American Institute of Mathematical Sciences

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

Modeling transcriptional co-regulation of mammalian circadian clock

Yanqin Wang 1,2, , Xin Ni 1, , Jie Yan 1, and  Ling Yang 1,,

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.

Keywords: Circadian clock, delay differential equations, stability, Hopf bifurcation.
Mathematics Subject Classification: Primary: 34D99; Secondary: 34C23.
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:
[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

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

Figure 1.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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 ($\tau_1, \, \tau_2$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[2]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, , () : -. doi: 10.3934/era.2020084
[3]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[4]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[5]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
[6]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[7]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[8]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[9]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[10]

Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395
[11]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[12]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[13]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[14]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[15]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[16]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[17]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[18]

Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[19]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[20]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (63)
  • HTML views (121)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences