University of Groningen. Why and how do we model circadian rhythms? Beersma, DGM. Published in: Journal of Biological Rhythms

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1 University of Groningen Why and how do we model circadian rhythms? Beersma, DGM Published in: Journal of Biological Rhythms DOI: / IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 005 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Beersma, D. G. M. (005). Why and how do we model circadian rhythms? Journal of Biological Rhythms, 0(4), DOI: / Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 / JOURNALOF Beersma / MODELING BIOLOGICALRHYTHMS CIRCADIAN RHYTHMS / August 005 MODELS Why and How Do We Model Circadian Rhythms? Domien G. M. Beersma 1 University of Groningen, Department of Behavioral Biology, The Netherlands Abstract In our attempts to understand the circadian system, we unavoidably rely on abstractions. Instead of describing the behavior of the circadian system in all its complexity, we try to derive basic features from which we form a global concept on how the system works. Such a basic concept is a model of reality. The author discusses why it is advantageous or even necessary to transform conceptual models into mathematical formulations. As examples to demonstrate those advantages, the author reviews 4 types of mathematical models: negative feedback models thought to operate within pacemaker cells, models on coupling between pacemaker cells to generate pacemaker output, oscillator models describing the behavior of the composite circadian pacemaker, and models describing how the circadian pacemaker influences behavior. Key words modeling mathematical models, circadian rhythms, concept modeling, computational WHAT S IN A MODEL? Scientific experiments are seldomly performed for the sake of the experiments alone. They are performed to find answers to questions on functions and mechanisms of relevant processes. The functions and mechanisms inferred are expected to be valid more generally than only in the experiments performed. Therefore, the ultimate goal of scientific experimentation is to construct models. Commonly, these are conceptual models. They represent a global description of how a process might be organized. Conceptual models are extremely valuable, as they represent a flow chart of the major elements thought to be involved in a process. What counts are the basic mechanisms. Yet, because of the lack of detail, conceptual models often do not allow much critical testing. Transforming conceptual models into mathematical form generates mathematical models, and these present the opportunity to derive specific, sometimes nonintuitive, predictions and also to carry out critical testing. The transformation requires the selection of adequate mathematical equations and the choice of proper parameter values. It forces the researcher to specify his or her global conceptual model into more rigid detail. Simulations will reveal whether the mathematical model is capable of reproducing the observed phenomena. They will yield insight into the extent to which parameters are crucial or not. It may happen that the phenomena thought to be explainable with the conceptual model turn out not to be predicted by the mathematical formulation of the model. This may sometimes lead to the conclusion that both the mathematical and the conceptual model are wrong. It may also happen that the model reveals possible response characteristics that had not been anticipated. These are some of the reasons why mathematical models deserve their own place in science. 1. To whom all correspondence should be addressed: Domien G. M. Beersma, University of Groningen, Department of Behavioral Biology, P.O. Box 14, 9750 AA Haren, The Netherlands; d.g.m.beersma@rug.nl. JOURNAL OF BIOLOGICAL RHYTHMS, Vol. 0 No. 4, August DOI: / Sage Publications 304

3 Beersma / MODELING CIRCADIAN RHYTHMS 305 There is no general recipe on how to construct a mathematical model. Most processes can be modeled in many different ways, all similar in the quality with which they simulate the data but different in terms of underlying equations. Models of the circadian pacemaker are examples of such different mathematical models. The behavior of the pacemaker can, for instance, be modeled by a limit-cycle oscillator. This deterministically predicts the state of the pacemaker in the near future from its current state, for instance, on the basis of a set of differential equations. Alternatively, the pacemaker can be modeled as a set of coupled cellular oscillators, the behavior of each of which is stochastic and only predictable in terms of averages. In spite of the fundamental differences between them, both models simulate a large set of circadian characteristics. In that sense, both models are correct. Given, however, that each model is just one out of many representations of the circadian pacemaker, the word correct is inappropriate. Models never correctly represent reality. They are models. What matters is whether a model is useful or not. This depends on the extent to which it aids in understanding the mechanism