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Mathematical and theoretical biology, or biomathematics, is a branch of
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
which employs theoretical analysis, mathematical models and abstractions of the
living organisms
In biology, an organism () is any life, living system that functions as an individual entity. All organisms are composed of cells (cell theory). Organisms are classified by taxonomy (biology), taxonomy into groups such as Multicellular o ...
to investigate the principles that govern the structure, development and behavior of the systems, as opposed to
experimental biology
Experimental biology is the set of approaches in the field of biology concerned with the conduction of experiments to investigate and understand biological phenomena. The term is opposed to theoretical biology which is concerned with the mathematic ...
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, even though the two terms are sometimes interchanged.
Mathematical biology aims at the mathematical representation and modeling of
biological process
Biological processes are those processes that are vital for an organism to live, and that shape its capacities for interacting with its environment. Biological processes are made of many chemical reactions or other events that are involved in the ...
es, using techniques and tools of
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
. It can be useful in both
theoretical
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
and
practical
Pragmatism is a philosophical tradition that considers words and thought as tools and instruments for prediction, problem solving, and action, and rejects the idea that the function of thought is to describe, represent, or mirror reality. ...
research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter. This requires precise
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s.
Because of the complexity of the
living systems
Living systems are open self-organizing life forms that interact with their environment. These systems are maintained by flows of information, energy and matter.
In the last few decades, some scientists have proposed that a general living systems ...
, theoretical biology employs several fields of mathematics, and has contributed to the development of new techniques.
History
Early history
Mathematics has been used in biology as early as the 13th century, when
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
used the famous
Fibonacci series
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
to describe a growing population of rabbits. In the 18th century
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
applied mathematics to describe the effect of smallpox on the human population.
Thomas Malthus
Thomas Robert Malthus (; 13/14 February 1766 – 29 December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography.
In his 1798 book '' An Essay on the Principle of Population'', Mal ...
' 1789 essay on the growth of the human population was based on the concept of exponential growth.
Pierre François Verhulst
Pierre François Verhulst (28 October 1804, Brussels – 15 February 1849, Brussels) was a Belgian mathematician and a doctor in number theory from the University of Ghent in 1825. He is best known for the logistic growth model.
Logistic equ ...
formulated the logistic growth model in 1836.
Fritz Müller
Johann Friedrich Theodor Müller (31 March 1822 – 21 May 1897), better known as Fritz Müller, and also as Müller-Desterro, was a German biologist who emigrated to southern Brazil, where he lived in and near the German community of Blumenau, ...
described the evolutionary benefits of what is now called
Müllerian mimicry
Müllerian mimicry is a natural phenomenon in which two or more well-defended species, often foul-tasting and sharing common predators, have come to mimic each other's honest warning signals, to their mutual benefit. The benefit to Müllerian ...
in 1879, in an account notable for being the first use of a mathematical argument in
evolutionary ecology
Evolutionary ecology lies at the intersection of ecology and evolutionary biology. It approaches the study of ecology in a way that explicitly considers the evolutionary histories of species and the interactions between them. Conversely, it can ...
to show how powerful the effect of natural selection would be, unless one includes
Malthus
Thomas Robert Malthus (; 13/14 February 1766 – 29 December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography.
In his 1798 book ''An Essay on the Principle of Population'', Malt ...
's discussion of the effects of
population growth
Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
that influenced
Charles Darwin
Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended fr ...
: Malthus argued that growth would be exponential (he uses the word "geometric") while resources (the environment's carrying capacity) could only grow arithmetically.
The term "theoretical biology" was first used as a monograph title by
Johannes Reinke
Johannes Reinke (February 3, 1849 – February 25, 1931) was a German botanist and philosopher who was a native of Ziethen, Lauenburg. He is remembered for his research of benthic marine algae.
Academic background
Reinke studied botany with h ...
in 1901, and soon after by
Jakob von Uexküll
Jakob may refer to:
People
* Jakob (given name), including a list of people with the name
* Jakob (surname), including a list of people with the name
Other
* Jakob (band), a New Zealand band, and the title of their 1999 EP
* Max Jakob Memorial Aw ...
in 1920. One founding text is considered to be
On Growth and Form
''On Growth and Form'' is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The book covers many topi ...
(1917) by
D'Arcy Thompson
Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology, travelled on expeditions to the Bering Strait an ...
, and other early pioneers include
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
,
Hans Leo Przibram
Hans Leo Przibram [] (7 July 1874 – 20 May 1944) was an Austrian people, Austrian biologist who founded the biological laboratory in Vienna.
Career
Hans was as elder son of Gustav and Charlotte Przibram. His mother was the daughter of Friedric ...
