Chemical oscillations and biological rhythms

Rhythms are among the most conspicuous properties of living systems. They occur at all levels of biological organization, from unicellular to multicellular organisms, with periods ranging from fractions of a second to years (Table 1.1). In humans, the cardiac and respiratory functions and the circadian rhythm of sleep and wakefulness point to the key role of periodic processes in the maintenance of life. In spite of their close association with physiology, however, periodic phenomena are by no means restricted to living systems. Since the discovery of the oscillatory chemical reaction of Belousov and Zhabotinsky (Belousov, 1959; Zhabotinsky, 1964; Tyson, 1976), it has become clear over the last three decades (Nicolis & Portnow, 1973; Noyes & Field, 1974; Pacault et al, 1976; Epstein, 1983, 1984; Field & Burger, 1985) that numerous chemical reactions studied in the laboratory display periodic behaviour in the course of time. Biological rhythms nevertheless remain more common than oscillations in purely chemical systems.

However, oscillatory behaviour does not always possess a simple periodic nature. Thus, both in chemistry and biology, oscillations sometimes present complex patterns of bursting, in which successive trains of high-frequency spikes are separated at regular intervals by phases of quiescence. Yet another mode of complex oscillatory behaviour, which has received increased attention over the last decade, is characterized by its aperiodic nature and sensitivity to initial conditions. Such chaotic oscillations have been observed in chemical reactions (Scott, 1991; Field & Gyorgi, 1993) and in a variety of biological contexts (Olsen & Degn, 1985; Holden, 1986; Glass & Mackey, 1988).

Experimental observations on cytosolic Ca2+ oscillations

Ca2+ oscillations are among the most significant findings of the last decade in the field of intracellular signalling. Together with the mitotic oscillator, which underlies the eukaryotic cell division cycle (examined in chapter 10), Ca2+ oscillations are also one of the most important periodic phenomena uncovered in recent years in the field of biochemical and cellular oscillators. Ca2+ oscillations are of interest for a variety of reasons. First, they occur in a large number of cell types, either spontaneously or after stimulation by hormones or neurotransmitters. Second, it is by now clear that they represent the oscillatory phenomenon that is the most widespread at the cellular level, besides the rhythms encountered in electrically excitable cells. Third, Ca2+ oscillations are often associated with the propagation of Ca2+ waves within the cytosol, and sometimes between adjacent cells; even though its physiological significance remains to be determined, this phenomenon has become one of the most important examples of spatiotemporal organization at the cellular level.

Since their first direct observation in fertilized mouse oocytes (Cuthbertson & Cobbold, 1985) and hormone-stimulated hepatocytes (Woods et al., 1986,1987), which followed their earlier, theoretical prediction (Rapp & Berridge, 1977; Kuba & Takeshita, 1981) and indirect characterization (Rapp & Berridge, 1981), the number of experimental reports on Ca2+ oscillations has mushroomed at an increasing pace in recent years; these experimental results have been examined in several reviews (Berridge & Galione, 1988; Berridge, Cobbold & Cuthbertson, 1988; Berridge, 1989, 1990; Cuthbertson, 1989; Rink & Jacob, 1989; Cobbold & Cuthbertson, 1990; Jacob, 1990a; Petersen & Wakui, 1990; Tsien & Tsien, 1990; Meyer & Stryer, 1991; Tsunoda, 1991; Fewtrell, 1993) and in a special issue of Cell Calcium (Cuthbertson & Cobbold, 1991).

Whereas the function of glycolytic oscillations, at least in yeast, remains puzzling, the same does not hold for the role of cAMP oscillations in Dictyostelium amoebae. After reviewing the different aspects of the function of these oscillations and the molecular bases for the efficiency of pulsatile signalling, a link is established here with the rhythmic, pulsatile secretion observed for an increasing number of hormones. Besides their circadian modulation (Van Cauter & Aschoff, 1989), many hormones are indeed secreted in the form of ultradian, highfrequency pulses (Crowley & Hofler, 1987; Negro-Vilar et al., 1987). Particular attention is devoted below to the pulsatile release of gonadotropic hormones by the pituitary in response to pulses of the gonadotropin-releasing hormone (GnRH), emitted by the hypothalamus with a frequency close to one pulse per hour.

