CONTENTS 1. Introduction 2. Oscillatory enzymes: simple periodic behaviour in an allosteric model for glycolytic oscillations 3. Birhythmicity: coexistence between two stable rhythms 4. From simple periodic behaviour to complex oscillations, including bursting and chaos 5. Models for the periodic synthesis and relay of cAMP signals in Dictyostelium discoideum amoebae 6. Complex oscillations and chaos in the cAMP signalling system of Dictyostelium 7. The onset of cAMP oscillations in Dictyostelium as a model for the ontogenesis of biological rhythms 8. Function of the rhythm of intercellular communication in Dictyostelium: Link with pulsatile hormone secretion 9. Oscillations and waves of intracellular calcium 10. Modelling the mitotic oscillator driving the cell division cycle 11. Towards a model for circadian oscillations in the Drosophila period protein (PER) 12. Conclusions and perspectives

This book addresses the molecular bases of some of the most important biochemical rhythms known at the cellular level. Clarifying the mechanism of these oscillatory phenomena is of key importance for understanding the origin as well as the physiological function of these rhythms, and the conditions in which simple periodic behaviour transforms into complex oscillations including bursting and chaos.

The approach rests on the analysis of theoretical models closely related to experimental observations. Among the main rhythms considered are glycolytic oscillations observed in yeast and muscle, oscillations of cyclic AMP in Dictyostelium amoebae, intracellular calcium oscillations observed in a variety of cell types, and the mitotic oscillator that drives the cell division cycle in eukaryotes. For each of these phenomena, experimental facts are reviewed and mathematical models presented. These models throw light on the mechanism of periodic behaviour at the molecular and cellular levels and show how enzyme regulation or receptor desensitization can give rise to oscillations. The theoretical models are also used to analyse the transition between simple periodic oscillations and more complex oscillatory phenomena such as birhythmicity (i.e. the coexistence between two stable rhythms), bursting oscillations and chaos. The model for cyclic AMP oscillations in Dictyostelium is further used to discuss the function of pulsatile signalling in intercellular communication. The frequency encoding of cyclic AMP signals is related to the fact that many hormones are secreted in pulses whose frequency governs their physiological effect.

This book which contains more than 1200 references, provides a wide survey of work on biochemical oscillations and cellular rhythms. The author has made numerous contributions to the subject since the early developments in the field.

The book will be of interest to life scientists such as biochemists, cell biologists, chronobiologists, medical scientists and pharmacologists. In addition, it will appeal to scientists studying nonlinear phenomena, including oscillations and chaos, in chemistry, physics, mathematics and theoretical biology.

'I [have begun] to appreciate the value of good modelling, especially now that we have learnt so much more about the basic biochemical details regulating cellular activity. For each system now being investigated, there are so many variables that it becomes impossible to use our intuition to assess how each parameter influences the oscillatory cycle. The only way to understand these biological rhythms, therefore, is to become more quantitative and to develop rigorous mathematical models. Albert Goldbeter is at the very forefront of this new approach.'

`Albert Goldbeter ... is a reliable guide through the intricacies of the biochemistry and mathematics ... The great strength of the book is his unshrinking dedication to understanding these processes from start to finish ... The book is conversational, clear and accurate ... The volume is loaded with experimental data, hypothetical mechanisms, differential equations, numerical simulations and careful comparison between theory and observations.' J. J. Tyson, Nature

`It is this coincidence of interpreting molecular details and unfolding common principles which makes the book by Albert Goldbeter a source of sustained intellectual joy.' A. Deutsch, Biomathematics Newsletter

`The book beautifully illustrates the art of modelling ... The author bas been studying the theory of oscillations and rhythms for 25 years and brings a great deal of experience to bear on this book ... It is an excellent read and will appeal to a wide-ranging audience, from students moving into the area to established researchers in the field, and from mathematical modellers to experimentalists in the life sciences.' P. Maini, Trends in Biochemical Sciences

`... will undoubtedly become a classic in the field of mathematical modelling of cellular processes. With all of its details, it is not a book that is easily read. On the other hand, I will enjoy having the book on my bookshelf for consultation, and I am confident that it will serve me well for many years.' L. F. Olsen, Bulletin of Mathematical Biology

`The focus of the book is as illuminating as its content. Here the focus is on defining and understanding the central components of oscillators and how they interact to produce oscillations. ... Goldbeter is to be congratulated for persevering in pulling together an enormous and disparate body of literature in a coherent and readable manner. ... I predict that this book will remain useful for a long time.' J. Dunlap, Endeavour

`A very impressive tome and a superb addition to the literature. It is quite clearly going to be the reference book on oscillations.' J. D. Murray (University of Washington, Seattle)

`His book will be essential reading for any investigator who wants a systematic introduction to this crucial area of biological research.' P. Rapp (Physiology, Medical College of Pennsylvania)