Cent. Eur. J. Biol.• 6(5) • 2011 • 712-729 DOI: 10.2478/s11535-011-0062-4

Central European Journal of Biology

Modeling circadian clocks: Roles, advantages, and limitations

Review Article

Didier Gonze1,2,*

1Unité de Chronobiologie Théorique, Service de Chimie Physique, Université Libre de Bruxelles, Campus Plaine, B-1050 Brussels, Belgium 2Laboratoire de Bioinformatique des Génomes et des Réseaux, Faculté des Sciences, Université Libre de Bruxelles Campus Plaine, B-1050 Brussels, Belgium Received 31 January 2011; Accepted 17 June 2011

Abstract: Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

Keywords: Mathematical modeling • Circadian rhythms • Interlocked feedback loops • Temperature compensation • Molecular noise • Synchronisation © Versita Sp. z o.o.

Introduction The major role of circadian rhythms is to coordinate physiological and behavioural processes with natural Did you already try to switch off your alarm clock daily variation, and in particular they allow an organism before going to bed? You will most probably wake up to anticipate daily variations of the environment [5,6]. at the usual time because your body has its own inbuilt This timekeeping mechanism has been shown to play clock. Since life exists on Earth, it is subject to the a crucial role in establishing the direction of migration periodic alternation of day and night and has developed in birds [7] and butterflies [8], to provide an adaptive mechanisms to adapt to this periodic change of the advantage to cyanobacteria [9] and to plants [10], and environment. Virtually all living organisms, ranging from to control the annual flowering time of Arabidopsis [11]. cyanobacteria to insects, plants, and mammals, exhibit In mammals, an obvious function of circadian clocks is daily changes in their behaviour. Remarkably, these to regulate the timing of our daily activity and sleeping rhythms have been demonstrated to persist even in the phases [12] through the regulation of hormone production absence of external clues (i.e. in constant darkness) [1], and metabolism [13]. Disruptions of the circadian clock indicating that they are generated endogenously by living have been related to sleep phase disorders [14], and to organisms. In constant conditions, biological rhythms are diseases such as cancer [15]. characterized by a free-running period of about 24h [2,3]. Genetic advances have demonstrated that circadian Franz Halberg qualified these rhythms as “circadian” oscillations originate at the molecular level through (from latin “circa”, around, and “dies”, day) [4]. the regulation of specific genes called “clock genes”

* E-mail: [email protected] 712 D. Gonze

[16,17]. Since the discovery of the first clock gene, named In mammals, the central circadian oscillator is per (for period), in Drosophila [18], many additional clock located in the suprachiasmatic nuclei (SCN) of the genes have been identified in several model organisms hypothalamus. Through the regulation of the expression including Neurospora, Drosophila, Arabidopsis and of clock-controlled genes in every SCN neuron, the mammals. Oscillations in the expression of these genes molecular clock controls the timing of physiological are caused by multiple regulatory feedback loops processes such as hormone release, blood pressure, involving a handful of key genes. The more we learn body temperature, or the sleep-wake cycle (Figure 2) [6]. about the molecular circuitry of the cellular circadian The circadian pacemaker is composed of a population clock, the more the picture resembles a complex but of cells which need to be coupled and synchronized carefully assembled system of cogs and gears of a to be able to generate a robust overt rhythm [19,20]. mechanical watch (Figure 1). As for the watch, biologists Feedback from the clock to the input pathways believe that the complexity of the molecular mechanism (sometimes called gating) and from the outputs to the should provide the biological clocks with useful clock make the understanding of the circadian system properties, like ensuring a high precision or a fine tuning even more challenging (Figure 2) [21,22]. of the clock-controlled processes. Although working Because of their involvement at different levels of autonomously, this molecular clock is also under the the organization, circadian rhythms are extensively influence of the environment. In particular it responds to studied by geneticists, molecular biologists, structural light, food, and temperature. biologists, physiologists and... mathematicians. Long

Figure 1. Cogs and gears of (A) a mechanical clock and (B) a biological clock. Circadian clocks are composed of mulitple genes and feedback loops which, as the cogs and gears of mechanical clocks, may be required for a high precision in the timing, or a reliable response to external signals.

Figure 2. Scheme of the circadian system. Three parts are commonly distinguished in this system: the inputs (environmental signals such as the light, the temperature, or the food), the core oscillator (or pacemaker) and the output (clock-controlled processes such as the blood pressure, hormones release and the sleep-wake cycle). The circadian pacemaker is composed of a population of cells (grey circles), each of them being able to produce circadian oscillations through a molecular mechanism involving a few clock genes and multiple feedback loops. Cells are synchronized through inter-cellular coupling. The complexity of the circadian system results from the integration of multiple inputs by the pacemaker, the tight regulation of various clock-controlled processes, the complexity of the molecular mechanism generating oscillations at the single cell level, the complexity of the cellular network, and the feedbacks from the pacemaker to the input pathways and from the outputs to the pacemaker.

