Mathematical biology or biomathematics is an interdisciplinary field of academic study which aims at modelling 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.

## Research

Below is a list of some areas of research in mathematical biology and links to related projects in various universities:

### 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.

### Modelling cell and molecular biology

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

• Modelling of neurons and carcinogenesis [1]
• Mechanics of biological tissues [2]
• Theoretical enzymology and enzyme kinetics [3]
• Cancer modelling and simulation [4]
• Modelling the movement of interacting cell populations [5]
• Mathematical modelling of scar tissue formation [6]
• Mathematical modelling of intracellular dynamics [7]

Check out the Gillespie algorithm J. Comput. Phys. 22, 403-434. (or a tutorial here: [8]) to simulate low-number chemical systems (like 100 copies of an mRNA, protein, or ribosome). This algorithm exactly simulates samples from the solution of the chemical master equation. (see also van Kampen's classic Stochastic Processes in Physics and Chemistry)

### 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.

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.

## Bibliographical references

• J.D. Murray, Mathematical Biology. Springer-Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An Introduction, 2002 ISBN 0387952233; Mathematical Biology: II. Spatial Models and Biomedical Applications, 2003 ISBN 0387952284.
• L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0075549506
• L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 052127477X
• F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0898710170
• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0471744468
• A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0521599466
• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0521448557
• P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0521406684
• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0198565623