table giving the dimensions of his representation is carved into the east face of the stone pillar, or belfry; it is barely legible now, but the details are preserved in the Uphall account. [ 14 ] Planet diameters and distances on the pillar are reproduced here, along with the values obtained by scaling inches up to miles, by a factor of 12,283.28. Modern values are shown for comparison. Details for the moons of Jupiter and Saturn have been omitted. [ i ]
Calculation of the values in the table starts from the new value of the AU calculated by Bryce. Kepler's Laws then give the solar distance (in miles) for every planet and therefore, given the actual dimensions of the orbits, it is straightforward to calculate the distance of any planet from Earth at the time of any observation. Then, using the observed angular sizes of the Sun and the planets he could deduce their diameters in miles. [ 15 ]
To fit the data on the table Bryce must have calculated the value for the AU to be 95,072,587 miles. This value is greater than the modern (average) value of 93,000,000 miles. This largely accounts for the discrepancies in Erskine's data for distances and diameters.
The third, fourth and fifth columns of the pillar are reproduced in a second table below. It shows that the eccentricities of the planets and their inclinations to the ecliptic were quite well known at the time. (In the table Erskine's eccentricity value 80)387( is simply the fraction 80/387 and this has been replaced by decimal 0.207 etc.). Eccentricity and inclination are the essential parameters for working out the motions of the planets. No values are given for the orbit inclinations to the ecliptic for Mars and Jupiter, the space on the table having been utilised for a comment on the moons of Jupiter. The last pair of columns refer to what Erskine terms the inclination; the planet rotation axis to the plane of the orbit. Nowadays the term axial tilt is used by astronomers: it defines the angle between the rotation axis and the normal to the plane of the orbit and it is equal to 90 degrees minus Erskine's inclination. The values for Mercury and Venus are omitted on the pillar.
The final column on the pillar is a prediction of where the planets will be on May 20, 2255. The heliocentric places within the zodiac constellations define an angle now termed the heliocentric ecliptic longitude . Both are measured from the point in the sky where Aries begins. Each constellation covers 30 degrees whereas the longitude covers the whole 360 degrees spanned by all 12 constellations. The order of zodiac constellations is Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces. Therefore 9°40′ in Sagittarius for Mercury becomes a (decimal) longitude of 249.667° etc.
The significance for the year 2255 specified in the prediction is that it is a year in which a transit of Venus occurs; the eighth after that of 1769. During such a transit the Earth, Venus and the Sun must be closely aligned, in other words the heliocentric [ j ] places (longitude) of the planets must be very close, as shown by the predictions for the actual transit on 9 June 2255. Therefore, since Erskine gives heliocentric places for Venus and Earth differing by about 35°, he was clearly not predicting a transit for 20 May. There is no astronomical phenomenon associated with that day but it must have had some significance for Erskine, as yet unexplained.
There are inscriptions on the four sides of the pillar but they are now difficult to read. Fortunately some are recorded in Erskine's history of Uphall [ 16 ] and others in the account of the same parish by James Primrose. [ 17 ] Most are in Latin, often abbreviated, but translations have been given by James Primrose in his chapter on Kirkhill.
This face has the table described in the previous section. Above the table is the quotation given at the beginning of the previous section where Erskine (Buchan) describes his construction and its scale.
An inscription in Latin: Jacobo Buchanano, Matheseos P. Glasg. Adolefcentiae meae Custod. incorruptissicno has Amoenitates Academicas
Manibus propriis dedicavi, inscripsi, sacraque esse volui. Anno ab ejus excessu XV. et a Christo natu MDCCLXXVII. Ille ego qui quondam patriae perculsus amore, Civibus oppressis, libertati succurrere aussim, Nunc Arva paterna colo, fugioque liruina regum.
Primrose gives the translation: To James Buchanan, Professor of Mathematics at Glasgow, [ k ] the most incorruptible guardian of my youth, have I dedicated, inscribed with my own hands, these Academic Amenities, and I wish them to be sacred. On the 15th year of his death and from the birth of Christ 1771, I who formerly animated by love of country, dared to succour liberty and oppressed citizens, now cultivate my paternal fields and shun the threshold of Kings.
James Buchanan was the tutor and mentor of Erskine's early years. He died in 1761.
A quotation from Vergil's Georgics : DIGNA MANET DIVINA GLORIA RURIS which may be translated as "Pay homage to the heavenly sent land" or "The worthy glory of the Divine Country is abiding" [ 13 ] Underneath the inscription is a large bow and arrow the significance of which is unknown, the sign for Scorpius, and an unidentified sign.
A long inscription gives abbreviated details of the location of the pillar and other points. Erskine gives a fuller version in his account of Uphall Parish. [ 18 ]
"The latitude of Kirkhill is 55°56'17" north, the west longitude in time from Greenwich Observatory is 13′ 59′′10′′′. [ l ] The variation of the compass 1778 in June was 22°, the dip of the north end of the needle at the same time was 71°33'. The elevation above high water mark at Lieth (sic) when there is 12 feet of water in the harbour 273 feet; it is lower than the top of Arthurs Seat, 546 feet, lower than the Observatory on Calton Hill 83, than the top of the Castle Rock 290. West longitude in time from Edinburgh Observatory, 1°8"; east longitude in time from Glasgow Observatory, 3′11′′50′′′ - distance from Kirknewton Manse in Midlothian, 20,108 feet; north from Kirknewton Manse, 17,005 feet or 2′47′′ (arc); west from Kirknewton Manse, 10,680 feet or 12′′30′′′ in time."
The mention of Kirknewton Manse links this inscription to its resident, Alexander Bryce, who provided the details of the epitome table. The latitude is in a conventional notation but the longitudes are defined in terms of time: 15 degrees of longitude corresponding to one hour. The Greenwich time separation from Kirkhill given as 13′ 59′′ 10′′′ (minutes, seconds, sixtieths) corresponds to longitude 3.496°W: the modern value is 3.46°W. Similarly time displacements of the observatories at Edinburgh and Glasgow should be read as 1′8′′ (not 1°8") and 3′11′′50′′′ respectively, corresponding to 17 and 48 arc minutes of longitude, or 11 and 31 miles. The distances from Kirknewton Manse to the pillar are direct, north and west: the latitude difference is 2′47″ (arc) and the longitude difference in time is 12′′30′′′ corresponding to 3.12 arc minutes of longitude. The height differences between the pillar and locations in Edinburgh are an interesting by-product of Bryce's survey of a canal from the city, past Kirkhill and on to Falkirk. [ 10 ] Since there were to be no locks between the city and Broxburn the height of the pillar was easily related to that of the canal terminus and hence other known Edinburgh locations.
There are a number of other inscriptions which were close to the pillar. The globe representing the sun was engraved, in large Hebrew letters, with the question "What is man?" A plinth showing the Moon orbiting the Earth was inscribed "Newtono Magno". A small building near the pillar was inscribed "Keplero Felici". The approach to Kirkhill was guarded by pillars inscribed "Libertate quietate". On a triangular equilateral stone in Erskine's garden, was the inscription, "Great are thy works, Jehovah, infinite thy power!"
In the years leading up to the 2012 transit a group of Scottish artists collaborated on an artistic realisation of the Solar System model of Erskine. The Kirkhill Pillar Project was commissioned under the auspices of Artlink Edinburgh. [ 19 ] The Sun is represented by a light box on the top of Broxburn academy, within a few hundred metres of the Erskine's own house. The artefacts representing the nine planets are distributed around the county of West Lothian at distances given by Erskine's scale. Mars and Jupiter are represented by small spheres mounted on plinths. Mercury is represented by a cast iron replica of the cratered surface of the predominantly iron planet. Venus is represented by a schematic version of its transit over the face of the Sun. Earth, inspired by the blue and white image seen on early space missions, is represented by two planters containing blue and white flowers. Mars is a distinctive red sculpture in community woodland. A cast acrylic clear block houses a painted model of the planet Jupiter. Saturn is represented by a technical image used by James Clerk Maxwell in his explanation of the structure and stability of the rings. [ 20 ] Uranus is represented by a band suspended from two trees: it houses seven opaque apertures which allow the light to shine through. Neptune is captured as a blue orb in a lantern above the doors of Kingscavil church. Pluto is carved into black polished granite placed in Beecraigs Country Park. Images, further details and a map of locations may be found on the website of the Kirkhill project. [ 19 ] | https://en.wikipedia.org/wiki/Kirkhill_Astronomical_Pillar |
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48). The problem states:
Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. [ 2 ]
A solution to this problem is an example of a Kirkman triple system , [ 3 ] which is a Steiner triple system having a parallelism , that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks. Such Steiner systems that have a parallelism are also called resolvable .
There are exactly seven non- isomorphic solutions to the schoolgirl problem, as originally listed by Frank Nelson Cole in Kirkman Parades in 1922. [ 4 ] The seven solutions are summarized in the table below, denoting the 15 girls with the letters A to O.
From the number of automorphisms for each solution and the definition of an automorphism group, the total number of solutions including isomorphic solutions is therefore:
The problem has a long and storied history. This section is based on historical work done at different times by Robin Wilson [ 5 ] and by Louise Duffield Cummings . [ 6 ] The history is as follows:
James Joseph Sylvester in 1850 asked if 13 disjoint Kirkman systems of 35 triples each could be constructed to use all ( 15 3 ) = 455 {\textstyle {15 \choose 3}=455} triples on 15 girls. No solution was found until 1974 when RHF Denniston at the University of Leicester constructed it with a computer. [ 22 ] Denniston's insight was to create a single-week Kirkman solution in such a way that it could be permuted according to a specific permutation of cycle length 13 to create disjoint solutions for subsequent weeks; he chose a permutation with a single 13-cycle and two fixed points like (1 2 3 4 5 6 7 8 9 10 11 12 13)(14)(15). Under this permutation, a triple like 123 would map to 234, 345, ... (11, 12, 13), (12, 13, 1) and (13, 1, 2) before repeating. Denniston thus classified the 455 triples into 35 rows of 13 triples each, each row being the orbit of a given triple under the permutation. [ 22 ] In order to construct a Sylvester solution, no single-week Kirkman solution could use two triples from the same row, otherwise they would eventually collide when the permutation was applied to one of them. Solving Sylvester's problem is equivalent to finding one triple from each of the 35 rows such that the 35 triples together make a Kirkman solution. He then asked an Elliott 4130 computer to do exactly that search, which took him 7 hours to find this first-week solution, [ 22 ] labeling the 15 girls with the letters A to O :
He stopped the search at that point, not looking to establish uniqueness. [ 22 ]
The American minimalist composer Tom Johnson composed a piece of music called Kirkman's Ladies based on Denniston's solution. [ 23 ] [ 24 ]
As of 2021, it is not known whether there are other non-isomorphic solutions to Sylvester's problem, or how many solutions there are.
The equivalent of the Kirkman problem for 9 schoolgirls results in S(2,3,9), an affine plane isomorphic to the following triples on each day:
The corresponding Sylvester problem asks for 7 different S(2,3,9) systems of 12 triples each, together covering all ( 9 3 ) = 84 {\textstyle {9 \choose 3}=84} triples. This solution was known to Bays (1917) which was found again from a different direction by Earl Kramer and Dale Mesner in a 1974 paper titled Intersections Among Steiner Systems (J Combinatorial Theory, Vol 16 pp 273-285). There can indeed be 7 disjoint S(2,3,9) systems, and all such sets of 7 fall into two non-isomorphic categories of sizes 8640 and 6720, with 42 and 54 automorphisms respectively.
Solution 1 has 42 automorphisms, generated by the permutations (A I D C F H)(B G) and (C F D H E I)(B G). Applying the 9! = 362880 permutations of ABCDEFGHI, there are 362880/42 = 8640 different solutions all isomorphic to Solution 1.
Solution 2 has 54 automorphisms, generated by the permutations (A B D)(C H E)(F G I) and (A I F D E H)(B G). Applying the 9! = 362880 permutations of ABCDEFGHI, there are 362880/54 = 6720 different solutions all isomorphic to Solution 2.
Thus there are 8640 + 6720 = 15360 solutions in total, falling into two non-isomorphic categories.
In addition to S(2,3,9), Kramer and Mesner examined other systems that could be derived from S(5,6,12) and found that there could be up to 2 disjoint S(5,6,12) systems, up to 2 disjoint S(4,5,11) systems, and up to 5 disjoint S(3,4,10) systems. All such sets of 2 or 5 are respectively isomorphic to each other.
In the 21st century, analogues of Sylvester's problem have been visited by other authors under terms like "Disjoint Steiner systems" or "Disjoint Kirkman systems" or "LKTS" (Large Sets of Kirkman Triple Systems), for n > 15. [ 25 ] Similar sets of disjoint Steiner systems have also been investigated for the S(5,8,24) Steiner system in addition to triple systems. [ 26 ]
In 1910 the problem was addressed using Galois geometry by George Conwell. [ 27 ]
The Galois field GF(2) with two elements is used with four homogeneous coordinates to form PG(3,2) which has 15 points, 3 points to a line, 7 points and 7 lines in a plane. A plane can be considered a complete quadrilateral together with the line through its diagonal points. Each point is on 7 lines, and there are 35 lines in all.
The lines of PG(3,2) are identified by their Plücker coordinates in PG(5,2) with 63 points, 35 of which represent lines of PG(3,2). These 35 points form the surface S known as the Klein quadric . For each of the 28 points off S there are 6 lines through it which do not intersect S . [ 27 ] : 67
As there are seven days in a week, the heptad is an important part of the solution:
When two points as A and B of the line ABC are chosen, each of the five other lines through A is met by only one of the five other lines through B. The five points determined by the intersections of these pairs of lines, together with the two points A and B we designate a "heptad". [ 27 ] : 68
A heptad is determined by any two of its points. Each of the 28 points off S lies in two heptads. There are 8 heptads. The projective linear group PGL(3,2) is isomorphic the alternating group on the 8 heptads. [ 27 ] : 69
The schoolgirl problem consists in finding seven lines in the 5-space which do not intersect and such that any two lines always have a heptad in common. [ 27 ] : 74
In PG(3,2), a partition of the points into lines is called a spread , and a partition of the lines into spreads is called a packing or parallelism . [ 28 ] : 66 There are 56 spreads and 240 packings. When Hirschfeld considered the problem in his Finite Projective Spaces of Three Dimensions (1985), he noted that some solutions correspond to packings of PG(3,2), essentially as described by Conwell above, [ 28 ] : 91 and he presented two of them. [ 28 ] : 75
The problem can be generalized to n {\displaystyle n} girls, where n {\displaystyle n} must be an odd multiple of 3 (that is n ≡ 3 ( mod 6 ) {\displaystyle n\equiv 3{\pmod {6}}} ), walking in triplets for 1 2 ( n − 1 ) {\textstyle {\frac {1}{2}}(n-1)} days, with the requirement, again, that no pair of girls walk in the same row twice. The solution to this generalisation is a Steiner triple system , an S(2, 3, 6 t + 3) with parallelism (that is, one in which each of the 6 t + 3 elements occurs exactly once in each block of 3-element sets), known as a Kirkman triple system . [ 29 ] It is this generalization of the problem that Kirkman discussed first, while the famous special case n = 15 {\displaystyle n=15} was only proposed later. [ 30 ] A complete solution to the general case was published by D. K. Ray-Chaudhuri and R. M. Wilson in 1968, [ 31 ] though it had already been solved by Lu Jiaxi ( Chinese : 陆家羲 ) in 1965, [ 32 ] but had not been published at that time. [ 33 ]
Many variations of the basic problem can be considered. Alan Hartman solves a problem of this type with the requirement that no trio walks in a row of four more than once [ 34 ] using Steiner quadruple systems.
More recently a similar problem known as the Social Golfer Problem has gained interest that deals with 32 golfers who want to get to play with different people each day in groups of 4, over the course of 10 days.
As this is a regrouping strategy where all groups are orthogonal, this process within the problem of organising a large group into smaller groups where no two people share the same group twice can be referred to as orthogonal regrouping. [ 35 ]
The Resolvable Coverings problem considers the general n {\displaystyle n} girls, g {\displaystyle g} groups case where each pair of girls must be in the same group at some point, but we want to use as few days as possible. This can, for example, be used to schedule a rotating table plan, in which each pair of guests must at some point be at the same table. [ 36 ]
The Oberwolfach problem , of decomposing a complete graph into edge-disjoint copies of a given 2- regular graph , also generalizes Kirkman's schoolgirl problem. Kirkman's problem is the special case of the Oberwolfach problem in which the 2-regular graph consists of five disjoint triangles. [ 37 ] | https://en.wikipedia.org/wiki/Kirkman's_schoolgirl_problem |
The Kirkwood–Buff (KB) solution theory , due to John G. Kirkwood and Frank P. Buff , links macroscopic (bulk) properties to microscopic (molecular) details. Using statistical mechanics , the KB theory derives thermodynamic quantities from pair correlation functions between all molecules in a multi-component solution. [ 1 ] The KB theory proves to be a valuable tool for validation of molecular simulations, as well as for the molecular-resolution elucidation of the mechanisms underlying various physical processes. [ 2 ] [ 3 ] [ 4 ] For example, it has numerous applications in biologically relevant systems. [ 5 ]
The reverse process is also possible; the so-called reverse Kirkwood–Buff (reverse-KB) theory, due to Arieh Ben-Naim , derives molecular details from thermodynamic (bulk) measurements. This advancement allows the use of the KB formalism to formulate predictions regarding microscopic properties on the basis of macroscopic information. [ 6 ] [ 7 ]
The radial distribution function (RDF), also termed the pair distribution function or the pair correlation function, is a measure of local structuring in a mixture. The RDF between components i {\displaystyle i} and j {\displaystyle j} positioned at r i {\displaystyle {\boldsymbol {r}}_{i}} and r j {\displaystyle {\boldsymbol {r}}_{j}} , respectively, is defined as:
where ρ i j ( R ) {\displaystyle \rho _{ij}({\boldsymbol {R}})} is the local density of component j {\displaystyle j} relative to component i {\displaystyle i} , the quantity ρ i j bulk {\displaystyle \rho _{ij}^{\text{bulk}}} is the density of component j {\displaystyle j} in the bulk, and R = | r i − r j | {\displaystyle {\boldsymbol {R}}=|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|} is the inter-particle radius vector. Necessarily, it also follows that:
Assuming spherical symmetry , the RDF reduces to:
where r = | R | {\displaystyle r=|{\boldsymbol {R}}|} is the inter-particle distance.
