Introduction

Voltage-gated ion channels are crucial ‘building blocks’ in various systems of signal transduction and processing in living organisms. They are ultimately responsible for information flow at several hierarchical levels of biological complexity that include signal sensing (Lu and Fishman, 1994) and generation of nerve action potentials (Hille, 1992), and are crucially important in synaptic transmission and other intercellular communications (Alberts et al., 1994). Preceding biologically inspired work on the role of external noise in electrical signal transduction concentrated on rather complex objects such as neurons (Bulsara et al., 1994; Pei et al., 1996; Chapeau-Blondeau et al., 1996; Longtin, 1997; Plesser and Tanaka, 1997) and neuronal ensembles (Gluckman et al., 1996; Chialvo et al., 1997). It was demonstrated that addition of random fluctuations, or noise to the input of these systems could improve the transmission efficiency for small input signals. Yet even more elaborate physiological systems showing similar properties include isolated sciatic nerves of a toad (Morse and Evans, 1996; Moss et al., 1996), rat SA1 cutaneous mechanoreceptors (Collins et al., 1996a), mechanosensory transduction pathways in arthropods (Douglass et al., 1993; Levin and Miller, 1996), and human sensory perception (Cordo et al., 1996; Collins et al., 1996b; Chiou-Tan et al., 1996; Simonoto et al., 1997). The counterintuitive phenomenon of noise-improved signal transduction, called ‘stochastic resonance’ – first introduced as a possible explanation for the periodic recurrences of the Earth's ice ages (Benzi et al., 1981) – has now been established empirically for many macroscopic systems and, for some of them, is understood theoretically (Wiesenfeld and Moss, 1995; Gammaitoni et al., 1998; see also Moss, Chapter 10, this volume).

The consideration of random fluctuations is central to any discussion of the limits of biological stimulus–response interactions. The conventional view holds that stochastic perturbations diminish the efficiency of biological signal detection and transduction pathways. Part III provides an overview of the recently found constructive role of noisy fluctuations in biochemical and biological processes. Two major concepts have emerged for describing such noise-facilitated processes, stochastic resonance and fluctuation-driven transport. In Chapter 10, Frank Moss opens with an overview of the history and of the physical principles of stochastic resonance. He then extends this discussion into the realm of biology and medicine, and offers experiments that demonstrate stochastic resonance in the detection by organisms of dynamical signals, including weak electric fields. The chapter concludes with new evidence for spatiotemporal stochastic resonance in two-dimensional systems. Sergey Bezrukov and Igor Vodyanoy take the discussion of stochastic resonance to the molecular level in Chapter 11. Their experiments demonstrate the noise-improved transduction of electrical signals by voltage-gated ion channels in a planar lipid bilayer.

Electromagnetic stimuli represent a special class of external perturbations that are discussed in almost all of the remaining book chapters. Part II therefore provides important information in regard to the biophysical foundations of interactions between biological processes and electric or magnetic fields. In addition, experimental examples are described that demonstrate the nonlinear sensitivity to electromagnetic stimuli of enzymes, single cells and tissues. The principles of electric field interactions and the functional role of bioelectric fields are reviewed in Chapter 6 by Paul Gailey. He discusses the remarkable electrosensitivity of selected biological systems and how oscillating electric fields may be detected and amplified by biological structures. The chapter concludes with the description of a model based on the concept of long-range coherence, which may explain how relatively weak electric fields may effectively interact with excitable cellular assemblies in the presence of noise. An electric field-sensitive cellular oscillator in cells of the immune system is the subject of Chapter 7 by Howard Petty. He discusses experiments that have led to the discovery of coherent metabolic oscillations in human neutrophils, and describes the response of these cells to time-varying chemical and electric fields.

Introduction

Biological systems provide many examples of well-studied, self-organized nonlinear dynamical behavior including biochemical oscillations, cellular or tissue-level oscillations or even dynamical diseases (Goldbeter, 1996). The latter include such phenomena as cardiac arrhythmias, Parkinson's disease and epilepsy. Part of the reason for progress in understanding these phenomena has been the willingness of investigators to communicate and share insights across disciplinary boundaries, even when this communication is hampered by differing jargon or concepts unfamiliar to the nonexpert. The common language of nonlinear systems theory has helped to facilitate this cross-disciplinary conversation as well as to provide a new definition of what it means to say that two things are dynamically ‘similar’ or even ‘the same’.

