Editing Mathematical and theoretical biology

{{Short description|Branch of biology}}
{{redirect|Biological theory|the scientific journal|Biological Theory (journal)}}
{{Math topics TOC}}
{{TopicTOC-Biology}}

[[File:FibonacciChamomile.PNG|250px|thumb|Yellow chamomile head showing the [[Fibonacci number]]s in spirals consisting of 21 (blue) and 13 (aqua). Such arrangements have been noticed since the [[Middle Ages]] and can be used to make mathematical models of a wide variety of plants.]]
'''Mathematical and theoretical biology''', or '''biomathematics''', is a branch of [[biology]] which employs theoretical analysis, [[mathematical models]] and abstractions of living [[organisms]] to investigate the principles that govern the structure, development and behavior of the systems, as opposed to [[experimental biology]] which deals with the conduction of experiments to test scientific theories.<ref>{{Cite web|url=http://www.bath.ac.uk/cmb/mathBiology/|title=What is mathematical biology {{!}} Centre for Mathematical Biology {{!}} University of Bath|website=www.bath.ac.uk|access-date=2018-06-07|archive-url=https://web.archive.org/web/20180923070442/http://www.bath.ac.uk/cmb/mathBiology/|archive-date=2018-09-23|url-status=dead}}</ref> The field is sometimes called '''mathematical biology''' or '''biomathematics''' to stress the mathematical side, or '''theoretical biology''' to stress the biological side.<ref>"There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in mathematical departments and to be a bit more interested in math inspired by biology than in the biological problems themselves, and vice versa." [http://life.biology.mcmaster.ca/~brian/biomath/careers.theo.biol.html Careers in theoretical biology] {{Webarchive|url=https://web.archive.org/web/20190914233407/http://life.biology.mcmaster.ca/~brian/biomath/careers.theo.biol.html |date=2019-09-14 }}</ref> 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  interchange; overlapping as [[Artificial immune system|Artificial Immune Systems]] of [[Amorphous computing|Amorphous Computation]].<ref>{{cite journal | vauthors = Longo G, Soto AM | title = Why do we need theories? | journal = Progress in Biophysics and Molecular Biology | volume = 122 | issue = 1 | pages = 4–10 | date = October 2016 | pmid = 27390105 | pmc = 5501401 | doi = 10.1016/j.pbiomolbio.2016.06.005 | url = https://www.di.ens.fr/users/longo/files/01_theories.pdf | series = From the Century of the Genome to the Century of the Organism: New Theoretical Approaches }}</ref><ref>{{cite journal | vauthors = Montévil M, Speroni L, Sonnenschein C, Soto AM | title = Modeling mammary organogenesis from biological first principles: Cells and their physical constraints | journal = Progress in Biophysics and Molecular Biology | volume = 122 | issue = 1 | pages = 58–69 | date = October 2016 | pmid = 27544910 | pmc = 5563449 | doi = 10.1016/j.pbiomolbio.2016.08.004 | series = From the Century of the Genome to the Century of the Organism: New Theoretical Approaches | arxiv = 1702.03337 }}</ref>

Mathematical biology aims at the mathematical representation and modeling of [[biological process]]es, using techniques and tools of [[applied mathematics]]. It can be useful in both [[basic science|theoretical]] and [[applied science|practical]] 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; requiring [[mathematical model]]s.

Because of the complexity of the [[organism|living systems]], theoretical biology employs several fields of mathematics,<ref>{{cite journal | vauthors = Robeva R, Davies R, Hodge T, Enyedi A | title = Mathematical biology modules based on modern molecular biology and modern discrete mathematics | journal = CBE: Life Sciences Education | volume = 9 | issue = 3 | pages = 227–40 | date = Fall 2010 | pmid = 20810955 | pmc = 2931670 | doi = 10.1187/cbe.10-03-0019 | publisher = The American Society for Cell Biology }}</ref> 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]] 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 François Verhulst]] formulated the logistic growth model in 1836.{{cn|date=May 2023}}