and/or function of the circadian pacemaker, as well as the extent to which it lets us predict the outcome of new experiments. Few models have been developed for the human circadian pacemaker exclusively. Therefore, I will review some fundamentally different modeling approaches that have been or can be applied to the human situation. Four levels at which models are used to enhance our understanding will be addressed: 1) the generation of circadian oscillations in single cells, ) the interaction between rhythmic cells, 3) the composite circadian pacemaker, and 4) the interaction between the pacemaker and behavior. These levels are depicted in Figure 1. PACEMAKER CELLS AS PACERS Several studies have demonstrated that cells taken from the mammalian SCN in vitro generate a circadian pattern in electrical activity and other functions (Welsh et al., 1995; Shirakawa et al., 000; Honma et al., 004). Hence the circadian output of the pacemaker as a whole does not exclusively result from the interaction between cells, but individual cells can generate circadian signals on their own, contributing to the pace of the pacemaker. Functionally, the pacemaker cells, therefore, are pacers, as surmised early on by Enright (1980a, 1980b). Much progress has been made in our knowledge of the bio-chemistry involved in the generation of these rhythms. The basic current concept (Fig. 1, top left panel) is that of a negativefeedback model (transcription-translation loop, or TTL), in which it is assumed that the proteins (with some time lag) attenuate their own production (Hardin et al., 1990; Reppert and Weaver, 00; but see Tan et al., 004, Tomita et al., 004). Lema et al. (000) transformed the conceptual negative feedback model into a mathematical model consisting of equations: with dp( t) = K G( t δ) K P( t) e 1 G( t δ) =. n 1 + [ P( t δ) K i ] These equations simply say that the rate of change of protein content is equal to the rate of protein production minus the rate of protein loss. The term dp(t)/ is the rate of change of the amount, P, of the protein. (Lema et al. do not specify which clock protein is considered). The term K e (t δ) denotes the increment of protein in response to the gene activity, G(t δ), that existed 1 time-lag interval, δ, ago. K e specifies how much protein results from each unit of gene activity. The term K d P(t) denotes the decrease of P, due to dispersion and chemical destruction. The decay is proportional to the amount of protein. The negative feedback arises from the protein that was present at that time as is specified in equation (). This equation is slightly more difficult to understand. It is easiest to 1st consider the situation that there is a lot of protein present. In that case, the term 1 in the denominator of equation can be neglected relative to [P(t δ)/k i ] n. This reduces equation () to: n Ki G( t δ) =. n [ P( t δ)] It means that the more protein is present, the less gene activity results. K i is reported by Lema et al. (000) to be the inhibition rate constant, denoting the efficacy of inhibition. Yet, from the perspective of gene d (1) () (3)

4 306 JOURNAL OF BIOLOGICAL RHYTHMS / August 005 (1) pacemaker cell models cell's contribution to pacemaker output light + protein activation state light + delay () coupled pacer models output circadian pacemaker (3) composite pacemaker models dx x + discriminator short light pulse threshold under pacemaker control sleep need sleep wake sleep wake (4) timing of sleep models Figure 1. Overview of types of models discussed in this article. Top left: Negative feedback within single pacer cells. Middle left: Mutual coupling between pacers. Middle right: Composite pacemaker models. Bottom: -process model gentle control of pacemaker over behavioral need need reduction processes. activation, one might argue that K i represents an activation factor, since at a certain protein concentration, gene activity increases with K i. The Hill coefficient of inhibition, n, represents the cooperation of protein molecules required for the inhibition. The more molecules are needed, the higher n. For smaller values of P(t δ), however, equation (3) faces the problem that G(t δ) tends to become infinitely large if P(t δ) approaches 0. To limit gene activity to realistic values, the denominator has, therefore, been extended with the addition of term 1. K e in equation (1) takes care of the proper scaling of gene activity relative to protein production. After specification of the values of the parameters and the dimension in which the variables are expressed, the model can be used to calculate the course of P (and G) as a function of time. This is commonly done by numerical integration. Lema et al. (000) showed that their model oscillates with a period close to 4 h for a wide range of parameter values as long as the delay time δ was chosen close to 8 h. Given the similarity between the simulations and the behavior of a real circadian pacemaker, one might conclude that the true delay in the negative feedback loop is about 8 h. This conclusion appears, however, to depend on specific details of the model.