713 Modeling circadian clocks: Roles, advantages, and limitations

before the discovery of the first clock genes, modeling In the accompanying paper, I have described approaches had been pursued. Before 1960, there were mathematical models developed in the circadian field already more than 600 modeling papers on circadian [25] and explained how these models are derived clocks [23]. Theoretical models are now widespread and and how they are analyzed. I have shown how to constitute a valuable tool to understand the complexity compute bifurcation diagrams, entrainment, and phase of the circadian organisation. response curves. In the present paper, I will illustrate Modeling and computer simulations can be used how mathematical models have shed light on key to test hypotheses and to make predictions but also questions raised in the circadian field. These examples to explain unintuitive dynamic behaviour. Some roles will also show that understanding the cogs and gears and advantages of modeling are listed in Table 1. As of circadian clocks requires an integrated view of the illustrated by this non-exhaustive list, the roles of circadian system in which various aspects (molecular modeling go much beyond the simple reproduction of circuitry, intercellular coupling, and stochasticity) may experimental data. Building a model implies gathering play complementary and inter-dependent roles. I will and synthesizing multiple and disparate information on also comment on several limitations of the modeling a system of interest, identification of key interactions approach, and identify challenges for the future. and connections between the different components of the system, and discrimination between essential and superfluous components. This information-gathering Key circadian problems and insights work thus provides a complete view of the system and from modeling highlights missing pieces in the puzzle. The scheme is then set up in a mathematical Origin of the 24-hour period framework and analyzed, most often, by numerical With a period of 24 hours, circadian oscillations represent simulations. Computer analyses have the advantage of the biological rhythm with the longest period known to being carried out quickly and without any cost and can lead be generated at the molecular level. The origin of this to predictable hypotheses, to the explanation of unintuitive particularly long period puzzled geneticists. Before the behavior, and can provide answers to questions that actual genetic components of the circadian clock were cannot be addressed easily by experiment. To understand identified, circadian biologists had already proposed a the “design principles” of circadian clocks, it is useful to “model” for the genetic circadian clock. In this so-called compare the actual structure of the genetic network with chronon model [26,3], a long polycistronic DNA region alternative architectures that would be very hard to build was placed at the center of the molecular mechanism. experimentally. Modeling is thus more than fitting observed This chronon was long enough that the time to be data [23]. Being conceptual or explicit, every model gives transcribed and translated covered a circadian period. new insights, or suggests a possible explanation of the The discovery of the clock genes in Drosophila (and observation. A model remains “a hypothesis about how their analogs in other organisms) rapidly discredited physical models work” [24]. the chronon model. The oscillations arise from a core

Roles and advantages of modeling

• Analyzing and understanding complex situations that are difficult to describe in verbal terms and for which sheer intuition becomes unreliable.

• Rapid and systematic exploration of alternative mechanisms and large ranges of conditions.

• Clarification and validation (or invalidation) of working hypotheses.

• Identification of key interactions and parameters, and their qualitative or quantitative influence on the system’s behaviour.

• To determine precisely the conditions in which different behaviours will occur.

• To address questions that are difficult or impossible to approach experimentally.

• To provide testable predictions and suggestions for new experiments, and sometimes counterintuitive explanations that may corroborate or not conclusions drawn from experimental observations.

• To provide a unified framework to account for the experimental observations and bring into light possible similarities between apparently unrelated processes.

Table 1. Roles and advantages of modeling biological systems (adapted from ref. [129]).

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negative feedback loop in which the clock protein The current picture of the circadian network suggests inhibits the transcription of its own gene. Nevertheless, that the actual mechanism is probably intermediary. the question of the origin of time delays in the molecular Indeed, the circadian network is composed of interlocked mechanism was still posed. Besides transcription multiple feedback loops, which potentially encompass and translation, which indeed require some time, several oscillators [34,35]. Relying on a proper balance additional time delays have been attributed to multiple of synthesis and degradation rates, this intricate network phosphorylations, nuclear transport, and protein probably emerged as a compromise to generate robust complex formation. Although these steps play a role in and long-period oscillations. The hypothesis that the timing circadian oscillations, analyses of mathematical circadian network evolved from the coupling of single- models show that it is not mandatory to incorporate feedback oscillators with ultradian and possibly damped multiple intermediary steps in the feedback loop. A oscillators was put forward and investigated through proper balance between synthesis and degradation mathematical modeling by Roenneberg and Merrow [36]. processes is already sufficient to induce long-period oscillations. This is exemplified by the model of Goodwin Temperature compensation which includes only three components [27,28]. These Temperature compensation is one of the most models predict that a single delayed negative feedback conspicuous properties of circadian clocks. The free- loop mechanism is able to generate 24-hour period limit- running period of circadian oscillations is remarkably well cycle oscillations (Figure 3). compensated. It remains nearly the same at different Another hypothesis suggested by mathematical temperatures, as long as the temperature is maintained modeling comes from the observation that the coupling constant, within the physiological range [3,37]. The between two ultradian-period oscillators can generate very origin of this phenomenon is still not fully understood. long period oscillations [29-31]. Indeed, oscillators with a Since temperature is expected to affect many kinetic period much smaller than 24 hours can, once mutually parameters of the system, understanding how the global coupled, generate 24-hour period oscillations. The period dynamics remains unaffected is not straightforward. of such coupled systems is sensitive to small perturbations Mathematical models have been proposed to test in the period of the individual oscillators [32,33]. Moreover, various hypotheses. although many genetic oscillators have been shown to One of the first models to address this issue was display ultradian periods, they were not shown to be proposed by Lakin-Thomas et al. [38]. Using a single- involved in the generation of circadian rhythms. variable delay model, the authors proposed that

A A Cell

negative feedback

degradation light * transcription P

clock gene Delay (phosphorylation, complex formation, nuclear transport, etc)

translation degradation P mRNA (M) protein

B C 14

12 12

10 10

8 P 8

6 6 P* 4 4

Concentration (nM) 2 2 M

0 protein concentration, P (nM) 0 0 12 24 36 48 60 72 84 96 0 0.5 1 1.5 2 2.5 Time (h) mRNA concentration, M (nM)

Figure 3. Example of limit-cycle oscillations. (A) Scheme of the core model for the circadian clock. (B) Times series of key variables: mRNA (M) and protein (P and P*) concentration. (C) Corresponding trajectory (limit cycle) in the phase space, i.e. the space defined by the variables. The limit cycle is shown as a projection in the (M, P) plane.