[[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 (he uses the word "geometric") while resources (the environment's [[carrying capacity]]) could only grow arithmetically.<ref name=Mallet2001>{{cite journal | vauthors = Mallet J | title = Mimicry: an interface between psychology and evolution | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 98 | issue = 16 | pages = 8928–30 | date = July 2001 | pmid = 11481461 | pmc = 55348 | doi = 10.1073/pnas.171326298 | bibcode = 2001PNAS...98.8928M | author-link = James Mallet | doi-access = free }}</ref>

The term "theoretical biology" was first used as a monograph title by [[Johannes Reinke]] in 1901, and soon after by [[Jakob von Uexküll]] in 1920. One founding text is considered to be [[On Growth and Form]] (1917) by [[D'Arcy Thompson]],<ref>Ian Stewart (1998), [https://www.worldcat.org/oclc/37211069 Life's Other Secret: The New Mathematics of the Living World], New York: John Wiley, {{isbn|978-0471158455}}</ref> and other early pioneers include [[Ronald Fisher]], [[Hans Leo Przibram]], [[Vito Volterra]], [[Nicolas Rashevsky]] and [[Conrad Hal Waddington]].<ref>{{cite book | vauthors = Keller EF | date = 2002 | url =  https://books.google.com/books?id=NdtbR_N_vKYC | title = Making Sense of Life: Explaining Biological Development with Models, Metaphors and Machines | publisher = Harvard University Press | isbn = 978-0674012509 }}</ref>

===Recent growth===
{{more citations needed|date=March 2020}}

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]] revolution, which are difficult to understand without the use of analytical tools<ref>{{Cite journal| vauthors = Reed M |date=November 2015|title=Mathematical Biology is Good for Mathematics|journal=Notices of the AMS|volume=62|issue=10|pages=1172–1176|doi=10.1090/noti1288|doi-access=free}}</ref>
* Recent development of mathematical tools such as [[chaos theory]] to help understand complex, non-linear mechanisms in biology
* An increase in [[computer|computing]] power, which facilitates calculations and [[simulation]]s not previously possible
* An increasing interest in [[in silico]] experimentation due to ethical considerations, risk, unreliability and other complications involved in human and non-human animal research

==Areas of research==
Several areas of specialized research in mathematical and theoretical biology<ref name=s10516-005-3973-8>{{cite journal |doi=10.1007/s10516-005-3973-8 |title=Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks |year=2006 |vauthors = Baianu IC, Brown R, Georgescu G, Glazebrook JF |journal=Axiomathes |volume=16 |issue=1–2 |pages=65–122|s2cid=9907900 }}</ref><ref>{{cite web | vauthors = Baianu IC | title = Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models | date = 2004 | url = http://cogprints.org/3701/01/ANeuralGenNetworkLuknTopos_oknu4.pdf/ |access-date=2011-08-07 |url-status=dead |archive-url=https://web.archive.org/web/20070713122039/http://cogprints.org/3701/01/ANeuralGenNetworkLuknTopos_oknu4.pdf |archive-date=2007-07-13 }}</ref><ref>{{cite journal | vauthors = Baianu I, Prisecaru V | title = Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis. | journal = Nature Precedings | date = April 2012 | doi = 10.1038/npre.2012.7101.1 | doi-access = free }}</ref><ref name="Research in Mathematical Biology">{{cite web|url=http://www.maths.gla.ac.uk/research/groups/biology/kal.htm |title=Research in Mathematical Biology |publisher=Maths.gla.ac.uk |access-date=2008-09-10}}</ref><ref>{{cite journal | vauthors = Jungck JR | title = Ten equations that changed biology: mathematics in problem-solving biology curricula. | journal = Bioscene | date = May 1997 | volume = 23 | issue = 1 | pages = 11–36 | url = http://acube.org/volume_23/v23-1p11-36.pdf | archive-url = https://web.archive.org/web/20090326215300/http://acube.org/volume_23/v23-1p11-36.pdf | archive-date = 2009-03-26 }}</ref> 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]] developed by [[Humberto Maturana|Maturana]] and [[Francisco Varela|Varela]], [[Stuart Kauffman|Kauffman]]'s Work-Constraints cycles, and more recently the notion of closure of constraints.<ref>{{cite journal | vauthors = Montévil M, Mossio M | title = Biological organisation as closure of constraints | journal = Journal of Theoretical Biology | volume = 372 | pages = 179–91 | date = May 2015 | pmid = 25752259 | doi = 10.1016/j.jtbi.2015.02.029 | bibcode = 2015JThBi.372..179M | s2cid = 4654439 | url = https://hal.archives-ouvertes.fr/hal-01192916/file/Montevil-Mossio_2015_Closure-of-constraints.pdf }}</ref>

