# Mathematical biology: Wikis

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Mathematical and theoretical biology is an interdisciplinary academic research field with a range of applications in biology, medicine and biotechnology.[1] The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical biology to stress the biological side.[2] It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing.[citation needed]

Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. In order to do this, precise mathematical models are required. By describing the systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.

## Importance

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include:

• the explosion 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, nonlinear mechanisms in biology,
• an increase in computing power which enables calculations and simulations to be performed that were not previously possible, and
• an increasing interest in in silico 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[3][4][5][6][7] 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. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers, engineers, biologists, physiologists, research physicians, biomedical researchers, oncologists, molecular biologists, geneticists, embryologists, zoologists, chemists, etc.

### Computer models and automata theory

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987,[8] 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, cancer modelling, neural nets, genetic networks, abstract relational biology, metabolic-replication systems, category theory[9] applications in biology and medicine,[10] automata theory,cellular automata, tessallation models[11][12] and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories.[13][14] This published report also includes 390 references to peer-reviewed articles by a large number of authors.[3][15][16]

Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology.[6]

• Mechanics of biological tissues[17]
• Theoretical enzymology and enzyme kinetics
• Cancer modelling and simulation[18][19]
• Modelling the movement of interacting cell populations[20]
• Mathematical modelling of scar tissue formation[21]
• Mathematical modelling of intracellular dynamics[22]
• Mathematical modelling of the cell cycle[23]

Modelling physiological systems

### Molecular set theory

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in Mathematical Medicine.[26] 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. 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.[26][27]

### Population dynamics

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The Lotka–Volterra predator-prey equations are a famous example. In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. 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.

### 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.

### 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 will always generate the same trajectory and no two trajectories cross in state space.

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.

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.

### Phylogenetics

Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics[33]

## Model example: the cell cycle

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 [34][35] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which 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 Michealis 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.
In order to fit the parameters the differential equations need to 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 proprieties of the equations are used to investigate the behavior of the system depending of 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 can handle the large number of variables and parameters is called a bifurcation diagram(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.

For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.

