Stability and bifurcation of genetic regulatory networks with small RNAs and multiple delays

In this paper, we study the stability, bifurcation and periodic oscillation of two-gene regulatory networks mediated by small RNAs (sRNAs) with multiple delays. We first show the effect of sRNAs on the stability and bifurcation of genetic regulatory networks. Then we present sufficient conditions for the local stability of two-gene genetic regulatory networks in the parameter space, and assess critical values of the Hopf bifurcation. Although such networks may have multiple delays, sRNAs, positive feedbacks and different connection strengths among two genes, their stability and bifurcation depend on the sum of all time delays among all elements (including both mRNAs and proteins). Furthermore, the period of oscillations increases with the time delay, and in the case of larger delay, the amplitude of oscillations is robust against the change in the delay. Two examples are employed to illustrate the theorems developed in this study.