Growth and depletion in linear stochastic reaction networks

Abstract

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.--- Significance Many biological, chemical, and social phenomena that involve the interaction of large numbers of substances or agents are modeled as reaction networks. Mathematically, reaction networks are high-dimensional dynamical systems, deterministic or stochastic. Numerical simulations have revealed a rich and diverse landscape, one that poses a nontrivial challenge for analysts. In this paper, we consider a class of linear stochastic reaction networks designed to capture certain salient characteristics of real biological networks, yet sufficiently idealized to permit analytical approaches. We present rigorous results on two network phenomena: exponential growth of network size and depletion of one of the substances involved. Both phenomena occur naturally and are known to have biological consequences.