Cellular Clocks and Oscillators

INTERNATIONAL REVIEW OF CYTOLOGY. VOL. 86

Cellular Clocks and Oscillators R. R. KLEVECZ,*S. A. KAUFFMAN,?AND R. M. S H Y M K O ~ Departments of *Cell Biology and $Radiation Research, City of Hope Research Institute, Duarte, California, and fDepartment of Biochemistry and Biophysics, University of Pennsylvania, Philadelphia, Pennsylvania Introduction: Temporal Order and Epigenetic Information.. . . . . . . . A. The Clock Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A Distinction between the Chromosome Replication Cycle, the Cell Growth Cycle, and the Cellular Clock.. . . . . . . . . . . . . . . . C. Sequentiality and Determinism . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Randomness and Probability in the Cell Cycle . . . . . . . . . . . . . . 11. Emanations of the Oscillator in the Cell Cycle . . . . . . . . . . . . . . . . . . A. Oscillator Dynamics. . . . . . . . . ........... B. Exponential Generation Time Cycle Oscillator . . . . . . . . . . . . . . . . . . . . . C. Periodic Gene Expression and Quantized Mammalian Cells as Manifestations of the Cellular Clock. . . . . D. Phase Properties of the Oscillator ................ E. Division Delay versus Phase Response .................... F. Phase Response to Sublethal Heat Shock. , . G. Phase Reset and Thermotolerance. . . . . . . . . . . . . . . . . . . . . . . . . H. Double Pulse Experiments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Effects of Size on the Expression of a Timekeeping Oscillator and the Quantizement of Generation Time. . . . . . . . . . . . . . . . . . J. An Explicit Model in Which Cell Division Is Gated by Cell Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Are Cellular Clocks an Essential Element of Circadian Rhythms . . . Vestiges of the Primitive Clock . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.

91 99 99 103 I04 I04 I04 105

109 112 1 I4

I15

I18 I19

120 121 123 124 I26

I. Introduction: Temporal Order and Epigenetic Information The work and viewpoint represented here will be concerned with the dynamic and temporal properties of cell growth and proliferation as they are manifested in cultured cells and intact organisms, and will encompass circadian rhythms, cellular clocks, and chemical oscillators. We will review the evidence for high frequency oscillations and will emphasize their manifestations and function as a clock in the cell cycle. It is our thesis that periodic, stable, and heritable trajectories in metabolic states specify not only the time sense of cycling cells but the proliferative potential and differentiable capacity of noncycling cells. The kinetic 91 Copyright 0 1984 by Acadcmic Press, Inc. All rights of reproducrion in any form reserved.

ISBN 0-12-364486-0

98

R. R. KLEVECZ ET AL.

and geometric properties of these oscillators, their mode of transmission, and their mechanisms of expression in development remain among the great unanswered problems in biology. While the cell cycle is one of the more accessible expressions of temporal organization, most summary views of the cycle have emphasized its linear or sequential aspects (Hartwell, 1974) while in other instances enthusiasm for what may be a peculiarity of the particular cellular system has yielded models which explain a single aspect of cellular behavior (Smith and Martin, 1973; Cooper, 1979; Pardee, 1974). If one has a longstanding familiarity with conventional views of the cell cycle, a certain flexibility of mind is required to accept the notion presented here that both cycling and noncycling cells can mainfest distinctive and yet nearly identical time-sense or phase properties with respect to a common cellular oscillator and that in either case the phase can be rapidly changed (Klevecz et al., 1982). We think that the benefits of such an understanding justify the effort of explanation. Several major points of contention will be introduced. They are (1) that it may be inappropriate to apply genetic analysis to what are essentially epigenetic, self-organizing systems; (2) that there is no benefit in overly simple representations of the cell cycle; (3) that most models of cell cycle behavior arise in response to the need to explain some particular phenomenon and generally ignore other cellular behaviors; and (4) that sampling and analysis must be performed with an understanding of the response of dynamic systems to perturbation. Because of the phase labile/period stable properties of dynamic systems, continuous monitoring by nonperturbing methods must be perfected and more extensively employed. For the past 30 years the attention of most biologists has been fixed on modes of storage, transmission, and expression of genetic information. Considerably less effort has been given over to understanding a second form of biological information; one that has been variously described as epigenetic, temporal, or talandic (Goodwin, 1963; Klevecz, 1969a). In part this must be due to the greater logistical difficulties encountered in studying the processes that stably specify and transmit this information. Dynamic components of biological organization are intrinsically more refractory to analysis since unlike genetics where information is resident in a covalent DNA sequence and stable over reasonably long time spans, temporal information exists only by virtue of stable trajectories and concentration gradients in an ensemble of interacting molecules. It cannot unfortunately be crystallized or put into the freezer when the work day is done and therefore work proceeds more readily along theoretical analytical as opposed to empirical paths. We would argue that the maintenance of stable temporal relationships in the ensemble of biological molecules is essential to the successful expression of the genome and that the gene is the passive partner in this biological congress. The theoretical basis of epigenetics begins with the work of Eigen ( 197 I) and

CELLULAR CLOCKS AND OSCILLATORS

99

Prigogine (1 947) on nonequilibrium thermodynamics and often borrows as well from the mathematics of stable attractors. The concept of self-organization implies that there is a directionality to time and that nonequilibrium situations may be a source of order. A more detailed discussion of these points is beyond the scope of this review but several articles can be recommended to the interested reader (Nicolis and Prigogine, 1977; Eigen et al., 1981; Eigen and Schuster, 1977). A. THECLOCKDEFINED

In order to proceed it is necessary for the reader to know what is meant by a “clock.” Most generally, a biological clock engenders time sense, and does not merely record time’s passage. It is a phase labile, period stable process, where period is to a large degree maintained independently of the local environment. Circadian rhythms, the best recognized expression of the biological clock, are defined as those rhythms whose T~ (free running period) is an approximation to the period of the earth’s rotation. According to Pittendrigh’s (1960) generalizations (with some modernization), circadian rhythms are ubiquitious, endogenous, self-sustaining, innate, occur at all levels of organization (integral), species specific (heritable), precise, temperature compensated, phase responsive, and entrainable. In addition they show aftereffects (inertial), light intensity dependence (obey Aschoff‘s Rule), phase lability but period stability (robust), and (at least where period is concerned) an intractability to chemical perturbation. A distinction can be made between a cellular clock and a chemical oscillator based on their capacity to maintain a reasonably constant sense of time. All the oscillators of which we have concrete knowledge via computer simulation (Prigogine and Lefever, 1968; Field and Noyes, 1974) can be continuously tuned and the period changed by the environment or some environmental parameter, whereas the period of cellular and organismic clocks appears to be heritably fixed in the chemistry of the organism and can only be tuned over a limited range, that is, the limits of entrainment. The question of whether a relatively high-frequency oscillator which behaves as a clock by all measurable criteria is also an approximation to some environmental variable is not readily answerable nor need it be in order to discuss timekeeping in cells.

B. A DISTINCTION BETWEEN THE CHROMOSOME REPLICATION CYCLE,THE CELLGROWTHCYCLE,AND THE CELLULAR CLOCK Cell cycle arrest, division delay, division setback and cell cycle phase response have variable and overlapping meanings to workers in the field. It seems that in no area is the conflict between old and new views of cellular dynamics

100

R. R. KLEVECZ ET AL.

greater and the communication between adherents of the conflicting viewpoints poorer than in the study of the kinetics of perturbed populations. Initially it may be helpful to distinguish between (1) the chromosome replication cycle, (2) the cell growth and division cycle, and (3) the cellular clock, though on occasion the distinctions are blurred and in proliferating cells at steady state these three elements can bear a fixed relationship to one another. The chromosome replication cycle (CRC) we equate loosely with the familiar cell cycle diagram derived many years ago from the experiments of Howard and Pelc (1953). It is sequential, deterministic, has only passive timekeeping and control properties, and can only be meaningfully spoken of in retrospect and only in cells which replicate their DNA and divide in the minimum cell cycle time. It is to this view of the cycle that much of the older descriptors of cell cycle behavior such as arrest can be accreted, and it is this concept with which cell cycle is most often identified, In contrast to what we have defined as the clock, the chromosome replication cycle can be stopped. The chromosome replication cycle was originally and operationally defined by autoradiographically determined measures of the fraction of labeled metaphases from a retrospective cell cycle. It can be argued that uncritical extrapolations of this model in the design and execution of a variety of cell kinetic studies have led to interpretive errors and confusion. The growth and division cycle (GDC) we equate with the accumulation of necessary cellular constituents, with cell mass, with size, and in general with the developmental processes of the cell. Inhibition of the CRC does not necessarily stop growth, though inhibitors which stop all measurable progress through the CRC will slow the accumulation of mass in the GDC and indeed its continued accumulation may lead to unbalanced growth (Rueckert and Mueller, 1960), and upon reversal to the alternation of lengths of generation times and a negative correlation in mother-daughter cell generation times (Jauker and Cleffman, 1970; Shymko and Klevecz, 1981). The concept of a cellular clock or limit cycle (CLC) as a means of describing behavior of cells is probably the least familiar to kineticists and will therefore be dealt with in greater detail and will be used contrapuntally in the interpretation of otherwise familiar experiments. Succinctly put, the cellular clock times and gates the GDC and the CRC and is in turn bounded by them. Within the limits of viability the clock cannot be stopped, though measurable progress toward DNA synthesis or mitosis may be absent. The precise time sense that cells express is thought to reside primarily in a macromolecular oscillator and to have properties of a stable attractor or limit cycle. The clock is phase labile and may respond to perturbations such as heat shock, antimetabolite treatment, and radiation by a change in phase. The period of the oscillator, once the transients induced by perturbation have died down, is stable and constant, though the expression of events in the CRC and GDC

