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C. R. Biologies 326 (2003) 189–203

Biological modelling / Biomodélisation

Stochastic models for circadian rhythms: effect of molecular non periodic and chaotic behaviour

Modèles stochastiques pour les rythmes circadiens : effet dumoléculaire sur les comportements périodiques et chaotiqu

Didier Gonze, José Halloy, Jean-Christophe Leloup, Albert Goldbeter∗

Unité de chronobiologie théorique, faculté des sciences, université libre de Bruxelles, Campus Plaine, CP 231, B1050 Brussels, B

Received 9 October 2002; accepted 17 December 2002

Abstract

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organismrhythms originate from the negative autoregulation of gene expression. Deterministic models based on such geneticprocesses account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkentrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers oand mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessaryto stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect ofnoise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations ofand TIM proteins and their mRNAs inDrosophila. The model is based on repression of theper andtim genes by a complebetween the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed bof the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic morespect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation pointwhich the system evolves to a stable steady state. Stochastic simulations indicate that robust circadian oscillations cat the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations arorder of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in cwhere the deterministic PER–TIM model admits chaotic behaviour. The difference between periodic oscillations of tcycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means osections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be disfrom the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the osciremain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good awith the predictions of the deterministic model for circadian rhythms.To cite this article: D. Gonze et al., C. R. Biologies 326(2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

* Corresponding author.E-mail address:[email protected] (A. Goldbeter).

1631-0691/03/$ – see front matter 2003 Académie des sciences/Édidoi:10.1016/S1631-0691(03)00016-7

tions scientifiques et médicales Elsevier SAS. Tous droits réservés.

http://

190 D. Gonze et al. / C. R. Biologies 326 (2003) 189–203

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Résumé

Les rythmes circadiens sont des oscillations endogènes qui se produisent avec une période proche de 24 h chezdes organismes vivants. Ces rythmes résultent de l’autorégulation négative de l’expression de gènes de l’horloge ciDes modèles déterministes fondés sur de tels processus de régulation génétique rendent compte de l’existencecircadiens dans des conditions d’environnement constant (par exemple, l’obscurité continue), de l’entraînement de cepar des cycles lumière–obscurité, et de leur déphasage par des impulsions de lumière. Lorsque le nombre de molécumessagers et de protéines impliquées dans le mécanisme des oscillations est faible, comme cela peut se produiconditions cellulaires, il devient nécessaire de recourir à des simulations stochastiques pour déterminer l’influencemoléculaire sur les rythmes circadiens. Nous étudions l’effet du bruit moléculaire en considérant la version stochastmodèle déterministe précédemment proposé pour les oscillations circadiennes des protéines PER et TIM et de lmessagers chez la drosophile. Ce modèle est fondé sur la répression des gènesper et tim par un complexe entre les protéinPER et TIM. Les simulations numériques de la version stochastique de ce modèle sont effectuées au moyen de la mGillespie. Les prédictions de l’approche stochastique sont en accord avec celles fournies par l’approche déterministece qui concerne les oscillations entretenues de type cycle limite que pour l’influence de la proximité d’un point de bifau-delà duquel le système évolue vers un état stationnaire stable. Les simulations stochastiques indiquent que dcircadiens robustes peuvent émerger au niveau cellulaire déjà lorsque le nombre maximum de molécules d’ARN mede protéines impliquées dans les oscillations est de l’ordre de quelques dizaines ou centaines seulement. Le modèle sreproduit également l’évolution vers un attracteur étrange dans les conditions où le modèle déterministe PER–TIMcomportement chaotique. La différence entre les oscillations périodiques de type cycle limite et les oscillations apérioc’est-à-dire le chaos – persiste en présence de bruit moléculaire, comme l’indiquent les sections de Poincaré obtenudeux types de comportement dynamique. La disparition progressive de la périodicité, qu’on observe à mesure quede molécules diminue, peut donc être distinguée de l’apériodicité résultant de la dynamique chaotique. Aussi longtemnombre de molécules impliquées dans les oscillations demeure suffisamment grand (de l’ordre de quelques dizaines,ou plus), les modèles stochastiques fournissent ainsi un bon accord avec les prédictions des modèles déterministrythmes circadiens.Pour citer cet article : D. Gonze et al., C. R. Biologies 326 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.

Keywords:circadian rhythms; stochastic simulations; molecular noise; robustness; chaos

Mots-clés :rythmes circadiens ; simulations stochastiques ; bruit moléculaire ; robustesse ; chaos

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1. Introduction

Most living organisms, from cyanobacteria to isects, plants and mammals, have developed thepability of generating autonomously sustained oslations with a period close to 24 h. These osciltions, known as circadian rhythms, are endogenouscause they can occur in constant environmental cotions, e.g., constant darkness [1,2]. Experimental sies during the last decade have shed much light onmolecular mechanism of circadian rhythms [3]. Intial studies pertained to the flyDrosophila [4,5] andthe fungusNeurospora[3]. Molecular studies of circadian rhythms have since been extended to cyanobaria, plants and mammals [6,7]. In all cases investigaso far, the molecular mechanism of circadian oscitions relies on the negative autoregulation exerteda protein on the expression of its gene [3–8]. Thus

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Drosophila, the proteins PER and TIM form a complex that indirectly represses the activation of theperandtim genes, while inNeurosporait is the FRQ pro-tein that represses the expression of its genefrq [3,6].The situation in mammals resembles that observeDrosophila, but instead of TIM it is the CRY proteithat forms a regulatory complex with a PER proteto inhibit the expression of theper genes [7]. Lightcan entrain circadian rhythms by inducing degradaof the TIM protein inDrosophila, and expression othefrq andpergenes inNeurosporaand mammals, respectively [3–7].

