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**Mathematical and theoretical biology** is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms in order to investigate the principles that govern the structure, development and behavior of the systems.^{ [1] } .The field is sometimes called **mathematical biology** or **biomathematics** to stress the mathematical side, or **theoretical biology** to stress the biological side.^{ [2] } Theoretical biology focuses 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.^{ [3] }^{ [4] }

**Biology** is the natural science that studies life and living organisms, including their physical structure, chemical processes, molecular interactions, physiological mechanisms, development and evolution. Despite the complexity of the science, there are certain unifying concepts that consolidate it into a single, coherent field. Biology recognizes the cell as the basic unit of life, genes as the basic unit of heredity, and evolution as the engine that propels the creation and extinction of species. Living organisms are open systems that survive by transforming energy and decreasing their local entropy to maintain a stable and vital condition defined as homeostasis.

In biology, an **organism** is any individual entity that exhibits the properties of life. It is a synonym for "life form".

- History
- Early history
- Areas of research
- Abstract relational biology
- Algebraic biology
- Complex systems biology
- Computer models and automata theory
- Computational neuroscience
- Evolutionary biology
- Mathematical methods
- Mathematical biophysics
- Molecular set theory
- Organizational biology
- Recent growth
- Model example: the cell cycle
- Societies and institutes
- See also
- Notes
- References
- Further reading
- External links

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 models. Mathematical biology employs many components of mathematics,^{ [5] } and has contributed to the development of new techniques. It can be contrasted with experimental biology (which deals with the conduction of experiments to prove and validate the scientific theories).^{ [1] }

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 and engineering disciplines, as well as in the social sciences.

**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 mathematical modelling and abstractions of the biological systems. Due to the complexity of the investigated systems, biology is primarily an experimental science. However, as a consequence of the modern increase in computational power, it is now becoming more feasible to find approximate solutions and validate mathematical models of complex living organisms.

Mathematical biology is relevant to the mathematical representation and modeling of biological processes using techniques and tools of applied mathematics. It has both theoretical and practical applications in biological, biomedical and biotechnology research.

**Biological processes** are the processes vital for a living organism to live. Biological processes are made up of many chemical reactions or other events that are involved in the persistence and transformation of life forms. Metabolism and homeostasis are examples.

**Applied mathematics** is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Mathematics has been applied to biology as early as the 12th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human populution was based on the concept of exponential growth. Pierre Francois Verhulst formulated the logistic growth model in 1836.

**Fibonacci** was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, "Fibonacci", was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for *filius Bonacci*. He is also known as **Leonardo Bonacci**, **Leonardo of Pisa**, or **Leonardo Bigollo** ("traveller") **Pisano**.

**Daniel Bernoulli** FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing.

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.^{ [6] }

**Johann Friedrich Theodor Müller**, 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, Santa Catarina. There he studied the natural history of the Atlantic forest south of São Paulo, and was an early advocate of Darwinism. He lived in Brazil for the rest of his life. Müllerian mimicry is named after him.

**Müllerian mimicry** is a natural phenomenon in which two or more unprofitable species, that may or may not be closely related and share one or more common predators, have come to mimic each other's honest warning signals, to their mutual benefit, since predators can learn to avoid all of them with fewer experiences. It is named after the German naturalist Fritz Müller, who first proposed the concept in 1878, supporting his theory with the first mathematical model of frequency-dependent selection, one of the first such models anywhere in biology.

**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 be seen as an approach to the study of evolution that incorporates an understanding of the interactions between the species under consideration. The main subfields of evolutionary ecology are life history evolution, sociobiology, the evolution of inter specific relations and the evolution of biodiversity and of communities.

The term "theoretical biology" was first used by Johannes Reinke in 1901. One founding text is considered to be On Growth and Form (1917) by D'Arcy Thompson,^{ [7] } and other early pioneers include Ronald Fisher, Hans Leo Przibram, Nicolas Rashevsky and Vito Volterra.^{ [8] }

**Johannes Reinke** was a German botanist and philosopher who was a native of Ziethen, Lauenburg. He is remembered for his research of benthic marine algae.

* 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.

**Sir Ronald Aylmer Fisher** was a British statistician and geneticist. For his work in statistics, he has been described as "a genius who almost single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, his work used mathematics to combine Mendelian genetics and natural selection; this contributed to the revival of Darwinism in the early 20th-century revision of the theory of evolution known as the modern synthesis. For his contributions to biology, Fisher has been called "the greatest of Darwin’s successors".

