UK Nonlinear News , February 1997.

Biology: a Fertile Field for Nonlinear Mathematics

John Brindley

The biological sciences, interpreted in the widest possible sense to include much medical science, and comprising all scales, from molecular to whole populations, abound with inherently nonlinear phenomena. Virtually all living organisms depend on repetitive cyclic behaviour at a number of levels, from basic biochemistry to organised and sophisticated structures like hearts and breathing mechanisms. Thus the existence and stability of finite amplitude oscillations is often literally a matter of life and death to an organism. Moreover, since oscillations usually take place in a distributed medium, they are often associated with travelling waves and the geometrical structures, such as spirals and scrolls, to which such waves give rise in particular geometries.

Other manifestations of nonlinearity occur (often as the final visible state of complex evolution processes) in, for example, the structure of protein and DNA molecules, or as the results of normal morphogenesis or abnormal and unwelcome tumour growth through the process of angiogenesis and vascularisation. Hysteresis is commonly seen in response to changing conditions, and there are good grounds for believing at least some forms of heart and brain behaviour to be chaotic.

The collective behaviour of populations of individuals, linked through a great variety of interactive and feedback processes, is universally nonlinear, often exhibiting rich internal dynamics as well as great sensitivity and sometimes counter-intuitive responses to external influences.

This, then, is a very small sample of the phenomena which display the characteristics of nonlinearity. Their incidence is controlled or influenced by complex internal biochemistry and biophysics, and by every sort of interaction with their environment. Typically the key influences and controls are poorly, if at all, understood, and conflicting theories or speculations exist.

On the other hand, many of the fundamental biochemical and biophysical processes are now well known, respectable experimental results can be obtained, and convincing numerical simulations are sometimes within the power of available computations. The fertile soil therefore exists in which mathematical modelling and exploitation of the dynamical systems methodology can flourish. Indeed, progress and understanding urgently need such adventurous modelling, performed honestly with a Popperian ruthlessness shown towards inadequate models. Collaborations, based on mutual respect and appreciation, between mathematician, numericist and biological experimentalists/observer is almost always needed, since rare indeed is the individual who combines appropriate strengths in even deductive argument and the maximum of rigour to the whole process of development and analysis of the mathematical model. This usually means starting from a model whose apparently crude simplicity may unnerve the biologist, brought up to observe and respect detail and complexity. It does however permit that rational building, step by step, from a base of firm understanding which is the hallmark of good applied mathematics; the surprise and bonus is that what is formally correct in some distant asymptotic limit is so often useful in understanding real phenomena.

So, nonlinear dynamicists, go forth and inhabit the biological world - it has great rewards to offer and you have much to give to it.


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