Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Life Sciences

formerly Central European Journal of Biology

Editor-in-Chief: Ratajczak, Mariusz


IMPACT FACTOR 2018: 0.504
5-year IMPACT FACTOR: 0.583

CiteScore 2018: 0.63

SCImago Journal Rank (SJR) 2018: 0.266
Source Normalized Impact per Paper (SNIP) 2018: 0.311

ICV 2017: 154.48

Open Access
Online
ISSN
2391-5412
See all formats and pricing
More options …
Volume 6, Issue 5

Issues

Volume 10 (2015)

Modeling circadian clocks: From equations to oscillations

Didier Gonze
  • Unité de Chronobiologie Théorique, Service de Chimie Physique, Université Libre de Bruxelles, Campus Plaine, B-1050, Brussels, Belgium
  • Laboratoire de Bioinformatique des Génomes et des Réseaux, Faculté des Sciences, Université Libre de Bruxelles Campus Plaine, B-1050, Brussels, Belgium
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2011-09-02 | DOI: https://doi.org/10.2478/s11535-011-0061-5

Abstract

Circadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

Keywords: Mathematical models; Differential equations; Circadian rhythms; Limit-cycle oscillations; Bifurcation diagram

  • [1] Murray, J.D., Mathematical Biology, Springer-Verlag, Berlin, 1993 http://dx.doi.org/10.1007/b98869CrossrefGoogle Scholar

  • [2] Goldbeter, A., Biochemical Oscillations and Cellular Rhythms: The molecular bases of periodic and chaotic behaviour, Cambridge University Press, Cambridge, United Kingdom, 1996 http://dx.doi.org/10.1017/CBO9780511608193CrossrefGoogle Scholar

  • [3] Shearman L.P., Sriram S., Weaver D.R., Maywood E.S., Chaves I., Zheng B. et al., Interacting molecular loops in the mammalian circadian clock, Science, 2002, 288, 1013–1019 http://dx.doi.org/10.1126/science.288.5468.1013CrossrefGoogle Scholar

  • [4] Hastings M.H., Herzog E.D. Clock genes, oscillators, and cellular networks in the suprachiasmatic nuclei. J. Biol. Rhythms, 2004, 19, 400–413 http://dx.doi.org/10.1177/0748730404268786CrossrefGoogle Scholar

  • [5] Reppert S.M., Weaver D.R. Coordination of circadian timing in mammals. Nature, 2002, 418, 935–941 http://dx.doi.org/10.1038/nature00965CrossrefGoogle Scholar

  • [6] Yamada Y., Forger D., Multiscale complexity in the mammalian circadian clock. Curr. Opin. Genet. Dev., 2010, 20, 626–633 http://dx.doi.org/10.1016/j.gde.2010.09.006CrossrefGoogle Scholar

  • [7] Roenneberg T., Chua E.J., Bernardo R, Mendoza E, Modelling biological rhythms, Curr Biol, 2008, 18, R826–R835 http://dx.doi.org/10.1016/j.cub.2008.07.017CrossrefGoogle Scholar

  • [8] Granada A.E., Herzel H., How to achieve fast entrainment? The timescale to synchronization. PLoS One. 2009, 4, e7057 http://dx.doi.org/10.1371/journal.pone.0007057CrossrefGoogle Scholar

  • [9] Roenneberg T., Dragovic Z., Merrow M., Demasking biological oscillators: properties and principles of entrainment exemplified by the Neurospora circadian clock, Proc Natl Acad Sci USA, 2005, 102, 7742–7747 http://dx.doi.org/10.1073/pnas.0501884102CrossrefGoogle Scholar

  • [10] Amdaoud M., Vallade M., Weiss-Schaber C., Mihalcescu I. Cyanobacterial clock, a stable phase oscillator with negligible intercellular coupling, Proc Natl Acad Sci USA, 2007, 104, 7051–7056 http://dx.doi.org/10.1073/pnas.0609315104CrossrefGoogle Scholar

  • [11] Rougemont J., Naef F., Collective synchronization in populations of globally coupled phase oscillators with drifting frequencies, Phys Rev E Stat Nonlin Soft Matter Phys, 2006, 73, 011104 http://dx.doi.org/10.1103/PhysRevE.73.011104CrossrefGoogle Scholar

