Time delay effects of stochastic resonance induced by multiplicative periodic signal in the gene transcriptional regulatory model

Based on the universal transcriptional regulatory model proposed by Smolen et al.in 1998, we have studied the combined effects of cross-correlation noise, multiplicative periodic signal and time delay through modifying the ordinary differential equation with time evolution in gene transcriptional regulatory system. The expression of signal-to-noise ratio (SNR) is derived by applying the two-state theory in adiabatic limit and approximate analytical method for small time delay. The effects of multiplicative noise intensity, additive noise intensity, cross-correlation noise intensity and time delay on SNR are discussed by numerical calculation and graphical analysis. The results indicate that when the SNR is a function of multiplicative noise intensity, the peak value of the SNR rises as time delay increases; when SNR is a function of additive noise intensity, the value of SNR ascends as time delay increases; when SNR is a function of cross-correlation noise intensity, the peak value of SNR goes up with time delay increased. So when SNR is the function of multiplicative noise intensity, additive noise intensity and cross-correlation noise intensity respectively, time delay enhances the stochastic resonance.