Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales

Abstract

We investigate a class of stochastic chemical reaction networks with \(n{\ge }1\) chemical species \(S_1\), ..., \(S_n\), and whose complexes are only of the form \(k_iS_i\), \(i{=}1\),..., 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.