  Dual phospho/dephosphorylation cycles, as well as covalent
enzymatic-catalyzed modifications of substrates, are widely diffused within
cellular systems and are crucial for the control of complex responses such as
learning, memory and cellular fate determination. Despite the large body of
deterministic studies and the increasing work aimed to elucidate the effect of
noise in such systems, some aspects remain unclear. Here we study the
stationary distribution provided by the two-dimensional Chemical Master
Equation for a well known model of a two step phospho/dephosphorylation cycle
using the quasi steady state approximation of the enzymatic kinetics. Our aim
is to analyze the role of fluctuations and the molecules distribution
properties in the transition to a bistable regime. When detailed balance
conditions are satisfied it is possible to compute equilibrium distributions in
a closed and explicit form. When detailed balance is not satisfied, the
stationary non-equilibrium state is strongly influenced by the chemical fluxes.
In the last case, we show how the Helmholtz decomposition of the external field
associated to the generation and recombination transition rates, into a
conservative and a rotational (irreversible) part, allows to compute the
stationary distribution via a perturbative approach . For a finite number of
molecules, there exists diffusion dynamics in a macroscopic region of the state
space, where a relevant transition rate between the two critical points is
observed. Further, the stationary distribution function can be approximated by
the solution of a Fokker-Planck equation. We illustrate the theoretical results
using several numerical simulations.
