Non-Markovian Effect on Gene Transcriptional Systems

The time delays in both synthesis and degradation reactions, reflecting the non-Markovian behavior, are introduced in the stochastic gene transcriptional dynamics. The effects of the time delays on the stationary probability distribution, mean first passage time and stochastic resonance are discussed in detail based on the delayed stochastic differential equation and the corresponding delay Fokker-Planck equation. The time delays in synthesis reactions and in degradation reactions play a completely opposite role. The time delay in synthesis (degradation) reaction enhances (reduces) the mean first passage time, and tends to reduce (enhance) the signal-to-noise ratio. Finally, the effect of Gauss-distributed time delays on the transcriptional system is explored to test whether the previous approximation of employing a certain delay time works well or not.