Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays.

To understand how a gene regulatory network functioning as an oscillator is built, a genetic regulatory network with two transcriptional delays is investigated. We show by mathematical analysis and simulation that autorepression of mRNA and protein can provide a mechanism for the intracellular oscillator. Based on the linear stability approach and bifurcation theory, sufficient conditions for the oscillation of the genetic networks are derived, and critical values of Hopf bifurcation are assessed. In particular, the genetic network can exhibit Hopf bifurcation(oscillation appears) as the sum of delays or transcriptional rate passes through some critical values. Moreover, the robustness of amplitudes against change in delay can also be obtained from the delayed genetic network; period of oscillation increases with the total time delay in an almost linear way. While it is exactly opposite for transcriptional rate, the amplitude of oscillations always increases as the transcriptional rate increases; the robustness of period against change in the transcriptional rate occurs. Some simple genetic regulatory networks are used to study the impact of delays and transcriptional rate on the system dynamics where there are delays.