In this chapter, we compare the dynamics of a well-studied biochemical oscillator, the peroxidase-oxidase reaction, with that of epilepsy, a dynamical disease (Milton and Black, 1995). We are so accustomed to the normal way of reasoning in science that it seems wrong, somehow, to point out the similarities between, on the one hand, the oscillations in substrate concentration during an enzyme-catalyzed reaction and, on the other, the regular oscillations in the electroencephalography (EEG) signal observed during certain types of epileptic seizure. While these two systems could not be more different in terms of their material nature, they are actually quite similar dynamically.

Introduction

A physicist who wants to get a rough estimate of the acoustics of a concert hall just claps his hands and listens to the echoes. With this simple experiment he can obtain valuable information from the acoustic answer of the hall (at least if his ears are well trained). In more technical terms, one perturbs an unknown system with a very short pulse that in the ideal case would be a delta-function describing a perturbation of infintesimal short duration. Regardless of its specific nature, external perturbations belong to the most important methods for analyzing unknown systems. In this context, the perturbation by a short pulse is only one example for enforcing a characteristic answer. Other approaches include periodic perturbations where systematic variations of amplitude and frequency open ample possibilities to interrogate the system.

Although external perturbations have long proven to be a valuable tool for science and engineering, many experiments on novel systems studied in young emerging fields of science are dedicated to pure observation as a starting point. Investigations on spatial pattern formation in chemical and biological systems are an excellent example for this characteristic development of a young branch of experimental science (Field and Burger, 1985). Most of the current research activities in this field were triggered by the early work of Zhabotinsky and Winfree, who reported the observation of chemical waves in a reaction system that today is known as the Belousov-Zhabotinsky (BZ) reaction (Zaikin and Zhabotinsky, 1970; Winfree, 1972).

Introduction

The transport of life-sustaining materials across the cell membrane consumes large fractions of the total energy expenditures of cells. As such, membrane transport is tightly regulated and controlled. Membrane proteins are mostly responsible for the transport of molecules and ions in and out of the cell. There are many classes of membrane transporters. A simple classification lists three types: channels, carriers and pumps. Channels and carriers are passive downhill transporters, whereas pumps are active uphill transporters. However, a channel may also be able to fulfill the role of an uphill transporter by coupling a downhill transport reaction to an uphill transport reaction: the transporter, however, will still be a downhill transporter overall. Beside proteins, a small number of transporters come from heterocyclic organic compounds. These small molecules can serve as ion channels or ion carriers, but only a protein with enzymatic activity can function as a pump. All transporters are embedded in lipid bilayers and, for a pump to function, some degrees of freedom must be hindered or restricted. As I shall discuss later, this feature of a pump is crucial for capturing energy from a periodically oscillating or randomly fluctuating force field (Tsong and Astumian, 1986, 1988; Tsong, 1990, 1992).

In order for a pump to capture energy another feature is essential: it must be able to interact effectively with a driving force, e.g., an electric field or an acoustic field (Tsong and Astumian, 1986, 1988).

A biochemical model with two instability mechanisms

In the two-variable models studied for glycolytic oscillations and birhythmicity, periodic behaviour originates from a unique instability mechanism based on the autocatalytic regulation of an allosteric enzyme by its reaction product. The question arises as to what happens when two instability-generating mechanisms are present and coupled within the same system: can new modes of dynamic behaviour arise from such an interaction?

An example of such a situation was considered at the end of the preceding chapter: the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables.

The coupling in series of two enzyme reactions with autocatalytic regulation (fig. 4.1) permits the construction of a three-variable biochemical prototype containing two instability-generating mechanisms (Decroly, 1987a,b; Decroly & Goldbeter, 1982). As in the model for glycolytic oscillations, the substrate S of the first enzyme is introduced at a constant rate into the system; this substrate is transformed by enzyme E1 into product P1, which serves as substrate for a second enzyme E2 that transforms P1 into P2. The two allosteric enzymes are both activated by their reaction product; P1 and P2 are thus positive effectors of enzymes Ex and E2, respectively.