5 Beersma / MODELING CIRCADIAN RHYTHMS 307 Scheper et al. (1999) used almost the same model equations as Lema et al. (000). There are 3 differences. One is that Scheper dealt with messenger RNA instead of gene activity. Apart from replacing G in equations (1) and () with M, this also replaced equation () with differential equation (5) describing the fate of mrna, similar to the fate of protein in equation (1). The other differences are the introduction of a power term, m, in equation (1) and a decay term, K d G(t δ), in equation (). These modifications result in the following equations: with dp( t) m = K M ( t ) K P( t) e δ d 1 dm( t δ) 1 = K M t n d ( δ). 1 + [ P( t δ) K ] There are good reasons to include those additional terms. The term K d G(t δ) accounts for the fact that not all mrna leads to the production of new protein but that some of it decays before reaching the ribosomes. The exponent m accounts for nonlinearities in the protein production cascade, such as, for instance, when multiple mrna molecules are the substrate in the production of a protein. From simulations with this model (after specification of the values of more parameters), it turns out that the delay required for circadian oscillation is around 4 h (Scheper et al., 1999), instead of the 8 h that were needed in Lema et al. s model. Although the delay terms in the models do not refer to exactly the same step in the process (one is between gene activation and protein production and the other between mrna and protein production), it is clear that the delay is sensitive to modifications of the model. In fact, replacing equation () with equation (5) implies a change of concept. Equation () links current gene activity to the currently present amount of protein. Equation (5), in contrast, implies a different (and more realistic) view on the biochemistry. A change in protein content now leads to a change in the rate of mrna production. While low values of P in equation () immediately lead to high values of G, low values of P in equation (5) lead to high rates of change of M, which i (4) (5) only later will lead to peak values of M. The change in concept therefore results in a change in timing of the peak in mrna, which is part of the delay that should be accounted for by the model. This is the reason why the delay term is smaller in Scheper et al. (1999). Given the complexity of the real circadian system, we deduce from the comparison of the models that it is quite likely that the delay required for the negative feedback is due to the kinetics of the biochemistry involved in the circadian oscillation. Both models can also be used to simulate the impact of light. It only requires specifying how variables will change in response to light. Both Lema et al. (000) and Scheper et al. (1999) have demonstrated that there are many ways by which the models can produce realistic PRC and realistic entrainment to LD cycles. This is in some sense trivial: as soon as an oscillator is sensitive to an external force, it will show a PRC, and as soon as it has a PRC, it will demonstrate the basic features of entrainment (Roenneberg et al., 003). The many ways by which entrainment can be achieved in the model thus do not yield clues about which aspect of the protein production cascade is the most likely for light to interact with the oscillation. Many studies based on even more detailed descriptions of the relevant cellular processes in the generation of circadian rhythms have been published. Leloup and Goldbeter (004) extended the model for mammals to include 4 proteins and a series of intertwined feedback loops. Forger and Peskin (003) even tried to account for the activity of every component of the intracellular clock. In those models, the delay term, δ, is no longer recognizable as a separate parameter. As expected, it is covered up in the kinetics of the various components. Because of the inclusion of all proteins known to participate in the clock machinery and the many chemical interactions possible, the extended models include up to dozens of coupled differential equations, each characterized by parameter values that must be specified. With this increasing complexity, we gradually shift from concept modeling to computational modeling. The underlying research questions shift from is the concept sufficient to understand the basic behavior of the clock toward how can we mimic the behavior of the clock. With such complex models, the step from computer simulation to better concepts becomes increasingly difficult. Nevertheless, it has been demonstrated with this kind of model that the noise in the system due to stochastic

6 308 JOURNAL OF BIOLOGICAL RHYTHMS / August 005 variations in molecular concentrations within cells in fact contributes to circadian variation rather than disturbing it (Forger and Peskin, 005). The rapid increase in knowledge of genes and proteins involved in the circadian system nourishes the expectation that further relevant molecules will be found. Tomita et al. (004) recently demonstrated that the circadian clock of cyanobacteria persists in its temperature-compensated oscillation independent of de novo transcription and translation processes. Apparently, oscillator networks other than transcriptiontranslation feedback loops exist in the circadian clock of cyanobacteria. Similar processes may occur in mammals as well. The original scheme of a single negative-feedback loop is being replaced by a set of intertwined negative- and positive-feedback loops, the number and molecular origin of which, however, may be more variable than currently imagined. Such considerations have led Roenneberg and Merrow (003) to propose an entirely different modeling approach. They constructed a network of coupled feedback loops, to study its characteristics. The feedback loops were not specified in terms of biochemistry but only in terms of global temporal feedback behavior. The interesting result from their studies is that a network of fast oscillators (each with a period in the order of a few hours) can easily yield output periods of the coupled system of near 4 h. This modeling approach is especially important because it provides a global understanding of mechanisms that are so complex that they can barely be understood on the basis of the underlying biochemistry. The modeling result demonstrates the importance of mathematical formulation and simulation by yielding a nonintuitive result that would never have emerged from solely verbal reasoning. PACERS IN A NETWORK For questions about the output of the circadian pacemaker, it may not matter so much by which chemical reactions individual pacer cells manage to generate a circadian oscillation. It may be sufficient to know that they do. Long before the discoveries that individual pacemaker cells are circadian oscillators by themselves and that biochemical feedback loops are at the heart of the circadian clock work, Jim Enright proposed in 1980 that the vertebrate circadian pacemaker is composed of a large set of coupled circadian pacers (see Fig. 1, middle left panel). He characterized each pacer by parameters drawn randomly from normal distributions. He presumed that the behavior of the pacemaker as a whole arises from mutual coupling in the pacer network. Enright simplified the behavior of each pacer by assuming that it has only behavioral states. One is the active state, in which the pacer generates action potentials and contributes to the output of the pacemaker as a whole. The other is the recovery state in which the pacer is silent. Enright argued that the duration of the active state would, on average, be proportional to the duration of the prior recovery state. Each individual pacer would sense the activity of the others mainly near the end of its own recovery interval. At that time, the activity of a sufficient number of other pacers increases the probability of the recovering pacer to become active. Without such feedback, slow pacers would lag the majority of the others with increasing lag times. Since the feedback mechanism is activated as soon as most pacers in the ensemble are active, in the presence of feedback, the slow pacers will likely speed up. This mechanism causes synchrony of many pacers in the ensemble, although a small fraction will not always be synchronized. Mutual synchrony between pacers alone is not sufficient for proper pacemaker function. It also needs to be synchronized to the environment. For most organisms, light is the most important zeitgeber. Enright modeled the action of the zeitgeber by assuming that the impact of light on a pacer is similar to the impact of other pacers: light exposure near the end of a pacer s recovery interval would increase the probability that it becomes active. Simulations with this model of coupled stochastic elements revealed that the coupled system simulates the behavior of real circadian pacemakers in many important respects. 1) In constant conditions, there are spontaneous long-term drifts in the free-running period. ) In constant conditions, there are aftereffects on the free-running period from prior LD cycles (see, e.g., Pittendrigh and Daan, 1976a). 3) In constant light conditions, sleep-wake rhythms can persist indefinitely but only within a restricted range of light intensities (see, e.g., Daan and Pittendrigh, 1976). 4) Aschoff s rule: For diurnal animals, the free-running period usually shortens with increasing light intensity; for nocturnal animals, the opposite is commonly observed (Aschoff, 1960). 5) Brighter constant light prolongs activity in diurnal animals, while the opposite applies to nocturnal ani-

7 Beersma / MODELING CIRCADIAN RHYTHMS 309 mals (see, e.g., Aschoff, 1964). All of these phenomena can be simulated with Enright s model (Enright, 1980a, 1980b). In spite of the fact that individual pacers in the model can only be accelerated by light, the coupled system responds to light according to a PRC, including both advances and delays. For his simulations, Enright had to estimate parameter values. These included the duration of the active state of the pacers (estimated to be ~7 h), within-pacer variability in cycle duration (~ h), and between-pacer variability in cycle duration (~1 h). SCN cell culture studies (Welsh et al., 1995; Shirakawa et al., 000; Honma et al., 004) show plots of electrical activity profiles of single cells from which very similar durations of the active state can be estimated. Honma et al. (004) reported that the between-pacer variability in cycle duration is 1. h. Data on within-pacer variability are not available. It is impressive to see how many of the major characteristics of the behavior of the circadian pacemaker can result purely from the interaction between pacer cells. This is, however, not to say that these are not due to the behavior of the pacer cells alone, but rather that the characteristics of the pace-maker may originate from either one of these levels. No matter how closely the estimates of Enright approach the in vitro values determined recently, major assumptions remain to be tested. For instance, is there a mechanism to count the number of active cells at any time, which then determines the presence or absence of feedback to