In certain cases, it is useful to quantify the intermolecular correlations in terms of free energy. Specifically, the RDF is related to the potential of mean force (PMF) between the two components by:
where the PMF is essentially a measure of the effective interactions between the two components in the solution.
The Kirkwood–Buff integral (KBI) between components i {\displaystyle i} and j {\displaystyle j} is defined as the spatial integral over the pair correlation function:
which in the case of spherical symmetry reduces to:
G i j = 4 π ∫ r = 0 ∞ [ g i j ( r ) − 1 ] r 2 d r {\displaystyle G_{ij}=4\pi \int _{r=0}^{\infty }[g_{ij}(r)-1]r^{2}\,dr}
KBI, having units of volume per molecule, quantifies the excess (or deficiency) of particle j {\displaystyle j} around particle i {\displaystyle i} .
It is possible to derive various thermodynamic relations for a two-component mixture in terms of the relevant KBI ( G 11 {\displaystyle G_{11}} , G 22 {\displaystyle G_{22}} , and G 12 {\displaystyle G_{12}} ).
The partial molar volume of component 1 is: [ 1 ]
where c {\displaystyle c} is the molar concentration and naturally c 1 V ¯ 1 + c 2 V ¯ 2 = 1 {\displaystyle c_{1}{\bar {V}}_{1}+c_{2}{\bar {V}}_{2}=1}
The compressibility, κ {\displaystyle \kappa } , satisfies:
where k {\displaystyle k} is the Boltzmann constant and T {\displaystyle T} is the temperature.
The derivative of the osmotic pressure , Π {\displaystyle \Pi } , with respect to the concentration of component 2: [ 1 ]
where μ 1 {\displaystyle \mu _{1}} is the chemical potential of component 1.
The derivatives of chemical potentials with respect to concentrations, at constant temperature ( T {\displaystyle T} ) and pressure ( P {\displaystyle P} ) are:
or alternatively, with respect to mole fraction :
The relative preference of a molecular species to solvate (interact) with another molecular species is quantified using the preferential interaction coefficient, Γ {\displaystyle \Gamma } . [ 8 ] Lets consider a solution that consists of the solvent (water), solute, and cosolute. The relative (effective) interaction of water with the solute is related to the preferential hydration coefficient, Γ W {\displaystyle \Gamma _{W}} , which is positive if the solute is "preferentially hydrated". In the Kirkwood-Buff theory frame-work, and in the low concentration regime of cosolutes, the preferential hydration coefficient is: [ 9 ]
where M W {\displaystyle M_{W}} is the molarity of water, and W, S, and C correspond to water, solute, and cosolute, respectively.
In the most general case, the preferential hydration is a function of the KBI of solute with both solvent and cosolute. However, under very simple assumptions [ 10 ] and in many practical examples, [ 11 ] it reduces to:
So the only function of relevance is G C S {\displaystyle G_{CS}} . | https://en.wikipedia.org/wiki/Kirkwood–Buff_solution_theory |
The Kirsanov reaction is a method for the synthesis of certain organophosphorus compounds . In this reaction a tertiary phosphine is combined with a halogen and then an amine to give the iminophosphines, which are useful ligands and useful reagents . [ 1 ] A typical reaction involves triphenylphosphine with bromine to give bromotriphenyl phosphonium bromide:
This salt is treated in situ with alkylamines to give the iminophosphorane:
The method is used when the conventional Staudinger reaction is not applicable, i.e. when the organic azide is not available to generate the iminophosphorane. Thus, it is used to make iminophosphoranes from alkyl amines. [ 2 ] | https://en.wikipedia.org/wiki/Kirsanov_reaction |
The Kirschner value or Kirschner number is a value determined when examining fat . The Kirschner value is an indicator of how much volatile fatty acid can be extracted from fat through saponification . It consists of the number of milliliters of 0.1 normal sodium hydroxide necessary for the neutralization of water-soluble silver salts made from the water-soluble volatile fatty acids distilled from 5 grams of a given fat. [ 1 ]
The Reichert value and Polenske value are related numbers based on similar tests. | https://en.wikipedia.org/wiki/Kirschner_value |
KisMAC is a wireless network discovery tool for Mac OS X . It has a wide range of features, similar to those of Kismet (its Linux / BSD namesake). The program is geared toward network security professionals, and is not as novice-friendly as similar applications. Distributed under the GNU General Public License , [ 2 ] KisMAC is free software .
KisMAC will scan for networks passively on supported cards - including Apple's AirPort , and AirPort Extreme , and many third-party cards, and actively on any card supported by Mac OS X itself.
Cracking of WEP and WPA keys, both by brute force , and exploiting flaws such as weak scheduling and badly generated keys is supported when a card capable of monitor mode is used, and packet reinjection can be done with a supported card ( Prism2 and some Ralink cards). GPS mapping can be performed when an NMEA compatible GPS receiver is attached. [ 3 ]
Kismac2 is a fork of the original software with a new GUI, new features and that works for OS X 10.7 - 10.10, 64-bit only. It is no longer maintained.
Data can also be saved in pcap format and loaded into programs such as Wireshark .
The project was created and led by Michael Rossberg until July 27, 2007, when he removed himself from the project due to changes in German law (specifically, StGB Section 202c) that "prohibits the production and distribution of security software". [ 4 ] On this date, project lead was passed on to Geoffrey Kruse , maintainer of KisMAC since 2003, and
active developer since 2001. KisMAC is no longer being actively being developed. [ citation needed ] Primary development, and the relocated KisMAC web site were offline as of September 2016. As of August 6, 2007, the former homepage now denounces the new German law.
KisMac2 was project to continue development but is no longer maintained as well. | https://en.wikipedia.org/wiki/KisMAC |
In organic chemistry , cyclopropanation refers to any chemical process which generates cyclopropane ( (CH 2 ) 3 ) rings . It is an important process in modern chemistry as many useful compounds bear this motif; for example pyrethroid insecticides and a number of quinolone antibiotics ( ciprofloxacin , sparfloxacin , etc.). However, the high ring strain present in cyclopropanes makes them challenging to produce and generally requires the use of highly reactive species, such as carbenes , ylids and carbanions . [ 1 ] Many of the reactions proceed in a cheletropic manner.
Several methods exist for converting alkenes to cyclopropane rings using carbene type reagents. As carbenes themselves are highly reactive it is common for them to be used in a stabilised form, referred to as a carbenoid . [ 2 ]
In the Simmons–Smith reaction the reactive carbenoid is iodomethylzinc iodide , which is typically formed by a reaction between diiodomethane and a zinc-copper couple . Modifications involving cheaper alternatives have been developed, such as dibromomethane [ 3 ] or diazomethane and zinc iodide . [ 4 ] The reactivity of the system can also be increased by exchanging the zinc‑copper couple for diethylzinc . [ 5 ] Asymmetric versions are known. [ 6 ]
Certain diazo compounds , such as diazomethane , can react with olefins to produce cyclopropanes in a 2 step manner. The first step involves a 1,3-dipolar cycloaddition to form a pyrazoline which then undergoes denitrogenation, either photochemically or by thermal decomposition , to give cyclopropane. The thermal route, which often uses KOH and platinum as catalysts, is also known as the Kishner cyclopropane synthesis after the Russian chemist Nikolai Kischner [ 7 ] [ 8 ] and can also be performed using hydrazine and α,β-unsaturated carbonyl compounds . [ 9 ] The mechanism of decomposition has been the subject of several studies and remains somewhat controversial, although it is broadly thought to proceed via a diradical species. [ 10 ] [ 11 ] In terms of green chemistry this method is superior to other carbene based cyclopropanations; as it does not involve metals or halogenated reagents, and produces only N 2 as a by-product. However the reaction can be dangerous as trace amounts of unreacted diazo compounds may explode during the thermal rearrangement of the pyrazoline.
Methyl phenyldiazoacetate and many related diazo derivatives are precursors to donor-acceptor carbenes , which can be used for cyclopropanation or to insert into C-H bonds of organic substrates. These reactions are catalyzed by rhodium(II) trifluoroacetate and related chiral derivatives. [ 12 ] [ 13 ] [ 14 ]
Free carbenes can be employed for cyclopropanation reactions, however there is limited scope for this as few can be produced conveniently and nearly all are unstable (see: carbene dimerization ). An exception are dihalocarbenes such as dichlorocarbene or difluorocarbene , which are reasonably stable and will react to form geminal dihalo-cyclopropanes. [ 15 ] These compounds can then be used to form allenes via the Skattebøl rearrangement .
The Buchner ring expansion reaction also involves the formation of a stabilised carbene.
Cyclopropanation is also stereospecific as the addition of carbene and carbenoids to alkenes is a form of a cheletropic reaction , with the addition taking place in a syn manner. For example, dibromocarbene and cis -2-butene yield cis -2,3-dimethyl-1,1-dibromocyclopropane, whereas the trans isomer exclusively yields the trans cyclopropane. [ 16 ]
Cyclopropanes can be generated using a sulphur ylide in the Johnson–Corey–Chaykovsky reaction , [ 17 ] however this process is largely limited to use on electron-poor olefines, particularly α,β-unsaturated carbonyl compounds.
Cyclopropanes can be obtained by a variety of intramolecular cyclisation reactions. A simple method is to use primary haloalkanes bearing appropriately placed electron withdrawing groups. Treatment with a strong base will generate a carbanion which will cyclise in a 3-exo-trig manner, with displacement of the halide. Examples include the formation of cyclopropyl cyanide [ 18 ] and cyclopropylacetylene [ 19 ] This mechanism also forms the basis of the Favorskii rearrangement .
A related process is the cyclisation of 1,3-dibromopropane via a Wurtz coupling . This was used for the first synthesis of cyclopropane by August Freund in 1881. Originally this reaction was performed using sodium, [ 20 ] however the yield can be improved by exchanging this for zinc . [ 21 ]
Although cyclopropanes are relatively rare in biochemistry, many cyclopropanation pathways have been identified in nature. The most common pathways involve ring closure reactions of carbocations in terpenoids . Cyclopropane fatty acids are derived from the attack of S-adenosylmethionine (SAM) on unsaturated fatty acids. The precursor to the hormone ethylene , 1-aminocyclopropane-1-carboxylic acid , is derived directly from SMM via intramolecular nucleophilic displacement of the SMe 2 group subsequent to condensation with pyridoxal phosphate . [ 23 ] Direct carbene transfer from diazoesters to olefins has also been achieved through in vitro biocatalysis using engineered variants of the cytochrome P450 enzyme from Bacillus megaterium that were optimized by directed evolution . [ 24 ] | https://en.wikipedia.org/wiki/Kishner_cyclopropane_synthesis |
A kit is a set of components that has to be assembled by the buyer or at the site of use to get the definitive product.
Examples:
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kit_(components) |
Kevin Kit Parker is a lieutenant colonel in the United States Army Reserve [ 1 ] and the Tarr Family Professor of Bioengineering and Applied Physics at Harvard University . [ 2 ] His research includes cardiac cell biology and tissue engineering, traumatic brain injury , and biological applications of micro- and nanotechnologies. Additional work in his laboratory has included fashion design, marine biology, and the application of counterinsurgency methods to countering transnational organized crime. [ 3 ]
Parker attended Boston University 's College of Engineering and graduated in 1989. He earned a Master of Science degree in 1993 and a doctoral degree in applied physics in 1998 from Vanderbilt University . [ 4 ] [ 5 ]
Parker is a paratrooper who has served in the United States Army since 1992. After the September 11 attacks , he served two tours of duty in Afghanistan . [ 6 ]
In addition to his combat tours, Parker conducted two missions into Afghanistan as part of the Gray Team in 2011. [ 7 ] [ 8 ]
Initially, at Harvard the focus of his research was heart muscle cells. He turned to traumatic brain injury in 2005 after realizing that an Army friend of his, who had received injuries in an IED blast in Iraq in 2005, was suffering from an undiagnosed medical condition rather than a psychological problem. [ 9 ] [ 10 ]
Other research of Parker's includes designing camouflage using skin cells of cuttlefish and the use of a cotton candy machine to make dressings for wounds. [ 11 ]
Parker served on the Defense Science Research Council for nearly a decade, [ 12 ] the Defense Science Board Task Force on Autonomy, and has consulted to other US government agencies as well as the medical device and pharma industry.
In 2011, Parker headed Harvard's committee for reintroducing ROTC at the university. [ 13 ]
In July 2016, it was announced that The Disease Biophysics Group at Harvard, led by Kit Parker, created a tissue-engineered soft-robotic ray that swims using wave-like fin motions, and turns according to externally applied light cues. [ 14 ]
In January 2021, students at the Harvard School of Engineering and Applied Sciences created a petition objecting to Parker's course on Counter-Criminal Continuum policing, or C3 policing . Titled "Data Fusion in Complex Systems: A Case Study," the course promised to engage graduate student researchers to analyze the efficacy of C3 techniques in Springfield, Massachusetts.
The petition objected to the lack of research into the potential harms of C3 policing, particularly the ethical implications for marginalized communities. [ 15 ] The Dean of the Engineering School soon announced the class was canceled, [ 16 ] and committed to reviewing the process of vetting class offerings. [ 17 ] | https://en.wikipedia.org/wiki/Kit_Parker |
In condensed matter physics , the Kitaev chain or Kitaev–Majorana chain is a simplified model for a topological superconductor . It models a one dimensional lattice featuring Majorana bound states . The Kitaev chain have been used as a toy model of semiconductor nanowires for the development of topological quantum computers . [ 1 ] [ 2 ] The model was first proposed by Alexei Kitaev in 2000. [ 3 ]
The tight binding Hamiltonian in of a Kitaev chain considers a one dimensional lattice with N site and spinless particles at zero temperature, subjected to nearest neighbour hoping interactions, given in second quantization formalism as [ 4 ]
where μ {\displaystyle \mu } is the chemical potential , c j † , c j {\displaystyle c_{j}^{\dagger },c_{j}} are creation and annihilation operators , t ≥ 0 {\displaystyle t\geq 0} the energy needed for a particle to hop from one location of the lattice to another, Δ = | Δ | e i θ {\displaystyle \Delta =|\Delta |e^{i\theta }} is the induced superconducting gap (p-wave pairing) and θ {\displaystyle \theta } is the coherent superconducting phase. This Hamiltonian has particle-hole symmetry, as well as time reversal symmetry . [ 5 ]
The Hamiltonian can be rewritten using Majorana operators, given by [ 4 ]
which can be thought as the real and imaginary parts of the creation operator c j = 1 2 ( γ j A + i γ j B ) {\displaystyle c_{j}={\tfrac {1}{2}}(\gamma _{j}^{\rm {A}}+i\gamma _{j}^{\rm {B}})} . These Majorana operator are Hermitian operators , and anticommutate ,
Using these operators the Hamiltonian can be rewritten as [ 4 ]
where ω ± = | Δ | ± t {\displaystyle \omega _{\pm }=|\Delta |\pm t} .
