Modeling circadian clocks: From equations to oscillations

Central European Journal of Biology

Volumn 6, Issue 5, 2011, Pages 699-711

  Gonze, Didier ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

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

Indexed keywords

EUKARYOTA;

EID: 80052281434     PISSN: 1895104X     EISSN: 16443632     Source Type: Journal    
DOI: 10.2478/s11535-011-0061-5     Document Type: Review

References (107)

1. Murray, J. D., Mathematical Biology, Springer-Verlag, Berlin, 1993.
2. Goldbeter, A., Biochemical Oscillations and Cellular Rhythms: The molecular bases of periodic and chaotic behaviour, Cambridge University Press, Cambridge, United Kingdom, 1996.
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.
4. Hastings M. H., Herzog E. D. Clock genes, oscillators, and cellular networks in the suprachiasmatic nuclei. J. Biol. Rhythms, 2004, 19, 400-413.
5. Reppert S. M., Weaver D. R. Coordination of circadian timing in mammals. Nature, 2002, 418, 935-941.
6. Yamada Y., Forger D., Multiscale complexity in the mammalian circadian clock. Curr. Opin. Genet. Dev., 2010, 20, 626-633.
7. Roenneberg T., Chua E. J., Bernardo R, Mendoza E, Modelling biological rhythms, Curr Biol, 2008, 18, R826-R835.
8. Granada A. E., Herzel H., How to achieve fast entrainment? The timescale to synchronization. PLoS One. 2009, 4, e7057.
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.
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.
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.
12. Rougemont J., Naef F., Stochastic phase oscillator models for circadian clocks, Adv Exp Med Biol, 2008, 641, 141-149.
13. Winfree A. T., Biological rhythms and the behavior of populations of coupled oscillators, J Theor Biol, 1967, 16, 15-42.
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.
15. van der Pol B., A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, 1920, 701-710, 754-762.
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.
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.
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.
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.
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.
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.
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.
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.
24. Cummings F. W., A biochemical model of the circadian clock, J Theor Biol, 1975, 55, 455-70.
25. Goodwin B. C., Oscillatory behavior in enzymatic control processes, Adv Enzyme Regul., 1965, 3, 425-438.
26. Pavlidis T., A model for circadian clocks, Bull Math Biophys., 1967, 29, 781-791.
27. Pavlidis T., Populations of interacting oscillators and circadian rhythms, J Theor Biol., 1969, 22, 418-436.
28. Pavlidis T., Kauzmann W., Toward a quantitative biochemical model for circadian oscillators, Arch Biochem Biophys, 1969, 132, 338-348.
29. Griffith J. S., Mathematics of cellular control processes. I. Negative feedback to one gene, J Theor Biol., 1968, 20, 202-208.
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.
31. Rensing L., Schill W., Perturbation by single and double pulsesas analytical tool for analysing oscillatory mechanisms, In Temporal Order, Eds. L. Rensing and N. I. Jaeger, Springer-Verlag, Berlin, 226-231.
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.
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.
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.
35. Ruoff P., Rensing L., The temperature-compensated Goodwin model simulates many circadian clock propoerties, J. Theor. Biol., 1996, 179, 275-285.
36. Goodwin B. C., Temporal organization and disorganization in organisms, Chronobiol Int., 1997, 14, 531-536.
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.
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.
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.
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.
41. Gonze D., Bernard S., Waltermann C., Kramer A., Herzel H., Spontaneous synchronization of coupled circadian oscillators, Biophys J., 2005, 89, 120-129.
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.
43. Komin N, Murza A. C., Hernandez-Garcia E., Toral R., Synchronization and entrainment of coupled circadian oscillators, Interface Focus, 2011, 1, 167-176.
44. Goldbeter A., A model for circadian oscillations in the Drosophila period protein (PER), Proc. R. Soc. Lond. B, 1995, 261, 319-324.
45. Cornish-Bowden, A. Fundamentals of enzyme kinetics, 1995, Portland Press, London.
46. Alon U., An Introduction to Systems Biology - Design Principles of Biological Circuits, 2007, CRC press.