  • [12] Rougemont J., Naef F., Stochastic phase oscillator models for circadian clocks, Adv Exp Med Biol, 2008, 641, 141–149 http://dx.doi.org/10.1007/978-0-387-09794-7_10CrossrefGoogle Scholar

  • [13] Winfree A.T., Biological rhythms and the behavior of populations of coupled oscillators, J Theor Biol, 1967, 16, 15–42 http://dx.doi.org/10.1016/0022-5193(67)90051-3CrossrefGoogle Scholar

  • [14] Yang Q., Pando B.F., Dong G., Golden S.S., van Oudenaarden A., Circadian gating of the cell cycle revealed in single cyanobacterial cells, Science, 2010, 327, 1522–1526 http://dx.doi.org/10.1126/science.1181759CrossrefGoogle Scholar

  • [15] van der Pol B., A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, 1920, 701–710,754–762 Google Scholar

  • [16] Pittendrigh, C.S., Bruce V.G., An oscillator model for biological clocks. In Rhythmic and Synthetic Processes in Growth, D. Rudnic, ed., pp. 75–109, Princeton University Press, Princeton, NJ, 1957 Google Scholar

  • [17] Forger D.B., Jewett M.E., Kronauer R.E., A simpler model of the human circadian pacemaker, J Biol Rhythms., 1999, 14, 532–537 http://dx.doi.org/10.1177/074873099129000867CrossrefGoogle Scholar

  • [18] Jewett M.E., Kronauer R.E. Refinement of a limit cycle oscillator model of the effects of light on the human circadian pacemaker, J Theor Biol., 1998, 192, 455–465 http://dx.doi.org/10.1006/jtbi.1998.0667CrossrefGoogle Scholar

  • [19] Jewett M.E., Forger D.B., Kronauer R.E. Revised limit cycle oscillator model of human circadian pacemaker, J Biol Rhythms., 1999, 14, 493–499 CrossrefGoogle Scholar

  • [20] Kronauer R.E., Czeisler C.A., Pilato S.F., Moore-Ede M.C., Weitzman E.D., Mathematical model of the human circadian system with two interacting oscillators, Am. J. Physiol., 1982, 242, R3–R17 Google Scholar

  • [21] Gonze D., Roussel M.R., Goldbeter A., A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle, J. Theor. Biol., 2002, 214, 577–597 http://dx.doi.org/10.1006/jtbi.2001.2476CrossrefGoogle Scholar

  • [22] Achermann P., Kunz H., Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves, J Biol Rhythms., 1999, 14, 460–468 http://dx.doi.org/10.1177/074873099129001028CrossrefGoogle Scholar

  • [23] Kunz H., Achermann P., Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators, J Theor Biol., 2003, 224, 63–78 http://dx.doi.org/10.1016/S0022-5193(03)00141-3CrossrefGoogle Scholar

  • [24] Cummings F.W., A biochemical model of the circadian clock, J Theor Biol, 1975, 55, 455–70 http://dx.doi.org/10.1016/S0022-5193(75)80093-2CrossrefGoogle Scholar

  • [25] Goodwin B.C., Oscillatory behavior in enzymatic control processes, Adv Enzyme Regul., 1965, 3, 425–438 http://dx.doi.org/10.1016/0065-2571(65)90067-1CrossrefGoogle Scholar

  • [26] Pavlidis T., A model for circadian clocks, Bull Math Biophys., 1967, 29, 781–791 http://dx.doi.org/10.1007/BF02476928CrossrefGoogle Scholar

  • [27] Pavlidis T., Populations of interacting oscillators and circadian rhythms, J Theor Biol., 1969, 22, 418–436 http://dx.doi.org/10.1016/0022-5193(69)90014-9CrossrefGoogle Scholar

  • [28] Pavlidis T., Kauzmann W., Toward a quantitative biochemical model for circadian oscillators, Arch Biochem Biophys, 1969, 132, 338–348 http://dx.doi.org/10.1016/0003-9861(69)90371-3CrossrefGoogle Scholar

  • [29] Griffith J.S., Mathematics of cellular control processes. I. Negative feedback to one gene, J Theor Biol., 1968, 20, 202–208 http://dx.doi.org/10.1016/0022-5193(68)90189-6CrossrefGoogle Scholar