CELLULAR CLOCKS AND OSCILLATORS

101

domains may only be expressed as multiples of the oscillator period. The limit cycle is stable to perturbation and its steady state amplitude and wave form are independent of initial conditions. Unlike conservative systems limit cycles are robust and change properties only slightly with small changes in mechanism. Attempts to understand the molecular biology of cellular and organismic timekeeping have only begun to be productive. Traditionally questions of clock mechanisms have been approached using mathematical analysis and the computer to simulate, for example, the response of individual oscillators and populations of oscillators to changing coupling strengths (Winfree, 1967; Pavlidis, 1969), to study the effects of changing values for diffusion, tissue extent, or oscillator domain (Nicolis and Prigogone, 1977; Kauffman et al., 1978), or to investigate how oscillator period might be specified stably in an oscillation involving a large number of variables (Higgins, 1967; Pavlidis, 1969, 1973; Winfree, 1967, 1970). The underlying notion in such studies would be the idea that organismic clocks might somehow occur as a result of interactions among many more localized, perhaps single cell, or organellar oscillators. The behavior of populations of interacting oscillators has been considered analytically by a number of workers (Goodwin, 1963; Winfree, 1967; Pavlidis, 1969). Indeed, this is perhaps the only biological discipline where empiricism lags behind theor y . Goodwin ( 1 963) who was the first to explore this problem rigorously calculated that a macromolecular oscillator in the epigenetic domain whose constituents had half lives on the order of a few minutes to an hour or two and which was controlled by negative feedback would have a period length of 3-4 hours. His notion that sustained oscillations could occur in the particular, conservative, linear oscillator may not, in the strictest sense, be correct, but this work has paved the way for much of what has followed. It is usually argued that unless significant delays, due perhaps to diffusion, are included in Goodwin’s scheme such an oscillator would not be self-sustaining. Stimulated by the observations of Chance et al. (1964) on high-frequency oscillations in glycolytic intermediates from yeast extracts, early analyses were directed toward producing frequency reduction in the observable product oscillations through frequency beats (Higgins, 1967). That such a mechanism for generating circadian or any low-frequency rhythm is fraught with difficulties has been pointed out on a number of occasions (Pavlidis, 1969; Winfree, 1967). If frequencies of the individual oscillator are high and coupling is strong, in an attempt to produce rhythms of circadian frequency, the mutual entrainment at a high frequency tends to occur. On the other hand, if they are coupled weakly to avoid mutual entrainment, phase is determined by initial conditions and will be very subject to perturbation (Pavlidis, 1969). More recent analyses of this problem have focused on nonlinear oscillations of the general form described by Prigogine and co-workers (Prigogine and Lefever, 1968; Nicolis and Prigogine, 1977), and less commonly on the Be-

102

R. R. KLEVECZ ET AL.

lousov-Zaikin-Zhabotinsky reaction (Belousov, 1958; Zaikin and Zhabotinsky, 1970; Field and Noyes, 1974). The properties that make them appropriate as biological timekeepers have been discussed elsewhere (Winfree, 1970; Pavlidis, 1973; Klevecz et al., 1980a). Pavlidis (1969, 1973) has considered in some detail the consequences of weak or strong inhibitory coupling in populations of oscillators. Though analytically his study dealt only with weakly nonlinear harmonic oscillators, he found that when coupling is strong individual oscillators synchronize to a new lower frequency. For weaker coupling some of the highfrequency oscillations are expressed along with the lower frequency output. To understand the applicability of limit cycles as descriptors of cellular behavior it may help to begin with a most general property of the cell cycle, namely, its considerable period stability combined with phase lability. In spite of the complexity of processes occurring during the cell cycle, it is striking that cells which are so readily perturbed from their normal cycle do typically return to that same cycle period after a time, even following rather large perturbations. This suggests that the overall dynamic state of the cell cycle can be described rather simply as one in which most cellular states, as long as they are consistent with viability, relax smoothly back to the normal cyclical sequence of states following perturbations. A natural representation for this kind of behavior comes from the concept of stable attractors in nonlinear dynamic systems. The simplest form for a stable cycle, with two interacting variables, is termed a limit cycle. The motivation behind using limit cycle models to represent cellular behavior is the idea that cells exploit a timekeeping oscillator, consisting of self-sustained oscillations in a large number of continuously varying constituents to maintain periodic behavior and coordinate the timing of cell cycle events. The limit cycle is the simplest system which can maintain an oscillation that will reestablish itself after some perturbation and return stably to the same period and amplitude. A minimum of two variables with nonlinear interactions is required and, since it is an open, nonequilibrium system, there is net energy or mass input to the system. Limit cycle models handle the following cellular behaviors without difficulty: ( 1 ) variable phase resetting curves with both advances and delays (Klevecz et al., 1978), and obviate awkward phrases like negative delay (Smith and Mitchison, 1976), (2) altered scheduling or periodic skipping of cellular events such as DNA synthesis or mitosis after perturbation or mutation, as for example, in the yeast cdc-4 mutant experiments (Hartwell, 1974), (3) phase compromise as in Physarum fusion experiments (Kauffman and Wille, 1975), (4) induced arrhythmicity when the system is driven to, or near the phaseless point or singularity (Winfree, 1974), (5) probabilistic generation time distribution (Klevecz et al., 1980a; Smith and Martin, 1973), and (6) alternation or ping-ponging of generation time after perturbation or due to random variance (Kubitschek, 1971; Shymko and Klevecz, 1981; Jauker and Cleffman, 1970).

CELLULAR CLOCKS AND OSCILLATORS

103

In the following sections we will consider each of these behaviors in detail along with alternative models invoked to explain them. C. SEQUENTIALITY AND DETERMINISM Considerable excitement was generated a few years ago by Hartwell’s (1971) application of conditional lethality to the analysis of the yeast cell cycle. It was hoped that the great advances, particularly in phage genetics, achieved using temperature-sensitive mutations could be applied to dissect out the sequence of events leading from mitosis to mitosis. Initial success was notable and a considerable collection of events was associated with temperature sensitivity. It was also possible to establish a sequentiality to these events and thus begin to construct a set of skeleton pathways (Hartwell, 1974). With equal enthusiasm a number of individuals attempted to apply these techniques to the mammalian cell cycle (Burstin et al., 1974; Liskay, 1974). The results have not been illuminating. Unless one intends to find conditional lethals of the whole of metabolic and macromolecular chemistry of the system, such an approach will lead to an understanding of which downstream event precedes another but the dynamics of control processes and of cell division remain encrypted. In general what can be blocked and observed as blocked are events remote from the core of cellular timekeeping mechanisms. The most exquisite demonstration of this is Hartwell’s cdc-4 bud emergence mutant in which a new progeny S . cerevisiae buds from the parent but fails to separate (Hartwell, 1971). At a fixed interval (3.5 hours) slightly more than one wild-type cell cycle later and in the absence of DNA synthesis or nuclear division or other apparent cell cycling, another bud forms and after a second such interval another. Such results are significant in demonstrating that events in the cycle beyond the site of action of the lesion continue to occur, and that the temporal schedule of cycling is itself unaffected. If this is true when a process as important as DNA replication is inhibited, then it would not be surprising if it were true for other processes as well, and one might suspect that a considerable number of such noncritical mutations have yet to be discovered. This example points up some difficulties associated with the otherwise powerful cdc mutant approach to analyzing the cell cycle. The class of mutants which are readily selected and useful in revealing the logical structure of the cell cycle have the property that a cell’s progress through the cycle is only affected at the critical site of action of the gene. That is, cells beyond that site progress normally around the cycle until they return to the site on the next cycle. In fact, this is the only possibility within the limited conceptual picture of the cell cycle as a closed, branched sequence of causally related events, with each event dependent on the

104

R. R. KLEVECZ ET AL.

function of a single gene. Even in this restricted view, however, the mutant search will only recover mutants in processes which are necessary conditions for subsequent events. Within its own framework, the cdc approach will readily uncover mutants only where the causal web is very simple and the deficient event measurable. In addition, such an approach should only work and in fact finds its best application if the cell cycle period is equal to the clock period, a point about which we will have more to add later. D. RANDOMNESS AND PROBABILITY IN THE CELLCYCLE Generation times in populations of cultured mammalian cells typically fall in a right-skewed distribution, with a significant number of cells showing much longer generation times than the mode. The shape of the generation time distribution has been characterized as reciprocal normal, derived from the assumption that cycling rates are normally distributed (Kubitschek, 1962, 1971; Pardee et al., 1979), or negative exponential, based on the transition probability model (Burns and Tannock, 1970; Smith and Martin, 1973). In this second model, the cell cycle is divided into two parts, an indeterminate “A-state’’ which cells enter after mitosis and leave randomly at a constant rate per unit time, and a second “B-phase” which is traversed in a constant time TB. The resulting generation time distribution has a right-skewed, and occasionally, negative exponential tail. Most discussions of this model have been in terms of the so-called a-curve (the plot of the fraction of cells undivided versus cell age) and the p-curve (the distribution of differences in sister cell generation times). In a semilogarithmic plot, both curves are predicted to have straight lines of equal negative slope (Minor and Smith, 1974; Shields, 1978). While the transition probability model does reasonably well in predicting some of the statistical properties of the cell cycle, it makes no predictions about phase response curves, phase averaging following fusion, alternation of generation time, or quantizement. It must also be said that a rigorous statistical test of goodness of fit in support of this model has not been performed and even by qualitative measures the rules of the model are observed more in the breach (Grove and Cristofalo, 1976; Pardee et al., 1979). Most grievously it abides awkwardly with all that we know of control in biochemical systems by positive and negative feedback.

