Quantitative Analysis of a Transient Dynamics of a Genetic Regulatory Network

Abstract

We make a quantitative analysis of a simple genetic regulatory network with positive feedback, and show that the proteins become extinct with certainty in the stationary distribution when the stochastic noise is present, contrary to the solution of the deterministic rate equation where the protein number is predicted to converge to a non-zero value. We show that as the stochastic noise approaches zero, the extinction time diverges exponentially. We also show that the leaky expression from the inactive gene ameliorates the effect of the stochastic noise. The order of magnitude estimates using biological parameters suggest that for a real gene regulatory network, the stochastic noise is sufficiently small so that not only is the extinction time much larger than biologically relevant time-scales, but also the effect of the leaky expression dominates over that of the stochastic noise, leading to the protection from the catastrophic rare event of protein extinction.