Bidletin o! Mathematical Biolog3, Vol, 40, pp. 625 635

{t007-4985 78/0901 0625 $02.0~)0

Pergamon Press Ltd. 1978. Printed in Great Britain © Society for Mathematical Biology

W H A T D O M A T H E M A T I C A L M O D E L S T E L L US ABOUT CIRCADIAN CLOCKS?

• THEO PAVLIDIS

Department of Electrical Engineering and Computer Science, Princeton University, Princeton, New Jersey 08540, U.S.A.

The paper reviews a series of models for circadian clocks and discusses their conclusions and predictions. Attention is focused on Pittendrigh's empirical model, two mathematical models by the author and Winfree's work.

I. Introduction. This paper has as its goal to summarize the development of a class of models for circadian clocks without the use of mathematical formalism. Our first order of business is to clarify the term model since it has been used to mean different things by different people. Basically a model is a hypothesis about how a physical system works. In general it must have the following two essential properties: (1) It must summarize the available experimental evidence so that the description of the physical system through the model must be more concise than the description through a table of experimental results. (2) It must predict the behavior of the system under new circumstances. However it need not say anything about the "deep" structure of the system. Strictly speaking this structure can never be known. In many cases it is customary to assume it to coincide with that of a very successful model but this is not really justified. For example the acceptance of the heliocentric over the geocentric model in physics is not due to any "real truth" but to the fact that the former gives a more compact description of the planetary system and has far more successful predictions than the latter. It is still theoretically possible that one day someone may come up with a superior geocentric model. This limitation of models has a certain implication for their value in the search for the circadian clock. It is highly unlikely that they will ever give any direct

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evidence about the biological structure of the system. On the other hand they can be quite helpful in aiding the development of "structural" theories as we shall see in the sequel. It is a good idea to distinguish between a model and its instances. We define the latter term to mean different formulations of a given model which provide both the same degree of "data compaction" and the same set of predictions. It should be emphasized that the c o m m o n practice in the literature is to refer to such instances as models but I believe that this tends to confuse the issues involved. The major emphasis in this paper will be on single unit models. The study of populations of such units will be the subject of a second paper. References to the literature are to those papers directly related to this one. Excellent sources for an overview of the field are the volumes of collected papers published after different symposia in the past (Aschoff, 1965; Menaker, 1971; Hastings and Schweiger, 1976). The last reference has the particularly attractive feature of containing "state of the art" group reports in addition to individual papers. There are also a number of sources treating the general subjects of circadian clocks and the mathematics of biological oscillators (Bunning, 1973; Pavlidis, 1973b).

11. Review of Models Dealing with Pulse lnputs. We will start our discussion with Pittendrigh's empirical model which was first proposed more than ten years ago (Pittendrigh, 1965) and is discussed in detail in (Pittendrigh, 1974, 1976). It was developed for the Drosophila pseudoobscura eclosion rhythm and its basic claim is that the 15 min high light intensity phase response curve (PRC) describes the behavior of the system for a combination of such pulses. Note that the model meets the criteria of both compaction (if one knows the effect of a single pulse then he also knows the effect of a combination thereof) and prediction (the behavior of the system under untried light pulse combinations is predicted). More specifically the model can be stated as following: The state of the system is described by its phase, i.e. subjective circadian time (abbreviated as CT). The change in the phase predicted by the PRC occurs "instantaneously". The observed transients are due to the secondary system driven by the circadian regulator (pacemaker). The major success of the model has been its precise quantitative prediction and subsequent experimental verification of the entrainment of the eclosion rhythm by brief (15 min) light pulses (ibid). Operationally the model consists of the PRC, giving the phaseshift in terms of the current phase of the rhythm, and the relation: (new phase) = (old phase) + (phaseshift).