11. Emanations of the Oscillator in the Cell Cycle A. OSCILLATOR DYNAMICS

Typical properties of a limit cycle are shown in Fig. 1. The concentrations of two substances, X and Y, hypothetically responsible for the timing of cellular

105

CELLULAR CLOCKS AND OSCILLATORS

events, are plotted at each instant in time in the XY state space of this oscillation. The normal cycling system progresses clockwise (in this model) around a closed orbit in the XY state space. If the system is released at a point off the stable trajectory it will wind inward or outward, and in the limit approach the cycle asymptotically. Inside the limit cycle is a unique unstable steady state, or singularity, with the property that if the system is released or placed at that state, it remains unmoved; but if perturbed incrementally off the steady state, it returns to the limit cycle, In most formulations of this model, cell cycle events are postulated to occur in specific regions of XY variable space; for example, with some cell cycle event assumed to be triggered when one of the variables exceeds a threshold value. The difference between this view and those involving a single control variable, in some explicit forms termed a relaxation oscillator (Tyson and Sachsenmaier, 1978), and in less explicit forms termed inhibitor dilution or activator accumulation models (Mitchison, 197 1; Nurse, 1975; Sudbery and Grant, 1976), is that the continued oscillation of the system need not depend on the completion of any triggered event. This could happen, for example, if some perturbation raises the threshold without affecting the limit cycle dynamics, or if a perturbation or random fluctuation brings the oscillator variables within the cycle such that the system passes outside the region of triggering values. In other words, limit cycle models admit the important possibility of subthreshold oscillations. Go might be so represented. It should be emphasized that relaxation oscillator and limit cycle oscillator models are not distinguished by their kinetic behaviors, that is, the slow rise and rapid fall of the relaxation oscillator variable versus the more sinusoidal oscillations possible in a limit cycle oscillator (Fig. 1). A limit cycle can easily be constructed which has rapid “relaxation” kinetics in some part of the cycle, making many of the predictions of both models similar. Given this similarity, the argument taken up by Tyson and Sachsenmaier ( 1 978) for timing of cell division by a relaxation oscillator is not particularly meaningful unless one holds to the biochemically incredible notion that timing of cell division is controlled by a single variable. B. EXPONENTIAL GENERATION TIMEDISTRIBUTIONS FOR CYCLEOSCILLATOR

A

NOISY LIMIT

Though random behavior in cell populations is readily observed there is evidence for concurrent accurate timekeeping in the cell cycle [see, for example, a recent review by Edmunds and Adams (1981)l. In our experiments using timelapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hour intervals in cultured hamster cells (Klevecz, 1976). Phase response curves constructed following application of heat shock, ionizing radiation, inhibitors of

R. R. KLEVECZ ET AL

106

6t

4

D

3

Y 2

1

I I

3

2

4

1

1

I

2

I

3

I

I

4

5

X

X

I

6

BI

C

4

Y

Y

2

1

1

I

1

2

3

X

2

4

X

FIG. I . Phase plane portraits of the Brusselator. Approaches to the stable trajectory were simulated using differing values of parameters A and B in the equations x = A - Ex + x2y - x ; y = Ex x2y are shown. Released at any point in x , y space, the system moves in a clockwise direction to the stable trajectory surrounding the steady state (indicated by “0”).Crossing the threshold is assumed to trigger specific cell cycle events. Crosses indicate intervals of 1/10 cycle. Note the rapid movement around the “fast arc” in the lower right-hand portion of the cycle in c in contrast to a. (A) A relatively sinusoidal form of the oscillator with A = 1.35 and B = 3.0. 0 = 2.22. Released at x = I ; y = 1. Approach to the stable trajectory requires more turns of the oscillator than in b or c. (B) Here A = 0.9, B = 2 and the approach to the stable trajectory is more rapid. (C) A more relaxation oscillator like behavior is achieved by setting A = 0.5, B = 2. Note the very rapid approach to the stable trajectory. (D)Phase plane trajectory of the oscillator used to simulate results in this study. Values of parameters A and B are the same as those in c but here a random element has been added. Threshold 0 is indicated by the horizontal line. All simulations were done using parameter values A = 0.5, B =

6

CELLULAR CLOCKS AND OSCILLATORS

107

CAMP phosphodiesterase, or serum pulses in each case show a pattern which is repeated twice in cells with 8-9 hour modal generation time. To account for such data, we have described a cell cycle model in which an independent cellular clock controls cell cycle events (Klevecz et al., 1980a). We show here that if a random component is included in such a model, the observed generation time distributions including the a-and p-curves can be predicted without the necessity for postulating a localized indeterminate state in the cell cycle. We have represented the random movement by adding a random walk component to the movement of the variables X and Y around the limit cycle. Each random step can have components both parallel and perpendicular to the cycle trajectory. A limit cycle has a restoring force opposing movement away from the trajectory, but is neutral with respect to perturbations along the trajectory. This means that the effect of the random walk is to drive cells a limited distance off the cycle trajectory, and also to irreversibly desynchronize cell populations by spreading them out along the trajectory. The extent of the spread perpendicular to the cycle determines the fraction of cells crossing threshold on each cycle and therefore determines the slope of the exponential envelope of the cycle time distribution. The random motion parallel to the cycle spreads individual clusters of generation times but does not directly affect the exponential. With random fluctuations added to the normal movement of the cells around the limit cycle, a population of cells moves around the cycle in a cloud whose dispersion depends upon the relative magnitudes of the fluctuations and the rate of return of the system to the normal cycle trajectory. If threshold crossing is required for gating into the next cycle stage, part of the cloud may fail to cross threshold and therefore those cells will fail to advance in their cell cycle. Furthermore, if the random fluctuations are large enough, the position of a cell within the cloud during one cycle will be essentially uncorrelated with its position one cycle later, so that its probability of achieving threshold is constant per cycle, or equivalently constant per unit time. In our simulations we have chosen the random walk parameters so as to match the rate of desynchronization observed in cultures synchronized by mitotic shakeoff and have used these values of the parameters in simulating &-curves (Fig. 2). It is necessary to point out that the random element introduced discretely into these simulations might as well be accomplished by a timing oscillator that is the sum of a set of coupled, mutually inhibitory oscillators of a nonidentical but similar period. Such higher order systems display quasiperiodic behavior (Pavlidis, 1969) which for choice of appropriate parameters can be indis2.0, and random walk parameters D,,= D, = 0.024. Time for two loops was set at 8.9 hours in all cases. Threshold was chosen at 4.7 in this figure. A random step was taken at each of 200 time steps per oscillator cycle. The unperturbed system would move clockwise around a closed trajectory, crossing threshold on each cycle. With random walk present as shown, the system can pass below threshold and can fail to exceed threshold for two or more cycles.

108

R. R. KLEVECZ ET AL.

FIG. 2. Experimental and simulated generation time distributions. To measure intermitotic times (IMT) of random cultures, cells were cultured at low densities and after 18 hours of growth, recording for analysis was begun using a time lapse video-tape recorder. Individual cells were tracked and a Markov chain or pedigree chart of generations, Tg, constructed for each mother cell, its daughters, etc., for up to 5 generations. a plots and intermitotic time distributions were calculated from IMT data. Alpha curves of V79 and WI38 cells and their simulation by the oscillator model. Alpha curves describing the undivided fraction of cells in generation time distribution curves are shown; 200 -t 50 generation times are represented in each curve. Triangles indicate distributions of generation times in V79 cells growing under suboptimal conditions, while squares indicate the distribution of V79 cells growing under optimal conditions. Simulation of these curves (solid lines) shows that the distribution of generation times is exponential but quantized within the exponential envelope. Under suboptimal conditions, threshold 0 = 4.7, under optimal conditions 0 = 4.5. All other parameters are unchanged from those in Fig. 1. Circles indicate generation time distributions of W138 cells and the lines give the simulated distributions. Here parameters are the same except that cycle time is increased by requiring two rather than four oscillator loops in one cycle. Note that with the long cell cycle generated by this model, the resulting distribution of generation times is smoother and approaches a straight line at long generation times.

tinguishable from a single oscillator operating in a noisy environment. For clarity in understanding the role of the putative oscillator in the cell cycle, the simpler form has been chosen in these simulations. In this system, we now have an explicit mechanism for generating an overall exponential cycle time distribution as in the transition probability model, but in addition, the cell cycle times are clustered at intervals corresponding to the limit cycle period, which according to our experiments should be about 4 hours. As in the transition probability model, the random component (i.e., the 4-hour limit cycle with noise) could be placed anywhere in the cell cycle, and both quantizement and the exponential distribution would still result. However, since the quantizement interval, and by hypothesis the limit cycle period, are a fraction of the total cell cycle time, an important question is whether the timekeeping behavior persists through more of the cycle than G , , where most of the vari-

CELLULAR CLOCKS AND OSCILLATORS

109

ability in cycle time typically occurs (Pardee et a / . , 1979), and therefore can account for Go and the G,-less cycle (Cooper, 1979). If longer generation times are built up of multiple loops of a fundamental 4hour subcycle, phase perturbation response in long Tg cells should show multiple repeats, and there should be 4-hour quantizement in the IMT distribution. Such phase perturbation data are unavailable, but there is some evidence for quantizement in long Tg cells (Absher er d . , 1974; Absher and Absher, 1976; Klevecz, 1976). However, in general, long Tg cells tend to have smoother IMT distributions and smoother a-curves than short Tg cells. In the model this occurs naturally as a result of the greater spreading in Tg over a larger number of oscillator cycles. Figure 2 shows a- and p-curves for WI38 cells, taken from data of Absher and Absher (1976). The simulation was done assuming four loops per cell cycle, with all parameters except threshold 0 the same as for the V79 cells. It is important to note that even without the smoothing associated with cell desynchronization, any detailed structure in the IMT distribution becomes difficult to visualize when displayed in a cumulative logarithmic plot such as the aor p-curve. A cell cycle timekeeping mechanism such as we have described here would produce undulations in these curves; therefore small deviations from a pure exponential may be significant, and should not be overlooked. Note also that even though quantizement arises as a consequence of random motion in oscillator variables it is possible, if the restoring force that tends to give the oscillator a constant trajectory is strong enough, for all cells in the population to stay sufficiently close to the stable trajectory to execute cellular events in the minimum time set by the oscillator and hence for the quantizement of generation times to disappear. C. PERIODIC GENEEXPRESSION AND QUANTIZED GENERATION TIMEIN MAMMALIAN CELLSAS MANIFESTATIONS OF THE CELLULAR CLOCK Fluctuations through the cell cycle in the rates of RNA and DNA synthesis and in the levels of a number of enzymes and proteins are a common if not universal feature of cultured mammalian cells (Klevecz and Forrest, 1976). These fluctuations are more apparent in diploid or cloned aneuploid cells with short cell cycles than they are in heteroploid and transformed cells, particularly when these cells have relatively long cell cycles. While the maxima in various processes and activities are out of phase with one another they often have a similar 4-hour periodicity. This work began as an attempt to establish whether the early replicating genes in mammalian cells were preferentially transcribed. We observed that the rate of total RNA synthesis as measured by pulse incorporation of [3H]uridine fluctuated through the cell cycle of hamster cells and that the fluctuations in rate appeared to bear some relationship to DNA synthesis. Because the rate of incorporation doubled soon after the beginning of S phase it suggested a gene dosage