A model for the response of target cells to pulsatile stimulation in the presence of receptor desensitization allows us to investigate, in a general manner, the encoding of the pulsatile signal in terms of its frequency. This analysis, applied to the cAMP signalling system and to the pusatile release of GnRH, suggests the existence, in both cases, of a frequency capable of inducing an optimum response. The model shows how this optimum frequency is related to the kinetics of reversible desensitization in target cells. The validity of the results obtained in the general model is corroborated by the fact that similar results are recovered in the specific model for cAMP signalling based on receptor desensitization.

Beyond the case of GnRH, the conclusions on the existence of an optimum frequency of pulsatile signalling probably extend to other hormones, as is discussed for the cases of insulin and growth hormone.

Biological rhythms occur only under precise conditions, and variations in a control parameter can bring about their disappearance. In a symmetrical manner, the variation of such a parameter can lead to the appearance of a rhythm in the course of development. There is no example as yet where the molecular basis of the ontogenesis of a biological rhythm is known in detail. The rhythm of intercellular communication in the slime mould Dictyostelium discoideum provides us with a prototype for the study of this question.

Appearance of excitability and oscillations in the course of development of Dictyostelium amoebae

Studies of aggregation of D. discoideum amoebae on agar have shown that cells are capable of relaying an artificial cAMP signal applied iontophoretically by means of a pipette, before being able to produce sustained oscillations in an autonomous manner (Robertson & Drage, 1975; Gingle & Robertson, 1976). These spontaneous oscillations of cAMP occur only a few hours after the beginning of starvation.

Studies of D. discoideum cells in suspensions confirm this observation (Gerisch et al., 1979). At the beginning of starvation, the amoebae are unable to amplify cAMP signals. Some 2 h later, the amoebae amplify cAMP pulses whose amplitude exceeds a threshold. Four hours after the beginning of starvation, spontaneous oscillations arise and are maintained for 3 h before giving way to excitable behaviour (fig. 7.1). During the hours preceding starvation, the amoebae thus undergo successive transitions that modify the dynamic properties of the signalling system that controls intercellular communication. The sequence of developmental transitions leads from the absence of excitability to relay, and from relay to autonomous oscillations of cAMP.

A two-variable biochemical model for birhythmicity

Birhythmicity: coexistence of two stable limit cycles

The model governed by eqns (2.7) or (2.30) admits at most a single limit cycle, i.e. only one type of periodic behaviour for a given set of parameter values. The analysis of a three-variable model considered in chapter 4 has led to the fortuitous observation of a phenomenon of coexistence between two types of oscillation corresponding to two simultaneously stable limit cycles (Decroly & Goldbeter, 1982). This phenomenon represents a particular type of Instability, much less common than the one involving the coexistence between two simultaneously stable steady states, of which several examples are known in chemistry (Pacault et al, 1976; Epstein, 1984) as well as biochemistry (Degn, 1968; Naparstek et al, 1974; Eschrich et al, 1980, 1990), or that involving the coexistence between a stable steady state and a stable limit cycle; the latter phenomenon is known as hard excitation (Minorsky, 1962). To differentiate it from these two types of bistability, Decroly & Goldbeter (1982) coined the term birhythmicity to denote the coexistence between two simultaneously stable limit cycles.

Birhythmicity has not yet been clearly demonstrated in biological systems, although some observations suggest that this type of dynamic behaviour might occur in cardiac tissue (Mines, 1913; Gilmour et al, 1983) and in a neuronal preparation (Hounsgaard et al, 1988). Following the theoretical predictions of the phenomenon by the model analysed in chapter 4, an experimental study of a chemical system involving two coupled oscillatory reactions permitted the demonstration of a coexistence between two simultaneously stable periodic regimes (Alamgir & Epstein, 1983; Citri & Epstein, 1988).