A number of mathematical models for circadirhythms have been proposed [9–15] on the basithese experimental observations. These models aa deterministic nature and take the form of a sysof coupled ordinary differential equations. The moels predict that in a certain range of parameter v

D. Gonze et al. / C. R. Biologies 326 (2003) 189–203 191

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ues the genetic control network undergoes sustaoscillations of the limit cycle type correspondingthe circadian rhythm, whereas outside this rangegene network operates in a stable steady state.question arises as to whether deterministic modare always appropriate for the description of cirdian clocks [16]. Indeed, the number of moleculesvolved in the regulatory mechanism producing cirdian rhythms at the cellular level may well be reducThis number could vary from a few thousands doto hundreds and even a few tens of protein or mesger RNA molecules in each rhythm-producing cell.such low concentrations it is more appropriate to reto a stochastic approach to study the molecular bof the oscillatory phenomenon.

In a previous publication [17], we compared tstochastic and deterministic versions of a core moular model for circadian oscillations based on the native regulation exerted by a protein on the expressof its gene. Stochastic simulations were performedmeans of the Gillespie algorithm [18,19] after decoposing the deterministic model into elementary steWe studied the effect of molecular noise by assesthe robustness of circadian oscillations as a functiothe number of interacting molecules. We showed trobust circadian rhythmicity could already occur whthe maximum numbers of mRNA and clock protemolecules are in the tens and hundreds, respectiCooperativity of repression and periodic forcinglight-dark cycles enhance the robustness of circadoscillations. In subsequent work [20], we compatwo stochastic versions of this core model, one fudeveloped into elementary steps, and the other ndeveloped. We showed that stochastic treatmenthese two versions of the model for circadian rhythyields similar results.

The purpose of the present paper is to extendcomparison of deterministic and stochastic modelscircadian oscillations, by considering a more detaimodel for theDrosophilacircadian clock incorporating the formation of a complex between the PER aTIM proteins. Although this model does not take inaccount other proteins such as CLOCK and CYCvolved in the circadian oscillatory mechanism, itnevertheless more realistic than the core modelaccounts for a larger number of experimental obsetions. Moreover, this extended model predicts the psibility of autonomous chaotic behaviour [21]. Such

property will allow us to assess the effect of molecunoise not only on periodic but also on chaotic oscitions.

We first present in Section 2 the deterministic astochastic versions of the molecular model for cirdian rhythms. In the stochastic version we introdumolecular noise without decomposing the determintic mechanism into detailed reaction steps. The resof stochastic simulations performed by means ofGillespie method [18,19] are presented in Sectiofor the case of periodic behaviour. We assess theof fluctuations by determining the effect of the numbof mRNA and protein molecules on circadian rhymicity. In Section 4 we examine how the proximifrom a bifurcation point influences the robustnessthe oscillations with respect to molecular noise. Hsuch a noise affects chaotic behaviour is examineSection 5. Section 6 is devoted to a comparativecussion of deterministic versus stochastic simulatio

2. Deterministic and stochastic versions of themolecular model for circadian oscillations

2.1. Deterministic model for circadian oscillations

We consider a ten-variable model previously pposed for circadian oscillations of the PER and Tproteins and ofper and tim mRNAs in Drosophila[9,10]. The model, schematised in Fig. 1, is baon the negative feedback exerted by the complextween the nuclear PER and TIM proteins on thepression of their genes. For each of these proteinsgene is first transcribed in the nucleus into messger RNA (mRNA). The latter is transported into thcytosol where it is degraded and translated intoprotein P0 (T0). The protein PER (TIM) undergoemultiple phosphorylation, from P0 into P1 (T0 intoT1) and from P1 into P2 (T1 into T2). These modi-fications, catalysed by a protein kinase, are reveby a phosphatase. The fully phosphorylated formthe proteins is marked up for degradation and fora complex (C), which is transported into the nuclein a reversible manner. The nuclear form of the PETIM complex (CN) represses the transcription of tperandtimgenes. Recent experiments indicate thatpression is in fact of indirect nature: the CLOCK aCYC proteins promote the expression of theper and