110

R. R. KLEVECZ ET AL.

effect or stimulated transcription following replication, although it could not have been simply a matter of gene dosage since marked fluctuations in rate appeared to be superimposed on the doubling (Klevecz and Stubblefield, 1967). In pursuing the possibility that gene expression might vary through the cycle in some relationship to gene dosage, we observed that lactate dehydrogenase showed 2- to 4-fold fluctuations in enzyme activity through the cell cycle of hamster cells (Klevecz and Ruddle, 1968; Klevecz, 1969a). Maxima in activity occurred at 3.5-hour intervals through the cycle and appeared to lag behind maxima in ribosomal RNA synthesis by a small amount (Klevecz and Ruddle, 1968). Subsequently it was observed that there were multiple peaks in [3H]thymidineincorporation in the S phase of these cells, and that the maxima in thymidine incorporation were out of phase with maxima in LDH activity (Klevecz, 1969b). Many enzymes have since been analyzed in the cell cycle of several Chinese hamster cell lines and oscillations with a 3-4 hour periodicity in activity are a common, if not universal, property of the many cell lines (Klevecz et al., 1975; Klevecz and Forrest, 1976). It often appeared that fixed phase relationships existed between different enzymes but confirmation of this was beyond the limits of resolution of this system. It must be said that in the early cell cycle literature, particularly that dealing with the cell cycle in cultured mammalian cells, the majority of reports did not find evidence for enzyme oscillations or multiple bursts in DNA synthesis within S phase (Mitchison, 1971). The prevailing view might have been summarized as one cell cycle, one enzyme maximum, and clearly the cycle was, and is still, commonly viewed as the unit of timekeeping. In an attempt to rationalize the disparate observations Klevecz (1969a,b) suggested that in many instances the failure to observe enzyme oscillations or multiple bursts of DNA synthesis within S phase was due to the choice of cell lines and in particular to a loss of temporal order and organization in heteroploid tumor cells. It has become apparent that this generalization is not correct in its simplest form because some established nondiploid lines such as V79 (Klevecz, 1976; Forrest and Klevecz, 1978), tumor lines such as L5178Y (Kapp and Okada, 1972), and EMT-6 (King et al., 1980) and even the very heterogeneous HeLa cell (Collins, 197s) nevertheless displayed enzyme oscillations (Forrest and Klevecz, 1978), bimodal phase response (King et al., 1980), or multiple bursts in DNA synthesis with a single S phase (Collins, 1978). In the instances noted, the period of the enzyme oscillations or the bursts in DNA synthesis within S phase were 3.5-4 hours. It has also been noted that the number of peaks/cycle is related to the length of the cycle with two enzyme maxima (but only one burst in DNA synthesis) in cells with 8-9 hour generation times such as V79, L5178Y, and EMT-6, and three in cells with a 12 hour cycle such as Don and CHO hamster cells (Fig. 3). Most interesting is the fact that V79-8 cells normally have an 8-hour cycle, but at 36°C have an 1 1.5-hour cycle

CELLULAR CLOCKS AND OSCILLATORS

111

Hours After Mitotic

Selection

FIG.3. Kinetics and temporal relationships of thymidine incorporation and enzyme activity in synchronous cultures of V79 (Tg = 8 hours), Don (Tg = 12 hours), CHO (Tg = 12 hours), and W138 (Tg = 20 hours). Cultures were synchronized by mitotic selection and assayed for DNA synthesis and lactate dehydrogenase activity at hourly or half-hourly intervals thereafter as described previously (Klevecz and Kapp, 1973). Note that V79 cells with an 8-hour cycle replicate the bulk of their DNA in a single burst and display two maxima in enzyme activity whereas Don and CHO cells with a 12-hour cell cycle replicate DNA in two bursts and display three maxima in enzyme activity. W138 cells with a 20-hour generation time replicate DNA in three bursts and display multiple maxima in enzyme activity.

and two peaks in DNA synthesis (Holmquist, 1983). Whether this generalization will hold for cells with longer generation times will require further study with very well synchronized cultures. It has been noted though that W138 with a 20hour generation time displayed 4 or 5 maxima in enzyme activity and 2 or 3 bursts in DNA synthesis in a 12-hour S phase (Klevecz and Kapp, 1973). More recently Kapp and Painter (1977) and Kapp et al. (1979) have found multiple peaks with an approximate 4-hour period in thymidine incorporation in the S phase of recently explanted primary human lines. These relationships are summarized in Fig. 3. The notion of quantizement of generation time and the search for a short period clock in the cell cycle did not begin in earnest until it was observed that the distribution of possible generation times in mammalian cells did not appear to be continuous within the limits of range for each cell type; rather, generation time was quantized in multiples of 3-4 hours. Synchronous cultures of Chinese hamster V79 cells were prepared using manual and automated methods to select and stage mitotic cells. All IMT studies were done using synchronous cultures,

112

R. R . KLEVECZ ET AL.

time-lapse video tape phase microscopy, and very low light level ( l o p 6 ft cdl) video cameras. It was possible to show that generation times within a population of mitotically selected cells normally disperse in a quantized fashion, with intervals of 3-4 hours occurring between bursts in division (Klevecz, 1976). In addition, at temperatures above 37"C, V79 cells have a 7.5-8.5 hour modal cell cycle, while at temperatures below 35°C the modal cell cycle is 11-12 hours long (Klevecz and King, 1982). A survey of the synchrony literature reveals that the tendency to preferred and prohibited generation times holds between cell lines. The distribution of modal generation times from a variety of different cell types forms a series with a similar interval but with a greater range of values than that observed for V79 cells (Fig. 4). In Tetruhyrnenu pyriforrnis, Jauker (1975) observed that cells grown for considerable periods of time in sublethal doses of actinomycin D accumulated protein at a rate considerably different from the controls and in addition displayed an oscillation or, as we would call it, a quantizement of generation time. Whereas the control cultures normally divide with a generation time of 220 minutes (3.66 hours), the treated cells displayed generation times of 220,430, and 620 minutes or 3.66, 7.17, and 10.3 hours (Fig. 4). These times are separated by an interval of approximately 3.5 hours. The Tetruhyrnenu system can be thought of as having a clock of invariant period which will gate out cells that have achieved the appropriate developmental maturity only at fixed intervals. In addition the lack of balance or coordination between the clock and growth rate caused by the inhibitor leads to the alternation of generation times and the negative correlation in mother-daughter cell generation times. To satisfy the existing data, Klevecz (1976) proposed a subcycle, Gq, which had a traverse time equal to the period of the cellular clock. The period appeared to be fixed at close to the same value in all somatic cells. It follows that the period of the cellular clock is not necessarily equal to the generation time of the cell, rather the clock is suggested to have a relatively short period of 3 to 4 hours. The cell cycle is envisioned as being built up of multiples of this fundamental period. A summary of the evidence for a fundamental clock in the cell cycle is shown in Fig. 4.

D. PHASEPROPERTIES OF

THE

OSCILLATOR

Given the difficulty in studying gene expression and cellular timesense without inadvertently perturbing the system, some insight may be gained from experiments in which the cell is intentionally perturbed in order to assess its response and hence the phase sense of the putative oscillator and the cell cycle. Our understanding of phase rests initially on perturbation experiments in which one hopes to obtain information about the cell cycle timekeeping mechanism by observing the phase change in some marker event, usually mitosis, after

113

CELLULAR CLOCKS AND OSCILLATORS Tg-cultured mammalian cells

I 11 lllllll 1 1 1 11 I I1 I r-mutants - D melanogaster I 1 r-mutants- N.crassa I )I( I

Y 11 I I I I I Ill I1111

rquantizement -T. weisflagii

I

I

r-mutants-C. reinhardii

Ill I I I 1 11

I #-mutants -D. pseudo obscura I #-mutants -C. reinhardii 11 I I Tg normal and perturbed Tpyriformis 1) synchronous S phase bursts -mammalian cells I I I enzyme maxima- mammalian cells I

4

I

I

12

8

I

16

I

20

I

24

I

28

Time (hr) FIG.4. Expression of a fundamental oscillator in circadian and ultradian rhythms. Each line represents a peak of occurrence. Starting from the top of the figure: Line I . Quantized variation in generation times (Tg) of cultured mammalian cell lines. Generation times were determined from the published data on cells synchronized by mitotic selection or from time lapse analyses (IMT) of random cultures. The list is not exhaustive but represents a sampling of papers published between 1961 and the present in which the stated generation time could be directly confirmed in the data. Wherever possible modal generation times were obtained, and reports stating only population doubling time were excluded (Klevecz, 1976). Line 2. Polymodal distribution of generation times in the marine diatom Thalassiosira weisflogii growing in the circadian mode with normal T also expressed (Chisholm and Costello, 1980). Line 3. Long and short period (7) mutants of Drosophila melanogaster isolated by Konopka and Benzer (1971). Line 4. 7-mutants of Neurospora crassa studied by Feldman and co-workers (Feldman and Hoyle, 1973; Feldman et a l., 1978; Feldman and Dunlap, 1983). Line 5. Phototaxic 7-mutant of Chlamydomonas reinhardii picked and isolated by Bruce (1972) from cultures treated with nitrosoguanidine. Line 6. Phase angle (+) early eclosion mutants selected by Pittendrigh (1967) by continuous selection through SO generations for early emerging Drosophila pseudoobscura. Line 7 . mutants of Chlamydomonas reinhardii isolated by Bruce (1972) from cultures treated and selected for period T changes. Line 8. Oscillatory variations in generation times (Tg) of Terrahymena pyriformis perturbed by continuous incubation in low levels of Actinomycin D. Normal Tg in these culture is 4-4.5 hours (Jauker and Cleffman, 1970). Line 9. Interval between synchronous bursts in DNA synthesis in the S phase of mammalian cells (Klevecz, 1969b, 1973; Collins, 1978; Kapp et a / . , 1979; Holmquist, 1983) scored from a maximum slope of ['Hlthymidine incorporation rate between peaks. Line 10. Intervals between peaks in maximum enzyme activity or levels in the cell cycle of synchronous hamster cells in culture (Klevecz and Ruddle, 1968; Klevecz and Kapp, 1974; Klevecz et a l . , 1982).

