Mathematical Biology

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 Simulation s 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

, 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 Neuron s and Carcinogen esis {Link without Title}

Mechanics of biological tissues {Link without Title}

Theoretical enzymology and Enzyme Kinetics {Link without Title}

Cancer modelling and simulation {Link without Title}

Modelling the movement of interacting cell populations {Link without Title}

Mathematical modelling of scar tissue formation {Link without Title}

Mathematical modelling of intracellular dynamics {Link without Title}

Modelling physiological systems

Modelling of Arterial disease {Link without Title}

Multi-scale modelling of the Heart {Link without Title}

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 System s)

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.

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 Process es (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 Time s 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 Equation s 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 {Link without Title}

Swarming behaviour {Link without Title}

The mechanochemical theory of Morphogenesis {Link without Title}

Biological pattern formation {Link without Title}

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.

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Mathematical Biology