Uncovering the design principles of circadian clocks: mathematical analysis of flexibility and evolutionary goals. (English) Zbl 1445.92015
Summary: In this paper, we present the mathematical details underlying both an approach to the flexibility of regulatory networks and an analytical characterization of evolutionary goals of circadian clock networks. A fundamental problem in cellular regulation is to understand the relation between the form of regulatory networks and their function. Circadian clocks present a particularly interesting instance of this. Recent work has shown that they have complex structures involving multiple interconnected feedback loops with both positive and negative feedback. We address the question of why they have such a complex structure and argue that it is to provide the flexibility necessary to simultaneously attain multiple key properties of circadian clocks such as robust entrainment and temperature compensation. To do this we address two fundamental problems: (A) to understand the relationships between the key evolutionary aims of the clock and (B) to ascertain how flexible the clock’s structure is. To address the first problem we use infinitesimal response curves (IRCs), a tool that we believe will be of general utility in the analysis of regulatory networks. To understand the second problem we introduce the flexibility dimension \(d\), show how to calculate it and then use it to analyse a range of models. We believe our results will generalize to a broad range of regulatory networks.
Keywords:
circadian clocks; gene expression; feedback loops; oscillations; mathematical models; flexibilityReferences:
| [1] | Albrecht, U.; Zheng, B.; Larkin, D.; Sun, Z. S.; Lee, C. C., Mper1 and mper2 are essential for normal resetting of the circadian clock, J. Biol. Rhythms, 16, 2, 100-104 (2001) |
| [2] | Cheng, P.; Yang, Y. H.; Liu, Y., Interlocked feedback loops contribute to the robustness of the neurospora circadian clock, Proc. Natl Acad. Sci. USA, 98, 13, 7408-7413 (2001) |
| [3] | Cyran, S. A.; Buchsbaum, A. M.; Reddy, K. L.; Lin, M. C.; Glossop, N. R.; Hardin, P. E.; Young, M. W.; Storti, R. V.; Blau, J., vrille, pdp1, and dclock form a second feedback loop in the drosophila circadian clock, Cell, 112, 3, 329-341 (2003) |
| [4] | Daan, S.; Albrecht, U.; van der Horst, G. T.J.; Illnerova, H.; Roenneberg, T.; Wehr, T. A.; Schwartz, W. J., Assembling a clock for all seasons: are there m and e oscillators in the genes?, J. Biol. Rhythms, 16, 2, 105-116 (2001) |
| [5] | Eriksson, M. E.; Hanano, S.; Southern, M. M.; Hall, A.; Millar, A. J., Response regulator homologues have complementary, light- dependent functions in the arabidopsis circadian clock, Planta, 218, 1, 159-162 (2003) |
| [6] | Forger, D. B.; Peskin, C. S., A detailed predictive model of the mammalian circadian clock, Proc. Natl Acad. Sci. USA, 100, 25, 14806-14811 (2003) |
| [7] | Garceau, N. Y.; Liu, Y.; Loros, J. J.; Dunlap, J. C., Alternative initiation of translation and time-specific phosphorylation yield multiple forms of the essential clock protein frequency, Cell, 89, 3, 469-476 (1997) |
| [8] | Glossop, N. R.J.; Lyons, L. C.; Hardin, P. E., Interlocked feedback loops within the drosophila circadian oscillator, Science, 286, 5440, 766-768 (1999) |
| [9] | Goldbeter, A., Computational approaches to cellular rhythms, Nature, 420, 6912, 238-245 (2002) |
| [10] | Goldbeter, A., Computational biology of circadian rhythms, Mol. Biol. Cell, 13, 57 (2002) |
| [11] | 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., 214, 4, 577-597 (2002) |
| [12] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied Mathematical Sciences (1983), Springer: Springer New York · Zbl 0515.34001 |