Several areas of specialized research in mathematical and theoretical biology^{ [9] }^{ [10] }^{ [11] }^{ [12] }^{ [13] } 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 (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 Maturana and Varela, Kauffman's Work-Constraints cycles, and more recently the notion of closure of constraints.^{ [14] }

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 structures and study of genes.^{ [15] }^{ [16] }^{ [17] }

An elaboration of systems biology to understanding the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.

A monograph on this topic summarizes an extensive amount of published research in this area up to 1986,^{ [18] }^{ [19] }^{ [20] } including subsections in the following areas: computer modeling in biology and medicine, arterial system models, neuron models, biochemical and oscillation networks, quantum automata, quantum computers in molecular biology and genetics,^{ [21] } cancer modelling,^{ [22] } neural nets, genetic networks, abstract categories in relational biology,^{ [23] } metabolic-replication systems, category theory ^{ [24] } applications in biology and medicine,^{ [25] } automata theory, cellular automata,^{ [26] } tessellation models^{ [27] }^{ [28] } and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories.^{ [15] }^{ [29] }

**Modeling cell and molecular biology**

This area has received a boost due to the growing importance of molecular biology.^{ [12] }

- Mechanics of biological tissues
^{ [30] } - Theoretical enzymology and enzyme kinetics
- Cancer modelling and simulation
^{ [31] }^{ [32] } - Modelling the movement of interacting cell populations
^{ [33] } - Mathematical modelling of scar tissue formation
^{ [34] } - Mathematical modelling of intracellular dynamics
^{ [35] }^{ [36] } - Mathematical modelling of the cell cycle
^{ [37] }

**Modelling physiological systems**

- Modelling of arterial disease
^{ [38] } - Multi-scale modelling of the heart
^{ [39] } - Modelling electrical properties of muscle interactions, as in bidomain and monodomain models

Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system.^{ [40] }^{ [41] }

Ecology and 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 alleles by mutation, the appearance of new genotypes by recombination, and changes in the frequencies of existing alleles and genotypes at a small number of gene loci. When infinitesimal effects at a large number of gene loci are considered, together with the assumption of 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 phylogenetics. Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics^{ [42] } 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 Malthus formulated the first principle of population dynamics, which later became known as the Malthusian growth model. 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 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 adaptive dynamics.

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.

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.

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 – continuous time, continuous state space, no spatial derivatives.
*See also:*Numerical ordinary differential equations. - Partial differential equations – continuous time, continuous state space, spatial derivatives.
*See also:*Numerical partial differential equations. - Logical deterministic cellular automata – discrete time, discrete state space.
*See also:*Cellular automaton.

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 – 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 – master equation – 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 – stochastic differential equations or a Fokker-Planck equation – continuous time, continuous state space, events occur continuously according to a random Wiener process.

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
^{ [43] } - Swarming behaviour
^{ [44] } - A mechanochemical theory of morphogenesis
^{ [45] } - Biological pattern formation
^{ [46] } - Spatial distribution modeling using plot samples
^{ [47] } - Turing patterns
^{ [48] }

Molecular set theory (MST) 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.^{ [49] } 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.^{ [49] }^{ [50] }

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)^{ [51] } 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.^{ [52] }

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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
- Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology
- An increase in computing power, which facilitates calculations and simulations not previously possible
- An increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results. Two research groups ^{ [53] }^{ [54] } 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 equations 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 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 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 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.

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, 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, 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 ]}

- National Institute for Mathematical and Biological Synthesis
- Society for Mathematical Biology
- ESMTB: European Society for Mathematical and Theoretical Biology
- The Israeli Society for Theoretical and Mathematical Biology
- Société Francophone de Biologie Théorique
- International Society for Biosemiotic Studies

- Biological applications of bifurcation theory
- Biostatistics
- Entropy and life
- Ewens's sampling formula
- Journal of Theoretical Biology
- Mathematical modelling of infectious disease
- Metabolic network modelling
- Molecular modelling
- Morphometrics
- Population genetics
- Statistical genetics
- Theoretical ecology
- Turing pattern

