  In a stochastic process, noise often modifies the picture offered by the mean
field dynamics. In particular, when there is an absorbing state, the noise
erases a stable fixed point of the mean field equation from the stationary
distribution, and turns it into a transient peak. We make a quantitative
analysis of this effect for a simple genetic regulatory network with positive
feedback, where the proteins become extinct in the presence of stochastic
noise, contrary to the prediction of the deterministic rate equation that the
protein number converges to a non-zero value. We show that the transient peak
appears near the stable fixed point of the rate equation, and the extinction
time diverges exponentially as the stochastic noise approaches zero. We also
show how the baseline production from the inactive gene ameliorates the effect
of the stochastic noise, and interpret the opposite effects of the noise and
the baseline production in terms of the position shift of the unstable fixed
point. 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 baseline production dominates
over that of the stochastic noise, leading to the protection from the
catastrophic rare event of protein extinction.
