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- For use of basic artimethics in Biology, see relevant topic, such as Serial dilution.

Mathematical biology, biological mathematical modeling, biomathematics or computational biomodeling is an interdisciplinary field of academic study which aims at modeling natural, biological processes using applied mathematical techniques and tools. It has both practical and theoretical applications in biological research: In cell biology, protein interactions are typically expressed as “cartoon” models, which, although easy to visualize, do not fully describe the systems studied: to do this, mathematical models are required, which, by describing the systems in a quantitative manner, can better simulate their behavior and hence predict unseen properties.

- 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.

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

- Mechanics of biological tissues
- Theoretical enzymology and enzyme kinetics
- Cancer modelling and simulation
- Modelling the movement of interacting cell populations
- Mathematical modelling of scar tissue formation
- Mathematical modelling of intracellular dynamics
- Mathematical modelling of the cell cycle

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 following is a list of mathematical descriptions and their assumptions.

- Difference equations 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.
- Maps (Discrete time. Continuous state space)

- 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 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 Continuous-time Monte Carlo which is also called kinetic Monte Carlo or the stochastic simulation 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.)

- Travelling waves in a wound-healing assay
- Swarming behaviour
- The mechanochemical theory of morphogenesis
- Biological pattern formation
- Spatial distribution modeling using plot samples

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.

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- N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
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- 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
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- L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8

- F. Hoppensteadt, Getting Started in Mathematical Biology. Notices of American Mathematical Society, Sept. 1995.
- M. C. Reed, Why Is Mathematical Biology So Hard? Notices of American Mathematical Society, March, 2004.
- R. M. May, Uses and Abuses of Mathematics in Biology. Science, February 6, 2004.
- J. D. Murray, How the leopard gets its spots? Scientific American, 258(3): 80-87, 1988.
- S. Schnell, R. Grima, P. K. Maini, Multiscale Modeling in Biology, American Scientist, Vol 95, pages 134-142, March-April 2007.
- Chen KC et al. Integrative analysis of cell cycle control in budding yeast. Mol Biol Cell. 2004 Aug;15(8):3841-62.
- Csikász-Nagy A et al. Analysis of a generic model of eukaryotic cell-cycle regulation. Biophys J. 2006 Jun 15;90(12):4361-79.
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- UCLA Biocybernetics Laboratory
- TUCS Computational Biomodelling Laboratory
- Nagoya University Division of Biomodeling
- Technische Universiteit Biomodeling and Informatics
- BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology
- The Collection of Biostatistics Research Archive
- Statistical Applications in Genetics and Molecular Biology
- The International Journal of Biostatistics
- Society for Mathematical Biology
- European Society for Mathematical and Theoretical Biology
- Biomathematics Research Centre at University of Canterbury
- Centre for Mathematical Biology at Oxford University
- Mathematical Biology at the National Institute for Medical Research
- Institute for Medical BioMathematics
- Mathematical Biology Systems of Differential Equations from EqWorld: The World of Mathematical Equations
- Systems Biology Workbench - a set of tools for modelling biochemical networks

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Last updated on Tuesday October 07, 2008 at 10:26:31 PDT (GMT -0700)

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