,
Vito Volterra
Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis.
Biography
Born in Anc ...
,
Nicolas Rashevsky
Nicolas Rashevsky (November 9, 1899 – January 16, 1972) was an American theoretical physicist who was one of the pioneers of mathematical biology, and is also considered the father of mathematical biophysics and theoretical biology. Rober ...
and
Conrad Hal Waddington
Conrad Hal Waddington (8 November 1905 – 26 September 1975) was a British developmental biologist, paleontologist, geneticist, embryologist and philosopher who laid the foundations for systems biology, epigenetics, and evolutionary devel ...
.
Recent growth
Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include:
* The rapid growth of data-rich information sets, due to the
genomics
Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes as well as its hierarchical, three-dim ...
revolution, which are difficult to understand without the use of analytical tools
* Recent development of mathematical tools such as
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
to help understand complex, non-linear mechanisms in biology
* An increase in
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
power, which facilitates calculations and
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
s not previously possible
* An increasing interest in
in silico
In biology and other experimental sciences, an ''in silico'' experiment is one performed on computer or via computer simulation. The phrase is pseudo-Latin for 'in silicon' (correct la, in silicio), referring to silicon in computer chips. It ...
experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research
Areas of 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 also 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 highly complex, nonlinear, and supercomplex mechanisms, as it is being increasingly recognised that the result of such interactions may only be understood through a combination of mathematical, logical, physical/chemical, molecular and computational models.
Abstract relational biology
Abstract relational biology (ARB) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957-1958 as abstract, relational models of cellular and organismal organization.
Other approaches include the notion of
autopoiesis
The term autopoiesis () refers to a system capable of producing and maintaining itself by creating its own parts.
The term was introduced in the 1972 publication '' Autopoiesis and Cognition: The Realization of the Living'' by Chilean biologist ...
developed by
Maturana and
Varela,
Kauffman
Kaufmann is a surname with many variants such as Kauffmann, Kaufman, and Kauffman. In German, the name means ''merchant''. It is the cognate of the English '' Chapman'' (which had a similar meaning in the Middle Ages, though it disappeared from ...
's Work-Constraints cycles, and more recently the notion of closure of constraints.
Algebraic biology
Algebraic biology (also known as symbolic systems biology) applies the algebraic methods of
symbolic computation
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
to the study of biological problems, especially in
genomics
Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes as well as its hierarchical, three-dim ...
,
proteomics
Proteomics is the large-scale study of proteins. Proteins are vital parts of living organisms, with many functions such as the formation of structural fibers of muscle tissue, enzymatic digestion of food, or synthesis and replication of DNA. In ...
, analysis of
molecular structure
Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determ ...
s and study of
gene
In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a ba ...
s.
Complex systems biology
An elaboration of systems biology to understand the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.
Computer models and automata theory
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
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...
in biology and medicine, arterial system models,
neuron
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...
models, biochemical and
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
network
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
s, quantum automata,
quantum computers
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
in
molecular biology
Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and physi ...
and
genetics
Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar wor ...
, cancer modelling,
neural net
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
s,
genetic network
A gene (or genetic) regulatory network (GRN) is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels of mRNA and proteins which, in turn, determine the fun ...
s, abstract categories in relational biology, metabolic-replication systems,
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
applications in biology and medicine,
automata theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματο ...
,
cellular automata
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
,
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
models and complete self-reproduction,
chaotic system
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
s in
organism
In biology, an organism () is any living system that functions as an individual entity. All organisms are composed of cells (cell theory). Organisms are classified by taxonomy into groups such as multicellular animals, plants, and ...
s, relational biology and organismic theories.
Modeling cell and molecular biology
This area has received a boost due to the growing importance of
molecular biology
Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and physi ...
.
*Mechanics of biological tissues
*Theoretical enzymology and
enzyme kinetics
Enzyme kinetics is the study of the rates of enzyme-catalysed chemical reactions. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's kinetics in th ...
*
Cancer
Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...
modelling and simulation
*Modelling the movement of interacting cell populations
*Mathematical modelling of scar tissue formation
*Mathematical modelling of intracellular dynamics
*Mathematical modelling of the cell cycle
*Mathematical modelling of apoptosis
Modelling physiological systems
*Modelling of
arterial
An artery (plural arteries) () is a blood vessel in humans and most animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). Most arteries carry oxygenated blood; the two exceptions are the pul ...
disease
*Multi-scale modelling of the
heart
The heart is a muscular organ in most animals. This organ pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste such as carbon dioxide t ...