===Algebraic biology===
Algebraic biology (also known as symbolic [[systems biology]]) applies the algebraic methods of [[symbolic computation]] to the study of biological problems, especially in [[genomics]], [[proteomics]], analysis of [[molecular structure]]s and study of [[gene]]s.<ref name="cogprints.org">{{cite book | vauthors = Baianu IC |year=1987 |chapter=Computer Models and Automata Theory in Biology and Medicine | veditors = Witten M |title=Mathematical Models in Medicine |volume=7 |publisher=Pergamon Press |location=New York |pages=1513–1577 |chapter-url=http://cogprints.org/3687/ }}</ref><ref>{{cite book | vauthors = Barnett MP |author-link=Michael P. Barnett |chapter=Symbolic calculation in the life sciences: trends and prospects |title=Algebraic Biology 2005 |series=Computer Algebra in Biology | veditors = Anai H, Horimoto K |publisher=Universal Academy Press |location=Tokyo |year=2006 |archive-url=https://web.archive.org/web/20060616135155/http://www.princeton.edu/~allengrp/ms/annobib/mb.pdf |archive-date=2006-06-16 |chapter-url=http://www.princeton.edu/~allengrp/ms/annobib/mb.pdf }}</ref><ref>{{cite book |url=http://library.bjcancer.org/ebook/109.pdf |archive-url=https://web.archive.org/web/20120310224428/http://library.bjcancer.org/ebook/109.pdf |url-status=dead |archive-date=March 10, 2012 | vauthors = Preziosi L |title=Cancer Modelling and Simulation |publisher=Chapman Hall/CRC Press |year=2003 |isbn=1-58488-361-8 }}</ref>

=== 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,<ref>{{cite book |chapter=Computer Models and Automata Theory in Biology and Medicine |year=1986 |title=Mathematical Modeling : Mathematical Models in Medicine |volume=7 |pages=1513–1577 | veditors = Witten M |publisher=Pergamon Press |location=New York |chapter-url=https://cdsweb.cern.ch/record/746663/files/COMPUTER_MODEL_AND_AUTOMATA_THEORY_IN_BIOLOGY2p.pdf }}</ref><ref>{{cite web | vauthors = Lin HC  |year=2004 |title=Computer Simulations and the Question of Computability of Biological Systems |url=https://tspace.library.utoronto.ca/bitstream/1807/2951/2/compauto.pdf }}</ref><ref>{{cite book |title=Computer Models and Automata Theory in Biology and Medicine |year=1986 }}</ref> including subsections in the following areas: [[computer modeling]] in biology and medicine, arterial system models, [[neuron]] models, biochemical and [[oscillation]] [[wikt:network|network]]s, quantum automata, [[quantum computers]] in [[molecular biology]] and [[genetics]],<ref>{{cite journal |title=Natural Transformations Models in Molecular Biology |volume=N/A |pages=230–232 |year=1983 |journal=SIAM and Society of Mathematical Biology, National Meeting |location=Bethesda, MD |url=http://cogprints.org/3675/ }}</ref> cancer modelling,<ref>{{cite journal | vauthors = Baianu IC |title=Quantum Interactomics and Cancer Mechanisms |year=2004 |journal=Research Report Communicated to the Institute of Genomic Biology, University of Illinois at Urbana |url=https://tspace.library.utoronto.ca/retrieve/4969/QuantumInteractomicsInCancer_Sept13k4E_cuteprt.pdf }}</ref> [[neural net]]s, [[genetic network]]s, abstract categories in relational biology,<ref>{{cite book | vauthors = Kainen PC |year=2005 |chapter=Category Theory and Living Systems |title=Charles Ehresmann's Centennial Conference Proceedings |pages=1–5 |location=University of Amiens, France, October 7–9th, 2005 | veditors = Ehresmann A |chapter-url=http://vbm-ehr.pagesperso-orange.fr/ChEh/articles/Kainen.pdf }}</ref> metabolic-replication systems, [[category theory]]<ref>{{cite web |url=http://planetphysics.org/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html |title=bibliography for category theory/algebraic topology applications in physics |publisher=PlanetPhysics |access-date=2010-03-17 |url-status=usurped |archive-url=https://web.archive.org/web/20160107152607/http://planetphysics.org/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html |archive-date=2016-01-07 }}</ref> applications in biology and medicine,<ref>{{cite web |url=http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysicsAndMathematicalMedicine.html |title=bibliography for mathematical biophysics and mathematical medicine |publisher=PlanetPhysics |date=2009-01-24 |access-date=2010-03-17 |url-status=usurped |archive-url=https://web.archive.org/web/20160107152607/http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysicsAndMathematicalMedicine.html |archive-date=2016-01-07 }}</ref> [[automata theory]], [[cellular automata]],<ref>{{Cite journal|title=Cellular Automata|journal=Los Alamos Science|volume=Fall 1983}}</ref> [[tessellation]] models<ref>{{cite book |title=Modern Cellular Automata | vauthors = Preston K, Duff MJ |url=https://books.google.com/books?id=l0_0q_e-u_UC |isbn=9780306417375 |date=1985-02-28 | publisher = Springer }}</ref><ref>{{cite web|url=http://mathworld.wolfram.com/DualTessellation.html |title=Dual Tessellation&nbsp;– from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2010-03-03 |access-date=2010-03-17}}</ref> and complete self-reproduction, [[chaotic system]]s in [[organism]]s, relational biology and organismic theories.<ref name="cogprints.org" /><ref>{{cite web |url=http://theorylab.org/node/56690 |title=Computer models and automata theory in biology and medicine &#124; KLI Theory Lab |publisher=Theorylab.org |date=2009-05-26 |access-date=2010-03-17 |url-status=dead |archive-url=https://web.archive.org/web/20110728100340/http://theorylab.org/node/56690 |archive-date=2011-07-28 }}</ref>