the recovering pacers? Antle and colleagues (003), apparently independently of Enright s work, similarly proposed that the function of the calbindin cells in the SCN of mammals is to count the number of active pacemaker cells to serve as a gating mechanism for synchronizing the pacers. Several other studies deal with models of the interactions between pacers. They differ in their assumptions about the mechanisms of interaction between pacer cells, and they differ in the presumed characteristics of those cells. Yet, they share the conclusions of Enright that many pacemaker characteristics may arise at the level of intercellular interactions (Winfree, 1967; Diez-Noguera, 1994; Kunz and Achermann, 003; Gario-Ojalvo et al., 004). THE PACEMAKER AS A SINE WAVE GENERATOR AND ITS ENTRAINMENT For many circadian functions at the level of the organism, the details of the coupling between pacer cells will not be crucial. For those functions, it would be sufficient to summarize the output of the pacemaker in one simple model. An oscillator generating a sine wave with a period of 4 h is the simplest model of the circadian pacemaker. Such a model could be used to describe aspects of, for instance, the circadian fluctuation in core body temperature. Since certain limit-cycle models of the circadian pacemaker evolved from a sine wave generator model, it is illustrative to show this evolution with the relevant mathematical equations. A sine wave oscillator generates a signal, F(t) = sin(αt), in which α must be chosen to yield a period length of 4 h. The same mathematical model of the circadian pacemaker can also be written in a different notation: d x + α x = 0. This can be verified by substituting A sin (αt) for x in equation (6) and applying the rules of differentiation. The advantage of writing the model of the pacemaker as in equation (6) is the automatic link it provides with the behavior of a pendulum. If x were to represent the excursion of the pendulum relative to its equilibrium point, then the nd derivative with respect to time (i.e., the first term of equation (6)) is the acceleration, which according to Newton s law is proportional to the force driving the pendulum back to equilibrium. So equation (6) can be read as the force driving the pendulum toward its equilibrium is proportional to the distance to equilibrium. This formulation describes the motion of a pendulum. Normally, a pendulum does not oscillate forever but shows a gradual reduction of its amplitude, due to friction, which is inherently present in all mechanical oscillators. The biological circadian pacemaker will not escape such ever-present friction processes. To account for the friction, an appropriate term must be added to the model: d x dx + β + α x = 0. The reason to take the 1st derivative of x with respect to time for the attenuation term is that the 1st derivative is the speed of the pendulum, and frictional forces are usually proportional to speed. (6) (7)

8 310 JOURNAL OF BIOLOGICAL RHYTHMS / August 005 The self-sustainment property of circadian pacemakers (no attenuation of amplitude in the absence of zeitgebers) is lost with the added term in equation (7). Additional forces acting on the system are required to compensate for the dampening, such as in the Van der Pol modification of equation (7): d x dx + β ( x 1) + α x = 0. The term (x 1) modifies the friction in the system. When x is large, (x 1) is positive, and friction will occur. When x is smaller than 1, the friction changes sign, meaning that at those small values of x, extra energy is put into the system, to make it swing. Obviously, the circadian pacemaker must provide the energy to overcome the friction. The choice of the term (x 1) is arbitrary. Many functions yielding negative friction at small values of x compensate for the dampening. The term (x 1) is just the simplest function doing this. While the model of equation (8) is capable of simulating a self-sustained circadian oscillation in constant darkness, it is not suitable to understand phase shifts in response to a zeitgeber. This is because no zeitgeber influence is described in the model equation. To do so, we have to add a zeitgeber term, Z(t), describing the influence of the zeitgeber as a function of time. d x dx + β ( x 1) + α x = Z( t). Equation (9) describes the relationship between the forces that act on the position, x, of the circadian oscillator. Each of these forces is represented by a mathematical function or operation, and scaling factors determine proper balance between them. Some of the terms are rather arbitrary. They deserve more experimental and modeling attention before they are sufficiently defined to be useful for simulations. Circadian processes are rhythmic. To display this characteristic graphically, it is useful to plot the state of the system in a phase-plane representation rather than plotting the output of the model as a function of the time as it is progressing (see Fig. 1, right-hand middle panel). Originally, in phase-plane representations, the current position (x) of the pendulum was plotted along the horizontal axis and the corresponding speed (8) (9) (dx/) along the vertical axis. The point representing the state of the pendulum rotates around the equilibrium point in each cycle. With appropriate choice of the dimensions of the variables, a sine function representing x (as in equation 6) appears as a circle in phaseplane representation. The point representing the state of the oscillation moves across the circle with constant velocity. If a line is drawn from the origin to the point representing the current state