Relation between the convective field and the stationary probability distribution of chemical reaction networks

TL;DR

This paper explores how the convective field derived from the chemical Fokker-Planck equation relates to the stationary probability distribution in chemical reaction networks, providing insights into bifurcations and system size effects.

Contribution

It establishes a connection between the convective field and the stationary distribution, offering an efficient method to analyze bifurcations and probability maxima in chemical systems.

Findings

Convective field fixed points correspond to distribution extrema.

Bifurcations in the convective field relate to phenomenological bifurcations.

System size influences the number and shape of probability maxima.

Abstract

We investigate the relation between the stationary probability distribution of chemical reaction systems and the convective field derived from the chemical Fokker-Planck equation (CFPE) by comparing predictions of the convective field to the results of stochastic simulations based on Gillespie's algorithm. The convective field takes into account the drift term of the CFPE and the reaction bias introduced by the diffusion term. For one-dimensional systems, fixed points and bifurcations of the convective field correspond to extrema and phenomenological bifurcations of the stationary probability distribution whenever the CFPE is a good approximation to the stochastic dynamics. This provides an efficient way to calculate the effect of system size on the number and location of probability maxima and their phenomenological bifurcations in parameter space. For two-dimensional systems, we study models that have saddle-node and Hopf bifurcations in the macroscopic limit. Here, the existence of two stable fixed points of the convective field correlates either with two peaks of the stationary probability distribution, or with a peak and a shoulder. In contrast, a Hopf bifurcation that occurs in the convective field for decreasing system size is not accompanied by the onset of a crater-shaped probability distribution; decreasing system size rather destroys craters and replaces them by local maxima.