In the limit t = | Δ | → 0 {\displaystyle t=|\Delta |\to 0} , we obtain the following Hamiltonian
where the Majorana operators are coupled on the same site. This condition is considered a topologically trivial phase. [ 5 ]
In the limit μ → 0 {\displaystyle \mu \to 0} and | Δ | → t {\displaystyle |\Delta |\to t} , we obtain the following Hamiltonian
where every Majorana operator is coupled to a Majorana operator of a different kind in the next site. By assigning a new fermion operator c ~ j = 1 2 ( γ j B + i γ j + 1 A ) {\displaystyle {\tilde {c}}_{j}={\tfrac {1}{2}}(\gamma _{j}^{\rm {B}}+i\gamma _{j+1}^{\rm {A}})} , the Hamiltonian is diagonalized, as
which describes a new set of N -1 Bogoliubov quasiparticles with energy t . The missing mode given by c ~ M = 1 2 ( γ N B + i γ 1 A ) {\displaystyle {\tilde {c}}_{\rm {M}}={\tfrac {1}{2}}(\gamma _{N}^{\rm {B}}+i\gamma _{1}^{\rm {A}})} couples the Majorana operators from the two endpoints of the chain, as this mode does not appear in the Hamiltonian, it requires zero energy. This mode is called a Majorana zero mode and is highly delocalized. As the presence of this mode does not change the total energy, the ground state is two-fold degenerate. [ 4 ] This condition is a topological superconducting non-trivial phase. [ 5 ]
The existence of the Majorana zero mode is topologically protected from small perturbation due to symmetry considerations . For the Kitaev chain the Majorana zero mode persist as long as μ < 2 t {\displaystyle \mu <2t} and the superconducting gap is finite. [ 6 ] The robustness of these modes makes it a candidate for qubits as a basis for topological quantum computer . [ 7 ]
Using Bogoliubov-de Gennes formalism it can be shown that for the bulk case (infinite number of sites), that the energy yields [ 6 ]
and it is gapped, except for the case μ = 2 t {\displaystyle \mu =2t} and wave vector k = 0 {\displaystyle k=0} . For the bulk case there are no zero modes. However a topological invariant exists given by
where p f [ x ] {\displaystyle \mathrm {pf} [x]} is the Pfaffian operation. For μ > 2 t {\displaystyle \mu >2t} , the invariant is strictly Q = 1 {\displaystyle Q=1} and for μ < 2 t {\displaystyle \mu <2t} , Q = − 1 {\displaystyle Q=-1} corresponding to the trivial and non-trivial phases from the finite chain, respectively. This relation between the topological invariant from the bulk case and the existence of Majorana zero modes in the finite case is called a bulk-edge correspondence. [ 6 ]
One possible realization of Kitaev chains is using semiconductor nanowires with strong spin–orbit interaction to break spin-degeneracy, [ 8 ] like indium antimonide or indium arsenide . [ 9 ] A magnetic field can be applied to induce Zeeman coupling to spin polarize the wire and break Kramers degeneracy . [ 8 ] The superconducting gap can be induced using Andreev reflection , by putting the wire in the proximity to a superconductor. [ 8 ] [ 9 ] Realizations using 3D topological insulators have also been proposed. [ 9 ]
There is no single definitive way to test for Majorana zero modes. One proposal to experimentally observe these modes is using scanning tunneling microscopy . [ 9 ] A zero bias peak in the conductance could be the signature of a topological phase. [ 9 ] Josephson effect between two wires in superconducting phase could also help to demonstrate these modes. [ 9 ]
In 2023 QuTech team from Delft University of Technology reported the realization of a poor man's Majorana, a Kitaev chain with two or three sites that produces a Majorana bound state that is not topologically protected and therefore only stable for a very small range of parameters. [ 1 ] [ 2 ] It was obtained in a Kitaev chain consisting of two quantum dots in a superconducting nanowire strongly coupled by normal tunneling and Andreev tunneling with the state arising when the rate of both processes match. [ 1 ] [ 2 ] Some researches have raised concerns, suggesting that an alternative mechanism to that of Majorana bound states might explain the data obtained. [ 2 ] [ 7 ]
In 2024, the first experiment in an optomechanical network was conducted to create a bosonic analogue of a Kitaev chain. [ 10 ] | https://en.wikipedia.org/wiki/Kitaev_chain |
Kitbashing or model bashing is the practice of making a new scale model by taking pieces out of kits. These pieces may be added to a custom project or to another kit. For professional modelmakers , kitbashing is used to create concept models for detailing movie special effects . Commercial model kits are a ready source of "detailing", providing any number of identical, mass-produced components that can be used to add fine detail to an existing model. Professionals often kitbash to build prototype parts which are then recreated with lightweight materials. [ 1 ]
For the hobbyist, kitbashing saves time that would be spent scratch building an entire model. Hobbyists may kitbash to create a model of a subject (real [ 2 ] or imaginary) for which there is not a commercial kit.
Although it has a long history, kitbashing came to the attention of a wider public via the fine modelwork seen in TV series such as Thunderbirds , Star Trek and the films 2001: A Space Odyssey and Star Wars Episode IV: A New Hope . Many of the spaceship models created for these programs incorporated details from tank, speedboat and car kits. Another example is the Batmobile from the 2005 film Batman Begins , as seen in the special features disc of the film's DVD.
It is not uncommon for parts to be cut and filed into shapes leaving gaps that are later filled with putty to hide defects. Textural details known as greebles may be added to enhance a model.
Sometimes, kitbashing has been used to create works of art . The Toronto sculptor Kim Adams has used HO gauge freight cars, containers, detail parts, figures and scenery to create artistic landscapes. American artist Kris Kuksi also uses kitbashing to detail his maximalist sculptures.
A popular venue for kitbashing is diecast emergency vehicles such as fire apparatuses. Kitbashers often use models from manufacturers such as Code 3 and Corgi [ citation needed ] . The kitbash in such cases can be as simple as painting or redecaling a model, or as complex as tearing the model down and adding scratch-built components, followed by custom decals.
An important aspect of kitbashing in model railroading is the reconfiguration of structure kits, most often to fit the geometry of a specific space. Walls can be shortened or lengthened, and corner angles can be changed, to fit a given location on the layout. Another application is to use the wall parts to create a "flat", or shallow relief model to be displayed against the backdrop. In this configuration the parts for the rear wall of a structure, often an industrial building, can instead be abutted to the front to double the length of the building. Plain sheet styrene or other material is typically added to the rear to strengthen the resulting model.
In model rocketry , kitbashing refers simply to using the pieces from one kit to build a different model. This is typically used to create unusual or especially complex models.
With radio-controlled aircraft , such kitbashing can be done to kitted aircraft as they are being built, or, more often, to so-called "almost-ready-to-fly" (ARF) aircraft to change their appearance or flight characteristics to suit the owner. This can even extend to "plans-bashing", where a plans-built model has its construction plans partially re-drawn by the builder, either by hand or with computer-aided design software before any part of the model's airframe has been fabricated from raw materials. | https://en.wikipedia.org/wiki/Kitbashing |
The Kitchen rudder is the familiar name for "Kitchen's Patent Reversing Rudders", a combination rudder and directional propulsion delivery system for relatively slow speed displacement boats which was invented in the early 20th century by John G. A. Kitchen of Lancashire, England. It turns the rudder into a directional thruster , and allows the engine to maintain constant revolutions and direction of drive shaft rotation while altering thrust by use of a control which directs thrust forward or aft. Only the rudder pivots; the propeller itself is on a fixed shaft and does not.
"Kitchener gear" or "Kitchener rudder" have been common misnomers for the Kitchen rudder.
It is held under British Provisional Patent No. 3249/1914 and US Patent No. 1186210 (1916) and has been improved with the design in US Patent 4895093 (1990).
The rudder consists of a pair of slightly conical (usually but not always - designs vary), semi-cones mounted on a pivot either side of the propeller with the long axis of the cone running fore and aft when the helm is midships. They are pivoted about a vertical axis such that the cone may close off the propeller thrust aft of the propeller, directing the thrust forwards and thus creating motion astern.
In addition to the "jaws" of the cone being controlled the direction of thrust is also controlled by rudder direction. In this way, it is unlike the azimuth thrusters used on many medium and large vessels, or the outboard motors or stern drives used by some small boats, since these all use the directed thrust to avoid the need of a rudder altogether.
Modern equivalent include certain types of pump jets or Kort nozzle .
While not strictly Kitchen rudder technology, the "bucket" on some aircraft jet engines is an aeronautical derivative of the device. [ citation needed ] When the deflectors are deployed, directing thrust forwards, they are equivalent to the Kitchen rudder in the "full astern" position.
The operation of the Kitchen rudder is performed with the propeller engaged, even when the boat is stationary. [ 1 ] The rudder is controlled by a small wheel on the tiller . [ 2 ]
The engine is brought up to speed with the drive to the propeller engaged and with the Kitchen rudder in the "neutral" position. This is a position where an equal quantity of thrust is aimed forward and aft. [ 2 ] Each vessel will have a unique "neutral" position.
The Kitchen gear is opened up to direct an increasing proportion of thrust aft. As the balance changes the vessel will move ahead. [ 2 ]
The Kitchen gear is closed to direct an increasing proportion of thrust forward. As the balance changes the vessel will move astern. [ 2 ] | https://en.wikipedia.org/wiki/Kitchen_rudder |
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Kitrinoviricota is a phylum of RNA viruses that includes all positive-strand RNA viruses that infect eukaryotes and are not members of the phylum Pisuviricota or Lenarviricota . [ 1 ] The name of the group derives from Greek κίτρινος ( kítrinos ), which means yellow (a reference to yellow fever virus ), and - viricota , which is the suffix for a virus phylum . [ 2 ]
The following classes are recognized:
This virus -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kitrinoviricota |
The Kiviter process is an above ground retorting technology for shale oil extraction .
The Kiviter process is based on the earlier vertical retort technology (Pintsch's generator). [ 1 ] This technology underwent a long process of development. The early concept of central inlet of the heat carrier was later replaced by a concept of heat carrier gas cross flow in the retort. [ 2 ]
The Kiviter technology has been used in Estonia since 1921, when first experimental Kiviter retorts were built. [ 1 ] The first commercial scale oil plant based on the Kiviter technology was built in 1924. [ 3 ]
From 1955 to 2003, Kiviter technology was used for oil shale processing also in Slantsy , Russia. [ 4 ] [ 5 ] [ 6 ]
The Kiviter process is classified as an internal combustion technology. [ 7 ] The Kiviter retort is a vertical cylindrical vessel that heats coarse oil shale with recycled gases, steam, and air. [ 8 ] To supply heat, gases (including produced oil shale gas ) and carbonaceous spent residue are burnt within the retort. Raw oil shale is fed into the top of the retort, and is heated by the rising gases, which pass laterally through the descending oil shale causing decomposition of the rock. Pyrolysis is completed in the lower section of the retort, where the spent shale contacted with more hot gas, steam and air is heated to about 900 °C (1,650 °F) to gasify and burn the residual carbon ( char ). Shale oil vapors and evolving gases are delivered to a condensing system, where condensed shale oil is collected, while non-condensable gases are fed back to the retort. Recycled gas enters the bottom of the retort and cools the spent shale, which then leaves the retort through a water-sealed discharge system. [ 2 ]
The Kiviter process uses large amounts of water, which is polluted during processing, and the solid waste residue contains water-soluble toxic substances that leach into the surrounding area. [ 9 ] [ 10 ]
The Kiviter process is used by the Estonian Viru Keemia Grupp 's subsidiary VKG Oil. [ 8 ] [ 11 ] The company operates several Kiviter retorts, the largest of them having a processing capacity of 40 tonnes per hour of oil shale feedstock. [ 2 ] [ 12 ] | https://en.wikipedia.org/wiki/Kiviter_process |
A Kiwi drive is a holonomic drive system of three omni-directional wheels (such as omni wheels or Mecanum wheels ), 120 degrees from each other, that enables movement in any direction using only three motors. This is in contrast with non-holonomic systems such as traditionally wheeled or tracked vehicles which cannot move sideways without turning first. [ 1 ] [ 2 ]
This drive system is similar to the Killough platform which achieves omni-directional travel using traditional non-omni-directional wheels in a three-wheel configuration. [ 3 ]
It was named for the flightless national bird of New Zealand , the Kiwi . [ 4 ]
When only the front wheel is powered, the chassis will turn and strafe at once. If the back wheels turn the same amount in the same direction, the strafe is cancelled out, so the chassis will only turn. If the back wheels turn half as much in the opposite direction, the turn is cancelled out, so the chassis will only strafe. If the front wheel is not powered, and the back wheels turn the same amount in opposite directions, the chassis will only drive. [ citation needed ]
This robotics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kiwi_drive |
The Kjeldahl method or Kjeldahl digestion ( Danish pronunciation: [ˈkʰelˌtɛˀl] ) in analytical chemistry is a method for the quantitative determination of a sample's organic nitrogen plus ammonia / ammonium (NH 3 /NH 4 + ). Without modification, other forms of inorganic nitrogen, for instance nitrate , are not included in this measurement. Using an empirical relation between Kjeldahl nitrogen and protein, it is an important method for indirectly quantifying protein content of a sample. This method was developed by the Danish chemist Johan Kjeldahl in 1883. [ 1 ] [ 2 ]
The method consists of heating a sample to 360–410 °C with concentrated sulfuric acid ( H 2 SO 4 ), which decomposes, or digests, the organic sample by oxidation to liberate the reduced nitrogen as stable ammonium sulfate : (NH 4 ) 2 SO 4 . [ 3 ]
Hot concentrated sulfuric acid oxidizes carbon (as bituminous coal ) and sulfur (see sulfuric acid's reactions with carbon ):
Most of organic carbon and sulfur are decomposed and eliminated as gaseous CO 2 and SO 2 .
In contrast to organic carbon and sulfur, the digested organic nitrogen remains preserved in the concentrated sulfuric acid as stable ammonium cation ( NH + 4 ). Ammonium does not further oxidize to gaseous N 2 , or a higher oxidized form of nitrogen, such as, e.g., N 2 O, NO, NO − 2 , NO 2 , or NO − 3 . If it was the case, the Kjeldahl method would not work.
Catalysts like selenium , Hg 2 SO 4 or CuSO 4 are often added to accelerate the digestion. Na 2 SO 4 or K 2 SO 4 is also added to increase the boiling point of H 2 SO 4 . Digestion is complete when the liquor clarifies with the release of fumes. [ 3 ]
After complete digestion of the sample, to recover ammonia ( NH 3 ) from the ammonium sulfate, sodium hydroxide (NaOH) is first added to the residual sulfuric acid to neutralize it and to convert the soluble ammonium ion into volatile ammonia :
Then, ammonia is recovered by distillation using the system below (right side of the figure).
The end of the condenser is dipped into a known volume of standard acid (i.e. acid of known concentration). A weak acid like boric acid (H 3 BO 3 ) in excess of ammonia is often used. Standardized HCl , H 2 SO 4 or some other strong acid can be used instead, but this is less commonplace. The sample solution is then distilled with a small excess of sodium hydroxide (NaOH). [ 3 ] NaOH can also be added with a dropping funnel . [ 4 ] NaOH converts dissolved ammonium (NH 4 + ) to gaseous ammonia (NH 3 ), which boils off the sample solution. Ammonia bubbles through the standard acid solution and reacts back to ammonium salts with the weak or strong acid. [ 3 ]
Ammonium ion concentration in the acid solution, and thus the amount of nitrogen in the sample, is measured via titration. If boric acid (or some other weak acid) was used, direct acid–base titration is done with a strong acid of known concentration. HCl or H 2 SO 4 can be used. Indirect back titration is used instead if strong acids were used to make the standard acid solution: a strong base of known concentration (like NaOH) is used to neutralize the solution. In this case, the amount of ammonia is calculated as the difference between the amount of HCl and NaOH. In the case of direct titration, it is not necessary to know the exact amount of weak acid (e.g. boric acid) because it does not interfere with the titration (it does have to be in excess of ammonia to trap it efficiently). Thus, one standard solution (e.g., HCl) is needed in the direct titration, while two are needed (e.g., HCl and NaOH) in the back- titration . One of the suitable indicators for these titration reactions is Tashiro's indicator . [ 3 ]
In practice, this analysis is largely automated; specific catalysts accelerate the decomposition. Originally, the catalyst of choice was mercuric oxide . However, while it was very effective, health concerns resulted in its replacement with cupric sulfate . Cupric sulfate was less efficient than mercuric oxide and yielded lower protein results. It was soon supplemented with titanium dioxide , the approved catalyst in all protein analysis methods in the Official Methods and Recommended Practices of AOAC International. [ 5 ]
The Kjeldahl method's universality, precision and reproducibility have made it the internationally recognized method for estimating the protein content in foods. It is the standard method against which all other methods are judged. It is also used to assay soils, waste waters, fertilizers and other materials. However, it does not allow for determining the true protein content, as it measures non-protein nitrogen in addition to the nitrogen in proteins. This is evidenced by the 2007 pet food incident and the 2008 Chinese milk powder scandal , when melamine , a nitrogen-rich chemical, was added to raw materials to fake high protein contents. Also, different correction factors are needed for different proteins to account for different amino acid sequences. Additional disadvantages, such as the need to use concentrated sulfuric acid at high temperature and the relatively long testing time (an hour or more), compare unfavorably with the Dumas method for measuring crude protein content. [ 6 ]
Total Kjeldahl nitrogen or TKN is the sum of nitrogen bound in organic substances, nitrogen in ammonia (NH 3 -N) and in ammonium (NH 4 + -N) in the chemical analysis of soil, water, or waste water (e.g. sewage treatment plant effluent).
Today, TKN is a required parameter for regulatory reporting at many treatment plants and for monitoring plant operations.