of the oscillation, then the angle between this line and the abscissa is the phase angle of the oscillation, hence the term phaseplane presentation. A dampening oscillation (equation 7) visualized in the same format yields a spiral curve contracting toward the equilibrium point. The Van der Pol model for a circadian pacemaker shows part of a contracting spiral at those phases where (x 1) is positive (i.e., when friction is present) and part of an expanding spiral when (x 1) is negative (i.e., when extra energy is provided to increase the amplitude of the oscillation). As a consequence, the phaseplane representation reveals a distorted circle. For the Van der Pol model, it can be shown that almost all initial conditions, after a sufficient number of cycles, result in the same distorted circle. Therefore, the circle is called a limit-cycle. Small perturbations, such as induced by short light pulses (Fig. 1), move the system away from the limit-cycle. When the stimulus is over, the system slowly returns to it (Winfree, 1980). No matter which type of model formulation is chosen, the terms need to be specified to allow useful simulations. The specific choice of the Van der Pol term, for instance, will have its influence on the time course of the output, x, of the model. The search for the terms and parameter values that lead to the best fits between model simulations and actual experimental data has led to a series of publications (see, for instance, Wever, 1966; Jewett et al., 1999; Kronauer et al., 1999; Forger et al., 1999). While the formulation of the general form of equation (9) is of the type that I have called concept modeling, the refinements of the Van der Pol term belong to the area of computational modeling. Hence, most of the work in this area has aimed at the best fit with real data with less emphasis on understanding underlying processes. This is quite different from the progress made concerning the zeitgeber influence. Since we know that the most important zeitgeber for entraining the human circadian pacemaker is light, it is necessary to think about how light interacts with the circadian system. Kronauer and colleagues have given much atten-

9 Beersma / MODELING CIRCADIAN RHYTHMS 311 tion to these questions. They accommodated fundamental photobiology into their model simulations (Kronauer et al., 1999). One of the elegant aspects of this work is that Kronauer and colleagues defined the light input pathway as a separate process in their model. Conceptually they view the system as a cascade of an input module in which actual light exposure is filtered according to the characteristics of the light input pathway. The filtered signal in turn is the input to the circadian pacemaker considered in the nd module. Apart from conceptual clarity, this approach allows for specific modifications of modules as soon as this is prompted by new experimental evidence. MODELING OF HUMAN CIRCADIAN BEHAVIOR Knowledge of the major characteristics of the circadian pacemaker will never be sufficient to understand circadian patterns of behavior. This is particularly evident in humans. Most human subjects organize their daily patterns of activity in response to the alarm clock rather than their biological clock; they let evening television programs influence their beimes, instead of internal requirements; many work night shifts; many take occasional naps. Since light is the most important zeitgeber for humans, the resulting changes in behavior modify the zeitgeber. In view of this, things are obvious: 1) human circadian patterns of behavior do not solely represent the output patterns of their circadian pacemaker and ) the patterns of behavior feed back onto the circadian pacemaker. Wever (1979) proposed the concept of interacting oscillators. One strong oscillator would represent the intrinsic circadian oscillations in core body temperature, while another, weaker oscillator would represent circadian fluctuations in a person s drive to be active. The oscillators were proposed to influence each other, and they would be influenced by external zeitgebers. In later years, Kronauer and colleagues (198) proposed that an external zeitgeber would exert its influence on the weak oscillator, which in turn would interact with the stronger oscillator. Earlier on, Pittendrigh (1958) had already proposed the interaction of multiple oscillators to underlie circadian patterns in fruit fly eclosion behavior. He thought that a central circadian pacemaker controls the phase of other, slave, oscillators. The models were meant to simulate patterns in activity and sleep under a variety of experimental circumstances. They remained unsatisfactory mainly because only 1 central circadian pacemaker was found. Beginning in the 1980s, both Alexander Borbély and Serge Daan proposed to integrate the concept of sleep homeostasis (Feinberg, 1974) with the concept of circadian rhythmicity into a concept of their interaction. This concept was elaborated independently by Borbély (198) and Daan and Beersma (1984). In addition, Daan and Beersma (1984) and Daan et al. (1984) transformed the concept into mathematical form, verified whether existing knowledge could be understood with the model, and predicted results of future experiments. This -process model of sleep regulation (Fig. 1, lower panel; Beersma, 1998; Achermann, 004) proposes that a need for sleep (called process S) accumulates during wakefulness. Endogenous mechanisms compare the momentary need for sleep with a threshold. When the threshold is reached, the subject goes to sleep, during which the need for sleep declines. Another