Mathematical and theoretical biology
Mathematical and theoretical biology is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems though the two terms are sometimes interchanged. Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics and it can be useful in both theoretical and practical research. Describing systems in a quantitative manner means their behavior can be better simulated, hence properties can be predicted that might not be evident to the experimenter.
This requires precise mathematical models. Because of the complexity of the living systems, theoretical biology employs several fields of mathematics, has contributed to the development of new techniques. Mathematics has been used in biology as early as the 12th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre Francois Verhulst formulated the logistic growth model in 1836. Fritz Müller described the evolutionary benefits of what is now called Müllerian mimicry in 1879, in an account notable for being the first use of a mathematical argument in evolutionary ecology to show how powerful the effect of natural selection would be, unless one includes Malthus's discussion of the effects of population growth that influenced Charles Darwin: Malthus argued that growth would be exponential while resources could only grow arithmetically.
The term "theoretical biology" was first used by Johannes Reinke in 1901. One founding text is considered to be On Growth and Form by D'Arcy Thompson, other early pioneers include Ronald Fisher, Hans Leo Przibram, Nicolas Rashevsky and Vito Volterra. Interest in the field has grown from the 1960s onwards; some reasons for this include: The rapid growth of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology An increase in computing power, which facilitates calculations and simulations not possible An increasing interest in in silico experimentation due to ethical considerations, risk and other complications involved in human and animal research Several areas of specialized research in mathematical and theoretical biology as well as external links to related projects in various universities are concisely presented in the following subsections, including a large number of appropriate validating references from a list of several thousands of published authors contributing to this field.
Many of the included examples are characterised by complex and supercomplex mechanisms, as it is being recognised that the result of such interactions may only be understood through a combination of mathematical, physical/chemical and computational models. Abstract relational biology is concerned with the study of general, relational models of complex biological systems abstracting out specific morphological, or anatomical, structures; some of the simplest models in ARB are the Metabolic-Replication, or --systems introduced by Robert Rosen in 1957-1958 as abstract, relational models of cellular and organismal organization. Other approaches include the notion of autopoiesis developed by Maturana and Varela, Kauffman's Work-Constraints cycles, more the notion of closure of constraints. Algebraic biology applies the algebraic methods of symbolic computation to the study of biological problems in genomics, analysis of molecular structures and study of genes. An elaboration of systems biology to understanding the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.
A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, including subsections in the following areas: computer modeling in biology and medicine, arterial system models, neuron models and oscillation networks, quantum automata, quantum computers in molecular biology and genetics, cancer modelling, neural nets, genetic networks, abstract categories in relational biology, metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular automata, tessellation models and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories. Modeling cell and molecular biology This area has received a boost due to the growing importance of molecular biology. Mechanics of biological tissues Theoretical enzymology and enzyme kinetics Cancer modelling and simulation Modelling the movement of interacting cell populations Mathematical modelling of scar tissue formation Mathematical modelling of intracellular dynamics Mathematical
In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, is simply called the "underlying." Derivatives can be used for a number of purposes, including insuring against price movements, increasing exposure to price movements for speculation or getting access to otherwise hard-to-trade assets or markets. Some of the more common derivatives include forwards, options and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter or on an exchange such as the New York Stock Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges. Derivatives are one of the three main categories of financial instruments, the other two being stocks and debt.
The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed a century ago, are a more recent historical example. Derivatives are contracts between two parties that specify conditions under which payments are to be made between the parties; the assets include commodities, bonds, interest rates and currencies, but they can be other derivatives, which adds another layer of complexity to proper valuation. The components of a firm's capital structure, e.g. bonds and stock, can be considered derivatives, more options, with the underlying being the firm's assets, but this is unusual outside of technical contexts. From the economic point of view, financial derivatives are cash flows, that are conditioned stochastically and discounted to present value; the market risk inherent in the underlying asset is attached to the financial derivative through contractual agreements and hence can be traded separately.
