Symmetry and Bifurcation in Biology

Interest in the application of mathematical methods to biology has been growing rapidly, worldwide, in recent years. The great advances in molecular biology have opened up major new areas, but it is becoming recognised that DNA sequences and other molecular information are only one aspect of the understanding of biological systems. Organisms make use of DNA in highly dynamic contexts. For example there is no doubt that DNA has a significant role in organising the development of an organism, but that very little information about the form of the organism and its developmental path can be 'read off' from its DNA sequence. Genetics acts in concert with dynamical physical and chemical processes. The more we learn about genes, the more evident becomes the need for a good understanding of dynamic effects in biology--- in growth, in development, in the regulation of genetic networks, in ecosystems, and in evolution.

The mathematics of dynamical systems has undergone its own revolution, as the need to consider nonlinear effects has become clear. The theory of dynamical systems is one of the major growth areas of today's mathematical research, and one of its strengths is a strong conenction with applied science.

There are two distinct ways to encourage interaction between mathematics and biology. 'Horizontal' programmes select specific problems in biology (such as protein-folding) and bring many different mathematical methods to bear. What we propose is the other kind of meeting: a 'vertical' program organized around a package of general methods that apply to many different biological problems. In this case, the package is the exploitation of symmetries in nonlinear dynamial systems, and the strong relation between symmetry and pattern formation.

Over the past 15 years, we and other authors have been exploring a far more active role for symmetry, in the context of nonlinear dynamical systems. It has become apparent that the symmetries of a system of nonlinear ordinary or partial differential equations can be used, in a systematic and unified way, to analyze, predict, and understand many general mechanisms of pattern-formation.

It is important to understand that 'pattern' here is not restricted to visual patterns such as shape or pigmentation. The structure and function of the visual cortex involves patterns, and can be modelled by a symmetric network of neurons. The formation of new species is a pattern: one group of organisms, a highly symmetric situation, splits into two groups--- a less symmetric one. Synchronous firing of neurons, which seems to be an important feature of brain function, is a pattern. Phase relations in biological oscillators are patterns.

In physics, patterns can often be understood by writing down very specific and accurate mathematical models --- equations. Few areas of biology are yet equipped with equations of comparable accuracy. It is here the the symmetry approach has major advantages: it is a general method that applies to a variety of models. It can lead to general conclusions even when specific models are unknown, or controversial, or of limited accuracy.

The point here is not the literal symmetry of a biological system, or an organism, or a process. Hardly anything in biology is exactly symmetric. But a huge range of biological systems possess approximate symmetries (for example all organisms in a species are approximately identical), and the best way to model such systems is to exploit the symmetry of an idealized model, and then consider what changes might occur to the conclusions if the symmetry is close, but not exact.

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