  A rigorous mathematical framework for analyzing the chemical master equation
(CME) with bistability, based on the theory of large deviation, is proposed.
Using a simple phosphorylation-dephosphorylation cycle with feedback as an
example, we show that a nonequilibrium steady-state (NESS) phase transition
occurs in the system which has all the characteristics of classic equilibrium
phase transition: Maxwell construction, discontinuous fraction of
phosphorylation as a function of the kinase activity, and Lee-Yang's zero for
the generating function. The cusp in nonlinear bifurcation theory matches the
tricritical point of the phase transition. The mathematical analysis suggests
three distinct time scales, and related mathematical descriptions, of (i)
molecular signaling, (ii) biochemical network dynamics, and (iii) cellular
evolution. The (i) and (iii) are stochastic while (ii) is deterministic.