The underlying asset does not have to be acquired. Derivatives therefore allow the breakup of ownership and participation in the market value of an asset; this provides a considerable amount of freedom regarding the contract design. That contractual freedom allows derivative designers to modify the participation in the performance of the underlying asset arbitrarily. Thus, the participation in the market value of the underlying can be weaker, stronger, or implemented as inverse. Hence the market price risk of the underlying asset can be controlled in every situation. There are two groups of derivative contracts: the traded over-the-counter derivatives such as swaps that do not go through an exchange or other intermediary, exchange-traded derivatives that are traded through specialized derivatives exchanges or other exchanges. Derivatives are more common in the modern era. One of the oldest derivatives is rice futures, which have been traded on the Dojima Rice Exchange since the eighteenth century.
Derivatives are broadly categorized by the relationship between the underlying asset and the derivative. Derivatives may broadly be categorized as "lock" or "option" products. Lock products obligate the contractual parties to the terms over the life of the contract. Option products provide the buyer the right, but not the obligation to enter the contract under the terms specified. Derivatives can be used either for speculation; this distinction is important because the former is a prudent aspect of operations and financial management for many firms across many industries. Along with many other financial products and services, derivatives reform is an element of the Dodd–Frank Wall Street Reform and Consumer Protection Act of 2010; the Act delegated many rule-making details of regulatory oversight to the Commodity Futures Trading Commission and those details are not finalized nor implemented as of late 2012. To give an idea of the size of the derivative market, The Economist has reported that as of June 2011, the over-the-counter derivatives market amounted to $700 trillion, the size of the market traded on exchanges totaled an additional $83 trillion.
For the fourth quarter 2017 the European Securities Market Authority estimated the size of European derivatives market at a size of €660 trillion with 74 million outstanding contracts. However, these are "notional" values, some economists say that this value exaggerates the market value and the true credit risk faced by the parties involved. For example, in 2010, while the aggregate of OTC derivatives exceeded $600 trillion, the value of the market was estimated much lower, at $21 trillion; the credit risk equivalent of the derivative contracts was estimated at $3.3 trillion. Still, eve
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is quantified numerically using the SI derived unit, the cubic metre; the volume of a container is understood to be the capacity of the container. Three dimensional mathematical shapes are assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, circular shapes can be calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space; the volume of a solid can be determined by fluid displacement. Displacement of liquid can be used to determine the volume of a gas; the combined volume of two substances is greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube. In the International System of Units, the standard unit of volume is the cubic metre; the metric system includes the litre as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = 3 = 1000 cubic centimetres = 0.001 cubic metres. Small amounts of liquid are measured in millilitres, where 1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where 1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, to liquids, grain, or the like, which take the shape of that which holds them". Capacity is not identical in meaning to volume, though related. Units of capacity are the SI litre and its derived units, Imperial units such as gill, pint and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are related: one litre is 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume, defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is an important parameter of a system being studied; the volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time. In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f = 1 and is written as: ∭ D 1 d x d y d z.
The volume integral in cylindrical coordinates is ∭ D r d r d θ d z, the volume integral in spherical coordinates has the form ∭ D ρ 2 sin ϕ d ρ d θ d ϕ. The above formulas can be used to show that the volumes of a cone and cylinder of the same radius and height are in the ratio 1: 2: 3, as follows. Let the radius be r and the height be h the volume of cone is 1 3 π r 2 h = 1 3 π r 2 = × 1, the volume of the sphere
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables; this contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained; these concepts are "marginal" because they can be found by summing values in a table along rows or columns, writing the sum in the margins of the table. The distribution of the marginal variables is obtained by marginalizing – that is, focusing on the sums in the margin – over the distribution of the variables being discarded, the discarded variables are said to have been marginalized out; the context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables.