(1)

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It is possible to plot the new phase as a function of the CT of the stimulus and obtain a phase transition curve. This was first used by Ottesen (Ottesen, 1965; Ottesen et al., 1976) in his analysis of the empirical model. Later was studied by Johnsson and Karlsson under the name of"transformation curve" (Johnsson and Karlsson, 1972). Phase transitions curves offer a mathematically convenient way of utilizing the information contained in the PRC's. It is proper to introduce at this point a fundamental mathematical concept, that of the state space. Any physical dynamical system involves certain variables which change with time and which describe its behavior. These are called state variables (Pavlidis, 1973b). Thus a chemical reaction can be described in terms of the concentrations of the chemicals involved. Note that these do not include parameters whose values affect the behavior of the system but which are not affected by the system itself. For a circadian clock light intensity is an obvious parameter. Unfortunately we are in the dark about the

/@

Figure 1.

A phase plane plot showing trajectories converging towards a limit cycle

nature of its state variables. If a system exhibits periodic behavior then after each period the values of the state variables repeat themselves. Let us think of a very simple system having only two state variables. (Typical examples are: A prey predator system where the two state variables are the populations of the prey and the predator. A mass spring system where the position and velocity of the mass are the state variables.) We may choose to make a plot where each of the coordinate axes corresponds to a state variable and plot a succession of points in the plane as time goes by. For a periodic system these will form a closed curve. In general any sequence of such points is called a trajectory. It turns out that physical systems which exhibit periodic behavior have often only one periodic trajectory and if the system starts outside such a trajectory it eventually reaches it in the manner shown in Figure 1. Such a trajectory is called a limit cycle (Pavlidis, 1973b; Tyson et al., 1976). The term phase plane is often used instead of "state space" when only two variables are involved.

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We may now describe some of the fundamental concepts of circadian rhythms in terms of limit cycles. If we assume that the circadian pacemaker is a limit cycle dynamical system, then the following are trivially true. Each point of the limit cycle corresponds to a phase or a subjective circadian time. The PRC describes a mapping of the limit cycle into itself since it puts into correspondence a new phase to an old phase according to (1). Pittendrigh's empirical model can then be expressed as following: The circadian regulator is a limit cycle system and the mapping described by the PRC for 15 min high intensity light pulses takes place "instantaneously". Note that the first assumption is not really a "drastic" one. All known physical oscillatory systems have a limit cycle behavior or they can be approximated closely by one with such behavior. Probably the most c o m m o n approximation to a limit cycle is a quasiperiodic trajectory, sometimes called a limit annulus (Pavlidis, 1965). Furthermore this commonality of limit cycles is to be expected on the basis of physical considerations (Andronov et al., 1966;

Figure 2.

Mapping of the limit cycle into itself

Pavlidis, 1973b). The experimental verification of Pittendrigh's empirical model and its expression in terms of limit cycles suggest the next step: To create a topological model based on an explicit mapping of the limit cycle. This was done by Pavlidis (1967a) and the mapping is shown in Figure 2. A major observation from the plot of Figure 2 is that the zone corresponding to circadian time (CT) 4 to 12 has a special meaning. Not only light (of duration over 15 min etc.) has no effect there but also if the system is exposed to light elsewhere it is driven in that zone, at least as a first approximation. This allows us now to predict the shape of PRC's for light duration longer than 15 min. Thus the topological model has an addmonal predictive value over the empirical model. It turned out that the predicted shapes were in close agreement with the ones observed experimentally (ibid). The next question that appears naturally is the prediction of the shape of PRC's for shorter light durations. It is obvious that if no-light leaves the system on the "'top" part of the limit cycle and 15 min light pulses drive the system all

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the way across then some intermediate values of light intensity and duration should leave the system within the limit cycle. (The only alternative is to assume a significant "quantum" effect on the part of light. Either no effect at all, or a very drastic one.) There are a number of major predictions from such an assumption. One is that after a perturbation by weak light the system will not return immediately to the limit cycle but gradually in the manner shown in Figure 1. A second prediction is derived from a fundamental theorem from the theory of dynamical system; that a limit cycle will in general surround a singularity, i.e. a point where if the system is placed at it, it will remain there indefinitely (Minorsky, 1962). Furthermore such a point is "phaseless" since no periodic motion occurs there. A corollary of the existence of such a point is the prediction of a qualitative change in the shape of the PRC's for light pulses leaving the system above and below the singularity. Those leaving it above would produce PRC's with a zero phase shift at the phase above the singularity, those leaving it below would yield PRC's with a discontinuity. Figure 3

