# American Institute of Mathematical Sciences

December  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:

show all references

##### References:
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
 [1] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and 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, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [3] Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493 [20] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

2018 Impact Factor: 1.313