TKN is often used as a surrogate for protein in food samples . The conversion from TKN to protein depends on the type of protein present in the sample and what fraction of the protein is composed of nitrogenous amino acids , like arginine and histidine . However, the range of conversion factors is relatively narrow. Example conversion factors, known as N factors, for foods range from 6.38 for dairy and 6.25 for meat, eggs, maize (corn) and sorghum to 5.83 for most grains; 5.95 for rice, 5.70 for wheat flour, and 5.46 for peanuts. [ 7 ] In practice, 6.25 is used for almost all food and feed regardless of applicability. The factor 6.25 is specifically required by US Nutrition Label regulations in the absence of another published factor. [ 8 ]
The Kjeldahl method is poorly sensitive in the original version. Other detection methods have been used to quantify NH 4 + after mineralisation and distillation, achieving improved sensitivity: in-line generator of hydride coupled to a plasma atomic emission spectrometer (ICP-AES-HG, 10–25 mg/L), [ 10 ] potentiometric titration (> 0.1 mg of nitrogen), zone capillary electrophoresis (1.5 μg/mL of nitrogen), [ 11 ] and ion chromatography (0.5 μg/mL). [ 12 ]
Kjeldahl method does not apply to compounds containing nitrogen in nitro and azo groups and nitrogen present in rings (e.g. pyridine , quinoline , isoquinoline ) as nitrogen of these compounds does not convert to ammonium sulfate under the conditions of this method. | https://en.wikipedia.org/wiki/Kjeldahl_method |
Klaas Wynne (also Wijnne ; born 1964) is a professor in the School of Chemistry at the University of Glasgow and chair of Chemical Physics . [ 1 ] [ 2 ] He was previously a professor in the Department of Physics at the University of Strathclyde (1996–2010). [ 3 ]
He received his BSc in chemistry from the University of Amsterdam in 1987 and his PhD in chemistry from the University of Amsterdam in 1990 under the supervision of Joop van Voorst. [ 4 ] He did his postdoctoral fellowship in the laboratory of Robin Hochstrasser at the University of Pennsylvania . [ 5 ]
Wynne has authored over 90 published scientific papers. [ 6 ] [ 7 ] [ 8 ] His work is focused on the structure and dynamics of liquids [ 9 ] [ 10 ] [ 11 ] [ 12 ] and solutions [ 13 ] [ 14 ] [ 15 ] [ 16 ] as well as peptides, [ 17 ] proteins, [ 18 ] [ 19 ] [ 20 ] and other biomolecules [ 21 ] [ 22 ] [ 23 ] treated as amorphous objects behaving much like liquids. He described the Mayonnaise Effect, which explains the anomalous increase of the viscosity of solutions with concentration in terms of a jamming transition. [ 24 ] He is particularly interested in phase behaviour such as "supercooling of liquids, folding transitions in peptides, [ 17 ] phase separation and nucleation using laser-tweezing, [ 25 ] nucleation of crystals from solution", [ 26 ] and liquid-liquid [ 27 ] [ 28 ] and liquid-crystalline transitions. [ 29 ] These phenomena are studied using femtosecond spectroscopies [ 30 ] such as ultrafast optical Kerr-effect spectroscopy, time-domain terahertz spectroscopy (THz-TDS) [ 31 ] [ 32 ] as well as optical microscopy and various other forms of spectroscopy. [ 33 ] [ 34 ] | https://en.wikipedia.org/wiki/Klaas_Wynne |
Klark Teknik is a company that designs and develops professional signal processing and audio equipment . Located in Kidderminster , Worcestershire, UK, the company was founded in 1974 by brothers Terence and Phillip Clarke. It developed a number of new types of equipment in the audio field, winning a Queen's Awards for Enterprise in 1986. It is now owned by Music Tribe .
In 1971, the brothers were running a company known as “Klark Equipment”, producing garage forecourt equipment: vending style vacuum cleaners , space heaters and other similar products.
Phillip was the entrepreneur having previously run a clothing business in Australia. Terry was the engineer: he had worked for Decca and was also an active local musician. With this background, the company soon branched into professional audio, at first upgrading and customising Decca multi-track tape recorders and later producing a very high quality ¼ inch 2-track tape machine of their own, the SM2. This tape machine was sold to the BBC and independent broadcasting companies under the brand name “Teknik”.
Terry also started to develop signal processing hardware including compressor/limiters and small graphic equalisers . These were produced in very low quantities and sold to (mostly local) recording studios to complement the relatively simple audio mixing console of the day. The graphic equalisers developed during 1973 onwards were called Teknik 7S and 9S, with seven and nine bands respectively, plus a stereo 11+11S with eleven bands of EQ.
In 1974 the brand name “Klark Teknik” came into being. 1976 saw the first KT product produced in real volume, the DN27 (later DN27A and previously DN27S). Approximately 6500 units shipped between 1977 and 1985.
In 1980 the company moved to its current purpose built premises at Walter Nash Road, Kidderminster. Up until this point the business had been run out of an old Nissen hut in a back corner of the Summerfield MOD solid fuel rocket motor factory with only 15 employees.
Klark Teknik PLC was formed in 1985. In the following year Klark Teknik acquired the Hounslow-based studio mixing company DDA and subsequently moved the manufacturing to Kidderminster. The company further diversified by developing active studio monitors under the brand “Klark Acoustic”, winning a Queen's Award for export. It also launched the DN772, a seven-second profanity delay with innovative time re-setting technology without pitch shifting artefacts, which featured in the BBC programme Tomorrow's World .
Klark Teknik purchased Midas Audio Systems Ltd in 1987. In 1992, the Clarke brothers sold Klark Teknik Group to Mark IV Audio. Two years later, Terry Clarke went on to set up MC2 Audio Ltd with co-founder Ian McCarthy, producing high quality power amplifiers for the commercial market.
In 1998, Mark IV Audio sold their holdings of EVI Audio (owners of Klark Teknik) and Telex Communications to venture capitalists Greenwich Street Partners, who merged the companies into a single corporation called Telex Communications . On 1 September 2006, Greenwich sold Telex Communications to the Bosch Group . In December 2009, Midas and Klark Teknik were acquired by Music Group , a holding company chaired by Uli Behringer . The parent company has since been rebranded as Music Tribe. [ 1 ] [ 2 ]
Klark Teknik landmark products: | https://en.wikipedia.org/wiki/Klark_Teknik |
Klaus Bechgaard (5 March 1945 – 7 March 2017 [ 1 ] ) was a Danish scientist and chemist , noted for being one of the first scientists in the world to synthesize a number of organic charge transfer complexes and demonstrate their superconductivity , therefore the name Bechgaard salt . These salts all exhibit superconductivity at low temperatures.
The first organic superconductor was discovered by Bechgaard and Denis Jérome in 1979. This discovery garnered attention in the international scientific community, and for a period he was one of the most cited scientists in the field of natural sciences. He also received a nomination for the Nobel Prize in chemistry for this discovery.
Klaus Bechgaard did research at the University of Copenhagen , where he also held a Professorship in organic chemistry until 1993. From 1993 until 2000 he was the chairman of the Department of Physics and Chemistry at Risø and in 2001 he was appointed head of the newly assigned Department of Polymer Research at Risø. From 2001 and onwards he was the head of Risø's nano technology programme, and The Danish Center of Polymers which is a joint venture between the Technical University of Copenhagen and Risø.
Bechgaard also conducted research in the field of polymers and nano technology at the University of Copenhagen .
Education:
Academic Appointments:
Other:
Honours:
Publications:
Awards: [ 2 ] | https://en.wikipedia.org/wiki/Klaus_Bechgaard |
Klaus Schmiegel (born June 28, 1939), is a German chemist best known for his work in organic chemistry , which led to the invention of Prozac , a widely used antidepressant .
Klaus Schmiegel was born in Chemnitz , Germany on June 28, 1939. After he immigrated to the United States in 1951, Schmiegel received a B.S. in chemistry from the University of Michigan , an A.M. in organic chemistry from Dartmouth College , and a Ph.D. in organic chemistry from Stanford University . [ 1 ] His strong educational background secured him a prestigious position as a senior organic chemist at Eli Lilly, a prominent pharmaceutical company .
At Eli Lilly in the 1960s, Schmiegel and Bryan Molloy , with the help of David Wong , searched for a compound to combat depression. Because depression and similar psychiatric disorders are associated with reduced serotonin levels [ citation needed ] , they focused their approach on prohibiting serotonin reuptake. During a regular nerve signal transmission, a neurotransmitter such as serotonin travels from a presynaptic neuron to a postsynaptic neuron; the neurotransmitter returns to the presynaptic neuron after fulfilling its function, the reuptake process. Therefore, slowing and diminishing serotonin reuptake boosts serotonin levels in the brain.
The scientists based their search on the template of the antihistamine drug diphenhydramine hydrochloride , commonly known as Benadryl . After many failures, the research team synthesized a group of compounds called aryloxyphenylpropylamines. Upon testing, a member of the group, fluoxetine hydrochloride , proved to affect only the neurotransmitter serotonin. This compound became the first selective serotonin reuptake inhibitor (SSRI) and the active ingredient in the vastly popular and effective drug Prozac . [ 2 ]
Eli Lilly recognized the potential of its new drug, but the company first tested it as a high blood pressure medication, an anti-obesity drug , and a remedy for severe depression. After those testing failures, Eli Lilly succeeded in treating five mildly depressed people; fluoxetine had found its niche. Eli Lilly announced its findings in 1974 and launched Prozac in 1987 after receiving FDA approval.
The “wonder drug” replaced earlier medications, tricyclic antidepressants , which were less effective with serious side effects such as headaches , blurred vision and hypertension . By 1999, Prozac was bringing in $2.5 billion per year, 25% of Eli Lilly's revenue. The drug helped erase the stigma of depression, inspiring celebrities and public figures to flaunt rather than hide their sufferings.
Prozac, which is recognized by Fortune magazine as a “Product of the Century,” has few side effects; it has been widely beneficial for those suffering from depression, obsessive compulsive disorders , panic disorders , eating disorders and premenstrual dysphoric disorders . However, some contend that Prozac has been doled out too liberally—it has even been prescribed for animals. Peer drugs, including Zoloft and Paxil , which are also SSRIs, have experienced similar successes.
Unfortunately for Eli Lilly, its patent on Prozac expired in 2001, causing massive revenue losses. The company hopes to bounce back with its newest drug, Cymbalta , a painkiller and an antidepressant combined. [ 3 ]
Schmiegel is listed as an inventor on all eighteen of his patents, and his patents are assigned to his company, Eli Lilly. Schmiegel's chemical work concentrated on supplements to bolster the health of animals (growth promotion), weight control agents, and antidepressants. [ 4 ]
In 1999, both Schmiegel and Molloy were inducted into the National Inventors Hall of Fame for their fluoxetine compound that revolutionized depression treatment. [ 5 ] In addition, the same year, the pair received the U.S. Department of Commerce 's Ronald H. Brown American Innovator award, honoring their great contribution to society. [ 6 ]
Schmiegel worked for Eli Lilly until his retirement in 1993. Though Schmiegel is retired, he still lives in Indianapolis near the Eli Lilly headquarters. | https://en.wikipedia.org/wiki/Klaus_Schmiegel |
Klavs Flemming Jensen [ 1 ] (born August 5, 1952) [ 2 ] is a chemical engineer who is currently the Warren K. Lewis Professor at the Massachusetts Institute of Technology (MIT). [ 2 ]
Jensen was elected a member of the National Academy of Engineering in 2002 for fundamental contributions to multi-scale chemical reaction engineering with important applications in microelectronic materials processing and microreactor technology.
From 2007 to July 2015 he was the Head of the Department of Chemical Engineering at MIT. [ 3 ]
Jensen received his chemical engineering education from the Technical University of Denmark ( M.Sc. , 1976) and University of Wisconsin–Madison ( PhD , 1980). [ 2 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] Jensen's PhD advisor was W. Harmon Ray . [ 7 ] In 1980, Jensen became assistant professor of chemical engineering and materials science at the University of Minnesota , before being promoted to associate professor in 1984 and full professor in 1988. [ 8 ] In 1989, he moved to the Massachusetts Institute of Technology . [ 8 ]
At the Massachusetts Institute of Technology , Professor Jensen has been the Joeseph R. Mares Career Development Chair in Chemical Engineering (1989–1994), the Lammot du Pont Professor of Chemical Engineering (1996–2007), and the Warren K. Lewis Professor of Chemical Engineering (2007– present). [ 9 ] Klavs served as Head of the MIT Department of Chemical Engineering from 2007 to 2015. [ 10 ] In 2015, Professor Jensen became the founding Chair of the scientific journal Reaction Chemistry and Engineering by the Royal Society of Chemistry focused on bridging the gap between chemistry and chemical engineering. [ 11 ]
Jensen's research revolves around reaction and separation techniques for on-demand multistep synthesis , methods for automated synthesis , and microsystems biological discovery and manipulation. [ 5 ] He is considered one of the pioneers of flow chemistry . [ 12 ]
Jensen, Armon Sharei and Robert S. Langer were the founders of SQZ Biotech. [ 13 ] [ 14 ] The trio, together with Andrea Adamo, developed the cell squeezing method in 2012. [ 15 ] It enables delivery of molecules into cells by a gentle squeezing of the cell membrane . [ 15 ] It is a high throughput vector-free microfluidic platform for intracellular delivery . [ 15 ] It eliminates the possibility of toxicity or off-target effects as it does not rely on exogenous materials or electrical fields. [ 15 ]
Jensen, along with Timothy F. Jamison , Allan Myerson and coworkers, designed a refrigerator-sized mini factory to make clinic-ready drug formulations. [ 16 ] The mini factory can make thousands of doses of a drug in about two hours. [ 16 ] The factory can allow sudden public health needs to be more easily addressed. [ 16 ] It can also be useful in developing countries and for making medicines with a short shelf life . [ 16 ] Chemical & Engineering News named the mini factory in their list of notable chemistry research advances from 2016. [ 16 ]
Cell Squeeze is the commercial name for a method for deforming a cell as it passes through a small opening, disrupting the cell membrane and allowing material to be inserted into the cell. [ 17 ] [ 18 ] It is an alternative method to electroporation or cell-penetrating peptides and operates similarly to a french cell press that temporarily disrupts cells, rather than completely bursting them. [ 19 ]
The cell-disrupting change in pressure is achieved by passing cells through a narrow opening in a microfluidic device . The device is made up of channels etched into a wafer through which cells initially flow freely. As they move through the device, the channel width gradually narrows. The cell's flexible membrane allows it to change shape and become thinner and longer, allowing it to squeeze through. As the cell becomes more and more narrow, it shrinks in width by about 30 to 80 percent [ 18 ] its original size and the forced rapid change in cell shape temporarily creates holes in the membrane, without damaging or killing the cell.
While the cell membrane is disrupted, target molecules that pass by can enter the cell through the holes in the membrane. As the cell returns to its normal shape, the holes in the membrane close. Virtually any type of molecule can be delivered into any type of cell. [ 20 ] The throughput is approximately one million per second. Mechanical disruption methods can cause fewer gene expression changes than electrical or chemical methods. [ 19 ] This can be preferable in studies that require the gene expression to be controlled at all times. [ 21 ]
Like other cell permeablisation techniques, it enables intracellular delivery materials, such as proteins, siRNA, or carbon nanotubes. The technique has been used for over 20 cell types, including embryonic stem cells and naïve immune cells. [ 22 ] Initial applications focused on immune cells, for example delivering:
The process was originally developed in 2013 by Armon Sharei and Andrea Adamo, in the lab of Langer and Jensen at Massachusetts Institute of Technology . [ 18 ] In 2014 Sharei founded SQZBiotech to demonstrate the technology. [ 25 ] That year, SQZBiotech won the $100,000 grand prize in the annual startup competition sponsored by Boston-based accelerator MassChallenge. [ 26 ]
Boeing and the Center for the Advancement of Science in Space (CASIS) awarded the company the CASIS-Boeing Prize for Technology in Space to support the use of Cell Squeeze on the International Space Station (ISS). [ 27 ]
Jensen was the recipient of a Guggenheim Fellowship in 1987 . [ 2 ] [ 4 ] [ 5 ] [ 28 ] Jensen became an Elected Fellow of the Royal Society of Chemistry in 2004 and American Association for the Advancement of Science in 2007. [ 2 ] [ 4 ] [ 29 ] [ 30 ] [ 31 ] [ 32 ] He also became a member of the National Academy of Engineering in 2002 and the American Academy of Arts and Sciences in 2008. [ 2 ] [ 4 ] [ 5 ] In May 2017, he was elected to the National Academy of Sciences in recognition of his "distinguished and continuing achievements in original research." [ 5 ] [ 7 ]
In 2008, Jensen was included as one of the "100 Chemical Engineers of the Modern Era" by the American Institute of Chemical Engineers ' (AIChE) Centennial Celebration Committee. [ 2 ] [ 33 ] [ 34 ] [ 35 ] In March 2012, he was the first recipient of the IUPAC -ThalesNano Prize in Flow Chemistry. [ 2 ] [ 12 ] [ 35 ] Jensen was named in Foreign Policy magazine's 2016 list of the leading global thinkers along with Timothy F. Jamison and Allan Myerson. [ 36 ] In 2016, he received the AIChE Founders Award for Outstanding Contributions to the Field of Chemical Engineering. [ 37 ] [ 38 ] Jensen has also received the National Science Foundation Presidential Young Investigator Award . [ 4 ] [ 5 ]
Klavs Jensen has authored numerous journal articles describing significant advances in flow chemistry , microfluidics , chemical vapor deposition, and chemical engineering which includes but is not limited to: | https://en.wikipedia.org/wiki/Klavs_F._Jensen |
In computational geometry , Klee's measure problem is the problem of determining how efficiently the measure of a union of ( multidimensional ) rectangular ranges can be computed. Here, a d -dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers , which is a subset of R d .
The problem is named after Victor Klee , who gave an algorithm for computing the length of a union of intervals (the case d = 1) which was later shown to be optimally efficient in the sense of computational complexity theory . The computational complexity of computing the area of a union of 2-dimensional rectangular ranges is now also known, but the case d ≥ 3 remains an open problem .
In 1977, Victor Klee considered the following problem: given a collection of n intervals in the real line , compute the length of their union. He then presented an algorithm to solve this problem with computational complexity (or "running time") O ( n log n ) {\displaystyle O(n\log n)} — see Big O notation for the meaning of this statement. This algorithm, based on sorting the intervals, was later shown by Michael Fredman and Bruce Weide (1978) to be optimal.
Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles , find the area of their union. He also obtained a complexity O ( n log n ) {\displaystyle O(n\log n)} algorithm, now known as Bentley's algorithm , based on reducing the problem to n 1 -dimensional problems: this is done by sweeping a vertical line across the area. Using this method, the area of the union can be computed without explicitly constructing the union itself. Bentley's algorithm is now also known to be optimal (in the 2-dimensional case), and is used in computer graphics , among other areas.