## Notes

1. ^ Mathematical and Theoretical Biology: A European Perspective
2. ^ "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
3. ^ a b c 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. doi:10.1007/s10516-005-3973-8.
4. ^ [1]
5. ^ [2]
6. ^ a b "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved 2008-09-10.
7. ^ J. R. Junck. Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula, Bioscene, (1997), 1-36
8. ^ http://en.scientificcommons.org/1857371
9. ^
10. ^ "bibliography for mathematical biophysics and mathematical medicine". PlanetPhysics. 2009-01-24. Retrieved 2010-03-17.
11. ^ Modern Cellular Automata by Kendall Preston and M. J. B. Duff http://books.google.co.uk/books?id=l0_0q_e-u_UC&dq=cellular+automata+and+tessalation&pg=PP1&ots=ciXYCF3AYm&source=citation&sig=CtaUDhisM7MalS7rZfXvp689y-8&hl=en&sa=X&oi=book_result&resnum=12&ct=result
12. ^ "Dual Tessellation - from Wolfram MathWorld". Mathworld.wolfram.com. 2010-03-03. Retrieved 2010-03-17.
13. ^ 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/
14. ^ "Computer models and automata theory in biology and medicine | KLI Theory Lab". Theorylab.org. 2009-05-26. Retrieved 2010-03-17.
16. ^ "bibliography for mathematical biophysics". PlanetPhysics. Retrieved 2010-03-17.
17. ^ Ray Ogden (2004-07-02). "rwo_research_details". Maths.gla.ac.uk. Retrieved 2010-03-17.
18. ^ Oprisan, Sorinel A.; Oprisan, Ana (2006). "A Computational Model of Oncogenesis using the Systemic Approach". Axiomathes 16: 155. doi:10.1007/s10516-005-4943-x.
19. ^ "MCRTN - About tumour modelling project". Calvino.polito.it. Retrieved 2010-03-17.
20. ^ "Jonathan Sherratt's Research Interests". Ma.hw.ac.uk. Retrieved 2010-03-17.
21. ^ "Jonathan Sherratt's Research: Scar Formation". Ma.hw.ac.uk. Retrieved 2010-03-17.
22. ^ http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf
23. ^ [3]
24. ^ Hassan Ugail. "Department of Mathematics - Prof N A Hill's Research Page". Maths.gla.ac.uk. Retrieved 2010-03-17.
25. ^ "Integrative Biology - Heart Modelling". Integrativebiology.ox.ac.uk. Retrieved 2010-03-17.
26. ^ a b "molecular set category". PlanetPhysics. Retrieved 2010-03-17.
27. ^ Representation of Uni-molecular and Multimolecular Biochemical Reactions in terms of Molecular Set Transformations http://planetmath.org/?op=getobj&from=objects&id=10770
28. ^ "Travelling waves in a wound". Maths.ox.ac.uk. Retrieved 2010-03-17.
29. ^ [4]
30. ^ "The mechanochemical theory of morphogenesis". Maths.ox.ac.uk. Retrieved 2010-03-17.
31. ^ "Biological pattern formation". Maths.ox.ac.uk. Retrieved 2010-03-17.
33. ^ Charles Semple (2003), Phylogenetics, Oxford University Press, ISBN 9780198509424
34. ^ "The JJ Tyson Lab". Virginia Tech. Retrieved 2008-09-10.
35. ^
36. ^ "abstract relational biology (ARB)". PlanetPhysics. Retrieved 2010-03-17.
37. ^ "Molecular Evolution and Protobiology | KLI Theory Lab". Theorylab.org. 2009-05-26. Retrieved 2010-03-17.
38. ^ Baianu, I. C.; Brown, R.; Glazebrook, J. F. (2007). "Categorical Ontology of Complex Spacetime Structures: the Emergence of Life and Human Consciousness". Axiomathes 17: 223. doi:10.1007/s10516-007-9011-2.
39. ^ a b Brown, R.; Glazebrook, J. F.; Baianu, I. C. (2007). "A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems". Axiomathes 17: 409. doi:10.1007/s10516-007-9010-3.
40. ^ a b c Băianu, I. (1970). "Organismic supercategores: II. On multistable systems". The Bulletin of Mathematical Biophysics 32: 539. doi:10.1007/BF02476770.
41. ^ Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0471735507 http://www.worldcat.org/oclc/101642
42. ^ Unreliable medical source?]
43. ^ Unreliable medical source?]
44. ^ Băianu I (December 1970). "Organismic supercategories. II. On multistable systems". The Bulletin of Mathematical Biophysics 32 (4): 539–61. doi:10.1007/BF02476770. PMID 4327361.
45. ^ "category of \$(M,R)\$ -systems". PlanetPhysics. Retrieved 2010-03-17.
46. ^ Organisms as Super-complex Systems http://planetmath.org/?op=getobj&from=objects&id=10890
47. ^ Unreliable medical source?]
48. ^ http://planetmath.org/?op=getobj&from=objects&id=10921
49. ^ Unreliable medical source?] "PlanetMath". PlanetMath. Retrieved 2010-03-17.
50. ^ "The KLI Theory Lab - authors - R". Kli.ac.at. Retrieved 2010-03-17.
51. ^ "KLI Theory Lab". Kli.ac.at. Retrieved 2010-03-17.