| [13] | Harmer, S. L.; Hogenesch, L. B.; Straume, M.; Chang, H. S.; Han, B.; Zhu, T.; Wang, X.; Kreps, J. A.; Kay, S. A., Orchestrated transcription of key pathways in arabidopsis by the circadian clock, Science, 290, 5499, 2110-2113 (2000) |
| [14] | Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102 |
| [15] | Hwang, J. T.; Dougherty, E. P.; Rabitz, S.; Rabitz, H., The green’s function method of sensitivity analysis in chemical kinetics, J. Chem. Phys., 69, 11, 5180-5191 (1978) |
| [16] | Johnson, C. H.; Elliott, J. A.; Foster, R., Entrainment of circadian programs, Chronobiol. Int., 20, 5, 741-774 (2003) |
| [17] | Lee, K.; Loros, J. J.; Dunlap, J. C., Interconnected feedback loops in the neurospora circadian system, Science, 289, 5476, 107-110 (2000) |
| [18] | Leloup, J. C.; Goldbeter, A., Toward a detailed computational model for the mammalian circadian clock, Proc. Natl Acad. Sci. USA, 100, 12, 7051-7056 (2003) |
| [19] | Leloup, J. C.; Gonze, D.; Goldbeter, A., Limit cycle models for circadian rhythms based on transcriptional regulation in drosophila and neurospora, J. Biol. Rhythms, 14, 6, 433-448 (1999) |
| [20] | Oster, H.; Yasui, A.; van der Horst, G. T.; Albrecht, U., Disruption of mcry2 restores circadian rhythmicity in mper2 mutant mice, Genes Dev., 16, 20, 2633-2638 (2002) |
| [21] | Ostlund, S.; Rand, D.; Sethna, J.; Siggia, E., Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica D, 8, 3, 303-342 (1983) · Zbl 0538.58025 |
| [22] | Preitner, N.; Damiola, F.; Lopez-Molina, L.; Zakany, J.; Duboule, D.; Albrecht, U.; Schibler, U., The orphan nuclear receptor rev-erbalpha controls circadian transcription within the positive limb of the mammalian circadian oscillator, Cell, 110, 2, 251-260 (2002) |
| [23] | Press, W. H.; Flanner, B. P.; Teukolsky, S. A.; Vetterling, W., Numerical Recipes in C (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0661.65001 |
| [24] | Reddy, A. B.; Field, M. D.; Maywood, E. S.; Hastings, M. H., Differential resynchronisation of circadian clock gene expression within the suprachiasmatic nuclei of mice subjected to experimental jet lag, J. Neurosci., 22, 17, 7326-7330 (2002) |
| [25] | Rensing, L.; Ruoff, P., Temperature effect on entrainment, phase shifting, and amplitude of circadian clocks and its molecular bases, Chronobiol. Int., 19, 5, 807-864 (2002) |
| [26] | Roenneberg, T., Merrow, M., 2003. The network of time: understanding the molecular circadian system. Curr. Biol. 13 (5), R198-207. |
| [27] | Ruoff, P., Temperature compensation in biological oscillators: a challenge for joint experimental and theoretical analysis, Comments Theor. Biol., 5, 6, 361-382 (2000) |
| [28] | Smolen, P.; Baxter, D. A.; Byrne, J. H., Modeling circadian oscillations with interlocking positive and negative feedback loops, J. Neurosci., 21, 17, 6644-6656 (2001) |
| [29] | Tyson, J. J.; Hong, C. I.; Thron, C. D.; Novak, B., A simple model of circadian rhythms based on dimerization and proteolysis of per and tim, Biophys. J., 77, 5, 2411-2417 (1999) |
| [30] | Ueda, H. R.; Hagiwara, M.; Kitano, H., Robust oscillations within the interlocked feedback model of drosophila circadian rhythm, J. Theor. Biol., 210, 4, 401-406 (2001) |
| [31] | Winfree, A., The Geometry of Biological Time (2001), Springer: Springer New York · Zbl 1014.92001 |
| [32] | Young, M. W.; Kay, S. A., Time zones: a comparative genetics of circadian clocks, Nature Rev. Genet., 2, 9, 702-715 (2001) |
| [33] | Zheng, B.; Albrecht, U.; Kaasik, K.; Sage, M.; Lu, W.; Vaishnav, S.; Li, Q.; Sun, Z. S.; Eichele, G.; Bradley, A.; Lee, C. C., Nonredundant roles of the mper1 and mper2 genes in the mammalian circadian clock, Cell, 105, 5, 683-694 (2001) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