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Fig. 1. Model for circadian rhythms inDrosophila. The model isbased on the negative regulation of theper and tim genes by acomplex between the PER and TIM proteins [11]. Theper (MP)andtim (MT) mRNAs are synthesised in the nucleus and transfeinto the cytosol, where they accumulate at the maximum ratesvsPandvsT, respectively; there they are degraded enzymatically amaximum ratesvmP and vmT, with the Michaelis constantsKmPand KmT. The rates of synthesis of the PER and TIM proteirespectively proportional toMP andMT, are characterised by thapparent first-order rate constantsksP and ksT. ParametersViP,ViT and KiP, KiT (i = 1, . . . ,4) denote the maximum rate anMichaelis constant of the kinase(s) and phosphatase(s) invoin the reversible phosphorylation of P0 (T0) into P1 (T1) and P1(T1) into P2 (T2), respectively. The fully phosphorylated form(P2 and T2) are degraded by enzymes of maximum ratevdP, vdTand Michaelis constantsKdP, KdT, and reversibly form a compleC (with the forward and reverse rate constantsk3, k4), which istransported into the nucleus at a rate characterised by the appfirst-order rate constantk1. Transport of the nuclear form of thPER–TIM complex (CN) into the cytosol is characterised by thapparent first-order rate constantk2. The negative feedback exerteby the nuclear PER–TIM complex onper and tim transcription isdescribed by an equation of the Hill type, in whichn denotes thedegree of cooperativity, andKIP andKIT the threshold constantfor repression.

tim genes and are prevented from exerting this acttion when forming a complex with PER and TIM [37]. In the model, the variables are the concentratiof the mRNAs (MP andMT), of the various forms othe PER and TIM proteins (P0, P1, P2, T0, T1, T2),and of the cytosolic (C) and nuclear (CN) forms of thePER–TIM complex. The temporal evolution of theconcentration variables is governed by a system okinetic equations that are listed in Appendix A (see10] for further details).

The deterministic PER–TIM model accounts fthe occurrence of sustained oscillations in continu

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darkness. When taking into account the controlerted by light on the maximum protein degradatratevdT, the model also accounts for entrainmentcircadian oscillations by light-dark cycles and for thphase shifting by pulses of light. Circadian oscillatiohave also been obtained in more detailed models bon indirect repression and involving additional clogene products such as CLOCK and CYC ([14,1J.-C. Leloup and A. Goldbeter, submitted for pubcation).

As indicated above, besides periodic oscillatiothe model of Fig. 1 is also capable of generatingtonomous chaos in conditions corresponding to ctinuous darkness. Although this behaviour is probanot of physiological significance [21], it provides uwith the rare opportunity of testing the effect of mocular noise on chaotic behaviour in a realistic mobased on genetic regulation.

2.2. Stochastic version of the model for circadianoscillations

To assess the effect of molecular noise, wescribe the reaction steps as stochastic birth and dprocesses [22]. Numerical simulations of the tempoevolution of the genetic control system are performby means of the Gillespie method [17,18]. Besidother approaches [23–25], this method has beento determine the dynamics of chemical [24,25] bchemical [26] or genetic systems [27] in the preseof molecular noise. The Gillespie method associatprobability with each reaction step; at each time sthe algorithm randomly determines the reaction ttakes place according to its probability, as well astime interval to the next reaction step. The numbermolecules of the different reacting species as welthe probabilities are updated at each time step. Inapproach [18,19], a parameter denotedΩ controls thenumber of molecules present in the system. UsingGillespie method we performed stochastic simulatiof the model described in section 2.1.

Our previous analysis of a core molecular mofor circadian rhythms showed [17] that similar resuare obtained when decomposing the determinmodel into elementary steps (developed model) orwhen the deterministic model is not decomposedsuch steps and non-linear kinetic functions are simincluded in the probabilities associated with the glo


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reaction steps (non-developed model). Because it ismuch easier to handle, we shall therefore restrictstochastic analysis to such a non-developed versiothe model schematised in Fig. 1.

The non-linear terms appearing in the kinetic eqtions (A.1) listed in Appendix A represent compakinetic expressions obtained after application of qusteady-state hypotheses on enzyme–substrate or grepressor complexes. The resulting expressions athe Michaelis–Menten type for enzyme reaction raor of the Hill type for cooperative binding of the repressor (here, the nuclear PER–TIM complex) togene promoter. We attribute to each linear or nlinear term of the kinetic equations a probabilityoccurrence of the corresponding reaction. These rtions and their associated probability are listed inble 1 in Appendix A. Thus reaction (1) correspondsthe transcription of theper gene intoper mRNA, MP;the occurrence of this reaction with a probabilityw1results in increasing by 1 the number of moleculesMP. Reaction (4) results in increasing by one the nuber of P1 molecules and decreasing by one the numof molecules of P0.

In contrast to the treatment presented in our prous work [17], here we do not decompose the bindof the repressor to the gene promoter into succeselementary steps, and rather retain the Hill functdescription for cooperative repression by the nucPER–TIM complex CN (steps (1) and (11)). A similaglobal approach is taken for describing degradatioper andtim mRNAs (reactions (2) and (12)), transltion of mRNAs into proteins P0 and T0 (reactions (3)and (13)), phosphorylation of P0 and T0 into P1 andT1 (reactions (4) and (14)) and of P1 and T1 into P2and T2 (reactions (6) and (16)), as well as dephosprylation of P1 and T1 into P0 and T0 (reactions 5 and15) and of P2 and T2 into P1 and T1 (reactions (7) and(17)), enzymatic degradation of P2 and T2 (reactions(10) and (18)), reversible formation of the complex(reactions 8 and 9), and reversible transport of coplex C into and out of the nucleus (reactions (19) a(20)). Steps (21)–(30) relate to non-specific degration of the various mRNA or protein species. Reations (2), (4)–(7), (10), (12), and (14)–(18) are of tMichaelian type; reaction (8) is of bimolecular natuwhile reactions (3), (9), (13) and (19)–(30) correspoto linear kinetics.

