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Interlocked multi-node positive and negative feedback loops facilitate oscillations

  • * Corresponding author: Tianshou Zhou

    * Corresponding author: Tianshou Zhou
This work was partially supported by the National Nature Science Foundation of China under Grant NO.91530320(T.Z.) and Grant NO.11775314(T.Z.) and 973 project of Science and Technology department of China under Grant NO.2014CB964703(T.Z.).
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  • Positive and negative feedback loops in biological regulatory networks appear often in a multi-node manner since regulatory processes are in general multi-step. Although it is well known that interlocked positive and negative feedback loops (iPNFLs) can generate sustained oscillations, how the number of nodes in each loop affects the oscillations remains elusive. By analyzing a model of iPNFLs with multiple nodes, we find that the node number of the negative loop mainly plays a role of amplifying oscillation amplitudes whereas that of the positive loop mainly plays a role of reducing oscillatory regions, both depending on the (competitive or noncompetitive) way of interaction between the two loops. We also find that given an iPNFL network of the same structure, the noncompetitive model is more likely to produce large-amplitude oscillations than the competitive model. These results not only indicate that multi-node iPNFLs are an effective mechanism of promoting oscillations but also are helpful for the design of synthetic oscillators.

    Mathematics Subject Classification: Primary: 92B05, 92C42; Secondary: 93C15.

    Citation:

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  • 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 $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$

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