Getting into the swing of things

Rhythms are an ineluctable feature of living organisms, from single-celled plants such as Acetabularia to mammals such as ourselves, and man has been fascinated by them throughout history.

All of us are aware of the circadian (i.e. roughly daily) rhythms of eating, sleeping and other activities in the world about us that appear to be coupled to the daily light/dark cycle, but many may not realise that we are more than mere slaves to the solar pacemaker. Indeed, 'tis true that biological systems "got rhythm" within them, as was shown for plants by the French geologist de Mairan in 1729, and since then found at almost every level of biological organisation. However, it has been a long trek from the early taxonomic description of rhythms, gained by perturbing an organism in various ways, to an understanding of endogenous oscillators at the molecular and cellular level. Not until the 1950s did the research paradigm shift from that of viewing the clock as a black box, whose only accessible output was the movement of its hands, to one of intrinsic biochemical oscillators whose mechanism we could hope to understand. Indeed, in the 1950s some theoretical chemists refused to believe that the kind of sustained oscillations needed to serve as a clock were even possible in reacting systems. In recent years significant progress has been made toward understanding the molecular components of the circadian clock in several systems, especially the fruitfly, Drosophila, but much remains to be done. In particular, little is known about how the output of the intrinsic clocks interacts with the neural networks that control behaviour. Unravelling this part will bring us full circle, for interest in circadian rhythms started with the behaviours, but when this stage is completed we will understand what makes the hands of the clock turn.

Experimental observations of sustained oscillations in chemically reacting systems date back at least to the work of W. C. Bray and others in the 1920s, but the real growth of interest in the subject has occurred in the past 40 years. In 1955 A. T. Wilson and M. Calvin demonstrated oscillatory transients in the photosynthetic cycle, and sustained oscillations in components in the glycolytic pathway (the biochemical reactions involved in the splitting of glucose) were first reported by L. N. M. Duysens and J. Amesz in 1959. At about the same time, the dramatic phenomena that occur in the inorganic system now known as the Belousov-Zhabotinskii reaction sparked the interest of experimentalists and theoreticians in chemistry and engineering.

There are simple models of two-component chemical systems that can oscillate, just as there are simple models of electronic oscillators such as the one due to Van der Pol. But in trying to understand oscillations in real biological systems one is confronted with the fact, already apparent in the context of biochemical networks, and even more important when spatial aspects come into play, that there are generally complex networks behind the observed biochemical oscillations. Fortunately, the theory of dynamical systems has developed to the point where the basic notions of bifurcations, oscillations and chaos are now taught at some level to many scientists. This provides many tools for analysing and understanding complex systems, though much remains to be done here before very large biochemical or neural networks are comprehensible.

In the absence of spatial variations in components, reacting systems are described by ordinary differential equations that encode how the amounts of each component changes due to chemical reactions. In order to write down such equations, one has to know or hypothesise a mechanism for each reaction and the rate constants associated with each elementary step. Once this is done, a complete qualitative description of the dynamics can be obtained from general principles in some cases. For example, some systems evolve toward a unique rest point at which there are no macroscopic changes in any of the components. Such systems clearly cannot oscillate, and facts such as this led many to disbelieve the existence of sustained oscillations when they were first reported. But biological systems are open, in that they are sustained by a continual flow of energy and mass through them, and these systems can show a far wider range of dynamical behaviour. Indeed, reacting systems can exhibit essentially all the known types of dynamical behaviour, from global convergence to a unique steady state to various kinds of chaotic dynamics. Thus the study of such systems is still more or less case by case because few general principles that limit the dynamics are known, and in their absence detailed studies of particular systems is the best we can do.

Albert Goldbeter, who has spent much of his career constructing and analysing models of complex biological systems, has now written a book that summarises some of his work. The book is divided into seven parts that begin at the molecular level and works up to the circadian oscillators that time us. His stated intention in writing the book is "to address the molecular bases of periodic and chaotic behaviour by considering theoretical models for some of the best-known examples of cellular oscillations of a biochemical nature''.

In the first couple of chapters the author systematically progresses from a description of the experimental observations to formulation and analysis of a model, and closes with a discussion of the possible function of oscillations in glycolysis. The next two chapters deal with more complicated dynamical behaviour, including the coexistence of distinct periodic solutions, bursting and chaotic dynamics. The transitions between these types of behaviour as parameters are changed are also investigated here. Chapters five to seven are devoted to models of oscillations of cyclic AMP (a universal messenger molecule in biological systems) in the cellular slime mould Dictyostelium discoideum, and part four of the book draws a parallel between the pulsatile release of cyclic AMP in the slime mould with the pulsatile release of hormones. Goldbeter introduces models for calcium dynamics in part five. Chapter ten deals with the mitotic oscillator and chapter 11 treats a model for circadian rhythms. The final chapter gives an overview of the common themes that emerge from the different systems.

The scope of the book is broad, and the treatment of individual topics is necessarily limited, even in a book of this length. What does emerge from the topics covered is a strategy for constructing and analysing mathematical models of complex biological problems and connecting the predictions made by the models to the experimental observations. From this point of view the book is a good model of how mathematics can be used to obtain important information about the behaviour of biological systems.

One major criticism is that the author has largely summarised his previously published work: there is little that is genuinely new, and there is no attempt to present a synthesis of his work and that of others in this area. This is disappointing, because Goldbeter is certainly qualified to do this and, had it been done, the book could have had an enormous impact. As it stands, the reader is left with the impression that the models presented are the best available, but his is not the case. Thus the book stands as a useful compendium of the author's work over the past 25 years, and it will provide an entree into the field for beginners, but a comprehensive treatise on the fascinating and important topic of biochemical oscillations and their role in various biological phenomena is still to be written.

Hans G. Othmer is in the department of mathematics, University of Utah.