'''Modeling cell and molecular biology'''

This area has received a boost due to the growing importance of [[molecular biology]].<ref name="Research in Mathematical Biology" />
* Mechanics of biological tissues<ref>{{cite web | vauthors = Ogden R |url=http://www.maths.gla.ac.uk/~rwo/research_areas.htm |title=rwo_research_details |publisher=Maths.gla.ac.uk |date=2004-07-02 |access-date=2010-03-17 |url-status=dead |archive-url=https://web.archive.org/web/20090202005852/http://www.maths.gla.ac.uk/~rwo/research_areas.htm |archive-date=2009-02-02 }}</ref><ref>{{cite journal | vauthors = Wang Y, Brodin E, Nishii K, Frieboes HB, Mumenthaler SM, Sparks JL, Macklin P | title = Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach | journal = Scientific Reports | volume = 11 | issue = 1 | pages = 1710 | date = January 2021 | pmid = 33462259 | pmc = 7813881 | doi = 10.1038/s41598-020-78780-7 | bibcode = 2021NatSR..11.1710W | url = }}</ref>
* Theoretical enzymology and [[enzyme kinetics]]
* [[Cancer]] modelling and simulation<ref>{{cite journal |doi=10.1007/s10516-005-4943-x |title=A Computational Model of Oncogenesis using the Systemic Approach |year=2006 | vauthors = Oprisan SA, Oprisan A |journal=Axiomathes |volume=16 |issue=1–2 |pages=155–163|s2cid=119637285 }}</ref><ref>{{cite web|url=http://calvino.polito.it/~mcrtn/ |title=MCRTN&nbsp;– About tumour modelling project |publisher=Calvino.polito.it |access-date=2010-03-17}}</ref>
* Modelling the movement of interacting cell populations<ref>{{cite web|url=http://www.ma.hw.ac.uk/~jas/researchinterests/index.html |title=Jonathan Sherratt's Research Interests |publisher=Ma.hw.ac.uk |access-date=2010-03-17}}</ref>
* Mathematical modelling of scar tissue formation<ref>{{cite web|url=http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html |title=Jonathan Sherratt's Research: Scar Formation |publisher=Ma.hw.ac.uk |access-date=2010-03-17}}</ref>
* Mathematical modelling of intracellular dynamics<ref>{{cite journal | vauthors = Kuznetsov AV, Avramenko AA | title = A macroscopic model of traffic jams in axons | journal = Mathematical Biosciences | volume = 218 | issue = 2 | pages = 142–52 | date = April 2009 | pmid = 19563741 | doi = 10.1016/j.mbs.2009.01.005 }}</ref><ref>{{cite journal | vauthors = Wolkenhauer O, Ullah M, Kolch W, Cho KH | title = Modeling and simulation of intracellular dynamics: choosing an appropriate framework | journal = IEEE Transactions on NanoBioscience | volume = 3 | issue = 3 | pages = 200–7 | date = September 2004 | pmid = 15473072 | doi = 10.1109/TNB.2004.833694 | s2cid = 1829220 }}</ref>
* Mathematical modelling of the cell cycle<ref>{{cite web |title=Tyson Lab |url=http://mpf.biol.vt.edu/Research.html |archive-date=July 28, 2007 |archive-url=https://web.archive.org/web/20070728093149/http://mpf.biol.vt.edu/Research.html }}</ref>
* Mathematical modelling of apoptosis<ref>{{cite journal | vauthors = Fussenegger M, Bailey JE, Varner J | title = A mathematical model of caspase function in apoptosis | journal = Nature Biotechnology | volume = 18 | issue = 2 | pages = 768–74 | date = July 2000 | pmid = 10888847 | doi = 10.1038/77589 | s2cid = 52802267 }}</ref>