All life is rhythmical. Just to feel the human pulse is to marvel at how the heart beats out its regular rhythm. It is a sobering thought that, for the whole of our lives, a small group of pacemaker cells dutifully send out a signal every second or so to drive each heart beat. For someone living to 70 years with a heart rate of 60 beats/minute, we can calculate that this oscillator will have faithfully transmitted well over two billion beats! Such rhythmicity occurs in many other cell types especially when they are being stimulated. For example, a liver cell generates regular pulses of calcium when it responds to the hormones that signal the breakdown of glycogen. Another striking example is the burst of spikes that occur in mammalian eggs following fertilization. In humans, as in other mammals, the egg remains quiescent for many years until it is coaxed into life by the train of spikes triggered by the arrival and attachment of the sperm. Exactly the same mechanisms are employed to generate the calcium spikes observed in liver cells and eggs. On a separate time scale, we are all aware of our circadian clock through its ability to drive our diurnal sleep-wake cycle. The strength of this circadian oscillator is all too obvious to jet travellers as they cross time zones.

The eukaryotic cell cycle is driven by a biochemical oscillator

Few cellular processes are as crucial as that governing cell division. The tight control of cell division indeed plays a prominent role in development and differentiation, while unrestrained proliferation is associated with the cancerous state. The control of the eukaryotic cell division cycle therefore represents a central issue in cell biology (for reviews of recent experimental advances, see the special issues of Science, 246:603-640 (1989) and of the Journal of Cell Science, Suppl. 12 (1989), as well as volume LVI of Cold Spring Harbor Symposium on Quantitative Biology (1991), and Cross et al., 1989; Hunter, 1992; Norbury & Nurse, 1992; Murray & Hunt, 1993; Heichman & Roberts, 1994; King, Jackson & Kirschner, 1994; Nurse, 1994; Peter & Herskowitz, 1994; Sherr, 1994). The cell cycle is classically portrayed as a sequence of phases (fig. 10.1): following mitosis (M) come, successively, the Gx phase leading to the S phase of DNA replication, and the G2 phase, which separates the latter from the next M phase; sometimes cells stay in a quiescent phase, Go, prior to their entry into Gt. The fact that in dividing cells mitosis recurs at regular intervals (whose duration, varying with the cell type, ranges from 10 min up to 24 h or even more) has for a long time raised the possibility that the cell cycle is driven by a continuous biochemical oscillator.

This book grew out of a passion for rhythmic phenomena in biological systems, which has impelled my work for 25 years. I had the chance of being introduced to that field when I joined the group of Ilya Prigogine at the University of Brussels, for what I thought would be a one-year experience at the end of my studies in chemistry. I was rapidly hooked, however, and still have the pleasure of pursuing my research there. I would like to seize this opportunity to express my gratitude to Ilya Prigogine, who kindly agreed to write a preface (reproduced here as a foreword) for the French edition of this book; I have been marked by the enthusiasm that drives his incessant questioning of the world. The atmosphere in Brussels has always been ideal for developing my work; in addition to Ilya Prigogine and Paul Glansdorff, I am indebted to Gregoire Nicolis and Rene Lefever for making this friendly intellectual climate so fruitful.

This book owes much to the graduate students and co-workers with whom I had the pleasure of collaborating over the years: Jean-Louis Martiel, Olivier Decroly, Federico Moran, Yue-Xian Li, Genevieve Dupont, and Jose Halloy. They made decisive contributions to the various topics considered in this book, by actually carrying out much of the research described. Special thanks are due to Thomas Erneux, with whom I am always happy to pursue new ventures.

Experimental observations on glycolytic oscillations

Glycolytic oscillations are the prototype for periodic phenomena in biochemistry, even if the function of these oscillations, if there is any, has for long remained obscure. Their study began some 30 years ago, and still continues. The first observation indicating the existence of oscillatory behaviour in glycolysis is due to Duysens & Amesz (1957), who reported, by studying the fluorescence of some glycolytic intermediates in yeast, that one of these underwent damped oscillations in the course of time (fig. 2.1).

This observation was pursued from 1964 on by Chance and coworkers (Chance, Schoener & Elsaesser, 1964; Ghosh & Chance, 1964; Pye & Chance, 1966), who showed that glycolytic oscillations could be maintained in yeast suspensions for relatively long periods of time. Quickly thereafter observations were conducted on yeast extracts, with very similar results. Regular oscillations were recorded over an extended period of time, provided that the glycolytic substrate was produced in a continuous manner (Chance, Hess & Betz, 1964; Hess, Brand & Pye, 1966; Pye & Chance, 1966; Pye, 1969, 1971). The latter condition was fulfilled when trehalose was used, this metabolite being transformed into a glycolytic substrate by the enzyme trehalase at a low, constant rate (fig. 2.2). The periodic phenomenon is also observed in a single yeast cell (Chance, Pye & Higgins, 1967). All glycolytic intermediates oscillate with the same frequency (fig. 2.3), but with different phases (Betz & Chance, 1965).

