  The behaviour of a high dimensional stochastic system described by a Chemical
Master Equation (CME) depends on many parameters, rendering explicit simulation
an inefficient method for exploring the properties of such models. Capturing
their behaviour by low-dimensional models makes analysis of system behaviour
tractable. In this paper, we present low dimensional models for the
noise-induced excitable dynamics in Bacillus subtilis, whereby a key protein
ComK, which drives a complex chain of reactions leading to bacterial
competence, gets expressed rapidly in large quantities (competent state) before
subsiding to low levels of expression (vegetative state). These rapid reactions
suggest the application of an adiabatic approximation of the dynamics of the
regulatory model that, however, lead to competence durations that are incorrect
by a factor of 2. We apply a modified version of an iterative functional
procedure that faithfully approximates the time-course of the trajectories in
terms of a 2-dimensional model involving proteins ComK and ComS. Furthermore,
in order to describe the bimodal bivariate marginal probability distribution
obtained from the Gillespie simulations of the CME, we introduce a tunable
multiplicative noise term in a 2-dimensional Langevin model whose stationary
state is described by the time-independent solution of the corresponding
Fokker-Planck equation.