715 Modeling circadian clocks: Roles, advantages, and limitations

temperature affects simultaneously the amplitude and dependant proportions [46]. A theoretical model was the speed of the system along the limit cycle in such a used to explore the possible role of this alternative way that at higher temperatures the system runs faster splicing in the temperature compensation [47]. but undergoes larger-amplitude oscillations. Because Although the question of temperature compensation they are perfectly balanced, these two effects result in of circadian clocks remains unresolved, theoretical a constant time duration to run through the limit cycle, models highlight the possible implication of different and therefore lead to a constant period. Unfortunately, mechanisms. These mechanisms, which can be although observed in other systems as well [39], this global or system-specific, have in common a reliance property usually does not hold for any non-linear system: on balancing antagonistic effects. By pointing to key an increase of the period is not always associated with elements of this balance, these models lead to the an increase of the speed along the limit cycle, and even testable prediction that altering or removing these if it is they are rarely perfectly balanced. elements would affect temperature compensation. Ruoff and colleagues used the Goodwin model to examine the effect of each parameter on the period. By Interlocked feedback loops describing explicitly the dependance of each reaction Recently, a large amount of evidence has shown that rate on temperature, they found that the period of the the regulation of circadian clock genes involves multiple oscillations increases when the speed of synthesis interlocked feedback loops: the core negative feedback reactions increases, while it decreases when the rate loop is usually interconnected with a positive loop, some of degradation steps increases. This observation led the genes/proteins being common to both circuits [48-50] authors to suggest that when the temperature is varied, (Figure 1B). In addition, feedback from the clock to all reactions are speeded up but their differential effects the light input exists. The reason(s) for this complexity on the period compensate each other. Ruoff called this has always intrigued circadian biologists (Figure 1B). phenomenon “antagonistic balance” [27,28,40]. A similar Based on intuitive considerations, it is often said that a conclusion was reached by Leloup and Goldbeter in single negative feedback loop would generate unstable a molecular model for the Drosophila circadian clock oscillations in the sense that they would eventually [41] and by Kurosawa and Iwasa who analyzed several damp out and be highly sensitive to noise [51]. In circadian clock models [42]. Interestingly, the loss of fact, numerous theoretical models show that delayed temperature compensation described in some mutants negative feedback loops are sufficient to produce could be explained by a change in a degradation self-sustained (limit-cycle) oscillations [28,52-54], parameter which destabilizes the antagonistic balance. which are relatively robust to molecular noise [55]. New theories are currently being developed to determine Whether the additional positive feedback loops further the conditions required for a proper compensation of the increase the robustness or provide additional properties differential effects of temperature [43,44]. to the circadian clockwork remains to be elucidated with Since a balance would require delicate tuning of theoretical approaches. parameter values, such mechanisms would not be The analysis of various designs of genetic oscillators robust to mutations and would therefore not explain by Novak and Tyson [56] showed that positive feedback the maintenance of temperature compensation in may lead to amplification of the oscillations. Positive many mutants. Besides these global properties that the feedbacks may also shape the oscillations: adding circadian system may have acquired, organism-specific positive feedbacks can be used to generate relaxation- molecular mechanisms and design principles have type oscillations, i.e. oscillations characterized by also been held responsible for period compensation different time scales associated with rapid/slow [45]. In Drosophila, a speculative explanation was variations of the variable [57,58]. Relaxation oscillators based on the observation that the central clock protein are more robust to noise [59] and easier to synchronize PER can form a dimer, and on the hypothesis that the [60] than sine-like oscillators. PER-PER dimer and the PER protein alone do not In an exhaustive comparison analysis, Tsai et al. [61] have the same properties: they are distinguished by showed that an oscillator based on interlocked positive different nuclear import rates, and temperature affects and negative feedback loops generates high amplitude the PER-PER association rate [45]. In this model, oscillations in a larger range of parameter values and temperature compensation occurs as a consequence that the period of the oscillations in these models is of a balance between the rate of nuclear entry and more tunable (i.e. it can be changed over several orders the rate of dimerization. In Neurospora, two forms of of magnitude without affecting much the amplitude of FRQ (the central clock gene) have been shown to be the oscillations) compared to models based on a single formed (through alternative splicing) with temperature- negative feedback loop.

