Growth and depletion in linear stochastic reaction networks

Peter Nandori, Lai Sang Young

Research output: Contribution to journalArticlepeer-review


This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on “enzymes,” which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω-limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the “phase transition” that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow–fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks.

Original languageEnglish (US)
Article numbere2214282119
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number51
StatePublished - Dec 20 2022


  • depletion
  • exponential growth
  • mean-field approximation
  • reaction networks

ASJC Scopus subject areas

  • General


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