+

114

R. R . KLEVECZ ET AL.

administration of a perturbing agent (Klevecz et al., 1980a). This approach is successful to the degree that the chosen perturbation affects the timekeeping mechanism directly, so that the phase response reflects the properties of the underlying clock. However, since the biochemical nature of the clock, and consequently the effects on this clock of any given perturbation, are largely unknown, it is difficult to identify confidently cellular phase responses with clock responses. Some or all of the observed phase shifts, for example, after any given perturbation may be due to defects in cellular functions normally under the control of an underlying clock, but not due to effects on the clock itself. In the absence of adequate methods for direct assessment of the clock, phase perturbation experiments yield results which taken in toto require at the least a reassessment of traditional views of the cell cycle.

E. DIVISION DELAYVERSUS PHASERESPONSE Excess division delay, or division set back, in response to a variety of chemical and physical agents is a well-characterized phenomenon in unicellular eukaryotic organisms (Rasmussen and Zeuthen, 1962). When synchronized cultures of Tetrahymena pyriformis or Schizosaccharomyces pombe are briefly exposed to heat or cycloheximide, they show a pattern of increasing division delay as the time of exposure occurs later and later in the cycle (Zeuthen, 1971; Polanshek, 1977). Late in the cell cycle there is a transition point beyond which no further delay or only a constant delay in division can be achieved. The heat shock and cycloheximide results have been discussed in terms of the accumulation of a protein structure which is needed for mitosis but which is unstable until completed at the transition point (Mitchison, 1971). According to this model, treatments such as heat shocks produce division synchrony because cells closest to mitosis are set back the farthest in the cell cycle. However, a paradox arises when an agent producing delays at one point in the cell cycle produces advances at another point (Smith and Mitchison, 1976), since destruction of a continuously accumulating cell component would be expected to uniformly cause delays. It was this paradox that first led us to consider the limit cycle model and the notion of phase response in discerning the underlying timekeeper in animal cells. Resetting is somewhat more complex in animal cells. Figure 5 shows the response of V79 Chinese hamster cells to serum pulses, heat shock, ionizing radiation, and CAMPphosphodiesterase inhibitors. Note the twice-repeated pattern of phase shifts through the modal 8.5-hour cycle. Since this work employed perturbations as mild as a transient increase in serum concentration to accomplish advances and delays in the time of division in Chinese hamster V79 cells, as well as severe heat shocks, and since the maximum delay in the cycle is as great as the maximum advance and both advances and delays are accomplished with the

115

CELLULAR CLOCKS AND OSCILLATORS

1

I

M1

M2

0

2 .

-2-

'

4

.

..

-4 -

2

6

8

6

8

I

.. .

.= 4

CELL CYCLE TIME (hr)

FIG. 5. Phase response of synchronous V79 cells to perturbation by serum, heat shock, and ionizing radiation and RO- 1724 a cAMP phosphodiesterase inhibitor. (a) cAMP phosphodiesterase inhibition. At 0.5-hour intervals following mitotic selection cells were exposed to RO-1724 at 50 )wM for 10 minutes. The medium was then replaced with conditioned medium. (b) Serum pulses. At intervals following mitotic selection, serum concentration in the medium was increased from 5 to 20%. Midpoints of first ( 0 )and second (0) mitotic waves. (c) Ionizing radiation. Synchronous V79 cells were exposed to 150 rads from a cobalt-60 source at 30-minute intervals through the first synchronous cell cycle. Analyses of division advance or delay were determined as described. (d) Heat shock. Midpoints of the first mitotic wave following synchronization and a 10-minute 45°C heat shock are compared for each pair of heat shocked and control cultures as described in Klevecz et a / . (1980a).

same agent, these results are difficult to reconcile with the division protein model but are a predicted outcome of timekeeping by a limit cycle oscillator.

F. PHASERESPONSETO SUBLETHAL HEATSHOCK Synchronous cultures of V79 cells show a cell cycle dependent variation in sensitivity to lethal heat shocks (King et a l . , 1980). Maximum sensitivity to single lethal heat shocks of 45°C 25 minutes occurs at 4 hours, a point close to the time of maximum phase reset resulting from a nonlethal pulse, while minimum sensitivity to lethal heat shock occurs at 1 hour in the cycle, a point close to the time of minimum reset for a nonlethal pulse. When mitotically selected V79

RMoToLERANcE

cLmmm/c/wTREATED CLONOGEN/C/TY CONTROL

g[ 8 ,

10 O;

,oo

,

PHASE DELAY (HOURS)

,

0

+

ADVANCE

-

0

RELATIVE RATE HSP SYNTHESIS

DELAY

PHASE RESPONSE A$ (HOURS)

CELLULAR CLOCKS AND OSCILLATORS

117

cells are given heat shocks at temperatures between 42 and 45°C through one synchronous cell cycle, resetting curves showing the advance or delay in mitosis are generated. In Fig. 6 the phase response of cells to 10 minute pulses at 42,43, and 45°C at 0.5-hour intervals through the cell cycle and a single response to 42°C 40 minutes are shown together with a simulated phase response curve predicted by hard resetting (90% of phase specifying molecules destroyed) of a limit cycle oscillator (Klevecz et al., 1980a; Shymko and Klevecz, 1981). Cells pulsed soon after mitosis are slightly delayed in the subsequent mitoses relative to the paired unshocked control. Minimum delay, and in some instances a slight advance, occurs when shocks are given 1.5 hours after mitosis. There follows a pattern of increasing delays up to 4.5 hours, when an abrupt shift in response occurs giving a second minimum in delay at about 5.5 hours of the cell cycle. Pulses given later than 6 hours in the cycle give a pattern of increasing delays up to the subsequent mitosis. The response curve appears as two parallel lines sloping downward to the right, with a small cluster of values between 4 and 6 hours of the cycle showing a constant 2 hour delay. In some experiments heat shocks given after 4 hours showed a splitting of the anaphase frequency histoFIG. 6. (A) Phase response to conditioning heat pulses in comparison with heat shock protein (HSP) synthesis in non-heat-shocked cells. Synchronized V79 cells were heat shocked by complete immersion of 25-cm2 plastic flasks in a Lauda model K2/R water bath. Water bath temperatures were maintained within 0.1"C of desired temperature during the heating interval and were calibrated against an NBS thermometer. Means of first mitotic wave following synchronization were compared for each pair of heat shocked and control cultures as a function of time in the cycle at which the heat shock was begun. Heat shocks of 45°C 10 minutes (A), 43°C 10 minute (a),42°C 10 minute (O), and 42" 40 minute ( X ) were employed. Positive values of A+ indicate that heated cultures divided sooner than controls; negative values, later than controls. Simulation of the phase change as specified by a limit cycle oscillator assuming destruction of 90% of both X and Y components is indicated by the dashed line. Heat shock proteins were identified in random cultures by treatment at 45°C 10 minutes followed by harvesting 3 hours later. Cyclic fluctuations in synthesis of the three major heat shock proteins in unperturbed synchronous cultures were measured by labeling for I hour with 200 pCi/ml [35S]methionine in standard medium. The midpoint of the labeling interval is indicated in the figure (-0). (B) Division delay following 45°C (A), 43°C (O),and 42OC (0) heat pulses for intervals from 2 to 90 minutes applied at 3.75 hours into the V79 cell cycle. Delays in the first synchronous division following heat shock are shown relative to paired controls as in (A) except that here delay is plotted as a positive value. Arrows indicate heat pulse duration necessary for maximum survival in (C). (C) Thermotolerance development following 45°C (TO), 43°C (a),or 42°C (0) heat pulses for durations of 2-90 minutes. Random V79 cells, 24 hours after subculture were trypsinized, counted, and appropriate numbers of cells were seeded in 25-cm2 flasks containing 10 ml of media which had been gassed for 24 hours previously in an atmosphere of 95% air and 5% C02. Inoculated flasks were then sealed and cells were allowed to attach by incubation at 37°C for 4 hours prior to heat treatment. Following pretreatment, cells were incubated for 4 hours at 37°C then reheated at 45°C for 24 minutes. Flasks were then returned to 37°C and assayed for clonogenicity 7-10 days later. Thermotolerance is expressed as the mean number of colonies in pretreated flasks divided by the mean number of colonies in unpretreated controls. A measurable enhancement of survival (2 to 4fold) appears within 10-15 minutes of conditioning heat treatment and plateaus by 4 hours. (From Klevecz et al., 1982.)

