### Mathematical Biosciences and Engineering

2017, Issue 5&6: 1247-1259. doi: 10.3934/mbe.2017064

# Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM

• Received: 19 May 2016 Published: 01 October 2017
• MSC : Primary: 34Cxx; Secondary: 92Bxx

• Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly $24$ hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

Citation: Jifa Jiang, Qiang Liu, Lei Niu. Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1247-1259. doi: 10.3934/mbe.2017064

### Related Papers:

• Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly $24$ hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.
 [1] [ R. Allada,N. E. White,W. V. So,J. C. Hall,M. Rosbash, A mutant Drosophila homolog of mammalian clock disrupts circadian rhythms and transcription of period and timeless, Cell, 93 (1998): 791-804. [2] [ K. Bae,C. Lee,D. Sidote,K-Y. Chuang,I. Edery, Circadian regulation of a Drosophila homolog of the mammalian clock gene: PER and TIM function as positive regulators, Mol. Cell. Biol., 18 (1998): 6142-6151. [3] [ T. K. Darlington,K. Wager-Smith,M. F. Ceriani,D. Staknis,N. Gekakis,T. D. L. Steeves,C. J. Weitz,J. S. Takahashi,S. A. Kay, Closing the circadian loop: CLOCK-induced transcription of its own inhibitors per and tim, Science, 280 (1998): 1599-1603. [4] [ A. Eskin,S. J. Yeung,M. R. Klass, Requirement for protein synthesis in the regulation of a circadian rhythm by serotonin, Proc. Natl. Acad. Sci. USA, 81 (1984): 7637-7641. [5] [ N. Gekakis,L. Saez,A.-M. Delahaye-Brown,M. P. Myers,A. Sehgal,M. W. Young,C. J. Weitz, Isolation of timeless by PER protein interaction: Defective interaction between timeless protein and long-period mutant $\mbox{PER}^{L}$, Science, 270 (1995): 811-815. [6] [ A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER), Proc. R. Soc. Lond. B, 261 (1995): 319-324. [7] [ P. E. Hardin,J. C. Hall,M. Rosbash, Feedback of the Drosophila Period gene product on circadian cycling of its messenger RNA levels, Nature, 343 (1990): 536-540. [8] [ M. W. Hirsch, Systems of differential equations which are competitive or cooperative. Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982): 167-179. [9] [ M. W. Hirsch, Systems of differential equations that are competitive and cooperative. Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 21 (1990): 1225-1234. [10] [ Z. J. Huang,K. D. Curtin,M. Rosbash, PER protein interactions and temperature compensation of a circadian clock in Drosophila, Science, 267 (1995): 1169-1172. [11] [ M. W. Karakashian,J. W. Hastings, The effects of inhibitors of macromolecular biosynthesis upon the persistent rhythm of luminescence in Gonyaulax, J. Gen. Physiol., 47 (1963): 1-12. [12] [ S. B. S. Khalsa,D. Whitmore,G. D. Block, Stopping the circadian pacemaker with inhibitors of protein synthesis, Proc. Natl. Acad. Sci. USA, 89 (1992): 10862-10866. [13] [ B. Kloss,J. L. Price,L. Saez,J. Blau,A. Rothenfluh,C. S. Wesley,M. W. Young, The Drosophila clock gene double-time encodes a protein closely related to human casein kinase l$\epsilon$, Cell, 94 (1998): 97-107. [14] [ R. J. Konopka,S. Benzer, Clock mutants of Drosophila melanogaster, Proc. Natl. Acad. Sci. USA, 68 (1971): 2112-2116. [15] [ J.-C. Leloup,A. Goldbeter, A model for circadian rhythms in Drosophila incorporating the formation of a complex between PER and TIM proteins, J. Biol. Rhythms, 13 (1998): 70-87. [16] [ J. L. Price,J. Blau,A. Rothenfluh,M. Abodeely,B. Kloss,M. W. Young, double-time is a novel Drosophila clock gene that regulates PERIOD protein accumulation, Cell, 94 (1998): 83-95. [17] [ P. Ruoff,L. Rensing, The temperature-compensated Goodwin model simulates many circadian clock properties, J. Theor. Biol., 179 (1996): 275-285. [18] [ J. E. Rutila,V. Suri,M. Le,M. V. So,M. Rosbash,J. C. Hall, CYCLE is a second bHLH-PAS clock protein essential for circadian rhythmicity and transcription of Drosophila period and timeless, Cell, 93 (1998): 805-814. [19] [ T. Scheper,D. Klinkenberg,C. Pennartz,J. van Pelt, A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19 (1999): 40-47. [20] [ A. Sehgal,J. L. Price,B. Man,M. W. Young, Loss of circadian behavioral rhythms and per RNA oscillations in the Drosophila mutant timeless, Science, 263 (1994): 1603-1606. [21] [ J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eqns., 38 (1980): 80-103. [22] [ H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eqns., 65 (1986): 361-373. [23] [ H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc. Providence, Rhode Island, 1995. [24] [ V. Suri,A. Lanjuin,M. Rosbash, TIMELESS-dependent positive and negative autoregulation in the Drosophila circadian clock, EMBO J., 18 (1999): 501-791. [25] [ W. R. Taylor,J. C. Dunlap,J. W. Hastings, Inhibitors of protein synthesis on 80s ribosomes phase shift the Gonyaulax clock, J. Exp. Biol., 97 (1982): 121-136. [26] [ J. J. Tyson,C. I. Hong,C. D. Thron,B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999): 2411-2417. [27] [ L. B. Vosshall,J. L. Price,A. Sehgal,L. Saez,M. W. Young, Block in nuclear localization of period protein by a second clock mutation, timeless, Science, 263 (1994): 1606-1609. [28] [ Y. Wang,J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns., 176 (2001): 470-493. [29] [ H. Zeng,Z. Qian,M. P. Myers,M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature, 380 (1996): 129-135. [30] [ H.-R. Zhu,H. L. Smith, Stable periodic orbits for a class of three dimensional competitive systems, J. Diff. Eqns., 110 (1994): 143-156.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3

Article outline

## Figures and Tables

Figures(5)  /  Tables(6)

• On This Site