These two problems are the 1- and 2-dimensional cases of a more general question: given a collection of n d -dimensional rectangular ranges, compute the measure of their union. This general problem is Klee's measure problem.
When generalized to the d -dimensional case, Bentley's algorithm has a running time of O ( n d − 1 log n ) {\displaystyle O(n^{d-1}\log n)} . This turns out not to be optimal, because it only decomposes the d -dimensional problem into n ( d-1 )-dimensional problems, and does not further decompose those subproblems. In 1981, Jan van Leeuwen and Derek Wood improved the running time of this algorithm to O ( n d − 1 ) {\displaystyle O(n^{d-1})} for d ≥ 3 by using dynamic quadtrees .
In 1988, Mark Overmars and Chee Yap proposed an O ( n d / 2 log n ) {\displaystyle O(n^{d/2}\log n)} algorithm for d ≥ 3. Their algorithm uses a particular data structure similar to a kd-tree to decompose the problem into 2-dimensional components and aggregate those components efficiently; the 2-dimensional problems themselves are solved efficiently using a trellis structure. Although asymptotically faster than Bentley's algorithm, its data structures use significantly more space, so it is only used in problems where either n or d is large. In 1998, Bogdan Chlebus proposed a simpler algorithm with the same asymptotic running time for the common special cases where d is 3 or 4.
In 2013, Timothy M. Chan developed a simpler algorithm that avoids the need for dynamic data structures and eliminates the logarithmic factor, lowering the best known running time for d ≥ 3 to O ( n d / 2 ) {\displaystyle O(n^{d/2})} .
The only known lower bound for any d is Ω ( n log n ) {\displaystyle \Omega (n\log n)} , and optimal algorithms with this running time are known for d =1 and d =2. The Chan algorithm provides an upper bound of O ( n d / 2 ) {\displaystyle O(n^{d/2})} for d ≥ 3, so for d ≥ 3, it remains an open question whether faster algorithms are possible, or alternatively whether tighter lower bounds can be proven. In particular, it remains open whether the algorithm's running time must depend on d . In addition, the question of whether there are faster algorithms that can deal with special cases (for example, when the input coordinates are integers within a bounded range) remains open.
The 1D Klee's measure problem (union of intervals) can be solved in O ( n log p ) {\displaystyle O(n\log p)} where p denotes the number of piercing points required to stab all intervals [ 1 ] (the union of intervals pierced by a common point can be calculated in linear time by computing the extrema).
Parameter p is an adaptive parameter that depends on the input configuration, and the piercing algorithm [ 2 ] yields an adaptive algorithm for Klee's measure problem. | https://en.wikipedia.org/wiki/Klee's_measure_problem |
Klee diagrams , named for their resemblance to paintings by Paul Klee , are false-colour maps that represent a way of assembling and viewing large genomic datasets . Contemporary research has produced genomic databases for an enormous range of life forms, inviting insights into the genetic basis of biodiversity . [ 1 ]
Indicator vectors are used to depict nucleotide sequences . This technique produces correlation matrices or Klee diagrams. Researchers Lawrence Sirovich , Mark Y. Stoeckle and Yu Zhang (2010) used their improved algorithm on a set of some 17000 DNA barcode sequences from 12 disparate animal taxa , finding that indicator vectors were a viable taxonomic tool, and that discontinuities corresponded with taxonomic divisions. | https://en.wikipedia.org/wiki/Klee_diagram |
In computability theory , Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 [ 1 ] and appear in his 1952 book Introduction to Metamathematics . [ 2 ] A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr. [ 3 ]
The recursion theorems can be applied to construct fixed points of certain operations on computable functions , to generate quines , and to construct functions defined via recursive definitions .
The statement of the theorems refers to an admissible numbering φ {\displaystyle \varphi } of the partial recursive functions , such that the function corresponding to index e {\displaystyle e} is φ e {\displaystyle \varphi _{e}} .
If F {\displaystyle F} and G {\displaystyle G} are partial functions on the natural numbers, the notation F ≃ G {\displaystyle F\simeq G} indicates that, for each n , either F ( n ) {\displaystyle F(n)} and G ( n ) {\displaystyle G(n)} are both defined and are equal, or else F ( n ) {\displaystyle F(n)} and G ( n ) {\displaystyle G(n)} are both undefined.
Given a function F {\displaystyle F} , a fixed point of F {\displaystyle F} is an index e {\displaystyle e} such that φ e ≃ φ F ( e ) {\displaystyle \varphi _{e}\simeq \varphi _{F(e)}} . Note that the comparison of in- and outputs here is not in terms of numerical values, but in terms of their associated functions.
Rogers describes the following result as "a simpler version" of Kleene's (second) recursion theorem. [ 4 ]
Rogers's fixed-point theorem — If F {\displaystyle F} is a total computable function, it has a fixed point in the above sense.
This essentially means that if we apply an effective transformation to programs (say, replace instructions such as successor, jump, remove lines), there will always be a program whose behaviour is not altered by the transformation. This theorem can therefore be interpreted in the following manner: “given any effective procedure to transform programs, there is always a program that, when modified by the procedure, does exactly what it did before”, or: “it’s impossible to write a program that changes the extensional behaviour of all programs”.
The proof uses a particular total computable function h {\displaystyle h} , defined as follows. Given a natural number x {\displaystyle x} , the function h {\displaystyle h} outputs the index of the partial computable function that performs the following computation:
Thus, for all indices x {\displaystyle x} of partial computable functions, if φ x ( x ) {\displaystyle \varphi _{x}(x)} is defined, then φ h ( x ) ≃ φ φ x ( x ) {\displaystyle \varphi _{h(x)}\simeq \varphi _{\varphi _{x}(x)}} . If φ x ( x ) {\displaystyle \varphi _{x}(x)} is not defined, then φ h ( x ) {\displaystyle \varphi _{h(x)}} is a function that is nowhere defined. The function h {\displaystyle h} can be constructed from the partial computable function g ( x , y ) {\displaystyle g(x,y)} described above and the s-m-n theorem : for each x {\displaystyle x} , h ( x ) {\displaystyle h(x)} is the index of a program which computes the function y ↦ g ( x , y ) {\displaystyle y\mapsto g(x,y)} .
To complete the proof, let F {\displaystyle F} be any total computable function, and construct h {\displaystyle h} as above. Let e {\displaystyle e} be an index of the composition F ∘ h {\displaystyle F\circ h} , which is a total computable function. Then φ h ( e ) ≃ φ φ e ( e ) {\displaystyle \varphi _{h(e)}\simeq \varphi _{\varphi _{e}(e)}} by the definition of h {\displaystyle h} .
But, because e {\displaystyle e} is an index of F ∘ h {\displaystyle F\circ h} , φ e ( e ) = ( F ∘ h ) ( e ) {\displaystyle \varphi _{e}(e)=(F\circ h)(e)} , and thus φ φ e ( e ) ≃ φ F ( h ( e ) ) {\displaystyle \varphi _{\varphi _{e}(e)}\simeq \varphi _{F(h(e))}} . By the transitivity of ≃ {\displaystyle \simeq } , this means φ h ( e ) ≃ φ F ( h ( e ) ) {\displaystyle \varphi _{h(e)}\simeq \varphi _{F(h(e))}} . Hence φ n ≃ φ F ( n ) {\displaystyle \varphi _{n}\simeq \varphi _{F(n)}} for n = h ( e ) {\displaystyle n=h(e)} .
This proof is a construction of a partial recursive function which implements the Y combinator .
A function F {\displaystyle F} such that φ e ≄ φ F ( e ) {\displaystyle \varphi _{e}\not \simeq \varphi _{F(e)}} for all e {\displaystyle e} is called fixed-point free . The fixed-point theorem shows that no total computable function is fixed-point free, but there are many non-computable fixed-point-free functions. Arslanov's completeness criterion states that the only recursively enumerable Turing degree that computes a fixed-point-free function is 0′ , the degree of the halting problem . [ 5 ]
The second recursion theorem is a generalization of Rogers's theorem with a second input in the function. One informal interpretation of the second recursion theorem is that it is possible to construct self-referential programs; see "Application to quines" below.
The theorem can be proved from Rogers's theorem by letting F ( p ) {\displaystyle F(p)} be a function such that φ F ( p ) ( y ) = Q ( p , y ) {\displaystyle \varphi _{F(p)}(y)=Q(p,y)} (a construction described by the S-m-n theorem ). One can then verify that a fixed-point of this F {\displaystyle F} is an index p {\displaystyle p} as required. The theorem is constructive in the sense that a fixed computable function maps an index for Q {\displaystyle Q} into the index p {\displaystyle p} .
Kleene's second recursion theorem and Rogers's theorem can both be proved, rather simply, from each other. [ 6 ] However, a direct proof of Kleene's theorem [ 7 ] does not make use of a universal program, which means that the theorem holds for certain subrecursive programming systems that do not have a universal program.
A classic example using the second recursion theorem is the function Q ( x , y ) = x {\displaystyle Q(x,y)=x} . The corresponding index p {\displaystyle p} in this case yields a computable function that outputs its own index when applied to any value. [ 8 ] When expressed as computer programs, such indices are known as quines .
The following example in Lisp illustrates how the p {\displaystyle p} in the corollary can be effectively produced from the function Q {\displaystyle Q} . The function s11 in the code is the function of that name produced by the S-m-n theorem .
Q can be changed to any two-argument function.
The results of the following expressions should be the same. φ {\displaystyle \varphi } p(nil)
Q(p, nil)
Suppose that g {\displaystyle g} and h {\displaystyle h} are total computable functions that are used in a recursive definition for a function f {\displaystyle f} :
The second recursion theorem can be used to show that such equations define a computable function, where the notion of computability does not have to allow, prima facie, for recursive definitions (for example, it may be defined by μ-recursion , or by Turing machines ). This recursive definition can be converted into a computable function φ F ( e , x , y ) {\displaystyle \varphi _{F}(e,x,y)} that assumes e {\displaystyle e} is an index to itself, to simulate recursion:
The recursion theorem establishes the existence of a computable function φ f {\displaystyle \varphi _{f}} such that φ f ( x , y ) ≃ φ F ( f , x , y ) {\displaystyle \varphi _{f}(x,y)\simeq \varphi _{F}(f,x,y)} . Thus f {\displaystyle f} satisfies the given recursive definition.
Reflexive, or reflective , programming refers to the usage of self-reference in programs. Jones presents a view of the second recursion theorem based on a reflexive language. [ 9 ] It is shown that the reflexive language defined is not stronger than a language without reflection (because an interpreter for the reflexive language can be implemented without using reflection); then, it is shown that the recursion theorem is almost trivial in the reflexive language.
While the second recursion theorem is about fixed points of computable functions, the first recursion theorem is related to fixed points determined by enumeration operators, which are a computable analogue of inductive definitions. An enumeration operator is a set of pairs ( A , n ) where A is a ( code for a) finite set of numbers and n is a single natural number . Often, n will be viewed as a code for an ordered pair of natural numbers, particularly when functions are defined via enumeration operators. Enumeration operators are of central importance in the study of enumeration reducibility .
Each enumeration operator Φ determines a function from sets of naturals to sets of naturals given by
A recursive operator is an enumeration operator that, when given the graph of a partial recursive function, always returns the graph of a partial recursive function.
A fixed point of an enumeration operator Φ is a set F such that Φ( F ) = F . The first enumeration theorem shows that fixed points can be effectively obtained if the enumeration operator itself is computable.
The first recursion theorem is also called Fixed point theorem (of recursion theory). [ 10 ] There is also a definition which can be applied to recursive functionals as follows:
Let Φ : F ( N k ) → ( N k ) {\displaystyle \Phi :\mathbb {F} (\mathbb {N} ^{k})\rightarrow (\mathbb {N} ^{k})} be a recursive functional. Then Φ {\displaystyle \Phi } has a least fixed point f Φ : N k → N {\displaystyle f_{\Phi }:\mathbb {N} ^{k}\rightarrow \mathbb {N} } which is computable i.e.
1) Φ ( f ϕ ) = f Φ {\displaystyle \Phi (f_{\phi })=f_{\Phi }}
2) ∀ g ∈ F ( N k ) {\displaystyle \forall g\in \mathbb {F} (\mathbb {N} ^{k})} such that Φ ( g ) = g {\displaystyle \Phi (g)=g} it holds that f Φ ⊆ g {\displaystyle f_{\Phi }\subseteq g}
3) f Φ {\displaystyle f_{\Phi }} is computable
Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator.
Consider the recursion equations for the factorial function f : f ( 0 ) = 1 f ( n + 1 ) = ( n + 1 ) ⋅ f ( n ) {\displaystyle {\begin{aligned}&f(0)=1\\&f(n+1)=(n+1)\cdot f(n)\end{aligned}}} The corresponding recursive operator Φ will have information that tells how to get to the next value of f from the previous value. However, the recursive operator will actually define the graph of f . First, Φ will contain the pair ( ∅ , ( 0 , 1 ) ) {\displaystyle (\varnothing ,(0,1))} . This indicates that f (0) is unequivocally 1, and thus the pair (0,1) is in the graph of f .
Next, for each n and m , Φ will contain the pair ( { ( n , m ) } , ( n + 1 , ( n + 1 ) ⋅ m ) ) {\displaystyle (\{(n,m)\},(n+1,(n+1)\cdot m))} . This indicates that, if f ( n ) is m , then f ( n + 1) is ( n + 1) m , so that the pair ( n + 1, ( n + 1) m ) is in the graph of f . Unlike the base case f (0) = 1 , the recursive operator requires some information about f ( n ) before it defines a value of f ( n + 1) .
The first recursion theorem (in particular, part 1) states that there is a set F such that Φ( F ) = F . The set F will consist entirely of ordered pairs of natural numbers, and will be the graph of the factorial function f , as desired.
The restriction to recursion equations that can be recast as recursive operators ensures that the recursion equations actually define a least fixed point . For example, consider the set of recursion equations: g ( 0 ) = 1 g ( n + 1 ) = 1 g ( 2 n ) = 0 {\displaystyle {\begin{aligned}&g(0)=1\\&g(n+1)=1\\&g(2n)=0\end{aligned}}} There is no function g satisfying these equations, because they imply g (2) = 1 and also imply g (2) = 0. Thus there is no fixed point g satisfying these recursion equations. It is possible to make an enumeration operator corresponding to these equations, but it will not be a recursive operator.
The proof of part 1 of the first recursion theorem is obtained by iterating the enumeration operator Φ beginning with the empty set . First, a sequence F k is constructed, for k = 0 , 1 , … {\displaystyle k=0,1,\ldots } . Let F 0 be the empty set. Proceeding inductively, for each k , let F k + 1 be F k ∪ Φ ( F k ) {\displaystyle F_{k}\cup \Phi (F_{k})} . Finally, F is taken to be ⋃ F k {\textstyle \bigcup F_{k}} . The remainder of the proof consists of a verification that F is recursively enumerable and is the least fixed point of Φ. The sequence F k used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem .
The second part of the first recursion theorem follows from the first part. The assumption that Φ is a recursive operator is used to show that the fixed point of Φ is the graph of a partial function. The key point is that if the fixed point F is not the graph of a function, then there is some k such that F k is not the graph of a function.
Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied. Rogers uses the term weak recursion theorem for the first recursion theorem and strong recursion theorem for the second recursion theorem. [ 3 ]
One difference between the first and second recursion theorems is that the fixed points obtained by the first recursion theorem are guaranteed to be least fixed points, while those obtained from the second recursion theorem may not be least fixed points.
A second difference is that the first recursion theorem only applies to systems of equations that can be recast as recursive operators. This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory . The second recursion theorem can be applied to any total recursive function.
In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering . [ 11 ] A Gödel numbering is a precomplete numbering on the set of computable functions so the generalized theorem yields the Kleene recursion theorem as a special case. [ 12 ]
Given a precomplete numbering ν {\displaystyle \nu } , then for any partial computable function f {\displaystyle f} with two parameters there exists a total computable function t {\displaystyle t} with one parameter such that
Footnotes | https://en.wikipedia.org/wiki/Kleene's_recursion_theorem |
In the mathematical areas of order and lattice theory , the Kleene fixed-point theorem , named after American mathematician Stephen Cole Kleene , states the following:
The ascending Kleene chain of f is the chain
obtained by iterating f on the least element ⊥ of L . Expressed in a formula, the theorem states that
where lfp {\displaystyle {\textrm {lfp}}} denotes the least fixed point.
Although Tarski's fixed point theorem does not consider how fixed points can be computed by iterating f from some seed (also, it pertains to monotone functions on complete lattices ), this result is often attributed to Alfred Tarski who proves it for additive functions. [ 1 ] Moreover, the Kleene fixed-point theorem can be extended to monotone functions using transfinite iterations. [ 2 ]
Source: [ 3 ]
We first have to show that the ascending Kleene chain of f {\displaystyle f} exists in L {\displaystyle L} . To show that, we prove the following:
As a corollary of the Lemma we have the following directed ω-chain:
From the definition of a dcpo it follows that M {\displaystyle \mathbb {M} } has a supremum, call it m . {\displaystyle m.} What remains now is to show that m {\displaystyle m} is the least fixed-point.
