Topological approaches to biological dynamics

The growth of computer power and biological data bases is revolutionizing the application of mathematics to biology. Mathematical models of biological systems are being developed at all levels of organization from the folding of protein molecules, to the development of cells and tissues, to the beating of a heart, to the evolution of ecosystems. In some cases, the mathematical model simulates the biological behavior at an extraordinary level of detail and accuracy. Yet, such models are invariably complicated and though the behavior might be similar to the biological system, the underlying principles behind the dynamics may be opaque. Of course, for some purposes it would make no difference whether or not we understand the observed behavior in the model. Thus, a sufficiently accurate model of the heart might enable us to test the effects of drugs in silico, rather than in vivo. Such a model would be useful in screening for drugs that could lead to fatal arrhythmias. There are many groups that are working towards realistic models and the potential benefit of such models will be huge. In this series of lectures, I will focus on mathematical approaches that are useful for thinking about the underlying principles for the organization of biological dynamics. The approach is topological and qualtitative, but in some cases, can make contact with and in fact require high precision data, as in the consideration of the experimental resetting of biological oscillators. The topological approach was championed in the 1970s and 1980s by giants like Art Winfree and René Thom, whose influence and contributions are gratefully acknowledged at the outset. Winfree’s masterpiece, The Geometry of Biological Time defines the approach and covers a huge range of topics [37]. I do not try to give a comprehensive review of all uses of topology in biology, but mostly focus on a series of results found by myself and colleagues related to fixed points and continuity in biological dynamics. The results I mention are all comparatively simple, and although some are well known, most are not. I will not attempt to be rigorous, but will try to present the material in a fashion that I hope can be understood and applied without advanced mathematics. I will not give elaborate proofs or definitions, and will assume the reader has some basic knowledge of mathematics. Whenever possible, I will give experimental or clinical biological examples to illustrate the ideas. All of the material is based on previous publications which I cite and which should be consulted for full details and extensive references to earlier work. This material will be in lecture note style. It is hard to get notation consistent from one section to the next, so try not to be bothered if the same symbol has different meanings in different sections. Some of the material presented here is repeated from [2] and some laboratory exercises from that book will be given. The downloadable matlab code for the exercises, which I hope will work on the PCMI computers is at http://www.cnd. mcgill.ca/ebook/. Since the background of different people will likely be quite different, a reference for basic concepts which contains some of the examples I will give is [28]. This book also has lots of pencil and paper exercises in 1 dimensional maps and 1 and 2 dimensional differential equations for those who need practice in doing those types of problems.