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Journal of Biological Physics logoLink to Journal of Biological Physics
. 2002 Dec;28(4):637–653. doi: 10.1023/A:1021286607354

Deterministic Versus Stochastic Models for Circadian Rhythms

D Gonze 1, J Halloy 1, A Goldbeter 1,
PMCID: PMC3456469  PMID: 23345804

Abstract

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression.Deterministic models based on genetic regulatory processes account for theoccurrence of circadian rhythms in constant environmental conditions (e.g.constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core modelwas previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens orhundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.

Keywords: circadian rhythms, molecular noise, robustness, stochastic simulations

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References

  • 1.Moore-Ede M.C., Sulzman F.M., Fuller C.A. The Clocks that Time Us. Physiology of the Circadian Timing System. Cambridge, MA: Harvard Univ. Press; 1982. [Google Scholar]
  • 2.Edmunds L.N., Jr. Cellular and Molecular Bases of Biological Clocks. Models and Mechanisms for Circadian Timekeeping. New York: Springer; 1988. [Google Scholar]
  • 3.Dunlap J.C. Molecular Bases for Circadian Clocks. Cell. 1999;96:271–290. doi: 10.1016/s0092-8674(00)80566-8. [DOI] [PubMed] [Google Scholar]
  • 4.Young M.W. Life's 24-hour Clock: Molecular Control of Circadian Rhythms in Animal Cells. Trends Biochem. Sci. 2000;25:601–606. doi: 10.1016/s0968-0004(00)01695-9. [DOI] [PubMed] [Google Scholar]
  • 5.Williams J.A., Sehgal A. Molecular Components of the Circadian System. Drosophila, Annu. Rev. Physiol. 2001;63:729–755. doi: 10.1146/annurev.physiol.63.1.729. [DOI] [PubMed] [Google Scholar]
  • 6.Young M.W., Kay S.A. Time Zones: A Comparative Genetics of Circadian Clocks. Nature Rev. Genetics. 2001;2:702–715. doi: 10.1038/35088576. [DOI] [PubMed] [Google Scholar]
  • 7.Reppert S.M., Weaver D.R. Molecular Analysis of Mammalian Circadian Rhythms. Annu. Rev. Physiol. 2001;63:647–676. doi: 10.1146/annurev.physiol.63.1.647. [DOI] [PubMed] [Google Scholar]
  • 8.Hardin P.E., Hall J.C., Rosbash M. Feedback of the Drosophila Period Gene Product on Circadian Cycling of its Messenger RNA Levels. Nature. 1990;343:536–540. doi: 10.1038/343536a0. [DOI] [PubMed] [Google Scholar]
  • 9.Goldbeter A. A Model for Circadian Oscillations in the Drosophila Period Protein (PER) Proc. R. Soc. Lond. B. 1995;261:319–324. doi: 10.1098/rspb.1995.0153. [DOI] [PubMed] [Google Scholar]
  • 10.Goldbeter A. Biochemical Oscillations and Cellular Rhythms. The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge, UK: Cambridge Univ. Press; 1996. [Google Scholar]
  • 11.Leloup J.-C., Goldbeter A. A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins. J. Biol. Rhythms. 1998;13:70–87. doi: 10.1177/074873098128999934. [DOI] [PubMed] [Google Scholar]
  • 12.Leloup J.-C., Gonze D., Goldbeter A. Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora. J. Biol. Rhythms. 1999;14:433–448. doi: 10.1177/074873099129000948. [DOI] [PubMed] [Google Scholar]
  • 13.Leloup J.-C., Goldbeter A. Modeling the Molecular Regulatory Mechanism of Circadian Rhythms in Drosophila. BioEssays. 2000;22:84–93. doi: 10.1002/(SICI)1521-1878(200001)22:1<84::AID-BIES13>3.0.CO;2-I. [DOI] [PubMed] [Google Scholar]
  • 14.Ueda H.R., Hagiwara M., Kitano H. Robust Oscillations within the Interlocked Feedback Model of Drosophila Circadian Rhythm. J. Theor. Biol. 2001;210:401–406. doi: 10.1006/jtbi.2000.2226. [DOI] [PubMed] [Google Scholar]
  • 15.Smolen P., Baxter D.A., Byrne J.H. Modeling Circadian Oscillations with Interlocking Positive and Negative Feedback Loops. J. Neurosci. 2001;21:6644–6656. doi: 10.1523/JNEUROSCI.21-17-06644.2001. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 16.Gillespie D.T. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys. 1976;22:403–434. [Google Scholar]
  • 17.Gillespie D.T. Exact Stochastic Simulation of Coupled Chemical Reactions. J. Phys. Chem. 1977;81:2340–2361. [Google Scholar]
  • 18.Nicolis G., Prigogine L. Self-Organization in Nonequilibrium Systems. New York: Wiley; 1977. [Google Scholar]
  • 19.Morton-Firth C.J., Bray D. Predicting Temporal Fluctuations in an Intracellular Signalling Pathway. J. Theor. Biol. 1998;192:117–128. doi: 10.1006/jtbi.1997.0651. [DOI] [PubMed] [Google Scholar]
  • 20.Baras F., Pearson J.E., Malek Mansour M. Microscopic Simulation of Chemical Oscillations in Homogeneous Systems. J. Chem. Phys. 1990;93:5747–5750. [Google Scholar]
  • 21.Baras F. Stochastic Analysis of Limit Cycle Behaviour. In: Schimansky-Geier L., Poeschel T., editors. Stochastic Dynamics, Lecture Notes in Physics (LNP484) Berlin: Springer-Verlag; 1997. pp. 167–178. [Google Scholar]
  • 22.McAdams H.H., Arkin A. Stochastic Mechanisms in Gene Expression. Proc. Natl. Acad. Sci. USA. 1997;94:814–819. doi: 10.1073/pnas.94.3.814. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 23.Arkin, A., Ross, J. and McAdams, H.H.: Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage ?-Infected Escherichia coli Cells, Genetics149 (1998), 1633- 1648. [DOI] [PMC free article] [PubMed]
  • 24.Gonze D., Halloy J., Goldbeter A. Robustness of Circadian Rhythms with Respect to Molecular Noise. Proc. Natl. Acad. Sci. USA. 2002;99:673–678. doi: 10.1073/pnas.022628299. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 25.Barkai N., Leibler S. Circadian Clocks Limited by Noise. Nature. 2000;403:267–268. doi: 10.1038/35002258. [DOI] [PubMed] [Google Scholar]
  • 26.von Hippel P.H., Berg O.G. Facilitated Target Location in Biological Systems. J. Biol. Chem. 1989;264:675–678. [PubMed] [Google Scholar]
  • 27.Kraus M., Lais P., Wolf B. Structured Biological Modelling: A Method for the Analysis and Simulation of Biological Systems Applied to Oscillatory Intracellular Calcium Waves. BioSystems. 1992;27:145–169. doi: 10.1016/0303-2647(92)90070-f. [DOI] [PubMed] [Google Scholar]
  • 28.Gonze, D., Halloy, J., Leloup, J.C. and Goldbeter, A.: Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behavior. C.R. Biologies (2003), in press. [DOI] [PubMed]

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