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3. Effect of molecular noise on periodicoscillations

Before dealing with the effect of molecular noislet us recall the predictions of the deterministic mogoverned by equations (A.1). Typical circadian oslations predicted by this model are shown in Fig.(middle panel). These oscillations correspond toevolution toward a limit cycle (Fig. 2A, left paneshown here as a projection onto the plane formedthe concentrations ofper mRNA (MP) and nuclearPER–TIM complex (CN). Because the behaviourperiodic and deterministic (i.e. oscillations occurthe absence of molecular noise), the histogram ofriods yields a single line corresponding to a circadperiod close to 24 h (Fig. 2A, right panel).

Turning to the effect of molecular noise, we noconsider the dynamic behaviour of the stochaversion of the PER–TIM model for circadian rhythmShown in panels B–D in Fig. 2 are the limit cycl(left panels), sustained oscillations ofper mRNA(middle panels), and histograms of periods (ripanels) obtained by stochastic simulations forΩ= 1000 (B), 100 (C) and 10 (D). For these valuesΩ , the numbers of molecules of nuclear PER–Tcomplex andper mRNA vary in the range 500–250and 0–3000, 50–300 and 0–300, and 0–60 and 0respectively.

The data presented in Fig. 2 show that the cirdian oscillatory behaviour predicted by the determintic model is recovered when using the stochastic mofor circadian rhythms. The mere effect of molecunoise is to increase the effective thickness of the licycle. The results further indicate that robust circadoscillations are still produced by the stochastic mowhen the maximum numbers of mRNA and protemolecules are in the order of hundreds. It is only whthese numbers decrease down to a few tens that nbegins to overcome rhythmicity, even though oscitions still subsist (Fig. 2D); the period histogramthen much wider but still presents a maximum cloto a circadian value.

Another way to illustrate the effect of moleculnoise is to draw a histogram of frequencies of passthrough the different points in the phase plane. Indeterministic case, such a plot would yield the safrequency of passage through all points of the limcycle, because these are all visited the same num


Fig. 2. Sustained oscillations predicted by the deterministic and stochastic versions of the model for circadian rhythms. (A) Limit cycle obtainedin the 10-variable deterministic model governed by equations (A.1), shown as a projection onto theMP–CN phase plane (left panel); the arrowindicates the direction of movement along the closed trajectory. Equations (A.1) and parameter values are listed in Appendix A. Middle panel:sustained oscillations of per mRNA (MP) corresponding to the limit cycle in the left panel. Right panel: histogram of periods. Here, in theabsence of molecular noise, the deterministic model yields a single line corresponding to the periodic circadian oscillations. (B)–(D) Limitcycle (left panel), sustained oscillations (represented by the time course of the number ofper mRNA molecules), and histograms of periodspredicted by the stochastic model for values of parameterΩ decreasing from 1000 (B) to 100 (C) and 10 (D). The results, obtained in thepresence of molecular noise, should be compared with those obtained with the deterministic model in (A). Numerical simulations in panels(B)–(D) were performed by means of the Gillespie method [16,17] with the stochastic model listed in Table 1 in Appendix A. As in thefollowing figures stochastic simulations were performed for 2500 h, which corresponds to some 100 successive cycles. For period histograms,the period was determined as the time interval separating two successive upward crossings of the mean level of mRNA or clock protein.


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Fig. 3. Probability of passage of a trajectory in a given regionthe phase space in the presence of molecular noise, in condcorresponding to the evolution to the limit cycle shown in Fig.(left panel). Molecular noise induces a dispersion of the trajectoaround the deterministic limit cycle (Fig. 2A, left panel); thprobability of passage is highest where the trajectories aredispersed. The bottom part of Fig. 3A shows contour linesthe probability of passage in different regions of the phase p(permRNA molecules, nuclear PER–TIM complex molecules). Tcontour plot contains 50, equidistant isoprobability lines betwminimum and maximum probability. Panel B shows an enlargemof the contour plot, rotated by 90◦ for the sake of comparison witthe limit cycles shown in the left panels of Fig. 2. Parameter vaare as in Fig. 2, withΩ = 100.

of times. In the presence of noise, some regions ofphase space are visited more often than others wdispersion is strong. This results in the occurreof peaks in the frequency of passage, as showFig. 3A. The bottom part of Fig. 3A shows a contoplot in the plane (permRNA molecules versus nuclePER–TIM complexes). The contour plot is obtainby projecting onto this plane the intersections ofhistogram with 50 parallel planes, correspondingequidistant frequencies. For the sake of claritycontour plot is rotated by 90◦ and shown enlargein Fig. 3B. The most frequently visited regionsthe phase plane correspond to the decrease in nuPER–TIM levels and rise inper mRNA. Largestdispersion occurs when mRNA decreases and PTIM levels rise.