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An advantage of modeling is that we can study The results reported in these various studies extend alternative regulatory network architecture. We can thus our intuitive ideas for the need of multiple feedback compare models with and without the positive feedback loops. Although it is hopeless to discriminate between loop (simply by removing a component or by setting these different advantages of complexity, this (non- a given kinetic parameter to zero). Comparing two exhaustive) list should at least convince biologists models for the Drosophila circadian clock, Ruoff et al. that the complex mechanism underlying circadian [62] observed that incoporating a positive feedback loop oscillations may provide the circadian system with many could account for the effect of gene dosage, amplify the properties, besides rendering the oscillations robust. oscillations, and favor temperature compensation. A detailed analysis of a mammalian circadian model Rhythm suppression by a light pulse [63] revealed that components of the positive feedback An intriguing property of circadian rhythms was reported loop are more sensitive to parameter variations. The by Engelmann et al. [67]. These authors observed fact that the output of the clock seems to be mediated that if a well-dosed pulse of light is administered at a by components of the positive feedback suggests that certain circadian phase, the petal movement rhythm regulation of the clock-controlled genes might be fine- of the plant Kalanchoe can be permanently stopped. tuned through this positive feedback loop. Indeed, the A second light pulse can then re-initiate the petal components of the positive feedback loop have more movement rhythm. Intuitive understanding of this variable phases and amplitudes, which would allow behavior is not easy. How could a transient perturbation them to modulate output pathways without disturbing induce a permanent, yet reversible, physiological circadian oscillations. Variations of parameters (such change? Analysis of the dynamic properties of non- as synthesis rates) of processes involved in the positive linear systems, as the regulatory mechanisms of the feedback loop would not affect the core oscillations, circadian clock, highlights possible mechanisms for the but would change the phase and strength of gene origin of this phenomenon. expression regulated by components of the positive Winfree explains this observation by the presence of feedback loop [63]. a “singularity point” [33,68]. The light pulse can bring the The mammalian circadian clock also comprises system near a singularity point, i.e. an unstable steady multiple negative feedback loops, which constitute state. The system can then be stopped for a potentially multiple potential sources for oscillations [34]. As a long time period. However, since the system cannot consequence, if a gene involved in one of the loops remain around this steady state forever, it will ultimately is mutated or removed, sustained oscillations in other return to the oscillatory regime. The effect of the second components can subsist. This redundancy of feedback light pulse would be to accelerate this recovery [33,68]. loops thus provides robustness to mutations and partial Using a molecular model for the Drosophila disruption of the circadian clock. circadian clock, Leloup and Goldbeter [69] provide Generating circadian oscillations in constant another possible explanation based on the coexistence conditions is not the primary task of the circadian clock. of a stable steady state and a limit cycle. In identical Oscillations must also be properly entrained by light- conditions the system can either reach the steady state dark cycles and adapt to various photoperiods. Using or oscillate. In these conditions, a light pulse could make a computational model for the mammalian circadian the system switch from the limit cycle to the steady clock, Geier et al. [64] reported that gating of the clock to state. Because the steady state is stable the system the light input pathway may be involved in control of the would remain definitively in this state unless a second phase locking. More recently, Troein et al. [65] applied light pulse returns the system to the oscillatory regime. a genetic algorithm to automatically generate numerous The model has been used to determinate the conditions clock models and to select the ones which exhibit the (phase, amplitude, duration of the pulse) in which such a best adaptation under various photoperiods. They found phenomenon might occur. Although not intuitive for most that models with several feedback loops could adapt biologists, the coexistence between several dynamic easily to a greater number of conditions. Saithong et al. behaviours (here a steady state and a limit cycle) for the [66] compared several variants of the Goodwin-type same set of parameter values is commonly encountered oscillators that incorporate one or several feedback when studying non-linear dynami systems. loops, as well as various versions of an Arabidopsis These theoretical results motivated experimental circadian clock model and showed, through sensitivity studies in the algae Gonyaulax [70] and in Neurospora analyses, that an interlocked multi-loop structure [71]. In each of these model organisms either a reinforces robustness and enhances the response to chemical or a light pulse can drive the system towards both external and internal signals. its singularity. These analyses further suggest that the