threshold determines the end of the sleep interval. Both thresholds (together called process C) are influenced by the single circadian pacemaker. Their levels are high during the day and low at night, but subject to changes, for instance, induced by conscious decisions (Daan et al., 1984). The thresholds are not interpreted as rigid predetermined values limiting the range of process S. Instead, they indicate at which values of process S the likelihood of a change of behavioral state increases, either from wakefulness to sleep or vice versa. Many experiments have demonstrated the usefulness of the -process model in a variety of conditions (Beersma, 1998; Achermann, 004). Most prominently it explains how internal desynchronization of behavioral and physiological rhythms may occur, even with only a single circadian pacemaker. The model shows how a stable circadian pacemaker can exert gentle control over major aspects of behavior. That principle of the model may be more generally applicable than just in the case of the timing of sleep and wakefulness. An important consequence of the -process concept is that it makes us realize that the circadian pattern of behavior does not directly represent the activity of the pacemaker. The pattern results from the interaction with one or more downstream processes. These downstream processes have their own time constants and temporal properties, and in the presence of the pacemaker may behave much like slave oscillators (Pittendrigh, 1958). As a consequence, characteristics of the pacemaker can only be derived from circadian activity data if it can be trusted that these data are not

10 31 JOURNAL OF BIOLOGICAL RHYTHMS / August 005 influenced by those downstream processes (see Pittendrigh, 1981). DISCUSSION The comparison of the 4 levels of modeling reveals interesting results. It is clear that characteristics of the pacemaker as a whole depend on the characteristics of circadian rhythmicity in single cells. Important characteristics may also arise from the coupling between cells, as shown by Enright s work. Modelers thus do not need to search for mechanisms at the cellular level to explain all the characteristics of the pacemaker. This aspect is important in the context of the renewed interest in the temporal structure of daily activity patterns. The observation that morning and evening peaks in activity in a variety of species change their temporal relationship across the year have long inspired chronobiologists to think about the underlying mechanisms (Aschoff, 1966; Pittendrigh and Daan, 1976b). Models have been proposed consisting of separate circadian oscillators, each regulating 1 of the activity peaks (Daan and Berde, 1978). In the lack of experimental evidence for anatomically separate pacemakers in mammals, and upon the discovery of apparent duplication in clock genes, the mammalian circadian pacemaker was proposed to have components at the genetic level (Daan et al., 001). The genes per1 and cry1 would be responsible for the morning component and per and cry for the evening component. The conceptual model did not specify whether each pacer cell would contain both types of oscillators, or whether there would be morning oscillator cells and evening oscillator cells. In Drosophila, the morning and evening peaks in locomotor activity appear to be at least partially regulated by distinct cell groups in the brain (Stoleru et al., 004; Grima et al., 004). In mammals, there are at least 3 possibilities: 1) functionally distinct components of the oscillator may occur in each pacer cell; ) the network of pacers may contain a continuum of fast and slow pacers, which by mutual interaction yield peaks in the output signal of the pacemaker, that change their timing in response to seasonal changes in light exposure; and 3) the output of the pacemaker may interact with another downstream process to yield the seasonal dependence of activity patterns, for instance, along the lines proposed in the -process model of sleep regulation. More experimental work needs to be done to find the responsible mechanisms. No matter how wrong models are, they can teach us a lot. The -process model of sleep regulation, for instance, tells us that it is quite possible that locomotor activity is just partially related to the output of the circadian pacemaker. This should make us aware of the pitfalls that exist whenever we interpret activity profiles of animals as being directly induced by their circadian pacemakers. More generally, mathematical modeling can prove intuition wrong in both directions: It can demonstrate that phenomena that seem logical according to a certain conceptual model are in fact impossible in that model. It can also produce phenomena that are intuitively unlikely. Therefore, mathematical modeling is not a mere hobby of a small group of whiz kids but an indispensable, undervalued, fundamental, and integrated aspect of science, and certainly of chronobiology. ACKNOWLEDGMENT I thank Martha Merrow, Serge Daan, Marty Zatz, and Melanie Rüger for their comments on an earlier version of the manuscript. REFERENCES Achermann P (004) The two-process model of sleep regulation revisited. Aviat Space Environ Med 75:A37-A43. Antle MC, Foley DK, Foley NC, and Silver R (003) Gates and oscillators: a network model of the brain clock. J Biol Rhythms 18: Aschoff J (1960) Exogenous and endogenous components in circadian rhythms. Cold Spring Harb Symp Quant Biol 5:11-8. Aschoff J (1964) Die Tagesperiodik licht- und dunkelaktiver Tiere. Rev suisse Zool 71: Aschoff J (1966) Circadian activity patterns with two peaks. Ecology 47: Beersma DGM (1998) Models of human sleep regulation. Sleep Med Rev : Borbély AA (198) A two-process model of sleep regulation. Hum Neurobiol 1: Daan S, Albrecht U, Van der Horst GTJ, Illnerova H, Roenneberg T, Wehr TA, and Schwartz WJ (001) Assembling a clock for all seasons: are there M and E oscillators in the genes? J Biol Rhythms 16: Daan S and Beersma DGM (1984) Circadian gating of human sleep and wakefulness. In Mathematical Models of the Circadian Sleep Wake Cycle, Moore Ede MC and Czeisler CA, eds, pp , New York, Raven Press. Daan S, Beersma DGM, and Borbély A (1984) Timing of human sleep: recovery process gated by a circadian pacemaker. Am J Physiol 46:R