'''Modelling physiological systems'''
* Modelling of [[artery|arterial]] disease<ref>{{cite book | vauthors = Noè U, Chen WW, Filippone M, Hill N, Husmeier D |year=2017 |chapter=Inference in a Partial Differential Equations Model of Pulmonary Arterial and Venous Blood Circulation using Statistical Emulation |title=13th International Conference on Computational Intelligence Methods for Bioinformatics and Biostatistics, Stirling, UK, 1–3 Sep 2016 |series=Lecture Notes in Computer Science |volume=10477 |pages=184–198 |isbn=9783319678337 |doi=10.1007/978-3-319-67834-4_15 |chapter-url=http://eprints.gla.ac.uk/129013/7/129013.pdf }}</ref>
* Multi-scale modelling of the [[heart]]<ref>{{cite web |url=http://www.integrativebiology.ox.ac.uk/heartmodel.html |title=Integrative Biology&nbsp;– Heart Modelling |publisher=Integrativebiology.ox.ac.uk |access-date=2010-03-17 |url-status=dead |archive-url=https://web.archive.org/web/20090113215021/http://www.integrativebiology.ox.ac.uk/heartmodel.html |archive-date=2009-01-13 }}</ref>
* Modelling electrical properties of muscle interactions, as in [[bidomain]] and [[monodomain model]]s

=== Computational neuroscience ===
[[Computational neuroscience]] (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system.<ref name=":0">{{Cite book|title=Fundamentals of Computational Neuroscience|url=https://archive.org/details/fundamentalscomp00ttra|url-access=limited| vauthors = Trappenberg TP |publisher=Oxford University Press Inc.|year=2002|isbn=978-0-19-851582-1|location=United States|pages=[https://archive.org/details/fundamentalscomp00ttra/page/n16 1]}}</ref><ref>{{cite book | vauthors = Churchland PS, Koch C, Sejnowski TJ | chapter = What Is Computational Neuroscience? | veditors = Gutfreund H, Toulouse G | title =  Biology And Computation: A Physicist's Choice | date = March 1994 | volume = 3 | pages = 25–34 | publisher = World Scientific | isbn = 9789814504140 | chapter-url = https://books.google.com/books?id=l7vsCgAAQBAJ&pg=PT42 }}</ref>