717 Modeling circadian clocks: Roles, advantages, and limitations

singularity behavior is due to the loss of rhythms in all oscillators can be observed [79,80]. A cluster is a group cells, and is not due to a desynchronisation of the cells. of oscillators which oscillate synchronously; but the There is another possible explanation for this light clusters are not synchronized between themselves. In pulse-induced suppression of circadian rhythms. One can free running conditions, they can even display different imagine that the light pulse induces a desynchronisation periods. The spontaneous apparition of clusters is of the individual cellular oscillators. When synchronized, commonly encountered in models of coupled oscillators. the coupled cells generate a global output which exhibits These results, as those discussed for the previous strong and stable oscillations, while desynchronized example (cf. “rhythm suppression by a light pulse”), show cells present out-of-phase oscillations in such a way that that in order to interpret the experimental observations the global output (average over all cells) does not exhibit it is important to keep in mind not only the cellular regular oscillations. This hypothesis was suggested and molecular mechanism which might rely on coupled tested by the group of Ueda [72] using mammalian cells feedback loops (and possibly coupled oscillators), but artificially modified to be responsive to light through also the inter-cellular coupling. a controllable melanopsin-dependent pathway. They show that a critical light pulse can drive cellular clocks Prediction of missing genes into a singularity state and that the loss of oscillations In all the examples illustrated above, the models were is due to the desynchronization of individual cellular mainly used to fit experimental data and to provide clocks. explanations of non-intuitive behavior. If a model cannot reproduce experimental data, it is clear that some Rhythm splitting in constant conditions key elements are missing in the model or that some The spontaneous splitting of circadian rhythms into two or processes are not accurately described (sometimes it more components in animals held in constant conditions may simply be that kinetic rates are wrongly estimated), was already reported by Pittendrigh and Daan in 1976 but it is hard to determine what exactly is missing. [73]. Splitting was manifest by the sudden apparition of This led Endy and Brent to formulate the following two bands of locomotor activity on the actogram. More criticism [81]: “By the 1840s, astronomical simulations recently, such spontaneous splitting was observed in were precise enough to allow prediction of Neptune. the rhythmic behaviour of SCN cells [75]. Such splitting By contrast, during the 1990s no biological model of was extensively studied in hamster [74,75] but was also circadian rhythm allowed prediction of the regulatory observed in rat [76] and in human [77]. Several aspects proteins Cycle or Clock”. characterize this phenomenon: (1) The period of the A notable exception which demonstrates that split components is shorter than the period of the unsplit modeling can be used to predict a key clock gene was (free-running) rhythms, (2) Synchronization occurs when provided by the lab of Millar, who studied the circadian the components reached a 180o phase-relation, and (3) clock of Arabidopsis [82,83]. A first mathematical model a reduction in light intensity entrains the re-fusion of the based on a single regulation of two central clock genes split components [76]. was shown to be sufficient to generate limit-cycle These results led Pittendrigh and Daan to suggest oscillations similar to those observed experimentally that the circadian pacemaker consists of two mutually [82]. This model, however, did not reproduce well coupled oscillators [73] that become uncoupled in the behaviour of a long-period mutant and did not constant conditions. This idea was recently reinvestigated respond accurately to long photoperiods. To tackle by Oda et al. [78], who used the same abstract model as these limitations, the authors extended their model by Pittendrigh and Daan and studied in a systematic way incorporating a new gene and postulating additional all possible coupling schemes. Assuming that the two regulations [83]. Analysis of this predictive model allows oscillators are not identical and that the coupling strength to fit the effect of light very accurately. Back tothe achieved in each individual animal is variable, this model lab, the authors then identified experimentally a good provides a unified picture of all different splitting patterns candidate of this additional clock gene and showed presented by the hamsters and explains why the phase that its regulations were consistent with the postulated difference achieved between the split components is regulations [83]. often near 180°. There are other examples showing that combining As proposed for the phenomenon of rhythm mathematical and experimental approaches may also suppression, another explanation can be hypothesized if be fruitful in predicting the role of clock genes in the we consider the behaviour of a population of oscillators. In circadian system. Gallego et al. [84] used a mathematical a heterogeneous population of oscillators, spontaneous model to suggest a role for a clock control kinase (tau) splitting of the population into two or several clusters of in the regulation of the phosphorylation and degradation

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of two circadian key regulators (PER1 and PER2) and coupling may easily lead to complex behaviours and these predictions were experimentally validated. thereby to physiologic disorders. Currently, such success stories are relatively uncommon in the field of circadian clocks because Robustness to molecular noise many models reproduce well the experimental data, In all the examples discussed above, the models were the latter being often qualitative. These models are of a deterministic nature. They describe an average mostly used to predict the behaviour of the system in behaviour as would be observed in an ideal system conditions not yet tested experimentally or to explain a perfectly mixed and with many molecules of each particular observation. Sometimes, ad hoc adaptations species. However, at the level of a single cell, the number of the model are done to fit the data or to obtain the of mRNA and protein molecules that take part in the desired behaviour. These adaptations, often arbitrary molecular clockwork is limited. To take into consideration and speculative, also point to aspects that deserve the molecular fluctuations that arise when the number of some attention and may open the door to interesting molecules involved in the regulatory mechanism is low, predictions. Moreover, the fact that oscillations can we need to resort to stochastic simulations [90,91]. be obtained in simple models, without involving all the Numerical simulations of the stochastic models components of the clock raises the question of their role show that robust circadian oscillations can already occur (see section “interlocked feedback loops”). with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively Exploring complex dynamics [55,92] (Figure 4). Such stochastic analyses also allow The fact that the circadian network relies on multiple us to identify various factors that affect the robustness interlocked feedback loops allows the generation of circadian oscillations with respect to molecular of complex dynamic behaviours such as chaotic noise. Besides an increase in the number of molecules, oscillations [85,86]. This kind of irregular behaviour entrainment by light-dark cycles and cooperativity in is often considered as pathologic. There are however repression enhance robustness, whereas the proximity theoretical analyses which show that such behaviour of a bifurcation point leads to less robust oscillations may provide the system with advantages, namely in [55]. The ability of the circadian oscillations to lock their the adaptation of the oscillator to various conditions, by phase to the external light-dark cycle as a function of the selecting an appropriate circadian period [87]. strength of the periodic forcing was examined in more Complex behaviour was also shown to be detail by Calander [93]. Another parameter that appears associated with sleep-phase disorders encountered to be crucial for the coherence of circadian rhythms is by some patients. Chaos and quasi-periodicity are the binding/unbinding rate of the inhibitory protein to the common behaviours obtained when an oscillator is not promoter of the clock gene [55,92]. Intercellular coupling properly entrained by a periodic forcing [88]. Quasi- further increases the robustness of circadian oscillations periodic oscillations are characterized by oscillations [94-96]. that are relatively regular but that are not phase-locked Using numerical simulations, one can also explore to the external periodic signal. Such behaviour has been alternative architectures of the circadian regulatory observed in a model for the mammalian circadian clock mechanism and quantify the robustness of the subject to a light-dark cycle when the levels of two clock oscillations to molecular noise. This comparison led proteins (CRY and PER) were not carefully balanced Leibler and his collaborators to predict that oscillators [89]. In this case quasi-periodicity was characterized by based on interlocked positive and negative feedback a drift of the phase of the circadian oscillations: each loops with different time scales would exhibit more day the phase was advanced (or delayed) by a few robust circadian oscillations [91,59]. hours. This behaviour can be related to a sleep-phase Besides molecular noise due to the stochastic disorder called the non-24h syndrome. Patients affected nature of the reactional steps involved in the core by this syndrome tend to go to sleep every day slightly mechanism, another source of noise, sometimes earlier (or later), as if their physiological rhythms were referred to as extrinsic noise, arises from unavoidable independent of the external light-dark cycle. fluctuations in parameter values and inputs of the clock. Chaos and quasi-periodicity are also common in Ullner et al. [97] investigated how the interplay between populations of coupled oscillators. Usually, synchronizing light fluctuations and inter-cellular coupling affects the a population of oscillators requires an appropriate dynamics of the collective rhythm in a population of balance between the stability of the individual oscillations globally coupled cellular clocks. Based on experimental and the inter-cellular coupling strength. We can thus considerations, they assumed an inverse dependence expect that defects of some clock components or in the of the cell-cell coupling strength on light intensity and

