Stochastic activation in a genetic switch model

We study a biological autoregulation process, involving a protein that enhances its own transcription, in a parameter region where bistability would be present in the absence of fluctuations. We calculate the rate of fluctuation-induced rare transitions between locally stable states using a path integral formulation and Master and Chapman-Kolmogorov equations. As in simpler models for rare transitions, the rate has the form of the exponential of a quantity S-0 (a "barrier") multiplied by a prefactor eta. We calculate S-0 and eta first in the bursting limit (where the ratio gamma of the protein and mRNA lifetimes is very large). In this limit, the calculation can be done almost entirely analytically, and the results are in good agreement with simulations. For finite gamma numerical calculations are generally required. However, S-0 can be calculated analytically to first order in 1/gamma, and the result agrees well with the full numerical calculation for all gamma > 1. Employing a method used previously on other problems, we find we can account qualitatively for the way the prefactor eta varies with gamma, but its value is 15-20% higher than that inferred from simulations.