First, we show that m {\displaystyle m} is a fixed point, i.e. that f ( m ) = m {\displaystyle f(m)=m} . Because f {\displaystyle f} is Scott-continuous , f ( sup ( M ) ) = sup ( f ( M ) ) {\displaystyle f(\sup(\mathbb {M} ))=\sup(f(\mathbb {M} ))} , that is f ( m ) = sup ( f ( M ) ) {\displaystyle f(m)=\sup(f(\mathbb {M} ))} . Also, since M = f ( M ) ∪ { ⊥ } {\displaystyle \mathbb {M} =f(\mathbb {M} )\cup \{\bot \}} and because ⊥ {\displaystyle \bot } has no influence in determining the supremum we have: sup ( f ( M ) ) = sup ( M ) {\displaystyle \sup(f(\mathbb {M} ))=\sup(\mathbb {M} )} . It follows that f ( m ) = m {\displaystyle f(m)=m} , making m {\displaystyle m} a fixed-point of f {\displaystyle f} .
The proof that m {\displaystyle m} is in fact the least fixed point can be done by showing that any element in M {\displaystyle \mathbb {M} } is smaller than any fixed-point of f {\displaystyle f} (because by property of supremum , if all elements of a set D ⊆ L {\displaystyle D\subseteq L} are smaller than an element of L {\displaystyle L} then also sup ( D ) {\displaystyle \sup(D)} is smaller than that same element of L {\displaystyle L} ). This is done by induction: Assume k {\displaystyle k} is some fixed-point of f {\displaystyle f} . We now prove by induction over i {\displaystyle i} that ∀ i ∈ N : f i ( ⊥ ) ⊑ k {\displaystyle \forall i\in \mathbb {N} :f^{i}(\bot )\sqsubseteq k} . The base of the induction ( i = 0 ) {\displaystyle (i=0)} obviously holds: f 0 ( ⊥ ) = ⊥ ⊑ k , {\displaystyle f^{0}(\bot )=\bot \sqsubseteq k,} since ⊥ {\displaystyle \bot } is the least element of L {\displaystyle L} . As the induction hypothesis, we may assume that f i ( ⊥ ) ⊑ k {\displaystyle f^{i}(\bot )\sqsubseteq k} . We now do the induction step: From the induction hypothesis and the monotonicity of f {\displaystyle f} (again, implied by the Scott-continuity of f {\displaystyle f} ), we may conclude the following: f i ( ⊥ ) ⊑ k ⟹ f i + 1 ( ⊥ ) ⊑ f ( k ) . {\displaystyle f^{i}(\bot )\sqsubseteq k~\implies ~f^{i+1}(\bot )\sqsubseteq f(k).} Now, by the assumption that k {\displaystyle k} is a fixed-point of f , {\displaystyle f,} we know that f ( k ) = k , {\displaystyle f(k)=k,} and from that we get f i + 1 ( ⊥ ) ⊑ k . {\displaystyle f^{i+1}(\bot )\sqsubseteq k.} | https://en.wikipedia.org/wiki/Kleene_fixed-point_theorem |
In descriptive set theory , the Kleene–Brouwer order or Lusin–Sierpiński order [ 1 ] is a linear order on finite sequences over some linearly ordered set ( X , < ) {\displaystyle (X,<)} , that differs from the more commonly used lexicographic order in how it handles the case when one sequence is a prefix of the other. In the Kleene–Brouwer order, the prefix is later than the longer sequence containing it, rather than earlier.
The Kleene–Brouwer order generalizes the notion of a postorder traversal from finite trees to trees that are not necessarily finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well-ordering if and only if the tree has no infinite branch. It is named after Stephen Cole Kleene , Luitzen Egbertus Jan Brouwer , Nikolai Luzin , and Wacław Sierpiński .
If t {\displaystyle t} and s {\displaystyle s} are finite sequences of elements from X {\displaystyle X} , we say that t < K B s {\displaystyle t<_{KB}s} when there is an n {\displaystyle n} such that either:
Here, the notation t ↾ n {\displaystyle t\upharpoonright n} refers to the prefix of t {\displaystyle t} up to but not including t ( n ) {\displaystyle t(n)} .
In simple terms, t < K B s {\displaystyle t<_{KB}s} whenever s {\displaystyle s} is a prefix of t {\displaystyle t} (i.e. s {\displaystyle s} terminates before t {\displaystyle t} , and they are equal up to that point) or t {\displaystyle t} is to the "left" of s {\displaystyle s} on the first place they differ. [ 1 ]
A tree , in descriptive set theory, is defined as a set of finite sequences that is closed under prefix operations. The parent in the tree of any sequence is the shorter sequence formed by removing its final element. Thus, any set of finite sequences can be augmented to form a tree, and the Kleene–Brouwer order is a natural ordering that may be given to this tree. It is a generalization to potentially-infinite trees of the postorder traversal of a finite tree: at every node of the tree, the child subtrees are given their left to right ordering, and the node itself comes after all its children. The fact that the Kleene–Brouwer order is a linear ordering (that is, that it is transitive as well as being total) follows immediately from this, as any three sequences on which transitivity is to be tested form (with their prefixes) a finite tree on which the Kleene–Brouwer order coincides with the postorder.
The significance of the Kleene–Brouwer ordering comes from the fact that if X {\displaystyle X} is well-ordered , then a tree over X {\displaystyle X} is well-founded (having no infinitely long branches) if and only if the Kleene–Brouwer ordering is a well-ordering of the elements of the tree. [ 1 ]
In recursion theory , the Kleene–Brouwer order may be applied to the computation trees of implementations of total recursive functionals . A computation tree is well-founded if and only if the computation performed by it is total recursive. Each state x {\displaystyle x} in a computation tree may be assigned an ordinal number | | x | | {\displaystyle ||x||} , the supremum of the ordinal numbers 1 + | | y | | {\displaystyle 1+||y||} where y {\displaystyle y} ranges over the children of x {\displaystyle x} in the tree. In this way, the total recursive functionals themselves can be classified into a hierarchy, according to the minimum value of the ordinal at the root of a computation tree, minimized over all computation trees that implement the functional. The Kleene–Brouwer order of a well-founded computation tree is itself a recursive well-ordering, and at least as large as the ordinal assigned to the tree, from which it follows that the levels of this hierarchy are indexed by recursive ordinals . [ 2 ]
This ordering was used by Lusin & Sierpinski (1923) , [ 3 ] and then again by Brouwer (1924) . [ 4 ] Brouwer does not cite any references, but Moschovakis argues that he may either have seen Lusin & Sierpinski (1923) , or have been influenced by earlier work of the same authors leading to this work. Much later, Kleene (1955) studied the same ordering, and credited it to Brouwer. [ 5 ] | https://en.wikipedia.org/wiki/Kleene–Brouwer_order |
In mathematics , the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent , in particular the version of Haskell Curry 's combinatory logic introduced in 1930, and Alonzo Church 's original lambda calculus , introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by Stephen Kleene and J. B. Rosser in 1935.
Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total, definable number-theoretic functions, which enabled them to construct a term that essentially replicates Richard's paradox in formal language.
Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox, and used this to construct a much simpler paradox, now known as Curry's paradox .
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kleene–Rosser_paradox |
In mathematics , a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program . More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G , which acts as the symmetry group of the geometry.
For background and motivation see the article on the Erlangen program .
A Klein geometry is a pair ( G , H ) where G is a Lie group and H is a closed Lie subgroup of G such that the (left) coset space G / H is connected . The group G is called the principal group of the geometry and G / H is called the space of the geometry (or, by an abuse of terminology, simply the Klein geometry ). The space X = G / H of a Klein geometry is a smooth manifold of dimension
There is a natural smooth left action of G on X given by
Clearly, this action is transitive (take a = 1 ), so that one may then regard X as a homogeneous space for the action of G . The stabilizer of the identity coset H ∈ X is precisely the group H .
Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X , we can construct an associated Klein geometry ( G , H ) by fixing a basepoint x 0 in X and letting H be the stabilizer subgroup of x 0 in G . The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G / H .
Two Klein geometries ( G 1 , H 1 ) and ( G 2 , H 2 ) are geometrically isomorphic if there is a Lie group isomorphism φ : G 1 → G 2 so that φ ( H 1 ) = H 2 . In particular, if φ is conjugation by an element g ∈ G , we see that ( G , H ) and ( G , gHg −1 ) are isomorphic. The Klein geometry associated to a homogeneous space X is then unique up to isomorphism (i.e. it is independent of the chosen basepoint x 0 ).
Given a Lie group G and closed subgroup H , there is natural right action of H on G given by right multiplication. This action is both free and proper . The orbits are simply the left cosets of H in G . One concludes that G has the structure of a smooth principal H -bundle over the left coset space G / H :
The action of G on X = G / H need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of G on X . It is given by
The kernel K may also be described as the core of H in G (i.e. the largest subgroup of H that is normal in G ). It is the group generated by all the normal subgroups of G that lie in H .
A Klein geometry is said to be effective if K = 1 and locally effective if K is discrete . If ( G , H ) is a Klein geometry with kernel K , then ( G / K , H / K ) is an effective Klein geometry canonically associated to ( G , H ) .
A Klein geometry ( G , H ) is geometrically oriented if G is connected . (This does not imply that G / H is an oriented manifold ). If H is connected it follows that G is also connected (this is because G / H is assumed to be connected, and G → G / H is a fibration ).
Given any Klein geometry ( G , H ) , there is a geometrically oriented geometry canonically associated to ( G , H ) with the same base space G / H . This is the geometry ( G 0 , G 0 ∩ H ) where G 0 is the identity component of G . Note that G = G 0 H .
A Klein geometry ( G , H ) is said to be reductive and G / H a reductive homogeneous space if the Lie algebra h {\displaystyle {\mathfrak {h}}} of H has an H -invariant complement in g {\displaystyle {\mathfrak {g}}} .
In the following table, there is a description of the classical geometries, modeled as Klein geometries. | https://en.wikipedia.org/wiki/Klein_geometry |
The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation ) is a relativistic wave equation , related to the Schrödinger equation . It is named after Oskar Klein and Walter Gordon . It is second-order in space and time and manifestly Lorentz-covariant . It is a differential equation version of the relativistic energy–momentum relation E 2 = ( p c ) 2 + ( m 0 c 2 ) 2 {\displaystyle E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2}\,} .
The Klein–Gordon equation can be written in different ways. The equation itself usually refers to the position space form, where it can be written in terms of separated space and time components ( t , x ) {\displaystyle \ \left(\ t,\mathbf {x} \ \right)\ } or by combining them into a four-vector x μ = ( c t , x ) . {\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)~.} By Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum obey the energy-momentum dispersion relation from special relativity . Here, the Klein–Gordon equation is given for both of the two common metric signature conventions η μ ν = diag ( ± 1 , ∓ 1 , ∓ 1 , ∓ 1 ) . {\displaystyle \ \eta _{\mu \nu }={\text{diag}}\left(\ \pm 1,\mp 1,\mp 1,\mp 1\ \right)~.}
x μ = ( c t , x ) {\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)\ }
ω = E ℏ , k = p ℏ {\displaystyle \ \omega ={\frac {\ E\ }{\hbar }},\quad \mathbf {k} ={\frac {\ \mathbf {p} \ }{\hbar }}\ }
p μ = ( E c , p ) {\displaystyle \ p^{\mu }=\left({\frac {\ E\ }{c}},\mathbf {p} \right)\ }
Here, ◻ = ± η μ ν ∂ μ ∂ ν {\displaystyle \ \Box =\pm \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\ } is the wave operator and ∇ 2 {\displaystyle \nabla ^{2}} is the Laplace operator . The speed of light c {\displaystyle \ c\ } and Planck constant ℏ {\displaystyle \ \hbar \ } are often seen to clutter the equations, so they are therefore often expressed in natural units where c = ℏ = 1 . {\displaystyle \ c=\hbar =1~.}
x μ = ( t , x ) {\displaystyle \ x^{\mu }=\left(\ t,\mathbf {x} \ \right)\ }
ω = E , k = p {\displaystyle \ \omega =E,\quad \mathbf {k} =\mathbf {p} \ }
p μ = ( E , p ) {\displaystyle \ p^{\mu }=\left(\ E,\mathbf {p} \ \right)\ }
time and space
Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of ω for each k : One positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes
which is formally the same as the homogeneous screened Poisson equation . In addition, the Klein–Gordon equation can also be represented as: [ 1 ]
where, the momentum operator is given as:
The equation is to be understood first as a classical continuous scalar field equation that can be quantized. The quantization process introduces then a quantum field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation . [ 2 ] The equation solutions include a scalar or pseudoscalar field [ clarification needed ] . In the realm of particle physics electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics , the practical utility for particles like pions is limited. [ nb 1 ] [ 3 ] There is a second version of the equation for a complex scalar field that is theoretically important being the equation of the Higgs Boson . In the realm of condensed matter it can be used for many approximations of quasi-particles without spin. [ 4 ] [ 5 ] [ nb 2 ]
The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. [ 6 ] The solutions have two components, reflecting the charge degree of freedom in relativity. [ 6 ] [ 7 ] It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude . The conserved quantity is instead interpreted as electric charge , and the norm squared of the wave function is interpreted as a charge density . The equation describes all spinless particles with positive, negative, and zero charge.
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Despite historically it was invented as a single particle equation the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory, any relativistic theory implies creation and annihilation of particles beyond a certain energy threshold. [ 8 ] [ nb 3 ]
Here, the Klein–Gordon equation in natural units, ( ◻ + m 2 ) ψ ( x ) = 0 {\displaystyle (\Box +m^{2})\psi (x)=0} , with the metric signature η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)} is solved by Fourier transformation. Inserting the Fourier transformation ψ ( x ) = ∫ d 4 p ( 2 π ) 4 e − i p ⋅ x ψ ( p ) {\displaystyle \psi (x)=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{-ip\cdot x}\psi (p)} and using orthogonality of the complex exponentials gives the dispersion relation p 2 = ( p 0 ) 2 − p 2 = m 2 {\displaystyle p^{2}=(p^{0})^{2}-\mathbf {p} ^{2}=m^{2}} This restricts the momenta to those that lie on shell , giving positive and negative energy solutions p 0 = ± E ( p ) where E ( p ) = p 2 + m 2 . {\displaystyle p^{0}=\pm E(\mathbf {p} )\quad {\text{where}}\quad E(\mathbf {p} )={\sqrt {\mathbf {p} ^{2}+m^{2}}}.} For a new set of constants C ( p ) {\displaystyle C(p)} , the solution then becomes ψ ( x ) = ∫ d 4 p ( 2 π ) 4 e i p ⋅ x C ( p ) δ ( ( p 0 ) 2 − E ( p ) 2 ) . {\displaystyle \psi (x)=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{ip\cdot x}C(p)\delta ((p^{0})^{2}-E(\mathbf {p} )^{2}).} It is common to handle the positive and negative energy solutions by separating out the negative energies and work only with positive p 0 {\displaystyle p^{0}} : ψ ( x ) = ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p 0 x 0 + i p i x i + B ( p ) e + i p 0 x 0 + i p i x i ) θ ( p 0 ) = ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p 0 x 0 + i p i x i + B ( − p ) e + i p 0 x 0 − i p i x i ) θ ( p 0 ) → ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) θ ( p 0 ) {\displaystyle {\begin{aligned}\psi (x)=&\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip^{0}x^{0}+ip^{i}x^{i}}+B(p)e^{+ip^{0}x^{0}+ip^{i}x^{i}}\right)\theta (p^{0})\\=&\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip^{0}x^{0}+ip^{i}x^{i}}+B(-p)e^{+ip^{0}x^{0}-ip^{i}x^{i}}\right)\theta (p^{0})\\\rightarrow &\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip\cdot x}+B(p)e^{+ip\cdot x}\right)\theta (p^{0})\\\end{aligned}}} In the last step, B ( p ) → B ( − p ) {\displaystyle B(p)\rightarrow B(-p)} was renamed. Now we can perform the p 0 {\displaystyle p^{0}} -integration, picking up the positive frequency part from the delta function only:
ψ ( x ) = ∫ d 4 p ( 2 π ) 4 δ ( p 0 − E ( p ) ) 2 E ( p ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) θ ( p 0 ) = ∫ d 3 p ( 2 π ) 3 1 2 E ( p ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) | p 0 = + E ( p ) . {\displaystyle {\begin{aligned}\psi (x)&=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}{\frac {\delta (p^{0}-E(\mathbf {p} ))}{2E(\mathbf {p} )}}\left(A(p)e^{-ip\cdot x}+B(p)e^{+ip\cdot x}\right)\theta (p^{0})\\&=\int \left.{\frac {\mathrm {d} ^{3}p}{(2\pi )^{3}}}{\frac {1}{2E(\mathbf {p} )}}\left(A(\mathbf {p} )e^{-ip\cdot x}+B(\mathbf {p} )e^{+ip\cdot x}\right)\right|_{p^{0}=+E(\mathbf {p} )}.\end{aligned}}}
This is commonly taken as a general solution to the free Klein–Gordon equation. Note that because the initial Fourier transformation contained Lorentz invariant quantities like p ⋅ x = p μ x μ {\displaystyle p\cdot x=p_{\mu }x^{\mu }} only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation. If one does not require Lorentz invariance, one can absorb the 1 / 2 E ( p ) {\displaystyle 1/2E(\mathbf {p} )} -factor into the coefficients A ( p ) {\displaystyle A(p)} and B ( p ) {\displaystyle B(p)} .
The equation was named after the physicists Oskar Klein [ 9 ] and Walter Gordon , [ 10 ] who in 1926 proposed that it describes relativistic electrons. Vladimir Fock also discovered the equation independently in 1926 slightly after Klein's work, [ 11 ] in that Klein's paper was received on 28 April 1926, Fock's paper was received on 30 July 1926 and Gordon's paper on 29 September 1926. Other authors making similar claims in that same year include Johann Kudar, Théophile de Donder and Frans-H. van den Dungen , and Louis de Broglie . Although it turned out that modeling the electron's spin required the Dirac equation, the Klein–Gordon equation correctly describes the spinless relativistic composite particles , like the pion . On 4 July 2012, European Organization for Nuclear Research CERN announced the discovery of the Higgs boson . Since the Higgs boson is a spin-zero particle, it is the first observed ostensibly elementary particle to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson observed is that of the Standard Model or a more exotic, possibly composite, form.