719 Modeling circadian clocks: Roles, advantages, and limitations

Figure 4. Example of stochastic oscillations obtained by simulation of the core circadian model (see Figure 3) using the stochastic Gillespie algorithm [90]. The level of noise depends on the number of molecules (symbolized by the dots on the left panel of each row). (A) When the number of molecules is high, the fluctuations are small. (B) When the number of molecules is low, the fluctuations are more pronounced and the distribution of the periods is broader. For details, see [55].

showed that noise-induced rhythms can be generated. we implicitly assume that all the cells are synchronized Interestingly, such improved coherence can be only and behave in a coordinated manner. Although circadian observed at the level of the overt rhythm and not at oscillations have indeed been observed at the level the level of the individual oscillators, thus suggesting a of single, isolated cells, the cells are usually coupled cooperative effect of noise, coupling, and the emerging together. Such inter-cellular coupling is required to keep synchronization between the oscillators. individual circadian oscillators in synchrony. Taken together, these stochastic studies allow The problem of inter-cellular coupling and the identification of the key factors that affect robustness of synchronization of a clock cell population was the oscillations, examination of alternative architectures, recognized a long time ago, namely by Pittendrigh [99], and elucidation of the intricate interplay between Pavlidis [32,100,101], and Winfree [102]. Early models individual clocks, intercellular coupling, and light input. revealed important generic properties resulting from Today, these questions cannot be answered on the the coupling between a population of circadian clocks. basis of experimental observations because this would The ability of synchronization depends on the coupling require accurate quantification of the effect of noise strength [101] and the idea that noise might play a role for several alternative cellular network architectures, in rhythm splitting or the spontaneous loss of synchrony which is beyond current experimental possibilities. was already evoked [101]. There is nevertheless a recent pioneer study that Today, the molecular origin of the coupling illustrates the possibility of measuring noise in single mechanisms is the object of numerous investigations, (uncoupled) cyanobacteria cells and shows how these but we are still far from understanding the molecular measurements could help explain the mechanism of the mechanisms allowing cells to tick synchronously, as circadian clock [98]. well as the reasons (if any) and consequences of the complexity of the cellular network [19]. Answering these Coupling and synchronization questions will be greatly helped by recent technological Most of the models cited above describe the molecular advances that allow us to monitor the expression of mechanisms responsible for the generation of oscillations clock genes in single cells [20,103]. These experiments at the level of a single cell. When such models are used reveal that the SCN is composed of a heterogeneous to relate genetic circadian oscillations with physiology, network of circadian clocks [20,103]: when cultured,

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these SCN cells display a large variability in their period SCN: efficient synchronization is achieved when the and amplitude (some cells present damped oscillations). average neurotransmitter concentration (which is the How such a heterogeneous ensemble of oscillators is coupling agent) dampens individual oscillators in such efficiently synchronized is not a trivial question [104]. a way that the individual oscillators behave like forced Both abstract and detailed molecular models have been damped oscillators [106,107]. Further works are currently proposed to identify the conditions required for efficient pursued to test different network topologies, including synchronization and to examine the effect of coupling local neighbour coupling and small world architectures mechanisms and network topologies (Figure 5). [108] (Hafner and Gonze, unpublished data). The first SCN models mainly focused on the effect More recently, detailed molecular models for the that network topology had on synchronizability. Kunz SCN have been published [109-111]. The model by and Achermann [79,105] showed that in a population of Bernard et al. [109] is based on a heterogeneous network locally coupled van der Pol oscillators, increasing the of circadian neuronal oscillators where individual number of oscillators results in a more stable rhythm oscillators are damped rather than self-sustained. The and a higher robustness to molecular noise. coupled circadian oscillators quickly synchronized and One of the first molecular models to address the produced a coherent output and in large populations question of the synchronization was proposed by Ueda such oscillators either synchronize or gradually lose et al. [94]. In this model, a population of Drosophila rhythmicity, but do not run out of phase, demonstrating circadian oscillators were globally coupled by a diffusible that rhythmicity and synchrony are codependent. The agent secreted periodically by each individual cell and number of oscillators and the connectivity are important acting back on each cell. Numerical simulation of the for these synchronization properties. Slow oscillators model in various conditions allowed the authors to show have a higher impact on the period in mixed populations that this mechanism is efficient in synchronizing the cells and coupled circadian oscillators can be efficiently and that the system is robust to molecular noise [94]. entrained by light-dark cycles. Another model, based on the coupling of a population Similarly, To et al. [110] showed that an ensemble of of Goodwin-like oscillators, highlights a possible heterogeneous oscillators (composed of a mixture of synchronization mechanism [106] in the mammalian self-sustained and damped oscillators) could efficiently