  • [30] Drescher K., Cornelius G., Rensing L., Phase response curves obtained by perturbing different variables of a 24 hr model oscillator based on translational control, J Theor Biol, 1982, 94, 345–353 http://dx.doi.org/10.1016/0022-5193(82)90315-0CrossrefGoogle Scholar

  • [31] Rensing L., Schill W., Perturbation by single and double pulsesas analytical tool for analysing oscillatory mechanisms, In Temporal Order, Eds. L. Rensing & N.I. Jaeger, Springer-Verlag, Berlin, 226–231 Google Scholar

  • [32] 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 http://dx.doi.org/10.1038/343536a0CrossrefGoogle Scholar

  • [33] Aronson B.D., Johnson K.A., Loros J.J., Dunlap J.C., Negative feedback defining a circadian clock: autoregulation of the clock gene frequency, Science, 1994, 263, 1578–1584 http://dx.doi.org/10.1126/science.8128244CrossrefGoogle Scholar

  • [34] Ruoff P., Mohsenzadeh S., Rensing L., Circadian rhythms and protein turnover: the effect of temperature on the period lengths of clock mutants simulated by the Goodwin oscillator, Naturwissenschaften, 1996, 83, 514–517 http://dx.doi.org/10.1007/BF01141953CrossrefGoogle Scholar

  • [35] Ruoff P., Rensing L., The temperature-compensated Goodwin model simulates many circadian clock propoerties, J. Theor. Biol., 1996, 179, 275–285 http://dx.doi.org/10.1006/jtbi.1996.0067CrossrefGoogle Scholar

  • [36] Goodwin B.C., Temporal organization and disorganization in organisms, Chronobiol Int., 1997, 14, 531–536 http://dx.doi.org/10.3109/07420529709001474CrossrefGoogle Scholar

  • [37] Ruoff P., Vinsjevik M., Monnerjahn C., Rensing L., The Goodwin oscillator: on the importance of degradation reactions in the circadian clock, J Biol Rhythms., 1999, 14, 469–479 http://dx.doi.org/10.1177/074873099129001037CrossrefGoogle Scholar

  • [38] Ruoff P., Vinsjevik M., Mohsenzadeh S., Rensing L., The Goodwin model: simulating the effect of cycloheximide and heat shock on the sporulation rhythm of Neurospora crassa, J Theor Biol., 1999, 196, 483–494 http://dx.doi.org/10.1006/jtbi.1998.0846CrossrefGoogle Scholar

  • [39] Ruoff P., Vinsjevik M., Monnerjahn C., Rensing L., The Goodwin model: simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa, J Theor Biol., 2001, 209, 29–42 http://dx.doi.org/10.1006/jtbi.2000.2239CrossrefGoogle Scholar

  • [40] Ruoff P., Loros J.J., Dunlap J.C., The relationship between FRQ-protein stability and temperature compensation in the Neurospora circadian clock. Proc Natl Acad Sci USA, 2005, 102, 17681–17686 http://dx.doi.org/10.1073/pnas.0505137102CrossrefGoogle Scholar

  • [41] Gonze D., Bernard S., Waltermann C., Kramer A., Herzel H., Spontaneous synchronization of coupled circadian oscillators, Biophys J., 2005, 89, 120–129 http://dx.doi.org/10.1529/biophysj.104.058388CrossrefGoogle Scholar

  • [42] Locke J.C., Westermark P.O., Kramer A., Herzel H., Global parameter search reveals design principles of the mammalian circadian clock, BMC Syst Biol, 2008, 2, 22 http://dx.doi.org/10.1186/1752-0509-2-22CrossrefGoogle Scholar

  • [43] Komin N, Murza A.C., Hernandez-Garcia E., Toral R., Synchronization and entrainment of coupled circadian oscillators, Interface Focus, 2011, 1, 167–176 http://dx.doi.org/10.1098/rsfs.2010.0327CrossrefGoogle Scholar

  • [44] Goldbeter A., A model for circadian oscillations in the Drosophila period protein (PER), Proc. R. Soc. Lond. B, 1995, 261, 319–324 http://dx.doi.org/10.1098/rspb.1995.0153CrossrefGoogle Scholar

