  In this paper we study noise-induced bistability in a specific circuit with
many biological implications, namely a single-step enzymatic cycle described by
Michaelis Menten equations with quasi-steady state assumption. We study the
system both with a Master Equation formalism, and with the Fokker-Planck
continuous approximation, characterizing the conditions in which the continuous
approach is a good approximation of the exact discrete model. An analysis of
the stationary distribution in both cases shows that bimodality can not occur
in such a system. We discuss which additional requirements can generate
stochastic bimodality, by coupling the system with a chemical reaction
involving enzyme production and turnover. This extended system shows a bistable
behaviour only in specific parameter windows depending on the number of
molecules involved, providing hints about which should be a feasible system
size in order that such a phenomenon could be exploited in real biological
systems.
