Stationary Distributions and Condensation in Autocatalytic Reaction Networks

Linard Hoessly,Christian Mazza

doi:10.1137/18m1220340

Abstract

We investigate a broad family of non weakly reversible stochastically modeled reaction networks (CRN), by looking at their steady-state distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first give explicitly product-form steady-state distributions for a class of non weakly reversible autocatalytic CRN of arbitrary deficiency. Examples of interest in statistical mechanics (inclusion process), life sciences and robotics (collective decision making in ant and robot swarms) are provided. The product-form nature of the steady-state then enables the study of condensation in particle systems that are generalizations of the inclusion process.

Highlights

Understanding the dynamics of reaction networks (CRNs) is of central importance in a variety of contexts in life sciences and complex systems, including molecular and cellular systems biology, which are some of the most vital areas in bioscience
The first is realized as a vector with concentrations of each molecular species as a state space governed by a system of ordinary differential equations (ODEs), whereas the second is described by a continuous-time Markov chain acting on discrete molecular counts of each molecular species
The stochastic model is used for cases with low molecular numbers where stochasticity is essential for the proper description of the dynamics

Summary

Introduction

Understanding the dynamics of reaction networks (CRNs) is of central importance in a variety of contexts in life sciences and complex systems, including molecular and cellular systems biology, which are some of the most vital areas in bioscience. Reaction networks, mass-action system, product-form stationary distributions, Markov process, inclusion process, condensation Product-form stationary distributions πN for a large class of autocatalytic mass preserving CRNs, including the models in [43, 40, 6, 5, 33, 34] (which were studied via simulation and approximations) and generalizing results of [28, 24].

Results

Conclusion