  • [45] Cornish-Bowden, A. Fundamentals of enzyme kinetics, 1995, Portland Press, London Google Scholar

  • [46] Alon U., An Introduction to Systems Biology — Design Principles of Biological Circuits, 2007, CRC press Google Scholar

  • [47] Becker-Weimann S., Wolf J., Herzel H., Kramer A., Modeling feedback loops of the mammalian circadian oscillator, Biophys. J, 2004, 87, 3023–3034 http://dx.doi.org/10.1529/biophysj.104.040824CrossrefGoogle Scholar

  • [48] Forger D.B., Peskin C.S., A detailed predictive model of the mammalian circadian clock, Proc Natl Acad Sci USA, 2003, 100, 14806–14811 http://dx.doi.org/10.1073/pnas.2036281100CrossrefGoogle Scholar

  • [49] Leloup J.C., Goldbeter A., Towards a detailed computational model for the mammalian circadian clock, Proc Natl Acad Sci USA., 2003, 100, 7051–7056 http://dx.doi.org/10.1073/pnas.1132112100CrossrefGoogle Scholar

  • [50] Mirsky H.P., Liu A.C., Welsh D.K., Kay S.A., Doyle F.J., A model of the cell-autonomous mammalian circadian clock, Proc Natl Acad Sci USA, 2009, 106, 11107–11112 http://dx.doi.org/10.1073/pnas.0904837106CrossrefGoogle Scholar

  • [51] Francois P., A model for the Neurospora circadian clock, Biophys J, 2005, 88, 2369–2383 http://dx.doi.org/10.1529/biophysj.104.053975CrossrefGoogle Scholar

  • [52] Hong C.I., Jolma I.W., Loros J.J., Dunlap J.C., Ruoff P. Simulating dark expressions and interactions of frq and wc-1 in the Neurospora circadian clock, Biophys J, 2008, 94, 1221–1232 http://dx.doi.org/10.1529/biophysj.107.115154Google Scholar

  • [53] Fathallah-Shaykh H.M., Bona J.L., Kadener S., Mathematical model of the Drosophila circadian clock: loop regulation and transcriptional integration, Biophys J, 2009, 97, 2399–2408 http://dx.doi.org/10.1016/j.bpj.2009.08.018CrossrefGoogle Scholar

  • [54] Leise T.L., Moin E.E., A mathematical model of the Drosophila circadian clock with emphasis on posttranslational mechanisms, J Theor Biol, 2007, 248, 48–63 http://dx.doi.org/10.1016/j.jtbi.2007.04.013CrossrefGoogle Scholar

  • [55] 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 http://dx.doi.org/10.1177/074873098128999934CrossrefGoogle Scholar

  • [56] 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 http://dx.doi.org/10.1006/jtbi.2000.2226CrossrefGoogle Scholar

  • [57] Xie Z., Kulasiri D., Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops, J Theor Biol, 2007, 245, 290–304 http://dx.doi.org/10.1016/j.jtbi.2006.10.028CrossrefGoogle Scholar

  • [58] Locke J.C., Millar A.J., Turner M.S., Modelling genetic networks with noisy and varied experimental data: the circadian clock in Arabidopsis thaliana, J Theor Biol., 2005, 234, 383–393 http://dx.doi.org/10.1016/j.jtbi.2004.11.038CrossrefGoogle Scholar

  • [59] Locke J.C., Southern M.M., Kozma-Bognar L., Hibberd V., Brown P.E., Turner M.S., Millar A.J., Extension of a genetic network model by iterative experimentation and mathematical analysis, Mol Syst Biol, 2005, 1, 383–393 http://dx.doi.org/10.1038/msb4100018CrossrefGoogle Scholar

  • [60] Locke J.C., Kozma-Bognar L., Gould P.D., Fehér B., Kevei E., Nagy F., Turner M.S., Hall A., Millar A.J., Experimental validation of a predicted feedback loop in the multi-oscillator clock of Arabidopsis thaliana, Mol Syst Biol, 2006, 2, 59 http://dx.doi.org/10.1038/msb4100102CrossrefGoogle Scholar

  • [61] Zeilinger M.N., Farré E.M., Taylor S.R., Kay S.A., Doyle F.J., A novel computational model of the circadian clock in Arabidopsis that incorporates PRR7 and PRR9, Mol Syst Biol, 2006, 2, 58 http://dx.doi.org/10.1038/msb4100101CrossrefGoogle Scholar