## References

• Nicolas Rashevsky. (1938)., Mathematical Biophysics. Chicago: University of Chicago Press.
• Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0471735507
• Israel, G., 2005, "Book on mathematical biology" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 936-44.
• Israel G (1988). "On the contribution of Volterra and Lotka to the development of modern biomathematics". History and Philosophy of the Life Sciences 10 (1): 37–49. PMID 3045853.
• Scudo FM (March 1971). "Vito Volterra and theoretical ecology". Theoretical Population Biology 2 (1): 1–23. doi:10.1016/0040-5809(71)90002-5. PMID 4950157.
• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus, 2001, ISBN 0-7382-0453-6
• N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.11 in M. Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York, (updated by Hsiao Chen Lin in 2004 ISBN 0080363776
• P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4
• L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6
• G. Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2
• A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6
• L.G. Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4
• F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0
• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3
• J.D. Murray, Mathematical Biology. Springer-Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 2003 ISBN 0-387-95228-4.
• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7
• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8
• L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X
• L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8.
Theoretical biology
• Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University Press.
• Hertel, H. 1963. Structure, Form, Movement. New York: Reinhold Publishing Corp.
• Mangel, M. 1990. Special Issue, Classics of Theoretical Biology (part 1). Bull. Math. Biol. 52(1/2): 1-318.
• Mangel, M. 2006. The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary Biology. Cambridge University Press.
• Prusinkiewicz, P. & Lindenmeyer, A. 1990. The Algorithmic Beauty of Plants. Berlin: Springer-Verlag.
• Reinke, J. 1901. Einleitung in die theoretische Biologie. Berlin: Verlag von Gebrüder Paetel.
• Thompson, D.W. 1942. On Growth and Form. 2nd ed. Cambridge: Cambridge University Press: 2. vols.
• Uexküll, J.v. 1920. Theoretische Biologie. Berlin: Gebr. Paetel.
• Vogel, S. 1988. Life's Devices: The Physical World of Animals and Plants. Princeton: Princeton University Press.
• Waddington, C.H. 1968-1972. Towards a Theoretical Biology. 4 vols. Edinburg: Edinburg University Press.

Lists of references

### Related societies

Mathematical biology is also called theoretical biology,[1] and sometimes biomathematics. It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing. It is an interdisciplinary academic research field with a wide range of applications in biology, medicine[2] and biotechnology.[3]

Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. In order to do this, precise mathematical models are required. By describing the systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.

## Importance

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include:

• the explosion 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, nonlinear mechanisms in biology,
• an increase in computing power which enables calculations and simulations to be performed that were not previously possible, and
• an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research.
For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.

## Areas of research

Several areas of specialized research in mathematical and theoretical biology[4][5][6][7][8][9] 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. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers, engineers, biologists, physiologists, research physicians, biomedical researchers,oncologists, molecular biologists, geneticists, embryologists, zoologists, chemists, etc.

### Computer models and automata theory

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987,[10] 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, cancer modelling, neural nets, genetic networks, abstract relational biology, metabolic-replication systems, category theory[11] applications in biology and medicine,[12] automata theory,cellular automata, tessallation models[13][14] and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories.[15][16] This published report also includes 390 references to peer-reviewed articles by a large number of authors.[17][18][19]

Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology.[20]

• Mechanics of biological tissues[21]
• Theoretical enzymology and enzyme kinetics
• Cancer modelling and simulation [22][23]
• Modelling the movement of interacting cell populations[24]
• Mathematical modelling of scar tissue formation[25]
• Mathematical modelling of intracellular dynamics[26]
• Mathematical modelling of the cell cycle[27]

Modelling physiological systems

### Molecular set theory

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in Mathematical Medicine.[30] 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. 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.[31][32]

### Population dynamics

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The Lotka–Volterra predator-prey equations are a famous example. In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed, and provide important results that may be applied to health policy decisions.

### 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.

### 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 will always generate the same trajectory and no two trajectories cross in state space.

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 – generalized master equation – continuous time with memory of past events, discrete state space, waiting times of events (or transitions between states) discretely occur and have a generalized probability distribution.
• Jump Markov processmaster 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 continuous-time Monte Carlo which is also called kinetic Monte Carlo or the stochastic simulation algorithm.
• Continuous Markov processstochastic differential equations or a Fokker-Planck equation – 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[33]
• Swarming behaviour[34]
• A mechanochemical theory of morphogenesis[35]
• Biological pattern formation[36]
• Spatial distribution modeling using plot samples[37]

### Phylogenetics

Phylogenetics is an area of mathematical biology that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics.Template:Fact The main mathematical concepts are trees, X-trees and maximum parsimony trees.Template:Fact

## Model example: the cell cycle

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 [38][39] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which 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 Michealis 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.
In order to fit the parameters the differential equations need to 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 proprieties of the equations are used to investigate the behavior of the system depending of 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 can handle the large number of variables and parameters is called a bifurcation diagram(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.
```

## Mathematical, theoretical and computational biophysicists

For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.
 [[Image:|32x28px]] Mathematics portal
• Biographies

## References

• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus, 2001, ISBN 0-7382-0453-6
• N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.11 in M. Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York, (updated by Hsiao Chen Lin in 2004[60],[61],[62] ISBN 0080363776 [63].
• P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4
• L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6
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### Lists of references

• A general list of Theoretical biology/Mathematical biology references, including an updated list of actively contributing authors[64].
• A list of references for applications of category theory in relational biology[65].
• An updated list of publications of theoretical biologist Robert Rosen[66]

## Notes: inline and online

1. Mathematical Biology and Theoretical Biophysics-An Outline: What is Life? http://planetmath.org/?op=getobj&from=objects&id=10921
2. http://www.kli.ac.at/theorylab/EditedVol/W/WittenM1987a.html
3. http://en.scientificcommons.org/1857372
4. http://www.kli.ac.at/theorylab/index.html
6. http://en.scientificcommons.org/1857371
7. http://cogprints.org/3687/
8. "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved on 2008-09-10.
9. http://acube.org/volume_23/v23-1p11-36.pdf J. R. Junck. Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula, Bioscene, (1997), 1-36
10. http://en.scientificcommons.org/1857371
11. http://planetphysics.org/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html
12. http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysicsAndMathematicalMedicine.html
13. Modern Cellular Automata by Kendall Preston and M. J. B. Duff http://books.google.co.uk/books?id=l0_0q_e-u_UC&dq=cellular+automata+and+tessalation&pg=PP1&ots=ciXYCF3AYm&source=citation&sig=CtaUDhisM7MalS7rZfXvp689y-8&hl=en&sa=X&oi=book_result&resnum=12&ct=result
14. http://mathworld.wolfram.com/DualTessellation.html
15. 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/
16. http://www.kli.ac.at/theorylab/EditedVol/W/WittenM1987a.html
19. http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysics.html
20. "Research in Mathematical Biology". Maths.gla.ac.uk. Retrieved on 2008-09-10.
21. http://www.maths.gla.ac.uk/~rwo/research_areas.htm
23. http://calvino.polito.it/~mcrtn/
24. http://www.ma.hw.ac.uk/~jas/researchinterests/index.html
25. http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html
26. http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf
27. http://mpf.biol.vt.edu/Research.html
28. http://www.maths.gla.ac.uk/~nah/research_interests.html
29. http://www.integrativebiology.ox.ac.uk/heartmodel.html
30. http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html
31. Representation of Uni-molecular and Multimolecular Biochemical Reactions in terms of Molecular Set Transformations http://planetmath.org/?op=getobj&from=objects&id=10770
32. http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html
33. http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm
34. http://www.math.ubc.ca/people/faculty/keshet/research.html
35. http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm
36. http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm
38. "The JJ Tyson Lab". Virginia Tech. Retrieved on 2008-09-10.
39. "The Molecular Network Dynamics Research Group". Budapest University of Technology and Economics.
40. http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html
41. http://www.kli.ac.at/theorylab/EditedVol/M/MatsunoKDose_84.html
42. 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://www.springerlink.com/content/w2733h7280521632/
46. Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0471735507 http://www.worldcat.org/oclc/101642
48. http://cogprints.org/3674/
49. http://cogprints.org/3829/
50. http://www.ncbi.nlm.nih.gov/pubmed/4327361
52. http://www.kli.ac.at/theorylab/ALists/Authors_R.html
53. Organisms as Super-complex Systems http://planetmath.org/?op=getobj&from=objects&id=10890
55. http://planetmath.org/encyclopedia/SupercategoriesOfComplexSystems.html
56. http://planetmath.org/?method=l2h&from=objects&name=NaturalTransformationsOfOrganismicStructures&op=getobj
57. http://www.kli.ac.at/theorylab/ALists/Authors_R.html
58. http://www.kli.ac.at/theorylab/index.html
59. http://www.worldcat.org/oclc/101642
60. http://cogprints.org/3718/1/COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf
63. http://www.bookfinder.com/dir/i/Mathematical_Models_in_Medicine/0080363776/
64. http://www.kli.ac.at/theorylab/index.html
65. http://planetmath.org/?method=l2h&from=objects&id=10746&op=getobj
66. Publications list for Robert Rosen http://www.people.vcu.edu/~mikuleck/rosen.htm