Abstract: More and more experiments show that microRNAs can regulate gene expression by stimulating degradation of mRNA or repression of translation of mRNA. In this paper, we incorporate the microRNA into a previous mathematical model of gene expression through forming a microRNA-induced silencing complex(RISC). Our findings demonstrate the dynamical behavior of the constructed system. By Hopf theories, we derive the theoretical results of globally asymptotical stability and provide the sufficient conditions for the oscillation of the simple gene regulatory system, and by numerical simulation further demonstrate how the amplitudes against the change of delay in the gene regulatory network. 

Key words: gene regulatory network, microRNA, time delay, Hopf bifurcation, oscillation