  • [62] Lakin-Thomas P.L., Brody S., Cote G.G., Amplitude model for the effects of mutations and temperature on period and phase resetting of the Neurospora, circadian oscillator, J Biol Rhythms., 1991, 6, 281–297 http://dx.doi.org/10.1177/074873049100600401CrossrefGoogle Scholar

  • [63] Lema M.A., Golombek D.A., Echave J., Delay model of the circadian pacemaker, J Theor Biol., 2000, 204, 565–573 http://dx.doi.org/10.1006/jtbi.2000.2038CrossrefGoogle Scholar

  • [64] Scheper T.O., Klinkenberg D., van Pelt J., Pennartz C., A model of molecular circadian clocks: multiple mechanisms for phase shifting and a requirement for strong nonlinear interactions, J Biol Rhythms., 1999, 14, 213–220 http://dx.doi.org/10.1177/074873099129000623CrossrefGoogle Scholar

  • [65] Scheper T.O., Klinkenberg D., Pennartz C., van Pelt J., A mathematical model for the intracellular circadian rhythm generator, J Neurosci., 1999, 19, 40–47 Google Scholar

  • [66] Sriram K., Gopinathan M.S., A two variable delay model for the circadian rhythm of Neurospora crassa, J Theor Biol, 2004, 231, 23–38 http://dx.doi.org/10.1016/j.jtbi.2004.04.006CrossrefGoogle Scholar

  • [67] Sriram K., Bernot G., Képès, Discrete Delay Model for the Mammalian Circadian Clock, ComPlexUs, 2006, 3, 185–199 http://dx.doi.org/10.1159/000095479CrossrefGoogle Scholar

  • [68] Mogilner A., Wollman R., Marshall W.F., Quantitative modeling in cell biology: what is it good for? Dev Cell, 2006, 11, 279–287 http://dx.doi.org/10.1016/j.devcel.2006.08.004CrossrefGoogle Scholar

  • [69] Forger D., Gonze D., Virshup D., Welsh D.K., Beyond intuitive modeling: combining biophysical models with innovative experiments to move the circadian clock field forward, J Biol Rhythms, 2007, 22, 200–210 http://dx.doi.org/10.1177/0748730407301823CrossrefGoogle Scholar

  • [70] Ermentrout, B., Simulating, analyzing and animating dynamical systems: A guide to XPPAUT for researchers and students, Philadelphia: SIAM, 2002 http://dx.doi.org/10.1137/1.9780898718195CrossrefGoogle Scholar

  • [71] Hoops S., Sahle S., Gauges R., Lee C., Pahle J., Simus N., et al., COPASI — a COmplex PAthway SImulator, Bioinformatics, 2006, 22, 3067–3074. (Availibility: http://www.copasi.org/) http://dx.doi.org/10.1093/bioinformatics/btl485CrossrefGoogle Scholar

  • [72] Dyson F., A meeting with Enrico Fermi, Nature, 2004, 427, 297 http://dx.doi.org/10.1038/427297aCrossrefGoogle Scholar

  • [73] Novak B., Tyson J.J., Design principles of biochemical oscillators, Nat Rev Mol Cell Biol, 2008, 9, 981–991 http://dx.doi.org/10.1038/nrm2530CrossrefGoogle Scholar

  • [74] Stelling J., Gilles E.D., Doyle F.J., Robustness properties of circadian clock architectures, Proc Natl Acad Sci USA, 2004, 101, 13210–13215 http://dx.doi.org/10.1073/pnas.0401463101CrossrefGoogle Scholar

  • [75] Leloup J.C., Goldbeter A., Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms, J Theor Biol, 2004, 230, 541–562 http://dx.doi.org/10.1016/j.jtbi.2004.04.040CrossrefGoogle Scholar

  • [76] Rand D.A., Shulgin B.V., Salazar J.D., Millar A.J., Uncovering the design principles of circadian clocks: mathematical analysis of flexibility and evolutionary goals. J Theor Biol., 2006, 238, 616–635 http://dx.doi.org/10.1016/j.jtbi.2005.06.026CrossrefGoogle Scholar

