Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales

TL;DR

This paper analyzes stochastic chemical reaction networks with hierarchical timescales, showing that under large scaling, certain scaled variables and occupation measures converge to deterministic limits, revealing the system's multiscale behavior.

Contribution

It introduces a hierarchical timescale analysis for stochastic CRNs with mass action kinetics, establishing convergence results for scaled variables and measures as the scaling parameter grows.

Findings

Scaled variables with $k_i=1$ converge to deterministic limits

Occupation measures of other coordinates also converge as $N$ increases

Hierarchy of timescales determines the limiting behavior

Abstract

We investigate a class of stochastic chemical reaction networks with $n{\ge}1$ chemical species $S_1$, \ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.