The Klein–Gordon equation was first considered as a quantum wave equation by Erwin Schrödinger in his search for an equation describing de Broglie waves . The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of 4 n / 2 n − 1 for the n -th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital-momentum quantum number l is replaced by total angular-momentum quantum number j . [ 12 ] In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure .
In 1926, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields , where forces were dependent on velocity , and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation . The Klein–Gordon equation for a free particle has a simple plane-wave solution.
The non-relativistic equation for the energy of a free particle is
By quantizing this, we get the non-relativistic Schrödinger equation for a free particle:
where
is the momentum operator ( ∇ being the del operator ), and
is the energy operator .
The Schrödinger equation suffers from not being relativistically invariant , meaning that it is inconsistent with special relativity .
It is natural to try to use the identity from special relativity describing the energy:
Then, just inserting the quantum-mechanical operators for momentum and energy yields the equation
The square root of a differential operator can be defined with the help of Fourier transformations , but due to the asymmetry of space and time derivatives, Dirac found it impossible to include external electromagnetic fields in a relativistically invariant way. So he looked for another equation that can be modified in order to describe the action of electromagnetic forces. In addition, this equation, as it stands, is nonlocal (see also Introduction to nonlocal equations ).
Klein and Gordon instead began with the square of the above identity, i.e.
which, when quantized, gives
which simplifies to
Rearranging terms yields
Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real-valued , as well as those that have complex values .
Rewriting the first two terms using the inverse of the Minkowski metric diag(− c 2 , 1, 1, 1) , and writing the Einstein summation convention explicitly we get
Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of
where
and
This operator is called the wave operator .
Today this form is interpreted as the relativistic field equation for spin -0 particles. [ 6 ] Furthermore, any component of any solution to the free Dirac equation (for a spin-1/2 particle) is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin due to the Bargmann–Wigner equations . Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation, [ 13 ] making the equation a generic expression of quantum fields.
The Klein–Gordon equation can be generalized to describe a field in some potential V ( ψ ) {\displaystyle V(\psi )} as [ 14 ]
Then the Klein–Gordon equation is the case V ( ψ ) = M 2 ψ ¯ ψ {\displaystyle V(\psi )=M^{2}{\bar {\psi }}\psi } .
Another common choice of potential which arises in interacting theories is the ϕ 4 {\displaystyle \phi ^{4}} potential for a real scalar field ϕ , {\displaystyle \phi ,}
The pure Higgs boson sector of the Standard model is modelled by a Klein–Gordon field with a potential, denoted H {\displaystyle H} for this section. The Standard model is a gauge theory and so while the field transforms trivially under the Lorentz group, it transforms as a C 2 {\displaystyle \mathbb {C} ^{2}} -valued vector under the action of the SU ( 2 ) {\displaystyle {\text{SU}}(2)} part of the gauge group. Therefore, while it is a vector field H : R 1 , 3 → C 2 {\displaystyle H:\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{2}} , it is still referred to as a scalar field, as scalar describes its transformation (formally, representation) under the Lorentz group. This is also discussed below in the scalar chromodynamics section.
The Higgs field is modelled by a potential
which can be viewed as a generalization of the ϕ 4 {\displaystyle \phi ^{4}} potential, but has an important difference: it has a circle of minima. This observation is an important one in the theory of spontaneous symmetry breaking in the Standard model.
The Klein–Gordon equation (and action) for a complex field ψ {\displaystyle \psi } admits a U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry. That is, under the transformations
the Klein–Gordon equation is invariant, as is the action (see below). By Noether's theorem for fields, corresponding to this symmetry there is a current J μ {\displaystyle J^{\mu }} defined as
which satisfies the conservation equation ∂ μ J μ ( x ) = 0. {\displaystyle \partial _{\mu }J^{\mu }(x)=0.} The form of the conserved current can be derived systematically by applying Noether's theorem to the U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry. We will not do so here, but simply verify that this current is conserved.
From the Klein–Gordon equation for a complex field ψ ( x ) {\displaystyle \psi (x)} of mass M {\displaystyle M} , written in covariant notation and mostly plus signature,
and its complex conjugate
Multiplying by the left respectively by ψ ¯ ( x ) {\displaystyle {\bar {\psi }}(x)} and ψ ( x ) {\displaystyle \psi (x)} (and omitting for brevity the explicit x {\displaystyle x} dependence),
Subtracting the former from the latter, we obtain
or in index notation,
Applying this to the derivative of the current J μ ( x ) ≡ ψ ∗ ( x ) ∂ μ ψ ( x ) − ψ ( x ) ∂ μ ψ ∗ ( x ) , {\displaystyle J^{\mu }(x)\equiv \psi ^{*}(x)\partial ^{\mu }\psi (x)-\psi (x)\partial ^{\mu }\psi ^{*}(x),} one finds
This U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry is a global symmetry, but it can also be gauged to create a local or gauge symmetry: see below scalar QED. The name of gauge symmetry is somewhat misleading: it is really a redundancy, while the global symmetry is a genuine symmetry.
The Klein–Gordon equation can also be derived by a variational method, arising as the Euler–Lagrange equation of the action
In natural units, with signature mostly minus , the actions take the simple form
S = ∫ d 4 x ( 1 2 ∂ μ ϕ ∂ μ ϕ − 1 2 m 2 ϕ 2 ) {\displaystyle S=\int d^{4}x\left({\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right)}
for a real scalar field of mass m {\displaystyle m} , and
S = ∫ d 4 x ( ∂ μ ψ ∂ μ ψ ¯ − M 2 ψ ψ ¯ ) {\displaystyle S=\int d^{4}x\left(\partial ^{\mu }\psi \partial _{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)}
for a complex scalar field of mass M {\displaystyle M} .
Applying the formula for the stress–energy tensor to the Lagrangian density (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is
and in natural units,
By integration of the time–time component T 00 over all space, one may show that both the positive- and negative-frequency plane-wave solutions can be physically associated with particles with positive energy. This is not the case for the Dirac equation and its energy–momentum tensor. [ 6 ]
The stress energy tensor is the set of conserved currents corresponding to the invariance of the Klein–Gordon equation under space-time translations x μ ↦ x μ + c μ {\displaystyle x^{\mu }\mapsto x^{\mu }+c^{\mu }} . Therefore, each component is conserved, that is, ∂ μ T μ ν = 0 {\displaystyle \partial _{\mu }T^{\mu \nu }=0} (this holds only on-shell , that is, when the Klein–Gordon equations are satisfied). It follows that the integral of T 0 ν {\displaystyle T^{0\nu }} over space is a conserved quantity for each ν {\displaystyle \nu } . These have the physical interpretation of total energy for ν = 0 {\displaystyle \nu =0} and total momentum for ν = i {\displaystyle \nu =i} with i ∈ { 1 , 2 , 3 } {\displaystyle i\in \{1,2,3\}} .
Taking the non-relativistic limit ( v ≪ c ) of a classical Klein–Gordon field ψ ( x , t ) begins with the ansatz factoring the oscillatory rest mass energy term,
Defining the kinetic energy E ′ = E − m c 2 = m 2 c 4 + c 2 p 2 − m c 2 ≈ p 2 2 m {\displaystyle E'=E-mc^{2}={\sqrt {m^{2}c^{4}+c^{2}p^{2}}}-mc^{2}\approx {\frac {p^{2}}{2m}}} , E ′ ≪ m c 2 {\displaystyle E'\ll mc^{2}} in the non-relativistic limit v = p / m ≪ c {\displaystyle v=p/m\ll c} , and hence
Applying this yields the non-relativistic limit of the second time derivative of ψ {\displaystyle \psi } ,
Substituting into the free Klein–Gordon equation, c − 2 ∂ t 2 ψ = ∇ 2 ψ − ( m c ℏ ) 2 ψ {\displaystyle c^{-2}\partial _{t}^{2}\psi =\nabla ^{2}\psi -({\frac {mc}{\hbar }})^{2}\psi } , yields
which (by dividing out the exponential and subtracting the mass term) simplifies to
This is a classical Schrödinger field .
The analogous limit of a quantum Klein–Gordon field is complicated by the non-commutativity of the field operator. In the limit v ≪ c , the creation and annihilation operators decouple and behave as independent quantum Schrödinger fields .
There is a way to make the complex Klein–Gordon field ψ {\displaystyle \psi } interact with electromagnetism in a gauge-invariant way. We can replace the (partial) derivative with the gauge-covariant derivative. Under a local U ( 1 ) {\displaystyle {\text{U}}(1)} gauge transformation, the fields transform as
where θ ( x ) = θ ( t , x ) {\displaystyle \theta (x)=\theta (t,{\textbf {x}})} is a function of spacetime, thus making it a local transformation, as opposed to a constant over all of spacetime, which would be a global U ( 1 ) {\displaystyle {\text{U}}(1)} transformation. A subtle point is that global transformations can arise as local ones, when the function θ ( x ) {\displaystyle \theta (x)} is taken to be a constant function.
A well-formulated theory should be invariant under such transformations. Precisely, this means that the equations of motion and action (see below) are invariant. To achieve this, ordinary derivatives ∂ μ {\displaystyle \partial _{\mu }} must be replaced by gauge-covariant derivatives D μ {\displaystyle D_{\mu }} , defined as
where the 4-potential or gauge field A μ {\displaystyle A_{\mu }} transforms under a gauge transformation θ {\displaystyle \theta } as
With these definitions, the covariant derivative transforms as
In natural units, the Klein–Gordon equation therefore becomes
Since an ungauged U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry is only present in complex Klein–Gordon theory, this coupling and promotion to a gauged U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry is compatible only with complex Klein–Gordon theory and not real Klein–Gordon theory.
In natural units and mostly minus signature we have
S = ∫ d 4 x ( − 1 4 F μ ν F μ ν + D μ ψ D μ ψ ¯ − M 2 ψ ψ ¯ ) {\displaystyle S=\int d^{4}x\,\left(-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D^{\mu }\psi D_{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)}
where F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is known as the Maxwell tensor, field strength or curvature depending on viewpoint.
This theory is often known as scalar quantum electrodynamics or scalar QED, although all aspects we've discussed here are classical.
It is possible to extend this to a non-abelian gauge theory with a gauge group G {\displaystyle G} , where we couple the scalar Klein–Gordon action to a Yang–Mills Lagrangian. Here, the field is actually vector-valued, but is still described as a scalar field: the scalar describes its transformation under space-time transformations , but not its transformation under the action of the gauge group.
For concreteness we fix G {\displaystyle G} to be SU ( N ) {\displaystyle {\text{SU}}(N)} , the special unitary group for some N ≥ 2 {\displaystyle N\geq 2} . Under a gauge transformation U ( x ) {\displaystyle U(x)} , which can be described as a function U : R 1 , 3 → SU ( N ) , {\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),} the scalar field ψ {\displaystyle \psi } transforms as a C N {\displaystyle \mathbb {C} ^{N}} vector
The covariant derivative is
where the gauge field or connection transforms as
This field can be seen as a matrix valued field which acts on the vector space C N {\displaystyle \mathbb {C} ^{N}} .
Finally defining the chromomagnetic field strength or curvature,
we can define the action.
S = ∫ d 4 x ( − 1 4 Tr ( F μ ν F μ ν ) + D μ ψ † D μ ψ − M 2 ψ † ψ ) {\displaystyle S=\int d^{4}x\,\left(-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+D^{\mu }\psi ^{\dagger }D_{\mu }\psi -M^{2}\psi ^{\dagger }\psi \right)}
In general relativity , we include the effect of gravity by replacing partial derivatives with covariant derivatives , and the Klein–Gordon equation becomes (in the mostly pluses signature ) [ 15 ]
or equivalently,
where g αβ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, ∇ μ is the covariant derivative , and Γ σ μν is the Christoffel symbol that is the gravitational force field .
With natural units this becomes
∇ a ∇ a Φ − m 2 Φ = 0 {\displaystyle \nabla ^{a}\nabla _{a}\Phi -m^{2}\Phi =0}
This also admits an action formulation on a spacetime (Lorentzian) manifold M {\displaystyle M} . Using abstract index notation and in mostly plus signature this is
S = ∫ M d 4 x − g ( − 1 2 g a b ∇ a Φ ∇ b Φ − 1 2 m 2 Φ 2 ) {\displaystyle S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-{\frac {1}{2}}g^{ab}\nabla _{a}\Phi \nabla _{b}\Phi -{\frac {1}{2}}m^{2}\Phi ^{2}\right)}
or
S = ∫ M d 4 x − g ( − g a b ∇ a Ψ ∇ b Ψ ¯ − M 2 Ψ Ψ ¯ ) {\displaystyle S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-g^{ab}\nabla _{a}\Psi \nabla _{b}{\bar {\Psi }}-M^{2}\Psi {\bar {\Psi }}\right)} | https://en.wikipedia.org/wiki/Klein–Gordon_equation |
In particle physics , the Klein–Nishina formula gives the differential cross section (i.e. the "likelihood" and angular distribution) of photons scattered from a single free electron , calculated in the lowest order of quantum electrodynamics . It was first derived in 1928 by Oskar Klein and Yoshio Nishina , constituting one of the first successful applications of the Dirac equation . [ 1 ] The formula describes both the Thomson scattering of low energy photons (e.g. visible light ) and the Compton scattering of high energy photons (e.g. x-rays and gamma-rays ), showing that the total cross section and expected deflection angle decrease with increasing photon energy.
In quantum field theory it is known as Klein–Nishina–Tamm formula , [ 2 ] adding the name of Igor Tamm who derived the formula from field quantization.
For an incident unpolarized photon of energy E γ {\displaystyle E_{\gamma }} , the differential cross section is: [ 3 ]
where
The angular dependent photon wavelength (or energy, or frequency) ratio is
as required by the conservation of relativistic energy and momentum (see Compton scattering ). The dimensionless quantity ϵ = E γ / m e c 2 {\displaystyle \epsilon =E_{\gamma }/m_{e}c^{2}} expresses the energy of the incident photon in terms of the electron rest energy (~511 keV ), and may also be expressed as ϵ = λ c / λ {\displaystyle \epsilon =\lambda _{c}/\lambda } , where λ c = h / m e c {\displaystyle \lambda _{c}=h/m_{e}c} is the Compton wavelength of the electron (~2.42 pm). Notice that the scatter ratio λ ′ / λ {\displaystyle \lambda '/\lambda } increases monotonically with the deflection angle, from 1 {\displaystyle 1} (forward scattering, no energy transfer) to 1 + 2 ϵ {\displaystyle 1+2\epsilon } (180 degree backscatter, maximum energy transfer).
In some cases it is convenient to express the classical electron radius in terms of the Compton wavelength: r e = α λ ¯ c = α λ c / 2 π {\displaystyle r_{e}=\alpha {\bar {\lambda }}_{c}=\alpha \lambda _{c}/2\pi } , where α {\displaystyle \alpha } is the fine structure constant (~1/137) and λ ¯ c = ℏ / m e c {\displaystyle {\bar {\lambda }}_{c}=\hbar /m_{e}c} is the reduced Compton wavelength of the electron (~0.386 pm), so that the constant in the cross section may be given as:
If the incoming photon is polarized, the scattered photon is no longer isotropic with respect to the azimuthal angle. For a linearly polarized photon scattered with a free electron at rest, the differential cross section is instead given by:
where ϕ {\displaystyle \phi } is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over cos 2 ( ϕ ) {\displaystyle \cos ^{2}(\phi )} .
For low energy photons the wavelength shift becomes negligible ( λ / λ ′ ≈ 1 {\displaystyle \lambda /\lambda '\approx 1} ) and the Klein–Nishina formula reduces to the classical Thomson expression :
which is symmetrical in the scattering angle, i.e. the photon is just as likely to scatter backwards as forwards. With increasing energy this symmetry is broken and the photon becomes more likely to scatter in the forward direction.
For high energy photons it is useful to distinguish between small and large angle scattering. For large angles, where ϵ ( 1 − cos θ ) ≫ 1 {\displaystyle \epsilon (1-\cos \theta )\gg 1} , the scatter ratio λ ′ / λ {\displaystyle \lambda '/\lambda } is large and
showing that the (large angle) differential cross section is inversely proportional to the photon energy.
The differential cross section has a constant peak in the forward direction:
independent of ϵ {\displaystyle \epsilon } . From the large angle analysis it follows that this peak can only extend to about θ c ≈ ϵ − 1 / 2 {\displaystyle \theta _{c}\approx \epsilon ^{-1/2}} . The forward peak is thus confined to a small solid angle of approximately π θ c 2 {\displaystyle \pi \theta _{c}^{2}} , and we may conclude that the total small angle cross section decreases with ϵ − 1 {\displaystyle \epsilon ^{-1}} .