A B 0.4 no coupling coupling 0.3

0.2 (i=1,2,..50) i

P 0.1

0 0 100 200 300 400 500 Time C 0.1

0.08

0.06

0.04 Mean field, F 0.02

0 0 100 200 300 400 500 Time

Figure 5. Synchronization of a population of circadian clocks. (A) Illustration of two types of coupling: global vs local coupling. Each circle symbolizes an oscillator (e.g. a SCN neuron), Σ denotes the mean field (e.g. the average neurotransmitter concentration in the extra-cellular medium). (B) Oscillations of 50 globally coupled oscillators. The inter-cellular coupling is switched on at t = 150. The curves represent the time series of the concentration of the clock protein. (C) Corresponding oscillations of the mean field. As soon as the coupling is on, the cells are rapidly synchronized and the mean field displays large amplitude circadian oscillations. For details, see [106].

721 Modeling circadian clocks: Roles, advantages, and limitations

be synchronized through the periodic release of parameter variation, or to molecular noise, this is likely neurotransmitters and that this efficiency depends not only to be the case for other genetic oscillators and therefore on the coupling strength but also on cell-specific factors these different systems may share common regulatory such as the mean and variability of the kinetic parameters. structures [56,61]. Many oscillators have been shown These results suggest that in order to understand to be composed of interlocked positive and negative the orchestration of timekeeping in the SCN, the feedback loops. A systematic survey of these systems intracellular circadian circuitry can not be isolated from allows their classification according to the topology of the intercellular communication and that the variability these positive- and negative-feedback loops [56] and (i.e. noise) of the parameters should also be taken into to associate general dynamic properties (sine-like vs account. These results support an integrated view of the relaxation oscillations, bifurcation structure, tunability of SCN circadian system. the period, robustness, etc.) to each class of topology [56,61]. Discussion Towards an integrated view of the circadian system Successful modeling achievements The various circadian problems described in this paper Because of the multiple roles of circadian rhythms in show that mathematical modeling enables us (1) to physiology, and because of the large amount of information understand the design of the circadian network by pertaining to the molecular mechanism of circadian clocks, comparing the actual clock mechanism with alternative, circadian biology is a fascinating topic and an active field hypothetical networks, (2) to assess the robustness to of research. There is an increasing number of publications noise through stochastic simulations, and (3) to explore related to circadian research, each adding a small piece to mechanisms of coupling and factors responsible for an the huge puzzle. Therefore a synthetic way to summarize efficient synchronization. these observations, to provide possible explanations - Although these various aspects are often treated sometimes counter-intuitive - for surprising experimental separately, recent modeling work suggests that these observations, and to suggest new experiments, is needed. different properties are interdependent: self-sustained To this end, theoretical modeling is of considerable help. oscillations and stability can be obtained through It provides a new approach to studying the dynamics of the coupling (and the resulting synchronization) of a complex systems. Mathematical models have been used population of heterogeneous (and possibly damped) to study both fundamental properties of the circadian clock oscillators. The synchronization efficiency depends [23,112,113] as well as its relations with physiology and not only on coupling mechanisms and strength, but behaviour [89,114]. also on cell-specific factors (variability of the kinetic Particularly intriguing is the complexity of the parameters, network architecture, etc.). Molecular noise circadian clocks. They originate at the molecular level may obliterate single-cell oscillations but inter-cellular from multiple interlocked genetic circuits. Since a synchronization can make the system more robust. simple delayed negative feedback loop-based oscillator All together, these observations lead us to consider is already capable of self-sustained oscillations the the circadian network as an integrated system in which question of the advantage of the additional positive and the design principles can be fully understood only by negative circuits comes naturally. Although no model taking into consideration the multiple aspects of the brings a definitive answer, they suggest possible roles system (Figure 6). It is likely that only the interplay for this complexity. Mathematical models can also between the cellular circuitry, the synchronization, explain intriguing behaviour such as the reversible and the robustness to noise can fully explain both rhythm suppression by a light pulse or the spontaneous the intricate cogs and gears of the cellular regulatory splitting of the circadian rhythms. Even less intuitive, mechanism and the complex inter-cellular organisation the question of the robustness of the oscillations to (e.g. in the SCN). molecular noise (and its link to the oscillator design) and the inter-cellular coupling mechanisms necessary Limitations of modeling to ensure synchrony, can be explored with the help of Although models give hints, ideas, and speculative theoretical modeling. These fundamental issues would explanations, they are subject to several caveats. indeed be hard to address experimentally. The most common criticism towards models is Another advantage of modeling is to highlight that both the type of equations and the values of general design principles of biological systems. If the parameters are arbitrary. Whereas molecular models circadian oscillations must be robust to temperature, to are based on well established genetic regulations,