A

PULSELEAVESSTATENEARB i~'k. PULSELEAVESSTATENEARC

Figure 3. Illustration of how a slight change in light duration changes the shape of a PRC illustrates this phenomenon. A is the "starting" phase, B and C are the points where weak light pulses bring the system when applied at A. The pulse bringing the state at C is slightly stronger than the one bringing it at B. Such a model was described for the first time by Pavlidis (1967b) and a computer simulation of an instance of it was performed. The weak light PRC's produced agreed not only in qualitative terms with the predictions but also in quantitative terms with the weak light PRC for the Drosophila pseudoobscura eclosion rhythm and somewhat surprisingly with the PRC's of various other organisms (ibid). In other words the results were compatible with the hypothesis of a "universal" circadian clock with the main difference among organisms being that of sensitivity to light. We shall refer to the model as the P-

2 model. What will happen if the system is brought exactly to the singularity by a light pulse? The answer depends on the details of the dynamical system involved. In general the return to the limit cycle will take longer than from other points and the increase in that time will be quite steep, the closer the system is brought to the singularity. In physical terms this would mean that if one could find the

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proper parameters of the light pulse to bring the system to that point it will require a long time to return to the original trajectory or in other words the amplitude of the oscillation will be reduced, at least temporarily. Macroscopically this could manifest itself as an arrhythmic behavior. If the singularity was stable then such arrythmic behavior would be permanent and therefore one would have the following surprising result: ".... while a strong light stimulus would fail to d a m p out the oscillation, a weak one (of the same duration) would damp it" (Pavlidis, 1968). At the time there was no experimental evidence in favor of such a statement and it was concluded that the singularity of the circadian oscillator was unstable. An effort to test this hypothesis was undertaken by Winfree (1970, 1971). It was first necessary to devise a systematic way of investigating the interior of the limit cycle. To this he developed the concept of the isochrone. If the state of the system is brought inside the limit cycle it will eventually return to it at some

Figure 4.

Plot of isochrones

phase. Thus interior points can be associated with phases. The locus of all points which return to a given phase is called an isochrone (ibid). All these curves converge to the singularity so that a typical plot of them in the state space would look like the one shown in Figure 4. Winfree used the property of the singularity as a limit point of the isochrones to determine the characteristics of the stimulus which would bring the system there and then he proceeded to perform the critical pulse experiment (ibid). The eclosion r h y t h m was indeed damped and in this way one of the predictions of the P-2 model was verified. However there is more to the story. The Drosophila pseudoobscura eclosion rhythm can be observed for at most nine cycles and because of the way the experiments must be performed there were only two periods available for observing the effects of the critical pulse. Thus although the existence of a singularity was verified experimentally not much could be said about its stability. Winfree proceeded to conduct a two pulse experiment, where the critical pulse was followed by a strong pulse. It turned out that after two days the state was still near the singularity (ibid). Furthermore if a noncritical weak stimulus was given, which left the system inside the limit cycle,

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there seemed to be no return to it within two days (Winfree 1973, 1975). Thus the circadian regulator behaves as a dynamical system where the state returns to the limit cycle quite slowly, if at all. Although a simple limit cycle model could be made to simulate such a behavior by a careful choice of its parameters this is not very desirable. It is difficult to imagine a product of evolution to be highly tuned in its response to a stimulus not encountered under normal environmental conditions. We leave the discussion of models dealing with pulse inputs in order to examine other types of stimuli. We note however that the slow return to the limit cycle has posed some important questions.

III. Review of Models Dealing with Continuous Light.

All the models discussed in the previous section dealt with the response of the system to pulses of light. What happens if light is left on for many days ? The models based on the Drosophila pseudoobscura eclosion rhythm PRC predict that the phase of the system will be ~frozen" at CT 12. On the other hand the P-2 model (Pavlidis, 1967b, 1968) as well as all other models based on continuous limit cycle

DIM LL

Figure 5.

