# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1447-1462. doi: 10.3934/mbe.2017075

## Modeling transcriptional co-regulation of mammalian circadian clock

 1 School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, China 2 School of Mathematics & Physics, Changzhou University, Changzhou 213164, Jiangsu, China

* Corresponding author: Ling Yang

Received  May 30, 2016 Accepted  January 20, 2017 Published  May 2017

Fund Project: The corresponding author is supported by National Natural Science Foundation of China grants 61271358, A011403 and the Priority Academic Program of Jiangsu Higher Education Institutions, the first author is supported by National Natural Science Foundation of China grant 11501055 and Changzhou University Research Fund (ZMF15020093).

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

Citation: Yanqin Wang, Xin Ni, Jie Yan, Ling Yang. Modeling transcriptional co-regulation of mammalian circadian clock. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1447-1462. doi: 10.3934/mbe.2017075
##### References:
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show all references

##### References:
 [1] M. Adimy, F. Crauste and S. G. Ruan, Periodic oscillations in leukopoiesis models with two delays, Journal of Theoretical Biology, 242 (2006), 288-299.  doi: 10.1016/j.jtbi.2006.02.020.  Google Scholar [2] M. P. Antoch, V. Y. Gorbacheva, O. Vykhovanets and A. Y. Nikitin, Disruption of the circadian clock due to the Clock mutation has discrete effects on aging and carcinogenesis, Cell Cycle, 7 (2008), 1197-1204.   Google Scholar [3] D. B. Forger and C. S. Peskin, A detailed predictive model of the mammalian circadian clock, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003), 14806-14811.  doi: 10.1073/pnas.2036281100.  Google Scholar [4] A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER), Proceedings. Biological sciences / The Royal Society, 261 (1995), 319-324.  doi: 10.1098/rspb.1995.0153.  Google Scholar [5] A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, 1996. doi: 10.1017/CBO9780511608193.  Google Scholar [6] C. I. Hong and J. J. Tyson, A proposal for temperature compensation of the circadian rhythm in Drosophila based on dimerization of the per protein, Chronobiology International, 14 (1997), 521-529.   Google Scholar [7] J. K. Kim and D. B. Forger, A mechanism for robust circadian timekeeping via stoichiometric balance, Molecular Systems Biology, 8 (2012), 630. doi: 10.1038/msb.2012.62.  Google Scholar [8] R. V. Kondratov, A. A. Kondratova and V. Y. Gorbacheva, Early aging and age-related pathologies in mice deficient in BMAL1, the core component of the circadian clock, Genes & Developoment, 20 (2006), 1868-1873.   Google Scholar [9] C. C. Lee, Tumor suppression by the mammalian Period genes, Cancer Causes Control, 17 (2006), 525-530 [PubMed: 16596306].  doi: 10.1007/s10552-005-9003-8.  Google Scholar [10] J. C. Leloup and A. Goldbeter, Toward a detailed computational model for the mammalian circadian clock: Sensitivity analysis and multiplicity of oscillatory mechanisms, J. Theoret. Biol., 230 (2004), 541-562.  doi: 10.1016/j.jtbi.2004.04.040.  Google Scholar [11] P. L. Lowrey and J. S. Takahashi, Mammalian circadian biology: elucidating genome-wide levels of temporal organization, Annual Review of Genomics and Human Genetics, 5 (2004), 407-441.  doi: 10.1146/annurev.genom.5.061903.175925.  Google Scholar [12] H. P. Mirsky, A. C. Liu, D. K. Welsh, S. A. Kay and F. J. Doyle, A model of the cell-autonomous mammalian circadian clock, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 11107-11112.  doi: 10.1073/pnas.0904837106.  Google Scholar [13] S. G. Ruan and J. J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 10 (2003), 863-874.   Google Scholar [14] F. A. Scheer, M. F. Hilton, C. S. Mantzoros and S. A. Shea, Adverse metabolic and cardiovascular consequences of circadian misalignment, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009), 4453-4458.  doi: 10.1073/pnas.0808180106.  