- 1 2 "What is mathematical biology | Centre for Mathematical Biology | University of Bath".
*www.bath.ac.uk*. Retrieved 2018-06-07. - ↑ "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." Careers in theoretical biology
- ↑ Longo, Giuseppe; Soto, Ana M. (2016-10-01). "Why do we need theories?" (PDF).
*Progress in Biophysics and Molecular Biology*. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches.**122**(1): 4–10. doi:10.1016/j.pbiomolbio.2016.06.005. - ↑ Montévil, Maël; Speroni, Lucia; Sonnenschein, Carlos; Soto, Ana M. (2016-10-01). "Modeling mammary organogenesis from biological first principles: Cells and their physical constraints".
*Progress in Biophysics and Molecular Biology*. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches.**122**(1): 58–69. arXiv: 1702.03337 . doi:10.1016/j.pbiomolbio.2016.08.004. - ↑ Robeva, Raina; et al. (Fall 2010). "Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics".
*CBE Life Sciences Education*. The American Society for Cell Biology.**9**(3): 227–240. doi:10.1187/cbe.10-03-0019. PMC 2931670 . PMID 20810955. - ↑ Mallet, James (July 2001). "Mimicry: An interface between psychology and evolution".
*PNAS*.**98**(16): 8928–8930. Bibcode:2001PNAS...98.8928M. doi:10.1073/pnas.171326298. PMC 55348 . PMID 11481461. - ↑ Ian Stewart (1998), Life's Other Secret: The New Mathematics of the Living World, New York: John Wiley, ISBN 978-0471158455
- ↑ Evelyn Fox Keller (2002) Making Sense of Life: Explaining Biological Development with Models, Metaphors and Machines, Harvard University Press, ISBN 978-0674012509
- ↑ Baianu, I. C.; Brown, R.; Georgescu, G.; Glazebrook, J. F. (2006). "Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks".
*Axiomathes*.**16**: 65–122. doi:10.1007/s10516-005-3973-8. - ↑ Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004) "Archived copy" (PDF). Archived from the original on 2007-07-13. Retrieved 2011-08-07.CS1 maint: Archived copy as title (link)
- ↑ Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis (2004) http://cogprints.org/3687/
- 1 2 "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved 2008-09-10.
- ↑ J. R. Junck. Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula,
*Bioscene*, (1997), 23(1):11-36 New Link (Aug 2010) - ↑ Montévil, Maël; Mossio, Matteo (2015-05-07). "Biological organisation as closure of constraints".
*Journal of Theoretical Biology*.**372**: 179–191. doi:10.1016/j.jtbi.2015.02.029. - 1 2 Baianu, I. C. 1987, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),
*Mathematical Models in Medicine*, vol.**7**., Ch.11 Pergamon Press, New York, 1513-1577. http://cogprints.org/3687/ - ↑ Michael P Barnett, "Symbolic calculation in the life sciences: trends and prospects, Algebraic Biology 2005" –
*Computer Algebra in Biology*, edited by H. Anai, K. Horimoto, Universal Academy Press, Tokyo, 2006. (on line .pdf format) - ↑ http://library.bjcancer.org/ebook/109.pdf%5B%5D L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8.
- ↑ Witten, M., ed. (1986). "Computer Models and Automata Theory in Biology and Medicine" (PDF).
*Mathematical Modeling : Mathematical Models in Medicine*.**7**. New York: Pergamon Press. pp. 1513–1577. - ↑ Lin, H. C. (2004). "Computer Simulations and the Question of Computability of Biological Systems" (PDF).
- ↑
*Computer Models and Automata Theory in Biology and Medicine*. 1986. - ↑ "Natural Transformations Models in Molecular Biology".
*SIAM and Society of Mathematical Biology, National Meeting*. Bethesda, MD. 1983. - ↑ Baianu, I. C. (2004). "Quantum Interactomics and Cancer Mechanisms" (PDF).
*Research Report communicated to the Institute of Genomic Biology, University of Illinois at Urbana*. - ↑ Kainen, P. C. (2005). "Category Theory and Living Systems" (PDF). In Ehresmann, A.
*Charles Ehresmann's Centennial Conference Proceedings*. University of Amiens, France, October 7-9th, 2005. pp. 1–5. - ↑ "bibliography for category theory/algebraic topology applications in physics". PlanetPhysics. Archived from the original on 2016-01-07. Retrieved 2010-03-17.