DD

Illustration of the shift of limit cycle location u n d e r LL

dynamics and using light as a parameter (Wever, 1965 ; Johnsson and Karlsson, 1972; Karlsson and Johnsson, 1972) predict that, depending on the light intensity, a new limit cycle will be followed or the oscillatory behavior will cease and the system will reach a new singular state. If the oscillations continue they will be, in general, with a different period than under DD. Thus such models conform to Aschofl's rule. How are the effects of light shown in the state space? Ira limit cycle still exists it will be in a different location and the same will be true for the singularity. Figure 5 shows such a "shift". Mathematical descriptions of such transformations have been described in detail elsewhere (Pavlidis, 1967b, 1968, 1973b). The important thing to notice is that the case of pulses can be treated easily under this model. Indeed suppose that the system is at A on the D D limit cycle when light is turned on. Under these conditions the steady state of the system is either the new limit cycle or the new singularity (depending on light intensity). However such states are not achieved instantaneously. Instead the system will follow a trajectory starting at A and tending towards the eventual steady state. One such trajectory is Shown in Figure 6. If the light is switched off then the system

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will be left at point B, inside the DD limit cycle (if the pulse is brief), or at point C, near the "opposite" site of the same cycle (if the pulse is longer), or at point D, outside of it (if the pulse is very long). Thus we have a reproduction of the phenomena discussed in the previous section. There are a few points worth summarizing. One is that the "velocity" of the motion of the state along the trajectories under different light intensities will not always be the same. However the change in the period or the phase shifts are not due only on such a change but also on the modification of the geometry of the motion under LL. As a matter of fact, it is possible to have a situation when the period is independent of light intensity, but the system exhibits "'normal" phase response curves and entrainment as well. An example of such a model has been described elsewhere (Pavlidis, 1973a). Physically that particular model is rather unlikely because of the need to choose a certain precise form for its equations but it demonstrates that a dependence of the period on LL is not a necessary condition for entrainment. In a mathematical sense the effect of light

Figure 6.

Effects of pulses under the "'continuous" model

is to change the velocity vector of the system. Under this definition all PRC's are in a sense also velocity response curves. However the latter term is usually reserved to denote the effects on the numerical value of the velocity vector only (Swade, 1969). Models with light intensity as a parameter would have no problem simulating the PRC's obtained by strong light pulses in rhythms free running under dim light. A number of instances of the P-2 model (Pavlidis, 1967b, 1968, 1973b) predict that the shape of the two PRC'S will be quite similar. Such models also predict the effect of dark pulses. Figure 7 shows a plot of the trajectories in the state space (phase plane) and the form of the resulting PRC. The latter has been derived on the basis of the isochrones where each trajectory intersects at the end of the dark period. Entrainment by slowly varying stimuli, like sine waves, is also predicted by the models since they are nonlinear oscillators which can be entrained by their input (Minorsky, 1962; Andronov et al., 1966). It should be emphasized that the mathematical form of all these models is such that light intensity is used indeed as an input. However when it is held at a constant value, then it can be incorporated into the equations as a parameter. One might then be tempted to use the term parametric entrainment for the one caused by slowly varying

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LIMIT CYCLE FOR DIM LL \ 18

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_/o2,,%p s Figure 7. Effects of dark pulses

