Interlocked multi-node positive and negative feedback loops facilitate oscillations

Figure 1.  An example of interlocked positive and negative feedback loops. (a) Network topology, where the homogenous negative feedback loop contains $N$ node with $N$ being an odd number, the homogeneous positive feedback loop contains $M+1$ node with $M$ being a positive integer, and node $X_N$ is common. (b, c) Two representative modes for the combinatorial regulation of two transcription factors, where (b) corresponds to noncompetitive binding whereas (c) to competitive binding

Figure 2.  Influence of parameter $\gamma$ on the real and imaginary parts of the root of the characteristic equation $f(\lambda)=0$:non-competition model. (a)$N=3$, $M=8$(corresponding to the case of $N<M+1$); (b)$N=3$, $M=2$ (corresponding to the case of$N=M+1$; (C)$N=5$, $M=2$) (corresponding to the case of $N>M+1$); (d) Bifurcation diagram of $x_3$ versus $\gamma$ for $N=3$, $M=2$.Here green solid and red dashed lines represent stable and unstable steady states respectively, and the symbol 'HB' represents the Hopf bifurcation point. Other parameter values are set as $\alpha_n=\alpha_p=3 $, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=\alpha$, and $K_1=K_2=1$

Figure 3.  The occurrence of oscillations in the system of interlocked multi-node positive and negative loops. (a, c) correspond to the noncompetitive model whereas (b, d) to the competitive model. (a, b) show time series of component $x_5$, where the inset is a phase trajectory in the $(x_4,x_5)$ plane. (c, d) show both a stable region for the fixed point (corresponding to 'nonoscillation' indicated in the diagram) and an oscillatory region (corresponding to 'oscillation' indicated in the diagram) in the $(N,M)$ plane, where the green das line is the border of the two regions. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=1$, and $K_1=0.5$, and $K_2=4$

Figure 4.  The effect of the node number of the negative feedback loop on the amplitude and period of oscillations, where $M=2$.(a, c) correspond to the noncompetitive model, (b, d) correspond to the competitive model. (a, b) for oscillation amplitude, (c, d) for oscillation period. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=1$, and $K_2=0.5$

Figure 5.  The effect of the node number of the positive feedback loop on the amplitude and period of oscillations, where $N=5$.(a, c) correspond to the noncompetitive model, (b, d) correspond to the competitive model. (a, b) for oscillation amplitude, (c, d) is related to period. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\gamma_n=\gamma_p=1$, and $K_2=0.5$

Figure 6.  Three dimensional pseudo-diagram in the $(\gamma,K)$ plane, where the color bar represents oscillation amplitude, where parameter $N$ is fixed at $N=3$.(a, b, c, d) correspond to non-competitive binding with $M=1,2,6,8$ from left to right (corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely); (e, f, g, h) correspond to competitive binding with $M=1,2,6,8$ from left to right(corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely). Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$

Figure 7.  Three dimensional pseudo-diagram in the $(\gamma,K)$ plane, where the color bar represents oscillation peroid, where parameter $N$ is fixed at $N=3$. (a, b, c, d) correspond to non-competitive binding with $M=1,2,6,8$ from left to right (corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely); (e, f, g, h) correspond to competitive binding with $M=1,2,6,8$ from left to right(corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely). Other parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$

Figure 8.  Comparison between influences of NFL node number and time delay on oscillating region, where the number of PFL nodes, $M$ is fixed at $M=2$. (a) displays the dependence of the maximum oscillation amplitude on the ratio of $K_1/K_2$; (b) shows the oscillation region of in the $(K_1,K_2)$ plane; (c) shows the oscillation region of $\tau=1.04$ also in the $(K_1,K_2)$ plane. Other parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$