Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model

TL;DR

This paper investigates stochastic chemical systems near a SNIPER bifurcation, revealing oscillatory behavior absent in the mean-field model, and derives explicit formulas for oscillation periods using Fokker-Planck analysis.

Contribution

It introduces a Fokker-Planck based framework to analyze stochastic effects near SNIPER bifurcations in chemical systems, including explicit period formulas.

Findings

Stochastic systems oscillate even when mean-field models do not.

Numerical solutions of the Fokker-Planck equation are provided.

Explicit formulas for oscillation periods are derived.

Abstract

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker-Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size.