722 D. Gonze

Figure 6. Integrated view of circadian clocks. the details of the molecular mechanisms are usually architecture of the SCN is heterogeneous (different unknown. Michaelis-Menten and Hill functions are parts of the SCN are distinguished by the neuron used because saturation and threshold kinetics are connectivity, response to light, etc.) and inter-cellular realistic descriptions of enzymatic processes and coupling is ensured by different coupling mechanisms because they provide the non-linearity necessary to (gap junctions, electrical, neurotransmitters, etc.) but obtain limit-cycle and other non-trivial behaviours, neither the topology of the neuronal network nor the but the hypotheses underlying these approximations molecular mechanisms of the signaling cascades are rarely shown to be satisfied. Theoretical models, are experimentally characterized. When establishing although qualified as molecular, should thus be a model, choices and simplifications are necessary. regarded as phenomenological models. They allow to Refinements will be possible as soon as more data will study qualitatively dynamical properties of circadian be available. clocks. Quantitative data (parameter values, absolute Researchers sometimes criticize models that mainly concentrations or numbers of molecules) are rarely reproduce already known data and miss predictions. It available. Therefore, it is impossible to develop is indeed true that most of the models would not directly accurate quantitative models for circadian clocks. predict missing genes or interactions. However, many Most of the models for circadian clocks are based on original (as well as unpublished) versions of the models ordinary differential equations. These models, as well as fail to reproduce data. Such failures are particularly their stochastic versions, neglect spatial aspects. They relevant and the way they are solved thus deserves implicitly assume that mRNA and protein molecules some attention, because they may point to an important freely move around in the cell. However, a cell is very (and perhaps missing) key piece in the circadian puzzle. crowded and cellular processes are highly organized in We can also recall here that models can achieve other space. It is likely that space and diffusion play critical goals than doing predictions (see Table 1). roles in the dynamics of cellular systems. Ignoring Finally, we have to be aware that none of the models, diffusion is probably a rough approximation, which also as detailed as they are, bring definitive answers. Rather, makes quantitative modeling challenging. Nevertheless, they provide elements of reflection. Obviously, the role a good fit between predicted behaviour and measured of positive feedback loops in the molecular mechanism data should support the credibility of the model and its of circadian clocks is not unraveled. However, different dynamic properties. models have provided clues to possible functions Analyses of multicellular systems also require of these additional loops: increasing robustness to the specification of the coupling type and coupling parameter variations, allowing the tunability of the topology. I have discussed above two classes of period, etc. Modeling has the merit of providing us with intercellular coupling, namely global and local. The real another view on the system.

723 Modeling circadian clocks: Roles, advantages, and limitations

Challenges for the future Ostreococcus [122] and in red blood cells [123]. Further The different examples presented in this review show surprises and challenges can be expected. that the links between the complex structure of the Since the molecular links between circadian rhythms clock and its dynamic properties are intricate. There are and various physiological disorders (sleep/wake cycle, still many open questions. Research on modeling the cell division, etc.) have been identified, they are now molecular mechanism of circadian clocks is still growing. the object of high interest for modelers. Modeling the Models evolve and new challenges arise. Molecular interaction of the circadian clock with other dynamic models are continuously extended to incorporate the systems can now be envisaged. All these systems, as well latest experimental data and hence tend to be more as their coupling mechanisms, are well characterized. and more detailed. They allow the study of very specific Also promising is the use of modeling approaches as aspects of the circadian system, such as specific a guide to develop chronotherapy strategies [124,125]. mutations or the role of a particular phosphorylation. In recent years, an increasing number of The most recent cellular models incorporate not only the experimental circadian research groups have performed regulatory network of clock genes but also associated experiments in conjunction with modeling [47,83]. Such components, such as neurotransmitters, signaling iterative procedures in which models are refined on the pathways, calcium and other ion transport processes basis of new experimental data, whereas model-based [115]. In parallel, simple (and sometimes abstract) experimental protocols will lead to new data, becomes models are still employed to investigate more generic more and more common in circadian research. questions of oscillator design and mechanisms of Modeling also constitutes a key step in synthetic synchronization. There is no doubt that future models, biology. Artificial biological clocks have been designed which will combine detailed molecular schemes with and constructed experimentally [126-128]. Designing realistic cellular network architectures, will provide new a robust and tunable biological clock clearly relies on insights to the functioning of circadian clocks. all the knowledge gained by the extensive modeling of The core circadian network is not yet fully natural biological clocks. understood. A big surprise arose when a circadian oscillator was shown to tick in vitro in the absence of a transcriptional feedback loop. The three cyanobacterial Acknowledgments clock proteins, KaiA, KaiB, and KaiC, were shown to produce robust circadian oscillations of KaiC I would like to thank Albert Goldbeter and Jean-Christophe phosphorylation in continuous darkness [116]. This Leloup for fruitful discussions over many years, Karoline discovery rapidly motivated the development of several Faust for careful reading of the manuscript, and Steffen original models [117-121]. The precise molecular origin Faust for the clock scheme shown in Figure 1A. This work of these oscillations, their interplay with the transcription was supported by grant #3.4636.04 from the Fonds de la feedback loop, as well as their synchronization at the cell Recherche Scientifique Médicale (F.R.S.M., Belgium), by population level still constitutes the subject of on-going the European Union through the Network of Excellence theoretical and experimental work. The persistence of BioSim (Contract No. LSHB-CT-2004-005137), and by circadian oscillations in the absence of transcription the Belgian Program on Interuniversity Attraction Poles, may be not limited to cyanobacteria. Circadian redox initiated by the Belgian Federal Science Policy Office, cycles have been observed in the unicellular alga project P6/25 (BioMaGNet).

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