Google Scholar [15] J. J. Tyson, C. I. Hong, C. D. Thron and B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys Journal, 77 (1999), 2411-2417.  doi: 10.1016/S0006-3495(99)77078-5.  Google Scholar [16] M. Ukai-Tadenuma, R. G. Yamada, H. Xu, J. A. Ripperger, A. C. Liu and H. R. Ueda, Delay in feedback repression by cryptochrome 1 is required for circadian clock function, Cell, 144 (2011), 268-281.  doi: 10.1016/j.cell.2010.12.019.  Google Scholar [17] J. Yan, G. Shi, Z. Zhang, X. Wu, Z. Liu, L. Xing, Z. Qu, Z. Dong, L. Yang and Y. Xu, An intensity ratio of interlocking loops determines circadian period length, Nucleic Acids Research, 42 (2014), 10278-10287.  doi: 10.1093/nar/gku701.  Google Scholar [18] X. Yang, M. Downes, R. T. Yu, A. L. Bookout, W. He, M. Straume, D. J. Mangelsdorf and R. M. Evans, Nuclear receptor expression links the circadian clock to metabolism, Cell, 126 (2006), 801-810.  doi: 10.1016/j.cell.2006.06.050.  Google Scholar [19] W. Yu, M. Nomura and M. Ikeda, Interactivating feedback loops within the mammalian clock: BMAL1 is negatively autoregulated and upregulated by CRY1, CRY2, and PER2, Biochemical and Biophysical Research Communications, 290 (2002), 933-941.  doi: 10.1006/bbrc.2001.6300.  Google Scholar [20] E. E. Zhang and S. A. Kay, Clocks not winding down: Unravelling circadian networks, Nature Reviews Molecular Cell Biology, 11 (2010), 764-776.  doi: 10.1038/nrm2995.  Google Scholar
The model of a mammalian circadian clock with two delays. Figure (a) is a schematic diagram of gene regulation in the mammalian circadian clock system, figure (b) is a schematic diagram of the simplified mathematical model of a mammalian circadian clock
Stability and Hopf bifurcation of system (4.1) for different $\tau_1\in [0, \, \infty)$ when $\tau_2=0$. The equilibrium point $x^{\ast}$ of (4.2) is locally asymptotically stable when $\tau_1=0.5$ in figure (a) and $\tau_1=1.5$ in figure (b), respectively. The equilibrium point $x^{\ast}$ of (4.2) losts its stability and stable bifurcation periodic solutions appear when $\tau_1=2.0$ in figure (c) and $\tau_1=4.0$ in figure (d), respectively
Stability of system (4.1) with different $\tau_2$ when $\tau_1^{\ast}=1.85 \in (0, \tau_1^0)$ and $0<\tau_2 < 4.05$. The equilibrium point $x^{\ast}$ of (4.1) is locally asymptotically stable when $\tau_2=0.5$ in figure (a), $\tau_2=1.5$ in figure (b), $\tau_2=3.5$ in figure (c), $\tau_2=4$ in figure (d), respectively
Instability of system (4.1) with different $\tau_2$ when $\tau_1^{\ast}=2.8 \in (\tau_1^0, \infty)$ and $0<\tau_2 < 2.1$. The equilibrium point $x^{\ast}$ of (4.1) is unstable when $\tau_2=0.5$ in figure (a), $\tau_2=1$ in figure (b), $\tau_2=1.5$ in figure (c), $\tau_2=2$ in figure (d), respectively
Bifurcation diagram of ($\tau_1, \, \tau_2$) for system (4.1). $S$ denotes stable regions, $US$ denotes oscillating regions. The black solid line is made up of critical bifurcation points for ($\tau_1, \, \tau_2$), the rest solid lines with different colours are lines consisting of critical bifurcation points when $\tau_2$ pluses different period respectively, and the marked six different points represent different values of ($\tau_1, \, \tau_2$)
Stability of system (4.1) with different $\tau_2$ when $\tau_1^{\ast}=2.4\in (\tau_1^0, \infty)$ and $\tau_2>0$. The equilibrium point $x^{\ast}$ of (4.1) is locally asymptotically stable when $\tau_2=2$ in figure (b), $\tau_2=9$ in figure (d), $\tau_2=15.5$ in figure (f), respectively, it is unstable when $\tau_2=0.5$ in figure (a), $\tau_2=5$ in figure (c), $\tau_2=12$ in figure (e), respectively
Oscillating range of ($\tau_1, \, \tau_2$) for system (4.1). Black regions represent oscillating solutions with periods for system (4.1) when ($\tau_1, \, \tau_2$) locates in the black region
The effect of time delays on the period of system (4.1). In figure (a), we fix $\tau_2=29,$ the black solid line represents the relation between $\tau_1$ and the period. In figure (b), we fix $\tau_1=10,$ the black solid line represents $\tau_2$ and the period
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