4. Influence of the vicinity of a bifurcation point

The question arises as to whether the proximfrom a bifurcation point may influence the robustneof circadian oscillations with respect to molecunoise. To address this issue, we first construcFig. 4 a bifurcation diagram showing the onsetlimit cycle oscillations as a function of the maximurate of TIM degradation,vdT, for the deterministicmodel governed by equations (A.1). The statethe system is represented in Fig. 4 by a sinvariable, the concentration ofper mRNA. BelowvdT = 0.46 nMh−1, the system evolves toward a stabsteady state. Above this critical bifurcation valulimit cycle oscillations occur. The diagram showthe envelope of the oscillations: the upper and lowbranches yield the maximum and minimum valuesper mRNA concentration as a function ofvdT in thecourse of oscillations.

The results of stochastic simulations performwith increasing values of parametervdT are shown inFig. 5 in the form of dynamics in the phase plane (ripanels) or time series (left panels). The steady stalimit cycle predicted by the deterministic model for tcorresponding parameter values is shown in the phplane by a white dot or a white curve, respectiveThe four panels in Fig. 5 correspond to the fourvdTvalues indicated by dashed vertical lines in Fig. 4.vdT = 0.1 nMh−1, the deterministic system evolvesa stable steady state far from the bifurcation point; s


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Fig. 4. Bifurcation diagram showing the onset of circadian oslations in the deterministic model as a function of parametervdTwhich measures the maximum rate of TIM degradation. The cushows the steady-state level ofper mRNA, stable (solid line) or unstable (dashed line), as well as the maximum and minimum contration of per mRNA in the course of sustained circadian osciltions. The vertical dashed lines refer to the four values considfor vdT in Fig. 5, in panels (A)–(D), respectively. The diagramestablished by means of the program AUTO [32] applied to eqtions (A.1) listed in Appendix A. Parameter values are as in Fig

chastic simulations show low-amplitude fluctuatioaround the deterministic steady state (Fig. 5A).vdT = 0.4 nMh−1, the deterministic system evolvesa stable steady state close to the bifurcation point;chastic simulations show fluctuations of larger amtude around the deterministic steady state (Fig. 5For vdT = 0.5 nMh−1, the deterministic system undergoes limit cycle oscillations of reduced amplitujust beyond the bifurcation point; stochastic simutions show small-amplitude, noisy oscillations arouthe deterministic limit cycle (Fig. 5C), which resemble the fluctuations shown in Fig. 5B. Finally, fovdT = 2.0 nMh−1, the deterministic system undergolimit cycle oscillations of large amplitude far from thbifurcation point; stochastic simulations show largamplitude, relatively less noisy oscillations arounddeterministic limit cycle (Fig. 5D).

The data in Fig. 5 indicate that stochastic simutions allow us to recover the dynamics predictedthe deterministic model. Below a critical paramevalue, the system displays low-amplitude fluctuatioaround a stable steady state, while above this vsustained oscillations occur. Moreover, the effectfluctuations is enhanced by the proximity from thefurcation point.

5. Effect of molecular noise on chaotic behaviour

We have previously reported the occurrence oftonomous chaos in the deterministic model of Fig[21]. Thus, for some parameter values, in conditiocorresponding to continuous darkness, sustainedriodic oscillations occur in this model, which corrspond to the evolution toward a strange attractothe phase space (see Fig. 6A). This phenomenonvides us with the rare opportunity to assess the efof molecular noise on chaotic behaviour in a realisbiochemical model. Shown in panels B and C of Figfor Ω = 1000 and 100, respectively, are the resultsstochastic simulations performed for parameter vacorresponding to those producing chaos in the deministic model in Fig. 6A.

The results indicate that chaos persists inpresence of noise, but the structure of the straattractor begins to be blurred when the numbermolecules decreases and the amplitude of fluctuatrises. Nevertheless the small curler-like substructhat characterises the strange attractor in the PTIM model (Fig. 6A, left panel) is still visible inthe attractors obtained by stochastic simulations (panels in Fig. 6B and C).

Because the PER–TIM model can produce perioas well as chaotic oscillations, we can use this moto compare the effect of molecular noise on the ttypes of dynamic behaviour. A convenient tool fsuch a comparison is provided by Poincaré sectiFor the case of periodic oscillations, the trajectin the (MP, MT, CN) plane takes the form oflimit cycle. Intersection of this trajectory with a plancorresponding to a given value ofMT generally yieldstwo points, one for whichMT is on the rise and thother for whichMT is on the decline. Shown in Fig. 7is the point intersection obtained for the determinismodel whenMT passes the value 1.5 nM upwarPanels B and C in Fig. 7 show the Poincaré sectobtained by stochastic simulations withΩ = 1000 and100, respectively, for the corresponding value ofnumber ofMT molecules (i.e., 1.5Ω). Instead of asingle point, we obtain a cloud of points surroundthe deterministic Poincaré section; the smallernumber of molecules, the more scattered the cloud

In the case of chaos, instead of a single pointintersection in the deterministic model takes the foof an open continuous curve (Fig. 7D). Stochas