===Evolutionary biology===
[[Ecology]] and [[evolution|evolutionary biology]] 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]]. Most population geneticists consider the appearance of new [[allele]]s by [[mutation]], the appearance of new [[genotype]]s by [[genetic recombination|recombination]], and changes in the frequencies of existing alleles and genotypes at a small number of [[gene]] [[locus (genetics)|loci]]. When [[infinitesimal]] effects at a large number of gene loci are considered, together with the assumption of [[linkage disequilibrium|linkage equilibrium]] or [[quasi-linkage equilibrium]], one derives [[quantitative genetics]]. [[Ronald Fisher]] made fundamental advances in statistics, such as [[analysis of variance]], via his work on quantitative genetics. Another important branch of population genetics that led to the extensive development of [[coalescent theory]] is [[Computational phylogenetics|phylogenetics]]. Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics<ref>{{cite book | vauthors = Semple C | date = 2003 | url = https://books.google.com/books?id=uR8i2qet | title = SAC Phylogenetics | publisher = Oxford University Press | isbn = 978-0-19-850942-4 }}</ref> Traditional population genetic models deal with alleles and genotypes, and are frequently [[stochastic]].

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]]. Work in this area dates back to the 19th century, and even as far as 1798 when [[Thomas Robert Malthus|Thomas Malthus]] formulated the first principle of population dynamics, which later became known as the [[Malthusian growth model]]. The [[Lotka–Volterra equation|Lotka–Volterra predator-prey equations]] are another famous example. Population dynamics overlap with another active area of research in mathematical biology: [[mathematical modelling of infectious disease|mathematical epidemiology]], the study of infectious disease affecting populations. Various models of the spread of [[infections]] have been proposed and analyzed, and provide important results that may be applied to health policy decisions.

In [[evolutionary game theory]], developed first by [[John Maynard Smith]] and [[George R. Price]], selection acts directly on inherited phenotypes, without genetic complications. This approach has been mathematically refined to produce the field of [[evolutionary invasion analysis|adaptive dynamics]].

=== Mathematical biophysics ===
The earlier stages of mathematical biology were dominated by mathematical [[biophysics]], 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|Difference equations/Maps]]&nbsp;– discrete time, continuous state space.
* [[Ordinary differential equations]]&nbsp;– continuous time, continuous state space, no spatial derivatives. ''See also:'' [[Numerical ordinary differential equations]].
* [[Partial differential equations]]&nbsp;– continuous time, continuous state space, spatial derivatives. ''See also:'' [[Numerical partial differential equations]].
* [[Cellular automaton|Logical deterministic cellular automata]]&nbsp;– discrete time, discrete state space. ''See also:'' [[Cellular automaton]].

====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]] with a corresponding [[probability distribution]].
* Non-Markovian processes&nbsp;– [[master equation|generalized master equation]]&nbsp;– continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur.
* Jump [[Continuous-time Markov process|Markov process]]&nbsp;– [[master equation]]&nbsp;– 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]] for numerical simulation methods, specifically [[dynamic Monte Carlo method]] and [[Gillespie algorithm]].
* Continuous [[Markov process]]&nbsp;– [[stochastic differential equation]]s or a [[Fokker–Planck equation]]&nbsp;– continuous time, continuous state space, events occur continuously according to a random [[Wiener process]].

====Spatial modelling====
One classic work in this area is [[Alan Turing]]'s paper on [[morphogenesis]] entitled ''[[The Chemical Basis of Morphogenesis]]'', published in 1952 in the [[Philosophical Transactions of the Royal Society]].
* Travelling waves in a wound-healing assay<ref>{{cite web|url=http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm|title=Travelling waves in a wound|publisher=Maths.ox.ac.uk|access-date=2010-03-17|archive-url=https://web.archive.org/web/20080606030634/http://www2.maths.ox.ac.uk/~maini/public/gallery/twwha.htm|archive-date=2008-06-06|url-status=dead}}</ref>
* [[Swarming behaviour]]<ref>{{Cite web |url=http://www.math.ubc.ca/people/faculty/keshet/research.html |title= Leah Edelstein-Keshet: Research Interests f |access-date=2005-02-26 |archive-url=https://web.archive.org/web/20070612135231/http://www.math.ubc.ca/people/faculty/keshet/research.html |archive-date=2007-06-12 |url-status=dead }}</ref>
* A mechanochemical theory of [[morphogenesis]]<ref>{{cite web|url=http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm|title=The mechanochemical theory of morphogenesis|publisher=Maths.ox.ac.uk|access-date=2010-03-17|archive-url=https://web.archive.org/web/20080606030629/http://www2.maths.ox.ac.uk/~maini/public/gallery/mctom.htm|archive-date=2008-06-06|url-status=dead}}</ref>
* [[Biological pattern formation]]<ref>{{cite web|url=http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm|title=Biological pattern formation|publisher=Maths.ox.ac.uk|access-date=2010-03-17|archive-date=2004-11-12|archive-url=https://web.archive.org/web/20041112100632/http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm|url-status=dead}}</ref>
* Spatial distribution modeling using plot samples<ref>{{cite journal|vauthors = Hurlbert SH |year=1990|title=Spatial Distribution of the Montane Unicorn|journal=Oikos|volume=58|issue=3|pages=257–271|doi=10.2307/3545216|jstor=3545216|bibcode=1990Oikos..58..257H }}</ref>
* [[Turing pattern]]s<ref>{{cite book | vauthors = Wooley TE, Baker RE, Maini PK | author-link2 = Ruth Baker | author-link3 = Philip Maini | chapter = Chapter 34: Turing's theory of morphogenesis | title=The Turing Guide| veditors = Copeland BJ, Bowen JP, Wilson R, Sprevak M | editor-link1=Jack Copeland| editor-link2=Jonathan Bowen| editor-link3=Robin Wilson (mathematician)|publisher=[[Oxford University Press]]|year=2017|isbn=978-0198747826|title-link=The Turing Guide}}</ref>

