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# Mathematical biology

Mathematical biology or biomathematics is an interdisciplinary field of academic study which aims at modeling natural, biological processes using mathematical techniques and tools. It has both practical and theoretical applications in biological research.

## 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 the complications involved in human and animal research.

## Areas of research

Below is a list of some areas of research in mathematical biology and links to related projects in various universities. These examples are characterised by complex, nonlinear mechanisms and it is being increasingly recognised that the result of such interactions may only be understood through mathematical and computational models. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, physicists, biologists, physicians, zoologists, chemists etc.

### 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 analysed, and provide important results that may be applied to health policy decisions.

### Modelling cell and molecular biology

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

• Modelling of neurons and carcinogenesis 
• 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 

### Modelling physiological systems

• Modelling of arterial disease 
• Multi-scale modelling of the heart 

## 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.

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.

• 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)

### 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 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.)

### 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 
• Swarming behaviour 
• The mechanochemical theory of morphogenesis 
• Biological pattern formation 
• Spatial distribution modeling using plot samples 

## Bibliographical 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
• 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
• 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

## External references

• 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.