Modeling transcriptional co-regulation of mammalian circadian clock

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