It is a privilege to introduce the book of my friend and colleague Albert Goldbeter.

Our century has witnessed basic changes in direction in science, and foremost among them I would place the ‘rediscovery of time’. Only a few decades ago, most of the research efforts in physical chemistry were devoted to equilibrium states. Not only did the notion of equilibrium dominate chemistry, but it was also considered to be the fundamental paradigm for the extension of scientific ideas to other fields, including economics and social sciences. The arrow of time was considered to be an artefact reflecting human ignorance. Today, the situation has radically changed. Everywhere, from astrophysics to biology, the essential role of nonequilibrium is well recognized.

It is indeed far from equilibrium that we encounter self-organization processes that lead to the breaking of temporal and spatial symmetries as well as to the appearance of coherent behaviour on a supramolecular scale. Chemical oscillations provide a particularly striking example. I remember my astonishment when I was shown for the first time a chemical oscillatory reaction, more than 20 years ago. I still think that this astonishment was justified, as the existence of chemical oscillations illustrates a quite unexpected behaviour. We are used to thinking of molecules as travelling in a disordered way through space and colliding with each other according to the laws of chance.

Circadian rhythms as ubiquitous biological clocks

Most living organisms exhibit circadian rhythms which allow them to adapt to an environment that varies with a periodicity of 24 h; such rhythms are ubiquitous and govern so many key physiological functions that they have become synonymous with biological clocks (Chovnik, 1960; Halberg, 1960; Pittendrigh, 1960, 1961; Menaker, 1971; Bunning, 1973; Aschoff, 1981; Moore-Ede et al, 1982; Edmunds, 1988). Likewise, for many, chronobiology refers specifically to the study of 24 h rhythms. In no way do we detract from the important role of circadian rhythms by stressing that such periodic phenomena comprise only one among many types of biological rhythm, besides those that have much shorter periods (see table 1.1).

Yet, circadian rhythms are of special interest because of both their prevalence among all eukaryotic organisms, unicellular or multicellular, and their immediate relevance to our human experience (Moore-Ede et al, 1982): the sleep-wake and nutrition cycles remind us every day of the fundamental, physiological role of circadian rhythms in allowing us to cope with our periodically changing environment.

Another reason why circadian rhythms remain so fascinating is that, although these rhythms have been known for so long, their mechanism remains largely unknown. The purpose of this introductory section is to present a brief overview of circadian oscillations and of the current understanding of their molecular bases, which at present is undergoing rapid progress, and to indicate how theoretical models have been used to investigate the properties of these rhythms.

In a somewhat surprising manner, numerical simulations of the model for cAMP signalling in D. discoideum revealed the possibility of complex modes of oscillatory behaviour similar to those studied in chapter 4 in the model for a multiply regulated enzyme system with two autocatalytic reactions coupled in series.

While the presence of two instability-generating mechanisms gives rise, in the latter model, to two endogenous oscillatory processes whose interaction results in complex oscillatory phenomena, the fact that such phenomena also occur in the model for cAMP signalling is, at first, rather surprising. This model, indeed, contains but a single instabilitygenerating mechanism based on the positive feedback exerted by extracellular cAMP on its own synthesis, through the binding of cAMP to the receptor and the ensuing activation of adenylate cyclase. Before addressing the origin of complex oscillatory behaviour, we shall first examine the different types of complex phenomena observed in the model for cAMP synthesis in D. discoideum.

Complex oscillations in a seven-variable model for cAMP signaling

The first indication of complex oscillations was that of the complex periodic behaviour of the ‘bursting’ type shown in fig. 6.1 (Martiel & Goldbeter, 1985). These oscillations resemble those obtained in the model for the multiply regulated enzyme system analysed in chapter 4. These complex periodic oscillations were observed in the model for cAMP signalling in D. discoideum by numerical integration of the differential equations, before reduction of the number of variables. As indicated in chapter 5, excitable and oscillatory behaviour occur in this model as a result of self-amplification in cAMP synthesis.