  • [77] Gunawan R., Doyle F.J., Phase sensitivity analysis of circadian rhythm entrainment, J Biol Rhythms, 2007, 22, 180–194 http://dx.doi.org/10.1177/0748730407299194CrossrefGoogle Scholar

  • [78] Hafner M., Koeppl H., Hasler M., Wagner A., ’Glocal’ robustness analysis and model discrimination for circadian oscillators, PLoS Comput Biol, 2009, 5, e1000534 http://dx.doi.org/10.1371/journal.pcbi.1000534CrossrefGoogle Scholar

  • [79] Rensing L., Meyer-Grahle U., Ruoff P., Biological timing and the clock metaphor: oscillatory and hourglass mechanisms, Chronobiol Int., 2001, 18, 329–369 http://dx.doi.org/10.1081/CBI-100103961CrossrefGoogle Scholar

  • [80] Sinha S., A linear oscillator model for circadian rhythms: implications for phase response curves, Chronobiologia, 1981, 8, 377–391 Google Scholar

  • [81] Sinha S., Are circadian oscillators structurally unstable? J Theor Biol, 1983, 101, 401–414 http://dx.doi.org/10.1016/0022-5193(83)90147-9CrossrefGoogle Scholar

  • [82] Winfree A.T., The geometry of biological time, Springer, New York, 2001 Google Scholar

  • [83] 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 http://dx.doi.org/10.1177/074873099129000948CrossrefGoogle Scholar

  • [84] Price J.L., Blau J., Rothenfluh A., Abodeely M., Kloss. B, Young. M.W., Double-time is a novel Drosophila clock gene that regulates PERIOD protein accumulation, Cell, 1998, 94, 83–95 http://dx.doi.org/10.1016/S0092-8674(00)81224-6CrossrefGoogle Scholar

  • [85] Kurosawa G., Iwasa Y., Saturation of enzyme kinetics in circadian clock models, J Biol Rhythms, 2002, 17, 568–577 http://dx.doi.org/10.1177/0748730402238239CrossrefGoogle Scholar

  • [86] Kurosawa G., Mochizuki A., Iwasa Y., Comparative study of circadian clock models, in search of processes promoting oscillation, J Theor Biol., 2002, 216, 193–208 http://dx.doi.org/10.1006/jtbi.2002.2546CrossrefGoogle Scholar

  • [87] Gérard C., Gonze D., Goldbeter A., Dependence of the period on the rate of protein degradation in minimal models for circadian oscillations, Philos Transact A Math Phys Eng Sci, 2009, 367, 4667–4683 http://dx.doi.org/10.1098/rsta.2009.0133CrossrefGoogle Scholar

  • [88] Gonze D., Goldbeter A., Entrainment versus chaos in a model for a circadian oscillator driven by lightdark cycles. J. Stat. Phys., 2000, 101, 649–663 http://dx.doi.org/10.1023/A:1026410121183CrossrefGoogle Scholar

  • [89] Oda G.A., Friesen W.O., A model for “splitting” of running-wheel activity in hamsters, J Biol Rhythms., 2002, 17, 76–88 http://dx.doi.org/10.1177/074873002129002357CrossrefGoogle Scholar

  • [90] Leloup J.C., Goldbeter A. Modeling the circadian clock: from molecular mechanism to physiological disorders, BioEssays, 2008, 30, 590–600 http://dx.doi.org/10.1002/bies.20762CrossrefGoogle Scholar

  • [91] Leloup J.C., Goldbeter A., Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila, BioEssays, 2000, 22, 84–93 http://dx.doi.org/10.1002/(SICI)1521-1878(200001)22:1<84::AID-BIES13>3.0.CO;2-ICrossrefGoogle Scholar

  • [92] Forger D.B., Kronauer R.E., Reconciling Mathematical Models of Biological Clocks by Averaging on Approximate Manifolds, SIAM, 2002, 62, 1281–1296 Google Scholar

  • [93] Indic P., Gurdziel K., Kronauer R.E., Klerman E.B., Development of a two-dimension manifold to represent high dimension mathematical models of the intracellular Mammalian circadian clock, J Biol Rhythms, 2006 21, 222–232 http://dx.doi.org/10.1177/0748730406287357CrossrefGoogle Scholar