*Modelling electrical properties of muscle interactions, as in
bidomain The bidomain model is a mathematical model to define the electrical activity of the heart. It consists in a continuum (volume-average) approach in which the cardiac mictrostructure is defined in terms of muscle fibers grouped in sheets, creating a ...
and
monodomain model
The monodomain model is a reduction of the bidomain model of the electrical propagation in myocardial tissue.
The reduction comes from assuming that the intra- and extracellular domains have equal anisotropy ratios.
Although not as physiologically ...
s
Computational neuroscience
Computational neuroscience
Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...
(also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system.
Evolutionary biology
Ecology
Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...
and
evolutionary biology
Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life fo ...
have traditionally been the dominant fields of mathematical biology.
Evolutionary biology has been the subject of extensive mathematical theorizing. The traditional approach in this area, which includes complications from genetics, is
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
. Most population geneticists consider the appearance of new
allele
An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is a variation of the same sequence of nucleotides at the same place on a long DNA molecule, as described in leading textbooks on genetics and evolution.
::"The chro ...
s by
mutation
In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...
, the appearance of new
genotype
The genotype of an organism is its complete set of genetic material. Genotype can also be used to refer to the alleles or variants an individual carries in a particular gene or genetic location. The number of alleles an individual can have in a ...
s by
recombination, and changes in the frequencies of existing alleles and genotypes at a small number of
gene
In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a ba ...
loci. When
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
effects at a large number of gene loci are considered, together with the assumption of
linkage equilibrium
In population genetics, linkage disequilibrium (LD) is the non-random association of alleles at different loci in a given population. Loci are said to be in linkage disequilibrium when the frequency of association of their different alleles is h ...
or
quasi-linkage equilibrium Quasi-linkage equilibrium (QLE) is a mathematical approximation used in solving population genetics problems. Motoo Kimura introduced the notion to simplify a model of Fisher's fundamental theorem. QLE greatly simplifies population genetic equati ...
, one derives
quantitative genetics
Quantitative genetics deals with phenotypes that vary continuously (such as height or mass)—as opposed to discretely identifiable phenotypes and gene-products (such as eye-colour, or the presence of a particular biochemical).
Both branches u ...
.
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
made fundamental advances in statistics, such as
analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...
, via his work on quantitative genetics. Another important branch of population genetics that led to the extensive development of
coalescent theory
Coalescent theory is a model of how alleles sampled from a population may have originated from a common ancestor. In the simplest case, coalescent theory assumes no recombination, no natural selection, and no gene flow or population structure, m ...
is
phylogenetics
In biology, phylogenetics (; from Greek language, Greek wikt:φυλή, φυλή/wikt:φῦλον, φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary his ...
. Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics Traditional population genetic models deal with alleles and genotypes, and are frequently
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
.
Many population genetics models assume that population sizes are constant. Variable population sizes, often in the absence of genetic variation, are treated by the field of
population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
History
Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
. Work in this area dates back to the 19th century, and even as far as 1798 when
Thomas Malthus
Thomas Robert Malthus (; 13/14 February 1766 – 29 December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography.
In his 1798 book '' An Essay on the Principle of Population'', Mal ...
formulated the first principle of population dynamics, which later became known as the
Malthusian growth model
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert ...
. The
Lotka–Volterra predator-prey equations are another famous example. Population dynamics overlap with another active area of research in mathematical biology:
mathematical epidemiology, the study of infectious disease affecting populations. Various models of the spread of
infections
An infection is the invasion of tissue (biology), tissues by pathogens, their multiplication, and the reaction of host (biology), host tissues to the infectious agent and the toxins they produce. An infectious disease, also known as a transmiss ...
have been proposed and analyzed, and provide important results that may be applied to health policy decisions.
In
evolutionary game theory
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Ma ...
, developed first by
John Maynard Smith
John Maynard Smith (6 January 1920 – 19 April 2004) was a British theoretical and mathematical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics und ...
and
George R. Price
George Robert Price (October 6, 1922 – January 6, 1975) was an American population geneticist. Price is often noted for his formulation of the Price equation in 1967.
Originally a physical chemist and later a science journalist, he moved ...
, selection acts directly on inherited phenotypes, without genetic complications. This approach has been mathematically refined to produce the field of
adaptive dynamics
Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of Phenotypic trait, traits in Asexual reproduction, asexually reprodu ...
.
Mathematical biophysics
The earlier stages of mathematical biology were dominated by mathematical
biophysics
Biophysics is an interdisciplinary science that applies approaches and methods traditionally used in physics to study biological phenomena. Biophysics covers all scales of biological organization, from molecular to organismic and populations. ...
, described as the application of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their components or compartments.
The following is a list of mathematical descriptions and their assumptions.
Deterministic processes (dynamical systems)
A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process always generates the same trajectory, and no two trajectories cross in state space.
*
Difference equations/Maps – discrete time, continuous state space.
*
Ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
– continuous time, continuous state space, no spatial derivatives. ''See also:''
Numerical ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also ...
.
*
Partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
– continuous time, continuous state space, spatial derivatives. ''See also:''
Numerical partial differential equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In principle, specialized methods for hyperbolic, parabolic or elliptic partia ...
.
*
Logical deterministic cellular automata – discrete time, discrete state space. ''See also:''
Cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
.
Stochastic processes (random dynamical systems)
A random mapping between an initial state and a final state, making the state of the system a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with a corresponding
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
* Non-Markovian processes –
generalized master equation – continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur.
* Jump
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
–
master equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determine ...
– continuous time with no memory of past events, discrete state space, waiting times between events discretely occur and are exponentially distributed. ''See also:''
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
for numerical simulation methods, specifically
dynamic Monte Carlo method In chemistry, dynamic Monte Carlo (DMC) is a Monte Carlo method for modeling the dynamic behaviors of molecules by comparing the rates of individual steps with random numbers. It is essentially the same as Kinetic Monte Carlo. Unlike the Metropol ...
and
Gillespie algorithm In probability theory, the Gillespie algorithm (or the Doob-Gillespie algorithm or ''Stochastic Simulation Algorithm'', the SSA) generates a statistically correct trajectory (possible solution) of a stochastic equation system for which the reaction ...
.
* Continuous
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
–
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...
s or a
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
– continuous time, continuous state space, events occur continuously according to a random
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
.
Spatial modelling
One classic work in this area is
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical com ...
's paper on
morphogenesis
Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of devel ...
entitled ''
The Chemical Basis of Morphogenesis
"The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from a homogeneous, uniform state. The theory, w ...
'', published in 1952 in the
Philosophical Transactions of the Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
.
*Travelling waves in a wound-healing assay
*
Swarming behaviour
*A mechanochemical theory of
morphogenesis
Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of devel ...
*
Biological pattern formation
The science of pattern formation deals with the visible, ( statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.
In developmental biology, pattern formation refers to the generation of ...
*Spatial distribution modeling using plot samples
*
Turing pattern
The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomousl ...
s
Mathematical methods
A model of a biological system is converted into a system of equations, although the word 'model' is often used synonymously with the system of corresponding equations. The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at
equilibrium. There are many different types of equations and the type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur.
Molecular set theory
Molecular set theory (MST) is a mathematical formulation of the wide-sense
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets. It was introduced by
Anthony Bartholomay Anthony Francis Bartholomay (1919–1975) was a mathematician who introduced molecular set theory, a topic on which he wrote books.
Life
Bartholomay was born on August 11, 1919. He would receive degrees from Hamilton College, Syracuse University, a ...
, and its applications were developed in mathematical biology and especially in mathematical medicine.
In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine.
Organizational biology
Theoretical approaches to biological organization aim to understand the interdependence between the parts of organisms. They emphasize the circularities that these interdependences lead to. Theoretical biologists developed several concepts to formalize this idea.
For example, abstract relational biology (ARB) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by
Robert Rosen in 1957-1958 as abstract, relational models of cellular and organismal organization.
Model example: the cell cycle
The eukaryotic
cell cycle
The cell cycle, or cell-division cycle, is the series of events that take place in a cell that cause it to divide into two daughter cells. These events include the duplication of its DNA (DNA replication) and some of its organelles, and subs ...
is very complex and is one of the most studied topics, since its misregulation leads to
cancer
Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...
s.
It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results. Two research groups
have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model that can represent a particular eukaryote depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et al., 2006).
By means of a system of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s these models show the change in time (
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
) of the protein inside a single typical cell; this type of model is called a
deterministic process (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
).
To obtain these equations an iterative series of steps must be done: first the several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential equations, such as
rate kinetics for stoichiometric reactions,
Michaelis-Menten kinetics for enzyme substrate reactions and
Goldbeter–Koshland kinetics
The Goldbeter–Koshland kinetics Zoltan Szallasi, Jörg Stelling, Vipul Periwal: ''System Modeling in Cellular Biology''. The MIT Press. p 108. describe a Steady state (chemistry), steady-state solution for a 2-state biological system. In this ...
for ultrasensitive transcription factors, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michaelis constants) must be fitted to match observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size.
To fit the parameters, the differential equations must be studied. This can be done either by simulation or by analysis. In a simulation, given a starting
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
(list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments.