In many applications, an analysis may start with a given collection of random variables first extend the set by defining new ones and reduce the number by placing interest in the marginal distribution of a subset. Several different analyses may be done, each treating a different subset of variables as the marginal variables. Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is the probability distribution of X averaging over information about Y, it is the probability distribution of X. This is calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can be written as Pr; this is Pr = ∑ y Pr = ∑ y Pr Pr, where Pr is the joint distribution of X and Y, while Pr is the conditional distribution of X given Y. In this case, the variable Y has been "marginalized out". Bivariate marginal and joint probabilities for discrete random variables are displayed as two-way tables.
For continuous random variables, the marginal probability density function can be written as pX. This is p X = ∫ y p X, Y d y = ∫ y p X ∣ Y p Y d y, where pX,Y gives the joint distribution of X and Y, while pX|Y gives the conditional distribution for X given Y. Again, the variable Y has been "marginalized out". A marginal probability can always be written as an expected value: p X = ∫ y p X ∣ Y p Y d y = E Y . Intuitively, the marginal probability of X is computed by examining the conditional probability of X given a particular value of Y, averaging this conditional probability over the distribution of all values of Y; this follows from the definition of expected value: E Y = ∫ y f p Y d y. Therefore marginalization provides the rule for the transformation of the probability distribution of a random variable Y and another random variable X = g: p X = ∫ y p X ∣ Y p Y d y = ∫ y δ p Y d y. Suppose that the probability that a pedestrian will be hit by a car, while crossing the road at a pedestrian crossing, without paying attention to the tr
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat, it is the two-dimensional analog of the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units, the standard unit of area is the square metre, the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, the area of any other shape or surface is a dimensionless real number. There are several well-known formulas for the areas of simple shapes such as triangles and circles.
Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape requires multivariable calculus. Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved. An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties: For all S in M, a ≥ 0. If S and T are in M so are S ∪ T and S ∩ T, a = a + a − a. If S and T are in M with S ⊆ T T − S is in M and a = a − a. If a set S is in M and S is congruent to T T is in M and a = a; every rectangle R is in M. If the rectangle has length h and breadth k a = hk. Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a ≤ c ≤ a for all such step regions S and T a = c, it can be proved that such an area function exists. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.
Thus areas can be measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, so forth. Algebraically, these units can be thought of as the squares of the corresponding length units; the SI unit of area is the square metre, considered an SI derived unit. Calculation of the area of a square whose length and width are 1 metre would be: 1 metre x 1 metre = 1 m2and so, a rectangle with different sides would have an area in square units that can be calculated as: 3 metres x 2 metres = 6 m2; this is equivalent to 6 million square millimetres. Other useful conversions are: 1 square kilometre = 1,000,000 square metres 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres 1 square centimetre = 100 square millimetres. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. 1 foot = 12 inches,the relationship between square feet and square inches is 1 square foot = 144 square inches,where 144 = 122 = 12 × 12.
Similarly: 1 square yard = 9 square feet 1 square mile = 3,097,600 square yards = 27,878,400 square feetIn addition, conversion factors include: 1 square inch = 6.4516 square centimetres 1 square foot = 0.09290304 square metres 1 square yard = 0.83612736 square metres 1 square mile = 2.589988110336 square kilometres There are several other common units for area. The are was the original unit of area in the metric system, with: 1 are = 100 square metresThough the are has fallen out of use, the hectare is still used to measure land: 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometresOther uncommon metric units of area include the tetrad, the hectad, the myriad. The acre is commonly used to measure land areas, where 1 acre = 4,840 square yards = 43,560 square feet. An acre is 40% of a hectare. On the atomic scale, area is measured in units of barns, such that: 1 barn = 10−28 square meters; the barn is used in describing the cross-sectional area of interaction in nuclear physics.