- ↑ "bibliography for mathematical biophysics and mathematical medicine". PlanetPhysics. 2009-01-24. Archived from the original on 2016-01-07. Retrieved 2010-03-17.
- ↑ "Cellular Automata".
*Los Alamos Science*. Fall 1983. - ↑ Preston, Kendall; Duff, M. J. B.
*Modern Cellular Automata*. - ↑ "Dual Tessellation – from Wolfram MathWorld". Mathworld.wolfram.com. 2010-03-03. Retrieved 2010-03-17.
- ↑ "Computer models and automata theory in biology and medicine | KLI Theory Lab". Theorylab.org. 2009-05-26. Archived from the original on 2011-07-28. Retrieved 2010-03-17.
- ↑ Ray Ogden (2004-07-02). "rwo_research_details". Maths.gla.ac.uk. Archived from the original on 2009-02-02. Retrieved 2010-03-17.
- ↑ Oprisan, Sorinel A.; Oprisan, Ana (2006). "A Computational Model of Oncogenesis using the Systemic Approach".
*Axiomathes*.**16**: 155–163. doi:10.1007/s10516-005-4943-x. - ↑ "MCRTN – About tumour modelling project". Calvino.polito.it. Retrieved 2010-03-17.
- ↑ "Jonathan Sherratt's Research Interests". Ma.hw.ac.uk. Retrieved 2010-03-17.
- ↑ "Jonathan Sherratt's Research: Scar Formation". Ma.hw.ac.uk. Retrieved 2010-03-17.
- ↑ Kuznetsov, A.V.; Avramenko, A.A. (2009). "A macroscopic model of traffic jams in axons".
*Mathematical Biosciences*.**218**(2): 142–152. doi:10.1016/j.mbs.2009.01.005. - ↑ Wolkenhauer, Olaf; Ullah, Mukhtar; Kolch, Walter; Cho, Kwang-Hyun (2004). "Modelling and Simulation of IntraCellular Dynamics: Choosing an Appropriate Framework".
*IEEE Transactions on NanoBioscience*.**3**(3): 200–207. doi:10.1109/TNB.2004.833694. - ↑ "Tyson Lab". Archived from the original on July 28, 2007.
- ↑ Noè, U.; Chen, W. W.; Filippone, M.; Hill, N.; Husmeier, D. (2017). "Inference in a Partial Differential Equations Model of Pulmonary Arterial and Venous Blood Circulation using Statistical Emulation".
*13th International Conference on Computational Intelligence Methods for Bioinformatics and Biostatistics, Stirling, UK, 1–3 Sep 2016*. pp. 184–198. doi:10.1007/978-3-319-67834-4_15. ISBN 9783319678337. - ↑ "Integrative Biology – Heart Modelling". Integrativebiology.ox.ac.uk. Archived from the original on 2009-01-13. Retrieved 2010-03-17.
- ↑ Trappenberg, Thomas P. (2002).
*Fundamentals of Computational Neuroscience*. United States: Oxford University Press Inc. p. 1. ISBN 978-0-19-851582-1. - ↑ What is computational neuroscience? Patricia S. Churchland, Christof Koch, Terrence J. Sejnowski. in Computational Neuroscience pp.46-55. Edited by Eric L. Schwartz. 1993. MIT Press "Archived copy". Archived from the original on 2011-06-04. Retrieved 2009-06-11.CS1 maint: Archived copy as title (link)
- ↑ Charles Semple (2003), Phylogenetics, Oxford University Press, ISBN 978-0-19-850942-4
- ↑ "Travelling waves in a wound". Maths.ox.ac.uk. Retrieved 2010-03-17.
- ↑
^{[ }*dead link*] - ↑ "The mechanochemical theory of morphogenesis". Maths.ox.ac.uk. Retrieved 2010-03-17.
- ↑ "Biological pattern formation". Maths.ox.ac.uk. Retrieved 2010-03-17.
- ↑ Hurlbert, Stuart H. (1990). "Spatial Distribution of the Montane Unicorn".
*Oikos*.**58**(3): 257–271. doi:10.2307/3545216. JSTOR 3545216. - ↑ Wooley, T. E., Baker, R. E. Maini, P. K., Chapter 34,
*Turing's theory of morphogenesis*. In Copeland, B. Jack; Bowen, Jonathan P.; Wilson, Robin; Sprevak, Mark (2017).*The Turing Guide*. Oxford University Press. ISBN 978-0198747826. - 1 2 "molecular set category". PlanetPhysics. Archived from the original on 2016-01-07. Retrieved 2010-03-17.
- ↑ Representation of Uni-molecular and Multimolecular Biochemical Reactions in terms of Molecular Set Transformations http://planetmath.org/?op=getobj&from=objects&id=10770%5B%5D
- ↑ Abstract Relational Biology (ARB) Archived 2016-01-07 at the Wayback Machine
- ↑ Rosen, Robert (2005-07-13).
*Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life*. Columbia University Press. ISBN 9780231075657. - ↑ "The JJ Tyson Lab". Virginia Tech. Archived from the original on 2007-07-28. Retrieved 2008-09-10.
- ↑ "The Molecular Network Dynamics Research Group". Budapest University of Technology and Economics. Archived from the original on 2012-02-10.