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Figure 8. Phasezones at transfers between dim LL and DD. When the organism is transferred from LL to DD it will appear that it had been held at a phase lyingin the arc marked by heavy line on the right. This will be so even if the oscillationhad been going on, on the trajectory shown at the left. A similar situation exists for transitions from DD to LL light stimuli. However the term "parametric entrainment" has a very specific meaning in the mathematical literature and it refers to the effects of periodic changes in the coefficients of the equations rather than to changes in a constant term added to them (ibid). It is relevant to mention here two recent experiments which conclude that the Drosophila pseudoobscura eclosion rhythm is shifted always to CT 12 when transferred from dim LL to D D (Pittendrigh, personal communication) and to about CT 23 when transferred from DD to dim LL (Pittendrigh, personal communication). Figure 8 shows a qualitative explanation of these experiments. Note that a major qualitative prediction is that the zone of phases at the dim LL to D D transfer should be narrower than those observed at the D D to dim LL transfer. We turn now our attention to a class of phenomena which the previous models have trouble in stimulating. We can lump all of them together under the general heading of Ji'ee run period lability. They include the rhythm splitting results, the spontaneous loss of rhythmicity and then its reappearance, the nonmonotonic transients., etc., (Pittendrigh, 1974). It is possible to modify the simple limit cycles models to simulate such experiments by allowing r a n d o m variations of parameters, noise, increase the number of state variables involved

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etc. All of these factors are feasible in a biological system but there seems to be another explanation which ties all these things together. Namely that the circadian clock consists of a population of coupled oscillators. This assumption is not at all radical in itself and there is considerable physical evidence in its favor. Its use to simulate some of these phenomena has been described in a number of papers (Pavlidis, 1971, 1973b, 1976) and it has also been shown that it can resolve the paradox of Winfree's experiments (Pavlidis, 1976). However we shall discuss these points in detail in a second paper. 1V. Discussion. A model of a somewhat different type than the above has been proposed by Johnsson and Karlsson (1972). Its major difference is the introduction of a time delay in its equations. This adds effectively another degree of freedom and it is certainly a realistic assumption for a biological system. On the other hand it is not clear whether this changes essentially its dynamics behavior in the aspects relevant to the simulation of circadian rhythms. The model has been successful in simulating the effects of both continuous and pulsed light as well as rhythm damping by a critical pulse (Englemann et al., 1973). An interesting illustration of the usefulness of mathematical models in biochemical research is offered by the development of the membrane model (Njus et al., 1974, Njus, 1976). There a concrete biochemical mechanism for the circadian clock has been suggested. However instead of being checked against a long list of experimental data it has made use of the P-2 model to check whether the proposed mechanism can create a limit cycle of the proper type. This work was supported by Grant GM20289 02 from Public Health Service (General Medical Sciences Institute, Biomedical Engineering Program). The writing of the paper was motivated during my attendance at two meetings. The Dahlem Conference in Berlin (November, 1975) and the NSF Conference in San Diego (January 1976). Many of the participants encouraged me to write a nonmathematical paper on models and I hope that they will not be disappointed with this effort. LITERATURE Andronov, A. A., A. A. Vitt and S. E. Khaikin. 1966. Theory of Oscillators. Oxford: Pergamon. Aschoff, J. (Ed.). 1965. Circadian Clocks. Amsterdam: North-Holland. Btinning, E. 1973. The Physiological Clock, 3rd ed. Springer, Berlin. Engelmann, W., H. G. Karlsson and A. Johnsson. 1973. "Phase Shifts in the Kalanchoe Petal Rhythm Caused by Light Pulses of Different Durations." Int. J. Chronobiol., 1, 147 156. Hastings, J. W. and H.-G. Schweiger (Eds.) 1976. The Molecular Basis of Circadian Rhythms. Berlin: Dahlem Konferenzen. Johnsson, A. and H. G. Karlsson. 1972. "A Feedback Model for Biological Rhythms: I. Mathematical Description and Properties of the Model." J. Theor. Biol., 36, 153-174.

