  Multiple phenotypic states often arise in a single cell with different
gene-expression states that undergo transcription regulation with positive
feedback. Recent experiments have shown that at least in E. coli, the gene
state switching can be neither extremely slow nor exceedingly rapid as many
previous theoretical treatments assumed. Rather it is in the intermediate
region which is difficult to handle mathematically.Under this condition, from a
full chemical-master-equation description we derive a model in which the
protein copy-number, for a given gene state, follow a deterministic mean-field
description while the protein synthesis rates fluctuate due to stochastic
gene-state switching. The simplified kinetics yields a nonequilibrium landscape
function, which, similar to the energy function for equilibrium fluctuation,
provides the leading orders of fluctuations around each phenotypic state, as
well as the transition rates between the two phenotypic states. This rate
formula is analogous to Kramers theory for chemical reactions. The resulting
behaviors are significantly different from the two limiting cases studied
previously.