The differential cross section may be integrated to find the total cross section :
In the low-energy limit there is no energy dependence, and we recover the Thomson cross section (~66.5 fm 2 ):
The Klein–Nishina formula was derived in 1928 by Oskar Klein and Yoshio Nishina , and was one of the first results obtained from the study of quantum electrodynamics . Consideration of relativistic and quantum mechanical effects allowed development of an accurate equation for the scattering of radiation from a target electron. Before this derivation, the electron cross section had been classically derived by the British physicist and discoverer of the electron , J.J. Thomson . However, scattering experiments showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments agreed perfectly with the predictions of the Klein–Nishina formula. [ citation needed ]
In 1930, Ivar Waller and Igor Tamm published their work independently on the field quantization of Compton scattering and reproduced Klein–Nishina formula. [ 4 ] | https://en.wikipedia.org/wiki/Klein–Nishina_formula |
A Klemperer rosette is a gravitational system of (optionally) alternating heavier and lighter bodies orbiting in a symmetrical pattern around a common barycenter . It was first described by W.B. Klemperer in 1962, [ 1 ] and is a special case of a central configuration .
Klemperer described rosette systems as follows:
Such symmetry is also possessed by a peculiar family of geometrical configurations which may be described as "rosettes". In these an even number of "planets" of two (or more) kinds, one (or some) heavier than the other, but all of each set of equal mass, are placed at the corners of two (or more) interdigitated regular polygons so that the lighter and heavier ones alternate (or follow each other in a cyclic manner). [ 1 ] (p 163)
The simplest rosette would be a series of four alternating heavier and lighter bodies, 90 degrees from one another, in a rhombic configuration [Heavy, Light, Heavy, Light], where the two larger bodies have the same mass, and likewise the two smaller bodies have the same mass, all orbiting their (empty) geometric center. The more general trojan system has unequal masses for the two heavier bodies, which Klemperer also calls a "rhombic" system, and is the only version that is not symmetric around the gravitational center.
The number of "mass types" can be increased, so long as the arrangement is symmetrical and cyclic pattern: e.g. [ 1,2,3 ... 1,2,3 ], [ 1,2,3,4,5 ... 1,2,3,4,5 ], [ 1,2,3,3,2,1 ... 1,2,3,3,2,1 ], etc.
Klemperer's article specifically analyzes regular polygons with 2–9 corners – dumbbell-shaped through nonagon – and non-centrally symmetric " rhombic rosettes" with three orbiting bodies, the outer two stationed at the middle orbiting body's triangular points (L4 and L5), which had already been described and studied by Lagrange in 1772. [ 2 ] Systems with an even number of 4 or more corners can have alternating heavy and light masses at the corners, though the possible range of mass ratios is constrained by para-stability requirements; systems with odd numbers of corners must have equal masses at every corner.
While Klemperer notes that all the rosettes and the rhombus are vulnerable to destabilization, the hexagonal rosette is the most nearly stable because the "planets" sit in each other's semi-stable triangular Lagrangian points , L4 and L5. [ 1 ] (p 165)
The regular polygonal configurations ("rosettes") do not require a central mass (a "sun" at the center is optional, and if present it may bobble above and below the orbital plane), although a Lagrange-type rhombus does. If a central body is present, its mass constrains the ranges for the mass-ratio between the orbiting bodies. [ 1 ]
The term "Klemperer rosette" (often misspelled " Kemplerer rosette") is used to mean a configuration of three or more equal masses, set at the points of an equilateral polygon and given an equal angular velocity about their center of mass . Klemperer does indeed mention this configuration at the start of his article, but only as an already known set of equilibrium systems before introducing the actual rosettes.
In Larry Niven 's novel Fleet of Worlds in the Ringworld series , the Puppeteers' eponymous "fleet of worlds" is arranged in such a configuration [ a ] that Niven calls a "Kemplerer rosette"; this (possibly intentional) misspelling is one viable source of the wider confusion. These fictional planets were maintained in position by large engines, in addition to gravitational force.
Both simple linear perturbation analysis and simulations of rosettes [ 4 ] demonstrate that such systems are unstable: Klemperer explains in his original article, any displacement away from the perfectly symmetrical geometry causes a growing oscillation, eventually leading to the disruption of the system. [ 1 ] (pp 165–166) The system is unstable regardless of whether the center of the rosette is in free space, or is in orbit around a central star.
The short-form reason for the instability is that any perturbation corrupts the geometric symmetry, which increases the perturbation, and further undermines the geometry, and so on. The longer explanation is that any tangential perturbation brings a body closer to one neighbor and further from another; the gravitational imbalance becomes greater towards the closer neighbor and less for the further neighbor, pulling the perturbed object more towards its closer neighbor, amplifying the perturbation rather than damping it. An inward radial perturbation causes the perturbed body to get closer to all other objects, increasing the force on the object and increasing its orbital velocity, which leads indirectly to a tangential perturbation and the argument above. [ b ] | https://en.wikipedia.org/wiki/Klemperer_rosette |
In biology, a klepton (abbreviated kl.) and synklepton (abbreviated sk.) is a species that requires input from another biological taxon (normally from a species which is closely related to the kleptonic species) to complete its reproductive cycle . [ 1 ] Specific types of kleptons are zygokleptons , which reproduce by zygogenesis ; gynokleptons which reproduce by gynogenesis , and tychokleptons , which reproduce by a combination of both systems. [ 2 ]
Kleptogenic reproduction results in three potential outcomes: [ 3 ]
The term is derived from the (Ancient or Modern) Greek κλέπτ(ης) , klépt(ēs) , 'thief' + -on, after taxon , or kleptein , 'to steal'. [ 4 ] A klepton "steals" from an exemplar of another species in order to reproduce. In a paper entitled "Taxonomy of Parthenogenetic Species of Hybrid Origin", Charles J. Cole argues that the thief motif closely parallels the behaviour of certain reptiles. [ 5 ]
In the wild, five species of Ambystoma salamanders contribute to a unisexual complex that reproduces via a combination of gynogenesis and kleptogenesis : A. tigrinum , A. barbouri , A. texanum , A. jeffersonium , and A. laterale . Over twenty genomic combinations have been found in nature, ranging from "LLJ" individuals (two A. laterale and an A. jeffersonium genome) to "LJTi" individuals (an A. laterale , A. jeffersonium , and an A. tigrinum genome). [ 3 ] Every combination, however, contains the genetic information from the A. laterale species, and analysis of mitochondrial DNA has indicated that these unisexual species most likely diverged from an A. barbouri individual some 5 million years ago, [ 6 ] making them the oldest known unisexual vertebrate species. [ 7 ]
The fact that these salamanders have persisted for so long is remarkable, as it contradicts the notion that a majority of asexual lineages arise when the conditions are right and quickly disappear. [ 8 ] It has been argued that this persistence is very much due to the aforementioned "genome replacement" strategy that accompanies kleptogenic reproduction—replacing a portion of the maternal genome with paternal DNA in offspring has allowed unisexual individuals to "refresh" their genetic material through time. This facet of kleptogenesis was recently ascertained from genetic research that indicates there is no ancestral A. laterale genome that is maintained from one unisexual to the next, and that there is not a specific "L" genome that is found more often than others. "L" genetic material found in these salamanders has also not evolved to be substantially unique from sexual genomes. [ 3 ]
In 2007 Bogart et al found that, within a population, unisexual and sexual individuals are able to co-exist; that the genetic makeup of the unisexuals is highly variable; and that unisexual individuals share alleles with sexual individuals. [ 6 ]
Other species exhibiting the property include European water frogs of the genus Pelophylax . [ 2 ]
The Amazon Molly ( Poecilia formosa ) exhibits gynogenesis. [ citation needed ] | https://en.wikipedia.org/wiki/Klepton |
Kleptopharmacophagy is a term used for describing the ecological relationship between two different organisms, where the first is stealing the second's chemical compounds and consuming them. [ 1 ] [ 2 ] This scientific term was proposed by Australian, Singaporean, and American biologists in September 2021 in an article that was published in the journal Ecology by the Ecological Society of America . [ 3 ] [ 4 ] The phenomenon was first noticed in milkweed butterflies that were attacking caterpillars and drinking their internal liquid, proposedly to obtain toxic alkaloids used for defense, as well as for mating purposes. [ 5 ]
Kleptopharmacophagy is a generic term and a scientific neologism , that is used to describe the phenomenon of chemical theft between living organisms. [ 1 ] This special type of behavior is something new for researchers, as it does not match the traditional descriptions of biological interaction . [ 5 ] Kleptopharmacophagy cannot be defined simply with a use of classical ecological relationships and their descriptions, such as predation , parasitism and mutualism . [ 1 ] Besides being an interspecific interaction, kleptopharmacophagy can also be seen as a cannibalizing act (hence an intraspecific act), [ 4 ] with adults attacking larvae of their own species. [ 6 ] [ 7 ]
Kleptopharmacophagy has only been reported for butterflies in the subfamily Danainae, more commonly known as the milkweed butterflies . The phenomenon was first described in butterflies that were observed scratching and feeding on live larvae of other milkweed butterflies. [ 3 ] The initial discovery was made in year 2019 in forests of North Sulawesi , in Indonesia , when two researchers noticed that milkweed butterflies of different species, [ 2 ] [ 6 ] well-known for their toxicity and bright warning colours ( aposematism ), [ 3 ] were interacting with the larval stages of other butterflies. [ 2 ] [ 6 ] The adult insects were attacking and harassing the caterpillars by scratching at them with tarsal claws on their legs. [ 2 ] The butterflies then imbibed the juices from the wounded caterpillars with their long and curved proboscis . [ 6 ] Caterpillars that were targeted appeared to range from living to dead and dying individuals. Live caterpillars were observed to contort their bodies in an attempt to deter scratching, but usually succumbed to the repeat harassment. [ 2 ] [ 6 ]
There is also one recorded example of kleptopharmachophagy occurring between two adult butterflies; a male Ideopsis vitrea vitrea was feeding on a liquid, oozing from the wings of the male I. blanchardii blanchardii . [ 8 ]
It is thought that such chemical theft serves at least two roles. By feeding on toxic caterpillars, the adult butterflies acquire additional toxins, which serve as a protection against predation. The stolen toxins, mostly pyrrolizidine alkaloids (PAs), are also of great importance for male butterflies, which use those compounds for producing mating pheromones . Additionally such chemicals are components of so-called nuptial gifts , consisting of male sperm and some nutrients, that are given from males to females during mating . Significance of the alkaloid use for male butterflies can be demonstrated through observation of male danaine butterflies congregating in large numbers at alkaloid producing plants, where they spend hours fervently scratching and liberating plant juices. [ 2 ]
Milkweed butterflies have also been previously seen feasting on some moribund pyrgormorphid grasshoppers , that are known to contain toxic alkaloids, [ 2 ] [ 8 ] [ 9 ] as well as obtaining desired chemicals from carcasses of dead insects of different taxa. Note that in this case, the term necropharmacophagy may be more appropriate, since butterflies feed on the dead animals. [ 8 ]
Similar behavior was observed with adult butterflies scratching leaves of different plants that possess various toxins in their vegetative organs (so-called leaf-scratching [ 6 ] ). Adult male danaine butterflies use their sharp tarsal claws to scratch and damage leaves of plants that contain high levels of pyrrolizidine alkaloids, thereby liberating the juices for consumption. [ 2 ] [ 10 ] Since kleptopharmacophagy was recorded and described only a few times, [ 4 ] it is thought that obtaining such chemicals by feeding on plants with alkaloids is a common and more frequent way of stocking toxins. [ 6 ] Kleptopharmacophagy is proposed to be an alternative way of acquiring these chemicals. [ 5 ] | https://en.wikipedia.org/wiki/Kleptopharmacophagy |
Kleptoplasty or kleptoplastidy is a process in symbiotic relationships whereby plastids , notably chloroplasts from algae , are sequestered by the host. The word is derived from Kleptes (κλέπτης) which is Greek for thief . The alga is eaten normally and partially digested, leaving the plastid intact. The plastids are maintained within the host, temporarily continuing photosynthesis and benefiting the host.
The word kleptoplasty is derived from Ancient Greek κλέπτης ( kléptēs ), meaning ' thief ' , and πλαστός ( plastós ), originally meaning formed or moulded, and used in biology to mean a plastid . [ 1 ]
Kleptoplasty is a process in symbiotic relationships whereby plastids , notably chloroplasts from algae , are sequestered by the host. The alga is eaten normally and partially digested, leaving the plastid intact. The plastids are maintained within the host, temporarily continuing photosynthesis and benefiting the host. [ 1 ] The term was coined in 1990 to describe chloroplast symbiosis. [ 2 ] [ 3 ]
Kleptoplasty has been acquired in various independent clades of eukaryotes , namely single-celled protists of the SAR supergroup and the Euglenozoa phylum, and some marine invertebrate animals .
Foraminifera
Dinoflagellata
Ciliata
Rapazida
Rhabdocoela
Sacoglossa
Nudibranchia
Some species of the foraminiferan genera Bulimina , Elphidium , Haynesina , Nonion , Nonionella , Nonionellina , Reophax , and Stainforthia sequester diatom chloroplasts. [ 4 ]
The stability of transient plastids varies considerably across plastid-retaining species. In the dinoflagellates Gymnodinium spp. and Pfiesteria piscicida , kleptoplastids are photosynthetically active for only a few days, while kleptoplastids in Dinophysis spp. , taken from cryptophytes , [ 5 ] can be stable for 2 months. [ 1 ] In other dinoflagellates, kleptoplasty has been hypothesized to represent either a mechanism permitting functional flexibility, or perhaps an early evolutionary stage in the permanent acquisition of chloroplasts. [ 6 ]
Mesodinium rubrum is a ciliate that steals chloroplasts from the cryptomonad Geminigera cryophila . [ 7 ] M. rubrum participates in additional endosymbiosis by transferring its plastids to its predators, the dinoflagellate planktons belonging to the genus Dinophysis . [ 8 ]
Karyoklepty is a related process in which the nucleus of the prey cell is kept by the host as well. This was first described in 2007 in M. rubrum . [ 9 ]
The first and only case of kleptoplasty within Euglenozoa belongs to the species Rapaza viridis , the earliest diverging lineage of Euglenophyceae . This microorganism requires a constant supply of a strain of Tetraselmis microalgae, which it ingests to extract chloroplasts. The kleptoplasts are then progressively transformed into ones that resemble the permanent chloroplasts of the remaining Euglenophyceae. Cells of Rapaza viridis can survive for up to 35 days with these kleptoplasts. [ 10 ]
Kleptoplasty is considered the mode of nutrition of the euglenophycean common ancestor. It is hypothesized that kleptoplasty allowed for various events of horizontal gene transfer that eventually allowed the establishment of permanent chloroplasts in the remaining Euglenophyceae. [ 10 ]
Two species of rhabdocoel marine flatworms, Baicalellia solaris and Pogaina paranygulgus, make use of kleptoplasty. The group was previously classified as having algal endosymbionts, though it was already discovered that the endosymbionts did not contain nuclei. [ 11 ]
While consuming diatoms, B. solaris and P. paranygulus , in a process not yet discovered, extract plastids from their prey, incorporating them subepidermally, while separating and digesting the frustule and remainder of the diatom. In B. solaris the extracted plastids, or kleptoplasts, continue to exhibit functional photosynthesis for a short period of roughly 7 days. As the two groups are not sister taxa, and the trait is not shared among groups more closely related, there is evidence that kleptoplasty evolved independently within the two taxa. [ 12 ]
Sea slugs in the clade Sacoglossa practise kleptoplasty. [ 13 ] Several species of Sacoglossan sea slugs capture intact, functional chloroplasts from algal food sources, retaining them within specialized cells lining the mollusc 's digestive diverticula . The longest known kleptoplastic association, which can last up to ten months, is found in Elysia chlorotica , [ 2 ] which acquires chloroplasts by eating the alga Vaucheria litorea , storing the chloroplasts in the cells that line its gut. [ 14 ] Juvenile sea slugs establish the kleptoplastic endosymbiosis when feeding on algal cells, sucking out the cell contents, and discarding everything except the chloroplasts. The chloroplasts are phagocytosed by digestive cells, filling extensively branched digestive tubules, providing their host with the products of photosynthesis. [ 15 ] It is not resolved, however, whether the stolen plastids actively secrete photosynthate or whether the slugs profit indirectly from slowly degrading kleptoplasts. [ 16 ]
Due to this unusual ability, the sacoglossans are sometimes referred to as "solar-powered sea slugs", though the actual benefit from photosynthesis on the survival of some of the species that have been analyzed seems to be marginal at best. [ 17 ] Studies have found that photosynthates from captured chloroplasts are able to influence growth in Elysia viridis . [ 18 ] How long a sacoglossan can live without food seems not to depend on the photosynthetic activity of its kleptoplasts, but rather on the ability of that sacoglossan species to manage starvation. [ 19 ]
Changes in temperature have been shown to negatively affect kleptoplastic abilities in sacoglossans. Rates of photosynthetic efficiency as well as kleptoplast abundance have been shown to decrease in correlation to a decrease in temperature. The patterns and rate of these changes, however, varies between different species of sea slug. [ 20 ]
Some species of another group of sea slugs, nudibranchs such as Pteraeolidia ianthina , sequester whole living symbiotic zooxanthellae within their digestive diverticula, and thus are similarly "solar-powered". [ 21 ] | https://en.wikipedia.org/wiki/Kleptoplasty |
A kleptoprotein is a protein which is not encoded in the genome of the organism which uses it, but instead is obtained through diet from a prey organism. Importantly, a kleptoprotein must maintain its function and be mostly or entirely undigested, drawing a distinction from proteins that are digested for nutrition, which become destroyed and non-functional in the process.
This phenomenon was first reported in the bioluminescent fish Parapriacanthus , which has specialized light organs adapted towards counter-illumination , but obtains the luciferase enzyme within these organs from bioluminescent ostracods, including Cypridina noctiluca or Vargula hilgendorfii . [ 1 ]
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Kleptoprotein |
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