  • [94] Taylor S.R., Doyle F.J., Petzold L.R., Oscillator model reduction preserving the phase response: application to the circadian clock, Biophys J, 2008, 95, 1658–1673 http://dx.doi.org/10.1529/biophysj.107.128678CrossrefGoogle Scholar

  • [95] Raj A., van Oudenaarden A., Nature, nurture, or chance: stochastic gene expression and its consequences, Cell, 2008, 135, 216–226 http://dx.doi.org/10.1016/j.cell.2008.09.050CrossrefGoogle Scholar

  • [96] Raser J.M., O’shea E.K., Noise in gene expression: origins, consequences, and control, Science, 2005, 309, 2010–2013 http://dx.doi.org/10.1126/science.1105891CrossrefGoogle Scholar

  • [97] Gillespie D.T., Exact Stochastic Simulation of Coupled Chemical Reactions, J Phys Chem, 1977, 81, 2340–2361 http://dx.doi.org/10.1021/j100540a008CrossrefGoogle Scholar

  • [98] Barkai N., Leibler S., Circadian clocks limited by noise, Nature, 2000, 403, 267–268 Google Scholar

  • [99] Calander N., Propensity of a circadian clock to adjust to the 24h day-night light cycle and its sensitivity to molecular noise, J Theor Biol, 2006, 241, 716–724 Google Scholar

  • [100] Forger D.B., Peskin C.S., Stochastic simulation of the mammalian circadian clock, Proc Natl Acad Sci USA, 2005, 102, 321–324 http://dx.doi.org/10.1073/pnas.0408465102CrossrefGoogle Scholar

  • [101] Gonze D., Halloy J., Goldbeter A., Robustness of circadian rhythms with respect to molecular noise, Proc. Natl. Acad. Sci. USA, 2002, 99, 673–678 http://dx.doi.org/10.1073/pnas.022628299CrossrefGoogle Scholar

  • [102] Gonze D., Halloy J., Gaspard P., Biochemical clocks and molecular noise: Theoretical study of robustness factors, J. Chem. Phys., 2002, 116, 10997–11010 http://dx.doi.org/10.1063/1.1475765CrossrefGoogle Scholar

  • [103] Gonze D., Goldbeter A., Circadian rhythms and molecular noise, Chaos, 2006, 16, 026110 http://dx.doi.org/10.1063/1.2211767CrossrefGoogle Scholar

  • [104] Bernard S., Gonze D., Cajavec B., Herzel H., Kramer A., Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus., PLoS Comput Biol., 2007, 3, e68 http://dx.doi.org/10.1371/journal.pcbi.0030068CrossrefGoogle Scholar

  • [105] To T.L., Henson M.A., Herzog E.D., Doyle F.J., A molecular model for intercellular synchronization in the mammalian circadian clock, Biophys J., 2007, 92, 3792–3803 http://dx.doi.org/10.1529/biophysj.106.094086CrossrefGoogle Scholar

  • [106] Vasalou C., Herzog E.D., Henson M.A. Small-world network models of intercellular coupling predict enhanced synchronization in the suprachiasmatic nucleus, J Biol Rhythms, 24, 243–254 Google Scholar

  • [107] Liu A.C., Lewis W.G., Kay S.A., Mammalian circadian signaling networks and therapeutic targets, Nat Chem Biol, 2007, 3, 630–639 http://dx.doi.org/10.1038/nchembio.2007.37CrossrefGoogle Scholar

About the article

Published Online: 2011-09-02

Published in Print: 2011-10-01


Citation Information: Open Life Sciences, Volume 6, Issue 5, Pages 699–711, ISSN (Online) 2391-5412, DOI: https://doi.org/10.2478/s11535-011-0061-5.

Export Citation

© 2011 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Svetlana Postnova
Clocks & Sleep, 2019, Volume 1, Number 1, Page 166
[2]
Rodrigo Laje, Patricia V. Agostino, and Diego A. Golombek
Frontiers in Integrative Neuroscience, 2018, Volume 12
[3]
Didier Gonze
Open Life Sciences, 2011, Volume 6, Number 5
[4]
R. El Cheikh, T. Lepoutre, and S. Bernard
Mathematical Modelling of Natural Phenomena, 2012, Volume 7, Number 6, Page 107

Comments (0)

Please log in or register to comment.
Log in