118

R. R. KLEVECZ ET AL.

gram, suggesting that at these values of time and perturbing stimulus member cells may be either slightly advanced or delayed, or quantally delayed by A+ + 4 hours. It is often the case in calculating the mean or midpoint of the population that the value will be found to lie between the two discrete peaks. This may serve to explain why in earlier division delay studies the results were described as showing a transition point with constant delays in response to perturbations late in the cycle. This capacity to phase jump is further shown when on occasion the midpoint of a population shocked at 7 hours is delayed, not by 0.5-1.0 hours, but by 5 hours, or a full subcycle. The phase response curves to 43 and 45°C are nearly identical while 10-minute pulses at 42°C through the cell cycle produce only a slight advancing or delaying effect on cell division. This sharp transition in phase delay between 42 and 43°C is of particular interest because of the fact that 42.5"C appears in the literature as a transition point in the induction of thermotolerance by acute vs continuous hyperthermia (Sapareto et al., 1978). Significantly in Tetrahymena which has a 3.7-hour cycle only a continuous smooth resetting curve is obtained as would be the case if the timekeeping oscillator and the cell cycle were of equal duration. The above results are consistent with a picture in which the perturbing agent affects the clock directly and cycling of the timekeeper continues through the entire cycle, in this case, repeating twice during the cell cycle.

G . PHASERESET AND THERMOTOLERANCE Phase reset and cell cycle redistribution are coupled to the development of thermotolerance. Induction of maximum thermotolerance occurs under pretreatment conditions just sufficient to cause full phase reset. When mitotically selected V79 cells are treated at 3.75 hours after mitosis with various durations of heating at 42, 43, and 45"C, division delay increases continuously with increasing duration of heating until a maximum continuous delay of 4 hours is achieved. Beyond that point, discrete, or quantized increments in delay occur with increasing exposure time. Similarly when random V79 cells are preheated for increasing durations at 42, 43, and 45°C and then assayed 2, 4, or 8 hours later for thermotolerance development, maximum thermotolerance is induced by that duration of heating at each temperature which is just sufficient to cause a full 4 hour reset (Fig. 6). Additional heat pretreatment beyond the minimum necessary to produce a full reset results in diminishing thermotolerance. The induction of thermotolerance behaves as though it were dependent upon the cell reaching a particular region in oscillator phase space. It is consistent to suggest that heat shock and stress proteins, which have been implicated in the acquisition of thermotolerance (Li et al., 1982), are preferentially translated phase-specific proteins and for that reason are not unique to heat-shocked cells. We think rather that they will be found at particular phases in synchronized cells

119

CELLULAR CLOCKS AND OSCILLATORS

or whenever a random, synchronous, or quiescent culture is driven to the appropriate phase space by any perturbation. This is supported by the observation that heat shock proteins appear in response to such phase-shifting agents as ethanol, anoxia, and a variety of chemical and mechanical stresses (Ashburner and Bonner, 1979; Guttman et al., 1980).

H. DOUBLEPULSEEXPERIMENTS By perturbing the cells with two heat pulses we have attempted to determine whether the results are consistent with a resetting of the putative timekeeping oscillator to a new phase or whether the results can be explained more prosaically by saying that the cells are simply arrested at the cycle phase in which the heat pulses were applied. From Fig. 6 it can be seen that treating synchronized cells at 3.75 hours after mitotic selection with 43°C 10-minute heat pulses causes a 4hour delay and treating with 43°C 20-minute heat pulses causes an 8-hour delay. If the 20-minute heat treatment is separated into two 10-minute heat pulses separated by an incubation at 37"C, the cells respond to the second pulse as though the first pulse had reset them to an oscillator phase roughly equivalent to a point 0 to 1 hour after mitosis, a point of relative heat insensitivity. This redistribution occurs instantaneously, or nearly so, for if the two pulses are separated by as little as 10 minutes at 37°C no delay beyond the 4-hour delay produced by a single pulse is manifest. According to our model the cells which were at 3.75 hour in the cell cycle have been reset to an oscillator phase equivalent to 0-1 hour in the cell cycle and respond to the second heat pulse as cells in this cycle phase respond to single heat pulses; that is, they show no further delay. Similar reset kinetics are seen in thermotolerance development. If randon V79 cells are pretreated with two 43°C 10-minuteheat pulses separated by 10 minutes at 37°C and then assayed 4 hours later for thermotolerance, subsequent thermotolerance development is identical to that produced by a single 43°C 10minute heat pretreatment and greater than that produced by a single 43°C 20minute pretreatment. One might argue from this that the intervening 10 minutes is sufficient to allow repair of sublethal damage or the simple elaboration and protective effect of heat shock proteins but, as the time at 37°C between the two pulses is increased to 4 hours, the delay produced by the two pulses increases beyond 4 hours to a maximum of 8 hours, suggesting the cells have again traversed into a sensitive phase and respond as would cells in the phase equivalent to 2-4 hours after mitosis. If, as appears to be the case, heat shock proteins are made in response to the first pulse, then either they are labile or their mere presence while conferring survival benefit has no effect on phase lability. From flow cytometric analysis of cell DNA content it would appear that oscillator phase is uncoupled from the chromosome replication cycle (CRC). Synchronous cell cultures were treated at 3.5-4 hours into the cycle at 43 or

-

120

R . R. KLEVECZ ET AL.

45°C for 10 minutes and simultaneously monitored for phase response and cell division using video microscopy and for DNA content using propidium iodide fluorescence and flow cytometry. At the time of heat treatment DNA content in these cells was slightly greater than the 2C (G,) amount. No further change in DNA content was detectable for the ensuing 4 hours. At the time when the phase response curve would predict that cells were again in early S phase with respect to the timekeeping oscillator, the normal increase in DNA content, expressed in control cells 4 hours earlier, resumed and was completed in the normal time interval and with apparently normal kinetics. An interesting prediction of the model is that a hard reset from G, to G, for example, by 45°C 12 minutes or 43°C 20 minutes should cause the cell to reinitiate DNA synthesis and become tetraploid as has recently been shown though with a differing interpretation (Read et al., 1982). I. EFFECTSOF SIZEON THE EXPRESSION OF A TIMEKEEPING OSCILLATOR AND THE QUANTIZEMENT OF GENERATION TIME

Size as an element of cellular timekeeping has much support in the literature. The evidence for such effects is strongest in nonmammalian cells such as yeast, where initiation of DNA synthesis and nuclear division both appear to depend on the cell’s attaining a critical size (Nurse, 1975; Nurse and Thuriaux, 1977). In mammalian cells, evidence for a size effect in the cell cycle is mixed. In early experiments on mouse fibroblasts, Killander and Zetterberg (1965) concluded that there is a critical size requirement for entry into S phase. More recently, however, Yen and co-workers, using percentage labeled mitoses methods (1975) and flow cytometry (Yen and Pardee, 1979), presented evidence suggesting that large cells, or cells with large nuclei, cycle faster than small ones, but concluded that this was not related to a critical size threshold for entry into S . We considered the possibility of interactive coupling between a timekeeping oscillator and some set of variables which are a function of size. For simplicity, size was taken to be a simple continuous function of cell age acting as a boundary condition; when triggering oscillator values were achieved a cell was considered to be capable of executing the event or process only if adequate size had previously or simultaneously been attained. Growth rate was in these simulations uncoupled from the oscillator dynamic system. In the strictest sense the model is flawed in this respect but leads to an adequate simulation of one of the more peculiar cellular behaviors, the ping-ponging of generation time, also described as quantizement and resulting in a negative correlation between mother and daughter cell generation times. As mentioned above if the restoring force which operates to produce the stable trajectory characteristic of a limit cycle is sufficiently strong then the system tends to suppress random movement away from the stable trajectory and hence to

CELLULAR CLOCKS AND OSCILLATORS

121

suppress the quantizement of generation time due to subthreshold oscillations. However, quantizement is reintroduced when the oscillator is bounded by size, because fluctuations in growth rate can alter the time at which threshold cell size is reached, thereby advancing or delaying triggering by one period of the limit cycle oscillator. In the next section, we will discuss the possibility that cell size is an appropriate general parameter for representing the causally related events in the cell cycle. As we will show, the flavor of the global behavior of cycling cells can be captured by superimposing a size threshold constraint on a continuous limit cycle timekeeper in the cell which gates the timing of cell cycle events. J . AN EXPLICIT MODELIN WHICHCELLDIVISION Is GATEDBY CELLSIZE Quantizement can arise as well from the interaction of size and oscillatory timekeeping and indeed quantizement cannot be readily generated by any model which uses some function of size alone to trigger mitosis. So long as generation time is a continuous function of growth rate or of the accumulation of some hypothetical mitogen, a continuous change in growth rate would be expected to yield a proportional but continuous change in generation time. If it can be shown that perturbation of the system, or manipulation of the growth rate to produce unbalanced growth, leads to quantizement or ping-ponging of generation time into discrete intervals then it is possible to simulate this behavior only by proposing that division is gated by a timekeeping mechanism that acts largely independently of size. That such phenomena occur is evidenced by a number of systems. We have carried out a theoretical analysis to assess the cmsequences of a size threshold superimposed on a limit cycle model of the cell cycle as described above (Shymko and Klevecz, 1981). In order to account for the quantizement effect, and phase perturbation results, we chose a two-loop model with a threshold crossing required for gating from one loop to the next, and further constrained the model by requiring that a critical size be reached before gating is implemented to exit from the second loop. Then, if cell growth is rapid enough size has no effect on the timing of mitosis, but with slow growth a cell remains within the second loop until the critical size is reached (Fig. 7). This kind of model, and in fact most models incorporating a size threshold, can be expected to have the qualitative property that sibling cell generation times are positively correlated, while those of mother/daughter cells are negatively correlated. The reason is that a cell which is larger than usual at division will have large daughters which more quickly reach the size threshold (if growth rates are unaltered) and therefore divide sooner on average than the daughters of a smaller parent. Conversely, if a cell randomly has a longer cycle time it will have a longer time to grow and will be larger than average at division. Its daughters will have a short cycle time as discussed above, and a pattern of negative correlation between cycle times in alternate generations will result. Positive