The amoeba Dictyostelium discoideum (Raper, 1935) is one of the most studied organisms in developmental biology (Bonner, 1967; Loomis, 1975, 1982). The main reason underlying the attractiveness of these amoebae is twofold. On one hand, they can switch from a unicellular to a multicellular stage during their life cycle (Raper, 1940a,b) by resorting to a chemical mechanism of intercellular communication (Bonner, 1947; Shaffer, 1956, 1962; Devreotes, 1982; Gerisch, 1982, 1987), which presents some similarities with hormonal communications in higher organisms (Newell, 1977). On the other hand, once the multicellular stage is reached, the amoebae differentiate into at least two distinct cell types, thus providing a simple model for the study of pattern formation (Nanjundiah & Saran, 1992) and cell differentiation in eukaryotes (Takeuchi, Noce & Tasaka, 1986; Gross, 1994). Recent advances in the manipulation of the Dictyostelium genome reinforce the importance of this organism in the study of the molecular bases of development.

It is for yet another reason that Dictyostelium amoebae draw our attention here. The mechanism of intercellular communication that governs the transition between the isolated and collective phases of their life cycle possesses a periodic nature (Gerisch, 1968,1971; Shaffer, 1962). After reviewing the experimental facts that make this phenomenon a prototype for biochemical oscillations in cell biology, theoretical models are developed in this chapter to account for this biochemical rhythm. A particularly interesting aspect of the Dictyostelium example, which will be highlighted here, is that it illustrates well the endless feedback between modelling and experimental studies. As the latter progress, sometimes stimulated or guided by the motivation of testing theoretical predictions, alterations to the initial model become necessary.

From the molecular properties of regulation to the temporal self-organization of biological systems

Glycolytic oscillations in yeast and muscle represent a classic example of periodic behaviour in biochemistry. Likewise, oscillations of cAMP in Dictyostelium amoebae are the prototype of intercellular communication by periodic signals of a nonelectrical nature. Besides these examples known for more than 20 years (Hess & Boiteux, 1971; Goldbeter & Caplan, 1976; Hess & Chance, 1978; Berridge & Rapp, 1979; Rapp, 1979), additional periodic phenomena in biochemistry have been characterized during the last decade. Conspicuous by their widespread nature are intracellular oscillations of Ca2+, which occur in a large variety of cell types, either spontaneously or as a result of external stimulation. The experimental demonstration of oscillations in cytosolic Ca2+, together with the unravelling of the molecular basis of the mitotic oscillator driving the cell division cycle in eukaryotes, represents the most significant advances in the field of biochemical oscillations in recent years. Also important is the progress currently made in elucidating the regulatory mechanisms underlying circadian rhythms. The study of theoretical models presented in this book allowed us to clarify the molecular mechanism of these oscillations that provide some of the best-known examples of biochemical and cellular rhythms.

The use of models based on experimental observations has shown how the regulatory properties at the level of enzymes and receptors can give rise to nonequilibrium, temporal self-organization in the form of sustained oscillations of the limit cycle type. In this, sustained oscillations represent examples of the dissipative structures described by Prigogine (1969).

Bounded and unbounded potentials

A molecular system is one in which the atoms are bound together by chemical bonds between specific pairs of atoms. To study the stability of molecular systems we have to study the stability of these individual chemical bonds. Often the bonds are not represented as a complex of electron orbitals but, rather, represented as attractive potentials between pairs of atoms that are functions only of interatom distance. All realistic models of these potentials are bounded. That is, if the two atoms involved are separated to large enough distances the interaction energy goes to zero, or depending on where one puts the origin in potential, to some finite value. Atoms are assumed to become independent of one another when separated by a large enough distance. In particular a bounded interaction potential does not go to infinity as the separation increases; if that happened atoms would be bound together the way quarks are and could never be found separated.

If one applies standard statistical mechanics to a molecule bound by such bounded potentials one always gets the result that the equilibrium state of the system is the dissociated state. In other words, the molecule is always melted. This is true at all temperatures, even infinitesimal temperatures, and is true for both classical and quantum analysis. In studying melting by statistical mechanics of systems with realistic potentials one is dealing with the contradictory situation of trying to study the melting of a molecule that is already melted.

