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Modeling transcriptional co-regulation of mammalian circadian clock

  • * Corresponding author: Ling Yang

    * Corresponding author: Ling Yang
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).
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  • 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.

    Mathematics Subject Classification: Primary: 34D99; Secondary: 34C23.

    Citation:

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  • 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 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 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 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 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 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 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 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

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