  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.