**Discrete mathematics** is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

**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 improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.

**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. Biophysical research shares significant overlap with biochemistry, molecular biology, physical chemistry, physiology, nanotechnology, bioengineering, computational biology, biomechanics, developmental biology and systems biology.

A **cellular automaton** is a discrete model studied in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling. Cellular automata are also called **cellular spaces**, **tessellation automata**, **homogeneous structures**, **cellular structures**, **tessellation structures**, and **iterative arrays**.

**Computational physics** is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science.

**Computational biology** involves the development and application of data-analytical and theoretical methods, mathematical modeling and computational simulation techniques to the study of biological, ecological, behavioral, and social systems. The field is broadly defined and includes foundations in biology, applied mathematics, statistics, biochemistry, chemistry, biophysics, molecular biology, genetics, genomics, computer science and evolution.

**Robert Rosen** was an American theoretical biologist and Professor of Biophysics at Dalhousie University.

**Systems biology** is the computational and mathematical modeling of complex biological systems. It is a biology-based interdisciplinary field of study that focuses on complex interactions within biological systems, using a holistic approach to biological research.

**Dynamical systems theory** is an area of mathematics used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called *continuous dynamical systems*. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called *discrete dynamical systems*. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

* In silico* is an expression meaning "performed on computer or via computer simulation" in reference to biological experiments. The phrase was coined in 1989 as an allusion to the Latin phrases

**Modelling biological systems** is a significant task of systems biology and mathematical biology. **Computational systems biology** aims to develop and use efficient algorithms, data structures, visualization and communication tools with the goal of computer modelling of biological systems. It involves the use of computer simulations of biological systems, including cellular subsystems, to both analyze and visualize the complex connections of these cellular processes.

A **biologist** is a scientist who has specialized knowledge in the field of biology, the scientific study of life. Biologists involved in fundamental research attempt to explore and further explain the underlying mechanisms that govern the functioning of living matter. Biologists involved in applied research attempt to develop or improve more specific processes and understanding, in fields such as medicine and industry.

In the mathematical area of bifurcation theory a **saddle-node bifurcation**, **tangential bifurcation** or **fold bifurcation** is a local bifurcation in which two fixed points of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a **fold bifurcation**. Another name is **blue skies bifurcation** in reference to the sudden creation of two fixed points.

**René Thomas** (14 May 1928 - 9 January 2017 was a Belgian scientist. His research included DNA biochemistry and biophysics, genetics, mathematical biology, and finally dynamical systems. He devoted his life to the deciphering of key logical principles at the basis of the behaviour of biological systems, and more generally to the generation of complex dynamical behaviour. He was professor and laboratory head at the Université Libre de Bruxelles, and taught and inspired several generations of researchers.

Creating a **cellular model** has been a particularly challenging task of systems biology and mathematical biology. It involves developing efficient algorithms, data structures, visualization and communication tools to orchestrate the integration of large quantities of biological data with the goal of computer modeling.

**Natural computing**, also called **natural computation**, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration from nature for the development of novel problem-solving techniques; 2) those that are based on the use of computers to synthesize natural phenomena; and 3) those that employ natural materials to compute. The main fields of research that compose these three branches are artificial neural networks, evolutionary algorithms, swarm intelligence, artificial immune systems, fractal geometry, artificial life, DNA computing, and quantum computing, among others.

**Complex systems biology** (**CSB**) is a branch or subfield of mathematical and theoretical biology concerned with complexity of both structure and function in biological organisms, as well as the emergence and evolution of organisms and species, with emphasis being placed on the complex interactions of, and within, bionetworks, and on the fundamental relations and relational patterns that are essential to life. CSB is thus a field of theoretical sciences aimed at discovering and modeling the relational patterns essential to life that has only a partial overlap with complex systems theory, and also with the systems approach to biology called systems biology; this is because the latter is restricted primarily to simplified models of biological organization and organisms, as well as to only a general consideration of philosophical or semantic questions related to complexity in biology. Moreover, a wide range of abstract theoretical complex systems are studied as a field of applied mathematics, with or without relevance to biology, chemistry or physics.

**Multi-state modeling of biomolecules** refers to a series of techniques used to represent and compute the behaviour of biological molecules or complexes that can adopt a large number of possible functional states.

The **Institute of Mathematical Problems of Biology** RAS is a research institute specializing in computational biology and bioinformatics. The objective of the institute is elaboration of mathematical and computational methods for biological research, as well as implementation of these methods directly addressing the problems of computational biology.

**Cell-based models** are mathematical models that represent biological cells as a discrete entities. They are used in the field of computational biology for simulating the biomechanics of multicellular structures such as tissues. Their main advantage is the easy integration of cell level processes such as cell division, intracellular processes and single-cell variability within a cell population.

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