Dynamics of conformation change

Biological macromolecules often undergo changes in shape carrying out their biological function. For example the action of muscle fibers involves a relative rotation of a section of a macromolecule. An important change in shape in DNA is the B to A conformation change that is caused by changes in the level of hydration of the helix. The A conformation is the low hydration form. In that transition no valence bonds, and likely no H-bonds, are even transiently disrupted. The change is mostly to the helicity of the helix, going from a complete turn in 10 base pairs to one in 11 base pairs. In the process the width of the helix increases, the length decreases, and the base pairs tilt away from the perpendicular to the helix axis (Saenger, 1984). Analysis of this transition has been carried out in the context of a soft mode displacive change transition (Eyster and Prohofsky, 1977; Lindsay et al., 1985). Displacive change is useful for transitions that involve continuous movement of atoms from one conformation to another, i.e. where the transition is second order. The mode that goes soft, in soft mode analysis, is related to the softening of the free energy barrier that separates the free energy minima as a function of order parameter, as discussed in a previous chapter.

Choice of methods

As discussed in earlier chapters, several advantages point to the use of an effective phonon approach to study dynamics in a large nonlinear system such as the DNA double helix. The first advantage is that in the effective phonon formulation unbounded interactions replace bounded interactions and this change allows one to apply statistical mechanical ensemble theory to the system without running into problems associated with infinite thermodynamic integrals. The second advantage is that the selfconsistent phonon approach divides the large nonlinear dynamics problem into two parts, 1) the incorporation of nonlinearities into an effective force constant, and 2) the solution of the large number of degrees of freedom by normal mode means. A third implicit advantage is related to the built in harmonic approximation, i.e. the advantage of not having to solve for the ground state conformation before being able to proceed to the dynamic solutions. All these theoretical and calculational advantages are present in the approach developed.

In addition to the theoretical advantages discussed above there is a very practical reason to formulate the dynamics as an effective phonon problem. Direct experimental observation of the vibrational excitations of double helices by Raman scattering and infrared absorption indicate that they are resonant rather than relaxational. The lines are broad but not so broad as to indicate that one is observing relaxational modes.

Biologically significant DNA

Most biologically significant DNA helices are very long, so long that it is a useful approximation to assume infinite length. Helical lattice dynamics can then be used for analysis without having to resort to large dimensional calculations if the systems have a repeating base sequence symmetry. The studies in earlier chapters dealt only with DNA which had repeating symmetry. One can learn much about the dynamics of native DNA from a study of repeating DNA because the polymer DNAs share much of the dynamics, and often bracket the behavior of native DNAs. For example, the melting temperature of native DNA falls between that of poly(dG)– poly(dC) and poly (dA–dT)–poly(dA–dT). The study of repeating DNA is, however, a study of the material science of the material DNA. It deals with ‘perfect DNA’ and not with the complex broken symmetry material of biological significance.

The departures from symmetry are of great importance as biological information is contained in them. A biologically oriented study then requires methods that can deal with departures from repeating symmetry but can still be applied to very long DNA. In this chapter we develop methods useful with symmetry breaking structures. We get around the difficulties of dealing with large or infinite systems by starting with initial repeat polymer solutions and applying the new methods to achieve appropriate solutions. The approach was initially developed to deal with defects in crystals, the mathematical method is a Green function approach that is detailed in Appendix 3.

Greater helix

The study of the dynamics of a helix with elements, such as a drug, attached to it is easiest when there is an excess of the element so that it forms a repeating pattern of attachment. This allows the use of lattice methods to solve the large dimensional problem associated with the long helix. When only one element (or a few elements) attaches to a long helix the study is best handled by methods developed in Chapters 11 and 13. One of the more extensively studied attaching drugs is daunomycin, probably because it is an important antitumor agent. It is one of a number of drugs that intercalate into the helix, i.e. part of it has a planar structure that enters between base pairs of the helix. The specific example studied is the repeating unit of daunomycin–poly(GCAT)–poly(ATGC) (Chen and Prohofsky, 1994). The calculation determines the probability of the drug dissociating from the DNA helix which can be converted to give the binding constant of the drug to the helix. The choice of the particular DNA sequence was dictated by the availability of X-ray conformational information (Wang et al., 1987). The helix is somewhat distorted by the intercalation and the calculation uses the distorted conformation and the correct daunomycin position. The bases are numbered in the unit cell by calling the first guanine G1 then proceeding down one strand in the 3′ to 5′ direction so that the next cytosine is C2, then comes A3 and T4.