11 Beersma / MODELING CIRCADIAN RHYTHMS 313 Daan S and Berde C (1978) Two coupled oscillators: simulations of the circadian pacemaker in mammalian activity rhythms. J Theor Biol 70: Daan S and Pittendrigh CS (1976) A functional analysis of circadian pacemakers in nocturnal rodents: III. Heavy water and constant light: homeostasis of frequency? J Comp Physiol 106: Diez-Noguera A (1994) A functional model of the circadian system based on the degree of intercommunication in a complex system. Am J Physiol 67:R1118-R1135. Enright JT (1980a) The Timing of Sleep and Wakefulness. New York: Springer. Enright JT (1980b) Temporal precision in circadian systems: a reliable neuronal clock from unreliable components? Science 09: Feinberg I (1974) Changes in sleep cycle patterns with age. J Psychiatr Res 10: Forger DB, Jewett ME, and Kronauer RE (1999) A simpler model of the human circadian pacemaker. J Biol Rhythms 14: Forger DB and Peskin CS (003) A detailed predictive model of the mammalian circadian clock. Proc Natl Acad Sci U SA 100: Forger DB and Peskin CS (005) Stochastic simulation of the mammalian circadian clock. Proc Natl Acad Sci U S A 10: Gario-Ojalvo J, Elowitz MB, and Strogatz SH (004) Modeling a synthetic cellular clock: repressilators coupled by quorum sensing. Proc Natl Acad Sci U S A 101: Grima B, Chélot E, Xia R, and Rouyer F (004) Morning and evening peaks of activity rely on different clock neurons of the Drosophila brain. Nature 431: Hardin PE, Hall JC, and Rosbash M (1990) Feedback of the Drosophila period gene product on circadian cycling of its messenger RNA levels. Nature 343: Honma S, Nakamura W, Shirakawa T, and Honma KI (004) Diversity in the circadian periods of single neurons of the rat suprachiasmatic nucleus depends on nuclear structure and intrinsic period. Neurosci Lett 358: Jewett ME, Forger DB, and Kronauer RE (1999) Revised limit cycle oscillator model of human circadian pacemaker. J Biol Rhythms 14: Kronauer RE, Czeisler CA, Pilato M, Moore-Ede MC, and Weitzman ED (198) Mathematical model of the human circadian system with two interacting oscillators. Am J Physiol 4:R3-R17. Kronauer RE, Forger DB, and Jewett ME (1999) Quantifying human circadian pacemaker response to brief, extended, and repeated light stimuli over the photopic range. J Biol Rhythms 14: Kunz H and Achermann P (003) Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators. J Theor Biol 4: Leloup JC and Goldbeter A(004) Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms. J Theor Biol 30: Lema MA, Golombek DA, and Echave J (000) Delay model of the circadian pacemaker. J Theor Biol 04: Pittendrigh CS (1958) On the significance of transients in daily rhythms. Proc Natl Acad Sci U S A 44: Pittendrigh CS (1981) Circadian organization and the photoperiodic clocks. In Biological Clocks in Seasonal Reproductive Cycles, Follett BK and Follett DE, eds, pp 1-35, Wright, Bristol, UK. Pittendrigh CS and Daan S (1976a) A functional analysis of circadian pacemakers in rodents: I. The stability and lability of spontaneous frequency. J Comp Physiol 106: 3-5. Pittendrigh CS and Daan S (1976b) A functional analysis of circadian pacemakers in rodents: V. Pacemaker structure: a clock for all seasons. J Comp Physiol 106: Reppert SM and Weaver DR (00) Coordination of circadian timing in mammals. Nature 418: Roenneberg T, Daan S, and Merrow M (003) The art of entrainment. J Biol Rhythms 18: Roenneberg T and Merrow M (003) The network of time: understanding the molecular circadian system. Curr Biol 13:R198-R07. Scheper TO, Klinkenberg D, van Pelt J, and Pennartz C (1999) A model of molecular circadian clocks: multiple mechanisms for phase shifting and a requirement for strong nonlinear interactions. J Biol Rhythms 14:13-0. Shirakawa T, Honma S, Katsuno Y, Oguchi H, and Honma KI (000) Synchronization of circadian firing rhythms in cultured rat suprachiasmatic neurons. Eur J Neurosci 1: Stoleru D, Peng Y, Agosto J, and Rosbash M (004) Coupled oscillators control morning and evening locomotor behaviour of Drosophila. Nature 431: Tan Y, Dragovic Z, Roenneberg T, and Merrow M (004) Entrainment dissociates transcription and translation of a circadian clock gene in Neurospora. Curr Biol 14: Tomita J, Nakajima M, Kondo T, and Iwasaki H (004) No transcription-translation feedback in circadian rhythm of KaiC phosphorilation. Science 307: Welsh DK, Logothetis DE, Meister M, and Reppert SM (1995) Individual neurons dissociated from rat suprachiasmatic nucleus express independently phased circadian firing rhythms. Neuron 14: Wever R (1966) Ein mathematisches Modell für die circadiane Periodik. Zeitschrift für angewane Mathematik und Mechanik 46:T148-T157. Wever R (1979) The Circadian System of Man. Berlin: Springer- Verlag. Winfree AT (1967) Biological rhythms and the behavior of populations of coupled oscillators. J Theor Biol 16:15-4. Winfree AT (1980) The Geometry of Biological Time. New York: Springer.