122

R. R . KLEVECZ ET AL.

x , Rc. 7. Schematic representation of a limit cycle model with a size threshold added. Limit cycle oscillations occur in X and Y while size increases in the third dimension. Crossing of the Y threshold (stippled top of the box) is ineffective in triggering mitosis until the threshold size (open face of the box) is reached. (From Shymko and Klevecz, 1981.)

correlations between sister cell generation times have been reported (McQuilkin and Earle, 1962; Minor and Smith, 1974), along with some reference to negative parent/daughter correlations (Killander and Zetterberg, 1965; Dawson et al., 1965; Jauker and Cleffman, 1970; Absher and Absher, 1976; Absher et al., 1974). Bimodal or quantized generation time distributions have also been reported (Van Wijk and Van de Pol 1979). A simulation observed of these results using the size bounded oscillator described above is shown in Fig. 7. The issue of generation time correlations has also been addressed in the context of the transition probability model. Brooks and co-workers (1980) were able to generate sister cell generation time correlations while retaining the random behavior inherent in the model, by proposing that a variable interval “L” begins in one cycle and is completed in the next cycle after division, after which follows a purely random interval as in the simple transition probability model. The generation-spanning “memory” resulting from the L-interval gives the positive sister-sister correlations while the random interval maintains the desired properties of the a-and p-curves. Cell size provides a more direct way to give exactly the same relationship between sister cells as does the L-interval because of the sharing of cell mass between dividing cells. Therefore in addition to its previously described predictive power, the limit cycle model with superimposed size threshold is able to account for that cell behavior addressed by the modified transition probability model, but now suggests a more physically based mechanism for that behavior.

CELLULAR CLOCKS AND OSCILLATORS

123

Whether cell size does indeed play a strong role in cell cycle control in mammalian cells remains to be proven by experiment. A perhaps more important property of the model we have described here is that it provides a paradigm for a system which has both an independent timekeeper and a set of causally connected cell cycle events, in this case corresponding to cell growth, either of which can dominate the timing of cell cycle events, depending on the growth conditions. 111. Are Cellular Clocks an Essential Element of Circadian Rhythms

The macroscopic properties of circadian rhythms have been characterized by Pittendrigh, Bunning, Aschoff, and their many students in the course of the past 30 years. But, while it is widely agreed that underlying the overt circadian rhythms there exists an endogenous, cellularly based oscillator or ensemble of oscillators with stable trajectories that specify the phase and period of the clock, identifying these oscillators with any particular element of cellular or organismic chemistry has proven elusive (Robertson, 1975). Even in the case where arrhythmic mutants of the D . melanogaster clock have been isolated (Konopka and Benzer, 1971) it is not clear whether these are mutants of “the” clock or merely the result of the uncoupling of the endogenous oscillator, or the disintegration of a set of oscillators, from one of the systems it drives. In other words, the cellular clock may be intact but not expressed in an organized way in the whole animal. In other cases clock mutants are exclusively period (7)mutants and the periods of these mutants appear to cluster around values that vary in quanta1 increments from the wild type. This is particularly true of D . melanogaster (Konopka and Benzer, 1971), excluding the arrhythmic mutant, and C . reinhardii (Bruce, 1972) but less perfectly so of N . crassa where periods cluster at both 2.5- and 5hour intervals (Feldman and Hoyle, 1976; Feldman et al., 1978). It is not yet universally agreed whether the stable period and phase relationships of circadian rhythms arise by differential strengths of coupling between endogenous rhythms that are themselves of circadian duration (Pittendrigh and Daan, 1976) or whether inhibitory coupling of higher frequency oscillations is involved (Pavlidis, 1973). In addition, an argument may still be joined with workers studying circadian rhythmicity in higher organisms regarding the cellular basis of organismic rhythmicity, or perhaps to put it more accurately, whether all cells of an organism are capable of clockedness. To some degree this is a matter of definition. In mammals for example, the capacity to sense and transduce temporal information appears to reside in the suprachiasmatic nucleus-hypothalamic-pituitary (SCN-H-P) axis. One presumes that in this system light is transduced through some collection of humoral factors, perhaps not unlike EGF, FGF, and

124

R. R. KLEVECZ ET AL

erythropoietin in the cases where proliferation is the desired end, to specify to cells and tissues a coordinating time sense. It appears that most cells in mammals do not have the capacity to respond directly or sensibly to light, nor has it been possible to show that they communicate time sense to other cells as can the SCN-H-P complex. Whether the circadian rhythm in bone marrow or other tissue proliferation would be sustained in the absence of entraining hormonal signals is not known. More significant for our discussion is the question of whether the circadian rhythm itself is built up from an intermediate frequency oscillation such as that described above for mammalian cells. To make this case requires that evidence for rhythmicity of period similar to that seen in animal cells be seen in systems normally expressing circadian rhythms. And in particular this must be seen under conditions where the integrative capacity of the circadian rhythm is purposely stressed or ablated. It would be surprising if such high-frequency components were readily seen in the undisturbed organism since the importance of precision in rhythmic expression would have led to fairly vigorous selection against a noisy system. One could only hope to find evidence for a fundamental oscillator in physiologically stressed systems, in unicellular eukaryotes growing in the ultradian mode, and possibly in mutants of the circadian integrative system. VESTIGES OF THE PRIMITIVE CLOCK Biological rhythmicities that qualify as clocks by virtue of being temperature compensated have been detected over a range of periodicities from a few seconds as in the courtship song of the male Drosophila (Kyriacou and Hall, 1980) through periodicities in cell cycle events approximating 4 hours (Klevecz, 1976; Klevecz et al., 1980a) to period mutants which show a clustering at 2.5 to 4-5 hour intervals (Bruce, 1972) and the natural quantizement of circadian rhythms at 4-6 hour intervals (Chisholm and Costello, 1980). Hence the argument that a particular subset of these rhythms represents the vestiges of the primordial oscillator is inclined to be specious. Nevertheless, we find some encouragement in the fact that the majority of these have a similar period. In Drosophila, Pittendrigh (1967) found by continuous selection for early eclosion applied for 50 generations yielded only phase mutant and that emergence of a mutant “early” strain of flies occurred at a phase 4 hours earlier than the wild type, T however was unchanged in these animals. By picking phototaxic variants of Chlamydomonas exposed to nitrosoguanidine, Bruce (1972) was able to isolate mutants with periods of 16, 21, 23.5, and 28 hours (wild type = 24 hours). Similarly Konopka and Benzer (1971) isolated long (28) and short (19 hour) period mutants of D . melanogaster. In the most complete study to date, Feldman and co-workers (Feldman and Hoyle, 1973, 1976; Feldman et al., 1978) have isolated and mapped a number of clock period mutants in

CELLULAR CLOCKS AND OSCILLATORS

125

Neurosporu which appear to cluster temporally at periods differing from wild type by 2.5 and 5 hours, with T ’ S of 16.5, 19, 21.5, 24, and 29 hours. When the suprachiasmatic nucleus (SCN) in hamsters is ablated and the integrating capacity of the animal is destroyed, no 24-hour rhythm endures. Rather, higher frequency oscillations with 8- and 12-hour periods are expressed in running wheel and drinking activity (Rusak, 1977). Jacklet and Geronimo (1971) observed that as the size of the Aplysiu retinal tissue was surgically diminished, so was periodic firing frequency. With only a slight reduction in the tissue size, free running period was reduced from 28 to 24 hours and remained unchanged until 80% of the tissue was removed at which time period declined first to 12 hours then to 6 hours, and then to 3 hours as additional tissue was removed. Similarly, in the circadian growth mode, the cell division rhythm of the autotrophic phytoplankton T. fluviutilis displays a naturally occurring quantizement of generation time with intermodal period of 4-6 hours (Chisholm and Costello, 1980). In mammalian cells as discussed in detail above, the distribution of cell cycle times of a number of cell types and the interval between modal generation times of differing cell types have been observed to show quantizement at nominal 4hour intervals (Klevecz, 1976). When the cells are intentionally perturbed their pattern of phase reset has a periodicity that again approximates 4 hours (Klevecz et al., 1980a). Cell cycle time appears to be controlled by a timekeeping oscillator of 4-hour period which is bounded in some way by cell size or tissue extent (Shymko and Klevecz, 1981). The questions of mechanism and the spatial properties and location of the timekeeping oscillator have been avoided. Several collections have attempted to present plausible mechanisms for generating time sense in cells and can be recommended to the interested reader (Hastings and Schweiger, 1976). It must be said though that the reductionist approach which has served molecular genetics so well has met with many disappointments when applied to the clock (Robertson, 1975). It often appears that the clock and the cell are one and that clock properties are diminished in proportion to the extent that the system being studied is reduced to a simpler form. In some systems the more emanant and immediate properties of the clock are cytoplasmic and membranous but in the longer term nuclear information is required, while in others change in nuclear function is immediately expressed. Moreover, since period mutants can be mapped, and clock period is species specific and heritable, it would seem ultimately to be a nuclear genetic property.