In India, 20 dhurki = 1 dhur 20 dhur = 1 khatha 20 khata = 1 bigha 32 khata = 1 acre In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of
Amino acids are organic compounds containing amine and carboxyl functional groups, along with a side chain specific to each amino acid. The key elements of an amino acid are carbon, hydrogen and nitrogen, although other elements are found in the side chains of certain amino acids. About 500 occurring amino acids are known and can be classified in many ways, they can be classified according to the core structural functional groups' locations as alpha-, beta-, gamma- or delta- amino acids. In the form of proteins, amino acid residues form the second-largest component of human muscles and other tissues. Beyond their role as residues in proteins, amino acids participate in a number of processes such as neurotransmitter transport and biosynthesis. In biochemistry, amino acids having both the amine and the carboxylic acid groups attached to the first carbon atom have particular importance, they are known as α-amino acids. They include the 22 proteinogenic amino acids, which combine into peptide chains to form the building-blocks of a vast array of proteins.
These are all L-stereoisomers, although a few D-amino acids occur in bacterial envelopes, as a neuromodulator, in some antibiotics. Twenty of the proteinogenic amino acids are encoded directly by triplet codons in the genetic code and are known as "standard" amino acids; the other two are selenocysteine, pyrrolysine. Pyrrolysine and selenocysteine are encoded via variant codons. N-formylmethionine is considered as a form of methionine rather than as a separate proteinogenic amino acid. Codon–tRNA combinations not found in nature can be used to "expand" the genetic code and form novel proteins known as alloproteins incorporating non-proteinogenic amino acids. Many important proteinogenic and non-proteinogenic amino acids have biological functions. For example, in the human brain and gamma-amino-butyric acid are the main excitatory and inhibitory neurotransmitters. Hydroxyproline, a major component of the connective tissue collagen, is synthesised from proline. Glycine is a biosynthetic precursor to porphyrins used in red blood cells.
Carnitine is used in lipid transport. Nine proteinogenic amino acids are called "essential" for humans because they cannot be produced from other compounds by the human body and so must be taken in as food. Others may be conditionally essential for medical conditions. Essential amino acids may differ between species; because of their biological significance, amino acids are important in nutrition and are used in nutritional supplements, fertilizers and food technology. Industrial uses include the production of drugs, biodegradable plastics, chiral catalysts; the first few amino acids were discovered in the early 19th century. In 1806, French chemists Louis-Nicolas Vauquelin and Pierre Jean Robiquet isolated a compound in asparagus, subsequently named asparagine, the first amino acid to be discovered. Cystine was discovered in 1810, although its monomer, remained undiscovered until 1884. Glycine and leucine were discovered in 1820; the last of the 20 common amino acids to be discovered was threonine in 1935 by William Cumming Rose, who determined the essential amino acids and established the minimum daily requirements of all amino acids for optimal growth.
The unity of the chemical category was recognized by Wurtz in 1865, but he gave no particular name to it. Usage of the term "amino acid" in the English language is from 1898, while the German term, Aminosäure, was used earlier. Proteins were found to yield amino acids after enzymatic acid hydrolysis. In 1902, Emil Fischer and Franz Hofmeister independently proposed that proteins are formed from many amino acids, whereby bonds are formed between the amino group of one amino acid with the carboxyl group of another, resulting in a linear structure that Fischer termed "peptide". In the structure shown at the top of the page, R represents a side chain specific to each amino acid; the carbon atom next to the carboxyl group is called the α–carbon. Amino acids containing an amino group bonded directly to the alpha carbon are referred to as alpha amino acids; these include amino acids such as proline which contain secondary amines, which used to be referred to as "imino acids". The alpha amino acids are the most common form found in nature, but only when occurring in the L-isomer.
The alpha carbon is a chiral carbon atom, with the exception of glycine which has two indistinguishable hydrogen atoms on the alpha carbon. Therefore, all alpha amino acids but glycine can exist in either of two enantiomers, called L or D amino acids, which are mirror images of each other. While L-amino acids represent all of the amino acids found in proteins during translation in the ribosome, D-amin