In analysis, the properties of the equations are used to investigate the behavior of the system depending on the values of the parameters and variables. A system of differential equations can be represented as a
vector field, where each vector described the change (in concentration of two or more protein) determining where and how fast the trajectory (simulation) is heading. Vector fields can have several special points: a
stable point
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an
unstable point, either a source or a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
, which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).
A better representation, which handles the large number of variables and parameters, is a
bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
using
bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
. The presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event (
Cell cycle checkpoint
Cell cycle checkpoints are control mechanisms in the eukaryotic cell cycle which ensure its proper progression. Each checkpoint serves as a potential termination point along the cell cycle, during which the conditions of the cell are assessed, wi ...
), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
and an
infinite period bifurcation
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
.
Societies and institutes
*
National Institute for Mathematical and Biological Synthesis
The National Institute for Mathematical and Biological Synthesis is a research institute focused on the science of mathematics and biology, located on the University of Tennessee, Knoxville, campus. Known by its acronym NIMBioS (pronounced NIM-bu ...
*
Society for Mathematical Biology The Society for Mathematical Biology (SMB) is an international association co-founded in 1972 in the United States by George Karreman, Herbert Daniel Landahl and (initially chaired) by Anthony Bartholomay for the furtherance of joint scientific ac ...
ESMTB: European Society for Mathematical and Theoretical BiologyThe Israeli Society for Theoretical and Mathematical BiologySociété Francophone de Biologie ThéoriqueInternational Society for Biosemiotic StudiesSchool of Computational and Integrative Sciences Jawaharlal Nehru University
Jawaharlal Nehru University (JNU) is a public major research university located in New Delhi, India. It was established in 1969 and named after Jawaharlal Nehru, India's first Prime Minister. The university is known for leading faculties and r ...
See also
*
Biological applications of bifurcation theory Biological applications of bifurcation theory provide a framework for understanding the behavior of biological networks modeled as dynamical systems. In the context of a biological system, bifurcation theory describes how small changes in an input p ...
*
Biophysics
Biophysics is an interdisciplinary science that applies approaches and methods traditionally used in physics to study biological phenomena. Biophysics covers all scales of biological organization, from molecular to organismic and populations. ...
*
Biostatistics
Biostatistics (also known as biometry) are the development and application of statistical methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experime ...
*
Entropy and life
Research concerning the relationship between the thermodynamic quantity entropy and the evolution of life began around the turn of the 20th century. In 1910, American historian Henry Adams printed and distributed to university libraries and histor ...
*
Ewens's sampling formula
In population genetics, Ewens's sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.
Definition
Ewens's sampling formula, introduced by Warren Ewens ...
*
Journal of Theoretical Biology
The ''Journal of Theoretical Biology'' is a biweekly peer-reviewed scientific journal covering theoretical biology, as well as mathematical, computational, and statistical aspects of biology. Some research areas covered by the journal include cell ...
*
Logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
*
Mathematical modelling of infectious disease
Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics alo ...
*
Metabolic network modelling
Metabolic network modelling, also known as metabolic network reconstruction or metabolic pathway analysis, allows for an in-depth insight into the molecular mechanisms of a particular organism. In particular, these models correlate the genome wi ...
*
Molecular modelling
Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials sci ...
*
Morphometrics
Morphometrics (from Greek μορϕή ''morphe'', "shape, form", and -μετρία ''metria'', "measurement") or morphometry refers to the quantitative analysis of ''form'', a concept that encompasses size and shape. Morphometric analyses are co ...
*
Population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
*
Spring school on theoretical biology
Estonian Naturalists' Society ( et, Eesti Looduseuurijate Selts, ELUS) is the oldest Estonia-based society of naturalists. It was founded in 1853, and since establishing has been the major scientific organisation focusing on natural history of E ...
*
Statistical genetics
Statistical genetics is a scientific field concerned with the development and application of statistical methods for drawing inferences from genetic data. The term is most commonly used in the context of human genetics. Research in statistical gen ...
*
Theoretical ecology
Theoretical ecology is the scientific discipline devoted to the study of ecological systems using theoretical methods such as simple conceptual models, mathematical models, computational simulations, and advanced data analysis. Effective models im ...
*
Turing pattern
The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomousl ...
Notes
References
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Biologist Salary , PayScale
"Biologist Salary , Payscale". Payscale.Com, 2021, https://www.payscale.com/research/US/Job=Biologist/Salary. Accessed 3 May 2021.
;Theoretical biology
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Further reading
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External links
The Society for Mathematical BiologyThe Collection of Biostatistics Research Archive
{{DEFAULTSORT:Mathematical And Theoretical Biology