ACKNOWLEDGMENTS

This work was supported in part by NIH Grants AGO3815 and GM31262 to RRK

126

R. R. KLEVECZ ET AL. REFERENCES

Absher, P. M., and Absher, R. G. (1976). Exp. Cell Res. 103, 247-255. Absher, P. M., Absher, R. G., and Bames, W. D. (1974). Exp. Cell Res. 88, 95-104. Ashbumer, M., and Bonner, J. J . (1979). Cell 17, 241-254. Belousov, B. P. (1958). Sh. Ref. Radiar. Med. Medgiz, Moscow p. 145. Brooks, R. F., Bennett, D. C., and Smith, J. A. (1980). Cell 19, 493-504. Bruce, V. G. (1972). Generics 70, 537-548. Bums, F. J., and Tannock, I. F. (1970). Cell Tissue Kinet. 3, 321-334. Burstin, S. J., Meiss, H. K., and Basilico, C. (1974). J. Cell. Physiol. 84, 397-408. Chance, B., Estabrook, R. W., and Ghosh, A. (1964). Proc. Narl. Acad. Sci. U.S.A. 51, 1244- 1251. Chisholm, S. W., and Costello, J. C. (1980). J. Phycol. 16, 375-383. Collins, J. M. (1978). J . B i d . Chem. 253, 8570-8577. Cooper, S. (1979). Narure (London) 280, 17-19. Dawson, K. B., Madoc-Jones, H., and Field, E. 0. (1965). Exp. Cell Res. 38, 75-84. Edmunds, L. N., and Adams, K. J. (1981). Science 211, 1002-1013. Eigen, M. (1971). Naturwissenschafien 58, 465-522. Eigen, M., and Schuster, P. (1977). Narurwissenschaften 64, 541-565. Eigen, M., Gardiner, W., Schuster, P., and Winkler-Oswatitsch, R. (1981). Sci. Am. 244, 88-118. Fantes, P., and Nurse, P. (1977). Exp. Cell Res. 107, 377-386. Feldman, J . F., and Dunlap, J. C. (1983). Phorochem. Phorobiol. Rev. 7, 319-368. Feldman, J . F., and Hoyle, M. N. (1973). Genetics 75, 605-613. Feldman, J . F., and Hoyle, M. N. (1976). Generics 82, 9-17. Feldman, J . F., Gardner, G . , and Denison, R. (1978). Naito Int. Symp. Tokyo. Field, R . J . , and Noyes, R. M. (1974). J. Chem. Phys. 60, 1877-1884. Forrest, G. L., and Klevecz, R. R. (1978). J . Cell Biol. 78, 441-450. Goodwin, B. C. (1963). “Temporal Organization in Cells,” p. 21. Academic Press, New York. Grove, G. L., and Cristofalo, V. J . (1976). Cell Tissue Kinet. 9, 395-399. Guttman, S. D., Glover, C. V. C., Allis, C. D., and Gorovsky, M. A. (1980). Cell 22, 299-307. Hartwell, L. H. (1971). J. Mol. Biol. 59, 183-194. Hartwell, L. H. (1974). Bacreriol. Rev. 38, 164-198. Hastings, J . W., and Schweiger, H.-G. (eds.). (1976). Dahlem Konf., Berlin. Higgins, J . (1967). Ind. Eng. Chem. 59, 19-62. Howard, A,, and Pelc, S. R. (1953). Herediry 6 , 261-264. Holmquist, G., Gray, M., Porter, T., and Jordan, J . (1982). Cell 31, 121-129. Jacklet, J. W., and Geronimo, J . (1971). Science 174, 299-302. Jauker, F. (1975). J. Biol. Chem. 67, 901-904. Jauker, F., and Cleffman, G. (1970). Exp. Cell Res. 62, 477-486. Kapp, L. N., and Okada, S. (1972). Exp. Cell Res. 72, 465. Kapp, L. N., and Painter, R. B. (1977). Exp. Cell Res. 107, 429-431. Kapp, L. N., Millis, A. J . T., and Pious, D. A. (1979). In Vitro 15, 669-672. Kauffman, S. A,, and Wille, J. J. (1975). J . Theor. Biol. 55, 47-93. Kauffman, S. A., Shymko, R. M., and Trabert, K. (1978). Science 199, 259. Killander, D., and Zetterberg, A. (1965). Exp. Cell Res. 40, 12-20. King, G. A., Archambeau, I. O., and Klevecz, R. R. (1980). Radiar. Res. 84, 290-300. Klevecz, R. R. (1969a). J . Cell Biol. 43, 207-219. Klevecz, R. R. (1969b). Science 166, 1536-1538. Klevecz, R. R. (1976). Proc. Narl. Acad. Sci. V.S.A. 73, 4012-4016. Klevecz, R . R., and Forrest, G. L. (1976). In “Growth, Nutrition, and Metabolism of Cells in Culture” (V. J. Cristofalo and G. H. Rothblat, eds.), Vol. 3, p. 149. Academic Press, New York.

CELLULAR CLOCKS AND OSCILLATORS

127

Klevecz, R. R., and Kapp, L. N. (1973). J . Cell Biol. 58, 564-573. Klevecz, R. R., and King, G. A. (1982). Exp. Cell Res. 140, 307-313. Klevecz, R. R., and Ruddle, F. H. (1968). Science 159, 634. Klevecz, R. R., and Stubblefield, E. (1967). J . Exp. Zool. 165, 259 Klevecz, R. R., Keniston, B. A., and Deaven, L. L. (1975). Cell 5, 195-203. Klevecz, R. R., Kros, J., and Gross, S. D. (1978). Exp. Cell Res. 116, 285-290. Klevecz, R. R., King, G. A,, and Shymko, R. M. (1980a). J. Supramol. Struct. 14, 329-342. Klevecz, R. R., Kros, J., and King, G. A. (1980b). Cytogenef. Cell Genet. 26, 236-243. Klevecz, R. R., King, G. A,, and Buzin, C. H. (1982). In “Heat Shock from Bacteria to Man.” (M. J. Schlesinger, M. Ashburner, and A. Tissieres, eds.), pp. 413-418. Cold Spring Harbor Laboratory, Cold Spring Harbor, New York. Konopka, R. S., and Benzer, S. (1971). Proc. Nutl. Acad. Sci. U.S.A. 68, 2112-2116. Kubitschek, H. E. (1962). Exp. Cell Res. 26, 439-450. Kubitschek, H. E. (1971). Cell Tissue Kine?. 4, 113-122. Kyriacou, C. P., and Hall, J. C. (1980). Proc. Natl. Acad. Sci. U.S.A. 77, 6729-6733. Li, G. C., Petersen, N. S., and Mitchell, H. K. (1982). In?. J. Radiut. Oncol. Biol. Phys. 8, 63. Liskay, M. (1974). J . Cell. Physiol. 84, 49-56. McQuilkin, W. T., and Earle, W. R. (1962). J. Natl. Cancer Inst. 28, 763-799. Minor, P. D., and Smith, J. A. (1974). Nature (London) 248,241-243. Mitchison, J. M. (1971). “The Biology of the Cell Cycle.” Cambridge Univ. Press, London and New York. Nicolis, G., and Prigogine, I. (1977). “Self-Organization in Nonequilibrium Systems.” Wiley, New York. Nurse, P. (1975). Nature (London) 256, 547-551. Nurse, P., and Thuriaux, P. (1977). Exp. Cell Res. 107, 365-375. Pardee, A. B. (1974). Proc. Natl. Acad. Sci. U.S.A. 71, 1286-1290. Pardee, A. B., Shilo, B.-Z., and Koch, A. L. (1979). In “Hormones and Cell Culture” (G. H. Sato and R. Ross, eds.), pp. 373-392. Cold Spring Harbor Laboratory, Cold Spring Harbor, New York. Pavlidis, T. (1969). J. Theor. Biol. 22, 418-436. Pavlidis, T. (1973). “Biological Oscillators: Their Mathematical Analysis.” Academic Press, New York. Pittendrigh, C. S. (1960). Cold Spring Harbor Symp. Quanr. Biol. 25, 159-184. Pittendrigh, C. S. (1967). Proc. Natl. Acad. Sci. U.S.A. 58, 1762-1767. Pittendrigh, C. S., and Daan, S. (1976). J. Comp. Physiol. 106, 223-252. Polanshek, M. (1977). J. Cell Sci. 23, 1-23. Prigogine, I. (1947). “Etude Thermodynamique des Processus Irreversibles.” Desoer, Liege. Prigogine, I., and LeFever, R. (1968). J. Chem. Phys. 48, 1695-1700. Rasmussen, L., and Zeuthen, E. (1962). C. R. Trav. Lab. Carlsberg 32, 333-358. Read, R. A., Fox, M. H., and Bedford, J. S. (1983). Radiaf. Res. 93, 93-106. Robertson, M. (1975). Nature (London) 258, 291. Rueckert, R. R., and Mueller, G. C. (1960). Cancer Res. 20, 1584-1591. Rusak, B. (1977). Mesocricetus J . Comp. Physiol. 118, 145-164. Sapareto, S. A,, Hopwood, L. E., Dewey, W. C., Raju, M. R., and Gray, J. W. (1978). Cancer Res. 38, 393-400. Shields, R. (1978). Nature (London) 273, 755-758. Shymko, R. M., and Klevecz, R. R. (1981). In “Biomathematics and Cell Kinetics” (M. Rotenberg, ed.), pp. 329-348. Elsevier, Amsterdam. Smith, H. T. B., and Mitchison, J. M. (1976). Exp. Cell Res. 99, 432-435. Smith, J. A., and Martin, L. (1973). Proc. Natl. Acad. Sci. U.S.A. 70, 1263-1267. Sudbery, P. E., and Grant, W. D. (1976). J . Cell Sci. 22, 59-65.

128

R. R. KLEVECZ ET AL.

Tyson, J . , and Sachsenrnaier, W. (1978). J . Theor.Biol. 73, 723-738. Van Wijk, R . , and Van de Pol, H. (1979). Cell Tissue Kiner. 12, 659-663. Winfree, A. T.(1967). J. Theor.Biol. 16, 15-42. Winfree, A. T. (1970). J . Theor.Biol. 28, 327-374. Winfree, A. T. (1974). Science 183, 970-972. Yen, A . , and Pardee, A. B. (1979). Science 204, 1315-1317. Yen, A., Fried, J . , Kitahara, T., Strife, A , , and Clarkson, B . D. (1975). Exp. Cell Res. 95, 295-302. Zaikin, A. N., and Zhabotinsky, A. M. (1970). Nature (London) 225, 535-537. Zeuthen, E. (1971). Exp. Cell Res. 68, 49-60.