Fig. 5. Effect of the proximity from a bifurcation point on the effect of molecular noise in the stochastic model for circadian rhythms. Thedifferent panels are established for the four increasing values of parametervdT shown in Fig. 4: 0.1 (A), 0.4 (B), 0.5 (C) and 2 (D); thesevalues, to be multiplied byΩ , are expressed here in molecules per hour. The right panels show the evolution in the phase plane while theleft panels represent the corresponding temporal evolution of the number ofper mRNA molecules. (A) Fluctuations around a stable steadystate. (B) Fluctuations around a stable steady state for a value ofvdT close to the bifurcation point which lies around 0.47 molecules per hour.Damped oscillations occur in these conditions when the system is displaced from the stable steady state. In (A) and (B), the white dot representsthe stable steady state predicted by the deterministic version of the model in corresponding conditions. (C) Oscillations observed close to thebifurcation point. (D) Oscillations observed further from the bifurcation point, well into the domain of sustained oscillations. In (C) and (D) thethick white curve represents the limit cycle predicted by the deterministic version of the model governed by equations (A.1), in correspondingconditions, for the same values ofvdT expressed in nMh

−1. The smaller amplitude of the limit cycle in (C) as compared to the limit cycle in(D) is associated with an increased influence of molecular noise. The curves are obtained by means of the Gillespie algorithm applied to themodel of Table 1 (see Appendix A). BesidesvdT, parameter values are as in Fig. 2.


Fig. 6. Effect of molecular noise on autonomous chaos. The strange attractor and the corresponding chaotic oscillations predicted by thedeterministic PER–TIM model for circadian rhythms are represented in (A). (B) and (C) Progressive dissolution of the strange attractor (leftpanels) and corresponding aperiodic oscillations (right panels) in the presence of molecular noise, forΩ = 1000 and 100, respectively. Thecurves in (A) are obtained by numerical integration of equations (A.1) listed in Appendix A. In (B) and (C), the curves are obtained by meansof the Gillespie algorithm applied to the model of Table 1 in Appendix A. Parameter values are given in Appendix A.


limit cyclettom

s the valued

Fig. 7. Poincaré sections in the absence or presence of molecular noise. The upper row corresponds to periodic oscillations of thetype in the deterministic model (A) and in the stochastic model forΩ = 1000 (B) and 100 (C) (see corresponding panels in Fig. 2). The borow shows the Poincaré sections obtained for chaos in the deterministic model (D) and in the presence of molecular noise forΩ = 1000 (E)and 100 (F); these curves correspond, respectively, to panels A–C in Fig. 6. In (A), the section is made when the trajectory crosseMT = 1.5 nM upward. In (D), the section corresponds to the upward crossing of the valueMT = 5.5 nM. For panels (B) and (C), and (E) an(F), the section corresponds, respectively, to the upward crossing of the valueMT = 1.5 Ω and 5.5 Ω .

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simulations produce a cloudy version of this curthe scattering of the points in this cloud increasesthe number of molecules decreases (compare paE and F in Fig. 7). Interestingly, in Fig. 7F, we caobserve a denser part of the cloud in the left uppart of the figure; this dense region correspondsnoise-induced visits to the curler-like sub-region ofstrange attractor (see Fig. 6, left panels).

6. Discussion

Circadian rhythms originate from the negative atoregulation of gene expression. Models based on sregulatory processes have been proposed for circarhythms inDrosophilaandNeurospora. These modelsare of a deterministic nature. When molecular nobecomes significant, however, the question arises

whether genetic control mechanisms can give riscoherent circadian oscillations at the cellular level.the presence of reduced numbers of molecules omRNA and protein species involved in the circadclock mechanism, stochastic simulations are neeto address this issue. In a previous publication [1we studied the stochastic version of a core modelcircadian oscillations based on negative autoregulaof a clock gene by its protein product. For stochasimulations, this deterministic five-variable model wfirst decomposed into a series of 30 elementary retion steps. Stochastic simulations performed by meof the Gillespie method [18,19] indicated that robcircadian oscillations, comparable to those predicby the deterministic approach, can occur in thisveloped stochastic model even when the maximnumbers of mRNA and clock protein molecules ain the tens and hundreds, respectively. The robust


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of circadian rhythms, quantified by the period distbution and the half-time of the autocorrelation funtion [17], is enhanced when the cooperativity of trepression process increases and when the numbemRNA and protein molecules involved in the osclatory mechanism become larger. Formation of coplexes between regulatory proteins is another stabing factor (D. Forger and C. Peskin, personal commnication). Stochastic simulations indicate, moreovthat forcing by a light-dark cycle stabilises the phaof the oscillations [17].

In a follow up study [20] we compared two stochatic versions of the core model for circadian osciltions. In one version (the developed stochastic moddescribed above, all reactions were decomposedelementary steps, while in the second version (ndeveloped model), the reactions were not decompointo elementary steps and non-linear rate expresswere simply included in the probability of occurrenof each reaction. A comparison of the two stochtic versions of the core model indicates that they byield similar results; the two approaches are equlent and corroborate the predictions of the determitic model. It is only when the maximum numbersmRNA and protein molecules are reduced to very lvalues, of the order of a few tens, that molecular nobegins to prevent the emergence of coherent circaoscillations. These numerical results are corroboraby a recent analytical study [28].

