  A simple stochastic model of a self regulating gene that displays bistable
switching is analyzed. While on, a gene transcribes mRNA at a constant rate.
Transcription factors can bind to the DNA and affect the gene's transcription
rate. Before an mRNA is degraded, it synthesizes protein, which in turn
regulates gene activity by influencing the activity of transcription factors.
Protein is slowly removed from the system through degradation. Depending on how
the protein regulates gene activity, the protein concentration can exhibit
noise induced bistable switching. An asymptotic approximation of the mean
switching rate is derived that includes the pre exponential factor, which
improves upon a previously reported logarithmically accurate approximation.
With the improved accuracy, a uniformly accurate approximation of the
stationary probability density, describing the gene, mRNA copy number, and
protein concentration is also obtained.