47. Becker-Weimann S., Wolf J., Herzel H., Kramer A., Modeling feedback loops of the mammalian circadian oscillator, Biophys. J, 2004, 87, 3023-3034.
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.
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.
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.
51. Francois P., A model for the Neurospora circadian clock, Biophys J, 2005, 88, 2369-2383.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
63. Lema M. A., Golombek D. A., Echave J., Delay model of the circadian pacemaker, J Theor Biol., 2000, 204, 565-573.
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.
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.
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.
67. Sriram K., Bernot G., Képès, Discrete Delay Model for the Mammalian Circadian Clock, ComPlexUs, 2006, 3, 185-199.
68. Mogilner A., Wollman R., Marshall W. F., Quantitative modeling in cell biology: what is it good for? Dev Cell, 2006, 11, 279-287.
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.
70. Ermentrout, B., Simulating, analyzing and animating dynamical systems: A guide to XPPAUT for researchers and students, Philadelphia: SIAM, 2002.
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/).
72. Dyson F., A meeting with Enrico Fermi, Nature, 2004, 427, 297.
73. Novak B., Tyson J. J., Design principles of biochemical oscillators, Nat Rev Mol Cell Biol, 2008, 9, 981-991.
74. Stelling J., Gilles E. D., Doyle F. J., Robustness properties of circadian clock architectures, Proc Natl Acad Sci USA, 2004, 101, 13210-13215.
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.
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.
77. Gunawan R., Doyle F. J., Phase sensitivity analysis of circadian rhythm entrainment, J Biol Rhythms, 2007, 22, 180-194.
78. Hafner M., Koeppl H., Hasler M., Wagner A., 'Glocal' robustness analysis and model discrimination for circadian oscillators, PLoS Comput Biol, 2009, 5, e1000534.
79. Rensing L., Meyer-Grahle U., Ruoff P., Biological timing and the clock metaphor: oscillatory and hourglass mechanisms, Chronobiol Int., 2001, 18, 329-369.
80. Sinha S., A linear oscillator model for circadian rhythms: implications for phase response curves, Chronobiologia, 1981, 8, 377-391.
81. Sinha S., Are circadian oscillators structurally unstable? J Theor Biol, 1983, 101, 401-414.
82. Winfree A. T., The geometry of biological time, Springer, New York, 2001.
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.
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.
85. Kurosawa G., Iwasa Y., Saturation of enzyme kinetics in circadian clock models, J Biol Rhythms, 2002, 17, 568-577.
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.
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.
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.
89. Oda G. A., Friesen W. O., A model for "splitting" of running-wheel activity in hamsters, J Biol Rhythms., 2002, 17, 76-88.
90. Leloup J. C., Goldbeter A. Modeling the circadian clock: from molecular mechanism to physiological disorders, BioEssays, 2008, 30, 590-600.
91. Leloup J. C., Goldbeter A., Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila, BioEssays, 2000, 22, 84-93.
92. Forger D. B., Kronauer R. E., Reconciling Mathematical Models of Biological Clocks by Averaging on Approximate Manifolds, SIAM, 2002, 62, 1281-1296.
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.
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.
95. Raj A., van Oudenaarden A., Nature, nurture, or chance: stochastic gene expression and its consequences, Cell, 2008, 135, 216-226.
96. Raser J. M., O'shea E. K., Noise in gene expression: origins, consequences, and control, Science, 2005, 309, 2010-2013.
97. Gillespie D. T., Exact Stochastic Simulation of Coupled Chemical Reactions, J Phys Chem, 1977, 81, 2340-2361.
98. Barkai N., Leibler S., Circadian clocks limited by noise, Nature, 2000, 403, 267-268.
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.
100. Forger D. B., Peskin C. S., Stochastic simulation of the mammalian circadian clock, Proc Natl Acad Sci USA, 2005, 102, 321-324.
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.
102. Gonze D., Halloy J., Gaspard P., Biochemical clocks and molecular noise: Theoretical study of robustness factors, J. Chem. Phys., 2002, 116, 10997-11010.
103. Gonze D., Goldbeter A., Circadian rhythms and molecular noise, Chaos, 2006, 16, 026110.
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.
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.
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.
107. Liu A. C., Lewis W. G., Kay S. A., Mammalian circadian signaling networks and therapeutic targets, Nat Chem Biol, 2007, 3, 630-639.