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Karlsson, H. G. and A. Jot-msson. 1972. "A Feedback Model for Biological Rhytlmls: II. Comparisons with Experimental Results• Especially on the Petal Rhythm ofKahmchoe.'" J. Theor. Biol., 36, 175 194. Menaker, M. (ed.). 1971• Biochronometry, Nat. Acad. Sci., Washington, D.C. Minorsky, N. 1962. Nonlinear Oscillations. Princeton, N.J.: Van Nostrand-Reinhold. Njus, D., F. M. Sulzman and J. W. Hastings, 1974• "Membrane Model for the Circadian Clock." Nature, 248, 116 120• Njus, D. 1976. "Experimental approaches to membrane models." In The Molecular Basis o/ Circadian Rhythms (J. W. Hastings and H.-G. Schweiger, Eds.), Dahlem Konferenzcn. Berlin, ptx 283 294• Ottesen, E. A. 1965. B.A. Thesis, Princeton University• • C. S. Pittendrigh and S. Daan. 1976. Unpublished manuscript. Pavlidis, T. 1965. "'A New Model for Simple Neural Nets and its Application in the Design of a Neural Oscillator." Bull. Math. Biophys., 27, 215 229. --. 1967a. "A Mathematical Model for the Light Affected System in the Drosophila Eclosion Rhythm•" Ball, Math. Biophys., 29, 291 310. -. 1967b. "'A Model for Circadian Clocks.'" Bull. Math. Biophys., 29, 781 791. • 1968. "Studies on Biological Clocks: A Model for the Circadian Rhythms of Nocturnal Organisms." In Lectures on Mathematics in L(]~" Sciences (M. Gerstenbaber, Ed.). American Math. Soc., Providence, R.I., pp. 88 112. •- - - . 1971. "Populations of Biochemical Oscillators as Circadian Clocks." J. Theor. Biol.. 33, 319 338. - - . 1973a. "The Free Run Period of Circadian Rhythms and Phase Response Curves." Am. Natur., 107, 524 530. • 1973b. Biological Oscillators: Their Mathematical ,4mdysis, New York: Academic Press. • 1976. "Spatial and Temporal Organization of Populations of Oscillators." In The Molecular Basis of Circadian Rhythms (J. W. Hastings and H.-G. Schweiger, Eds.L Dahlem Konferenzen, Berlin• Pittendrigh, C. S. 1965. ~'The Circadian Oscillation in Drosophila Pseudoobscura Pupae: A Model for the Photoperiodic Clock." Z. Pfhmtenphysiol., 54, 275 307. • 1974. "Circadian Organization in Cells and the Circadian Organization of Multicellular Systems." In Neurosciences Third Study Program (F. O. Schmitt and F. G. Worden, Eds.t, Cambridge, Mass.: M.I.T. Press• .1976. "Circadian Clocks: What are They?" In The Molecular Basis qlCircadia Rhythms (J. W. Hastings and H.-G. Schweiger, Eds.), Dahlem Konferenzen, Berlin, pp. 11 48. Swade, R. H. 1969. "Circadian Rhythms in Fluctuating Light Cycles: Toward a New Model of Entrainment." J. Theor. Biol., 24, 227 239. Tyson, J. I., S. G. A. Alivisatos, F. Grfin, T. Pavlidis, O. Richter and F. W. Schneider. 1976. "Mathematical Background Group Report." In The Molecular Basis o[ Circadia~ Rhythms (J. W. Hastings and H.-G. Schweiger, Eds.), Dahlem Konferenzen, Berlin, pp. 85 108. Wever, R. 1965. "A Mathematical Model for Circadian Rhythms." In Circadian Clocks (Aschoff, J., Ed.) Amsterdam: North-Holland, pp. 47 63. Winfree, A. T. 1970. "~Integrated View of resetting a Circadian Clock." J. Theor. Biol., 28, 327 374. ---. 1971. "Corkscrews and Singularities in Fruitflies: Resetting Behavior of the Circadian Eclosion Rhythm." In Biochronometry (M. Menaker, Ed.), Nat. Acad. Sci., Washington, D.C., pp. 81-109. • 1973. "Resetting the Amplitude of Drosophila's Circadian Chronometer." J. Comp. Physiol., 85, 105 140. .... . 1975. "Unclocklike Behavior of Biological Clocks." Nature, 253, 315 319. RECEIVL, D 10-26-76 REVISED 9 - 1 2 - 7 7

What do mathematical models tell us about circadian clocks?

Bidletin o! Mathematical Biolog3, Vol, 40, pp. 625 635 {t007-4985 78/0901 0625 $02.0~)0 Pergamon Press Ltd. 1978. Printed in Great Britain © Society...
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