Convergence of stochastic gene networks to hybrid piecewise deterministic processes

TL;DR

This paper rigorously analyzes how complex stochastic gene network models simplify into various types of hybrid piecewise deterministic processes depending on system scales, aiding in understanding biological systems.

Contribution

It provides a rigorous mathematical framework for the convergence of multiscale stochastic gene networks to four distinct types of hybrid processes, clarifying their asymptotic behavior.

Findings

Four types of limit processes identified and justified

Convergence results applicable to simplifying gene network models

Framework aids in understanding biological stochastic dynamics

Abstract

We study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes. Depending on the time and concentration scales of the system we distinguish four types of limits: continuous piecewise deterministic processes (PDP) with switching, PDP with jumps in the continuous variables, averaged PDP, and PDP with singular switching. We justify rigorously the convergence for the four types of limits. The convergence results can be used to simplify the stochastic dynamics of gene network models arising in molecular biology.