Here we have extended the comparison of deministic and stochastic models for circadian rhythby considering a more extended molecular model pposed for circadian oscillations of the PER and Tproteins and their mRNAs inDrosophila. The ten-variable model is based on the negative autoregtion of theper and tim genes by a complex betweethe PER and TIM proteins. Although additional prteins are known to be at work in the molecular mecnism of the circadian clock inDrosophilaand mam-mals, the PER–TIM model appears as more reatic than the core mechanism that we previously csidered for deterministic [9] and stochastic simutions [17]. Because the non-developed and develostochastic versions of the core model for circadrhythms gave similar results, we focused here oncomparison of the deterministic PER–TIM model wits non-developed stochastic version, which is eato handle than the developed one. An advantage o

f

PER–TIM model is that, in addition to periodic osclations, it also admits, for appropriate parameter vues, chaotic behaviour. This allowed us to assesseffect of molecular noise both on periodic and chaooscillations.

Stochastic simulations of the PER–TIM model croborate the conclusions reached for the simpler cmechanism for circadian rhythms. Sustained circadoscillations predicted by the deterministic modelrecovered by stochastic simulations. Robust circadoscillations closely related to those obtained withdeterministic model can already occur when the mimum numbers of protein and mRNA molecules ain the hundreds or tens, respectively (the mean nbers of these molecules in the course of oscillationssmaller, because the levels of both protein and mRcan go down to very low values at the trough of thecillations). Robustness increases with the numbermolecules involved, as reflected by the period distution (see the period histograms in Fig. 2). It is onlyvery low numbers of molecules of protein and mRN(reached, for example, forΩ = 10) that noise beginto obliterate circadian rhythmicity [17]. In both the dterministic and stochastic versions of the PER–Tmodel, critical parameter values separate the evolutoward a stable steady state from the evolution towsustained oscillations. The effect of noise is merelyinduce fluctuations around the stable steady stataround the stable limit cycle predicted by the detministic model. As shown here (see Fig. 5), and ewhere on the core model [20], the effect of molecunoise is enhanced in the vicinity of a bifurcation poi

A previous study of models based on negativetoregulation of gene expression reported a lack ofbustness of circadian oscillations with respect to mocular noise [16]. The difference with respect to tresults presented here and elsewhere [17,20] is lidue to the lower values considered for bimolecular rconstants characterising the association of the repsor to the gene promoter [17,29]. When these parater values remain too small, the analysis of the deministic model indicates that the steady state becostable but displays the property of excitability:the stochastic model, instead of sustained oscillatifluctuations give then rise to irregular, large-amplituexcursions away from the stable steady state.

Autonomous chaos has previously been descrin the deterministic PER–TIM model for circadia


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rhythms inDrosophila[21]. As for periodic circadianoscillations, chaotic behaviour is recovered when schastic simulations are performed for parameterues corresponding to chaos in the deterministic moBeyond a noisy appearance due to fluctuations inpresence of reduced numbers of mRNA and promolecules, the structure of the strange attractormains discernible (Fig. 6), in agreement with resuobtained in models for chemical oscillations [30,3The use of Poincaré sections (Fig. 7) allowed usplace on firmer ground the comparison of periodic achaotic behaviour in the presence of molecular noThe results lead to a clear distinction between nolimit cycle oscillations and chaotic oscillations sujected to noise. The agreement between the stochand deterministic approaches therefore holds for podic oscillations as well as for chaotic behaviour.

Acknowledgements

This work was supported by Grant No. 3.4607from theFonds de la recherche scientifique médic(FRSM, Belgium). J.-C.L. isChargé de recherches dFNRS(Belgium). Support from theFondation David& Alice Van Buurento D.G. is also acknowledged. Wwish to thank F. Baras, G. Dupont, D. Forger, andGaspard for fruitful discussions.

Appendix A

A.1. Kinetic equations of the deterministic model fcircadian oscillations

The time evolution of the 10 concentration vaables in the model of Fig. 1 is governed by the flowing system of ordinary differential equations:

dMPdt

= vsP KnIP

KnIP + CnN− vmP MP

KmP+ MP − kdMPdP0dt

= ksPMP − V1P P0K1P+ P0

+ V2P P1K2P+ P1 − kdP0

dP1 = V1P P0 − V2P P1 − V3P P1

dt K1P+ P0 K2P+ P1 K3P+ P1

+ V4P P2K4P+ P2 − kdP1

dP2dt

= V3P P1K3P+ P1 − V4P

P2

K4P+ P2 − k3P2T2

+ k4C − vdP P2KdP+ P2 − kdP2

dMTdt

= vsT KnIT

KnIT + CnN− vmT MT

KmT + MT − kdMT

dT0dt

= ksTMT − V1T T0K1T + T0 + V2T

T1

K2T + T1− kdT0

dT1dt

= V1T T0K1T + T0 − V2T

T1

K2T + T1 − V3TT1

K3T + T1+ V4T T2

K4T + T2 − kdT1dT2dt

= V3T T1K3T + T1 − V4T

T2

K4T + T2 − k3P2T2

+ k4C − vdT T2KdT + T2 − kdT2

dC

dt= k3P2T2 − k4C − k1C + k2CN − kdCC

(A.1)dCNdt

= k1C − k2CN − kdNCN

For the case of periodic oscillations in the determistic model (Figs. 2A and 7A), the following set oparameter values were used:n = 4, vsP= 1.0 nMh−1, vsT = 1.0 nMh−1, vmP = 0.7 nMh−1,vmT = 0.7 nMh−1, vdP = 2.0 nMh−1, vdT= 2.0 nMh−1, ksP = ksT = 0.9 h−1, k1 = 0.6 h−1,k2 = 0.2 h−1, k3 = 1.2 nM−1 h−1, k4 = 0.6 h−1,KmP = KmT = 0.2 nM, KIP = KIT = 1.0 nM, KdP= KdT = 0.2 nM, K1P = K1T = K2P = K2T = K3P= K3T = K4P= K4T = 2 nM,V1P= V1T = 8 nMh−1,V2P= V2T = 1 nMh−1, V3P = V3T = 8 nMh−1, V4P= V4T = 1 nMh−1, kd = kdC = kdN = 0.01 nMh−1.These values were also used for stochastic simulatin Figs. 2B–D and 7B–C. For chaotic behaviourthe deterministic model (Figs. 6A and 7D) and in schastic simulations (Figs. 6B and C and 7E andthe same parameter values were considered, exvmT = 0.35 nMh−1, vdT = 5.3 nMh−1.


Table 1Stochastic model for circadian rhythms, corresponding to the mechanism schematised in Fig. 1

Reaction number Reaction step Probability of reaction

1vsP−→ MP w1 = (vsP× Ω) (KIP × Ω)

n

(KIP × Ω)n + CnN2 MP

vmP−→ w2 = (vmP× Ω) MP(KmP× Ω) + MP

3 MPksP−→ P0 w3 = ksP× MP

4 P0V1P−→ P1 w4 = (V1P× Ω) P0

(K1P× Ω) + P05 P1

V2P−→ P0 w5 = (V2P× Ω) P1(K2P× Ω) + P1

6 P1V3P−→ P2 w6 = (V3P× Ω) P1

(K3P× Ω) + P17 P2

V4P−→ P1 w7 = (V4P× Ω) P2(K4P× Ω) + P2

8 P2 + T2k3−→ C w8 = k3 × P2 × T2/Ω

9 Ck4−→ P2 + T2 w9 = k4 × C

10 P2VdP−→ w10 = (VdP× Ω) P2

(KdP× Ω) + P211

VsT−→ MT w11 = (vsT × Ω) (KIT × Ω)n

(KIT × Ω)n + CnN12 MT

VmT−→ w12 = (VmT × Ω) MT(KmT × Ω) + MT

13 MTVsT−→ T0 w13 = ksT × MT

14 T0V1T−→ T1 w14 = (V1T × Ω) T0

(K1T × Ω) + T015 T1

V2T−→ T0 w15 = (V2T × Ω) T1(K2T × Ω) + T1

16 T1V3T−→ T2 w16 = (V3T × Ω) T1

(K3T × Ω) + T117 T2

V4T−→ T1 w17 = (V4T × Ω) T2(K4T × Ω) + T2

18 T2VdT−→ w18 = (VdT × Ω) T2

(KdT × Ω) + T219 C

k1−→ CN w19 = k1 × C20 CN

k2−→ C w20 = k2 × CN21 MP

kd−→ w21 = kd × MP22 P0

kd−→ w22 = kd × P023 P1

kd−→ w23 = kd × P124 P2

kd−→ w24 = kd × P225 MT

kd−→ w25 = kd × MT26 T0

kd−→ w26 = kd × T027 T1

kd−→ w27 = kd × T128 T2

kd−→ w28 = kd × T229 C

kdC−→ w29 = kdC × C30 CN

kdN−→ w30 = kdN × CN


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A.2. Stochastic version of the model for circadianoscillations

For stochastic simulations, the model schematiin Fig. 1 and governed by the deterministic systof equations (A.1) is presented as a sequence oaction steps in Table 1. Steps (21) to (30) refer to nspecific degradation reactions, of relatively reduimportance, which are not indicated in Fig. 1. The sond column in Table 1 lists the sequence of reacti(see section 2.2). The probability of each reactiongiven in the third column. Parameter values aresame as those listed above for the deterministic mobut units of nM have to be replaced by numbersmolecules, after multiplication of the parameter byΩ ,if need be, as indicated in the last column of Table

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Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviourModèles stochastiques pour les rythmes circadiens : effet du bruit moléculaire sur les comportements périodiques et chaotiquesIntroductionDeterministic and stochastic versions of the molecular model for circadian oscillationsDeterministic model for circadian oscillationsStochastic version of the model for circadian oscillations

Effect of molecular noise on periodic oscillationsInfluence of the vicinity of a bifurcation pointEffect of molecular noise on chaotic behaviourDiscussionAcknowledgementsReferences

stochastic models for circadian rhythms: effect of …...c. r. biologies 326 (2003) 189–203...

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