=== 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 point|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 is a mathematical formulation of the wide-sense [[chemical kinetics]] 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]], and its applications were developed in mathematical biology and especially in mathematical medicine.<ref name="planetphysics.org">{{cite web|url=http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html|title=molecular set category|publisher=PlanetPhysics|archive-url=https://web.archive.org/web/20160107152607/http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html|archive-date=2016-01-07|url-status=usurped|access-date=2010-03-17}}</ref>
In a more general sense, Molecular set theory 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.<ref name="planetphysics.org" />

===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)<ref>{{cite web | title = Abstract Relational Biology (ARB) | url = http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html | url-status = usurped | archive-url = https://web.archive.org/web/20160107152607/http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html | archive-date=2016-01-07 }}</ref> 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 (theoretical biologist)|Robert Rosen]] in 1957–1958 as abstract, relational models of cellular and organismal organization.<ref>{{Cite book|title=Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life| vauthors = Rosen R |date=2005-07-13|publisher=Columbia University Press|isbn=9780231075657|language=en}}</ref>

== Model example: the cell cycle ==
{{Main|Cellular model}}
The eukaryotic [[cell cycle]] is very complex and has been the subject of intense study, since its misregulation leads to [[cancer]]s.
It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results.  Two research groups<ref>{{cite web|url=http://mpf.biol.vt.edu/Tyson%20Lab.html |archive-url=https://web.archive.org/web/20070728091004/http://mpf.biol.vt.edu/Tyson%20Lab.html |url-status=dead |archive-date=2007-07-28 |title=The JJ Tyson Lab|publisher=[[Virginia Tech]]|access-date=2008-09-10 }}</ref><ref>{{cite web|url=http://cellcycle.mkt.bme.hu/|title=The Molecular Network Dynamics Research Group|publisher=[[Budapest University of Technology and Economics]]|url-status=dead|archive-url=https://web.archive.org/web/20120210210021/http://cellcycle.mkt.bme.hu/|archive-date=2012-02-10}}</ref> 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]]s these models show the change in time ([[dynamical system]]) of the protein inside a single typical cell; this type of model is called a [[deterministic system|deterministic process]] (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a [[stochastic process]]).

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 [[Reaction rate|rate kinetics]] for stoichiometric reactions, [[Michaelis-Menten kinetics]] for enzyme substrate reactions and [[Goldbeter–Koshland kinetics]] 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 [[Array data structure|vector]] (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.

[[Image:Cell cycle bifurcation diagram.jpg|thumb|500px]] 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 [[Lyapunov stability|stable point]], called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an [[Lyapunov stability|unstable point]], either a source or a [[saddle point]], 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]] using [[bifurcation theory]]. 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]]), 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]] and an [[infinite period bifurcation]].{{citation needed|date=May 2011}}

== See also ==
{{Div col|colwidth=20em}}
* [[Biological applications of bifurcation theory]]
* [[Biophysics]]
* [[Biostatistics]]
* [[Entropy and life]]
* [[Ewens's sampling formula]]
* [[Journal of Theoretical Biology]]
* [[List of bioinformatics software]]
* [[Logistic function]]
* [[Mathematical modelling of infectious disease]]
* [[Metabolic network modelling]]
* [[Molecular modelling]]
* [[Morphometrics]]
* [[Population genetics]]
* [[Spring school on theoretical biology]]
* [[Statistical genetics]]
* [[Theoretical ecology]]
* [[Turing pattern]]
{{div col end}}

==Notes==
{{Reflist|colwidth=30em}}

== References ==
{{Refbegin}}
* {{cite book | vauthors = Edelstein-Keshet L|title=Mathematical Models in Biology |publisher=SIAM |year=2004 |isbn=0-07-554950-6 }}
* {{cite book | vauthors = Hoppensteadt F |title=Mathematical Theories of Populations: Demographics, Genetics and Epidemics |publisher=SIAM |location=Philadelphia |orig-year=1975 |edition=Reprinted |year=1993 |isbn=0-89871-017-0 }}
* {{cite book | vauthors = Renshaw E |title=Modelling Biological Populations in Space and Time |publisher=C.U.P. |year=1991 |isbn=0-521-44855-7 |url-access=registration |url=https://archive.org/details/modellingbiologi0000rens }}
* {{cite book | vauthors =Rubinow SI |title=Introduction to Mathematical Biology |publisher=John Wiley |year=1975 |isbn=0-471-74446-8 }}
* {{cite book | vauthors =Strogatz SH |title=Nonlinear Dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering |publisher=Perseus |year=2001 |isbn=0-7382-0453-6 |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }}
{{Refend}}
* "Biologist Salary | Payscale". Payscale.Com, 2021, [https://www.payscale.com/research/US/Job=Biologist/Salary Biologist Salary | PayScale]. Accessed 3 May 2021.
;Theoretical biology
{{Refbegin}}
* {{cite book | vauthors = Bonner JT |year=1988 |title=The Evolution of Complexity by Means of Natural Selection |url=https://archive.org/details/evolutionofcompl0000bonn |url-access=registration |location=Princeton |publisher=Princeton University Press |isbn=0-691-08493-9 }}
* {{cite book | vauthors = Mangel M |year=2006 |title=The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary Biology |publisher=Cambridge University Press |isbn=0-521-53748-7 }}
{{Refend}}

== Further reading ==
{{Refbegin}}
* {{Cite journal | vauthors = Hoppensteadt F |url=https://www.ams.org/notices/199509/hoppensteadt.pdf |title=Getting Started in Mathematical Biology |journal=[[Notices of the American Mathematical Society]] |date=September 1995 }}
* {{cite journal | vauthors = May RM | title = Uses and abuses of mathematics in biology | journal = Science | volume = 303 | issue = 5659 | pages = 790–3 | date = February 2004 | pmid = 14764866 | doi = 10.1126/science.1094442 | s2cid = 24844494 | bibcode = 2004Sci...303..790M }}
* {{Cite journal | vauthors = Murray JD |url=http://www.resnet.wm.edu/~jxshix/math490/murray.doc |title=How the leopard gets its spots? |journal=Scientific American |volume=258 |issue=3 |pages=80–87 |year=1988 |doi=10.1038/scientificamerican0388-80 |bibcode=1988SciAm.258c..80M }}
* {{Cite journal | vauthors = Reed MC | author-link1 = Michael C. Reed |url=http://www.resnet.wm.edu/~jxshix/math490/reed.pdf |title=Why Is Mathematical Biology So Hard? |journal=[[Notices of the American Mathematical Society]] |date=March 2004 }}
* {{Cite journal | vauthors = Kroc J, Balihar K, Matejovic M | title = Complex Systems and Their Use in Medicine: Concepts, Methods and Bio-Medical Applications | url = https://www.researchgate.net/publication/330546521 | doi = 10.13140/RG.2.2.29919.30887 | year = 2019 }}
{{Refend}}

== External links ==
{{Commons category}}
* [http://www.smb.org/ The Society for Mathematical Biology]
* [https://web.archive.org/web/20080827161431/http://www.biostatsresearch.com/repository/ The Collection of Biostatistics Research Archive]

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