Stochastic Reaction Networks Within Interacting Compartments with Content-Dependent Fragmentation

TL;DR

This paper extends stochastic reaction network models to include content-dependent compartment fragmentation, providing new conditions for system stability and non-explosivity in biologically relevant scenarios.

Contribution

It introduces a framework for content-dependent compartment dynamics and establishes conditions for non-explosivity and positive recurrence, advancing the mathematical understanding of compartmentalized systems.

Findings

Content-dependent fragmentation can lead to non-explosive behavior under certain conditions.

The previous explosivity characterization does not hold in this new setting.

Linear Lyapunov functions help establish stability criteria.

Abstract

Stochastic reaction networks with mass-action kinetics provide a useful framework for understanding processes -- biochemical and otherwise -- in homogeneous environments. However, cellular reactions are often compartmentalized, either at the cell level or within cells, and hence non-homogeneous. We investigate a model of compartmentalization in which the rate of fragmentation of a compartment depends on the abundance of some designated species inside that compartment. The particular model of study is part of a general framework for compartmentalized chemistry with dynamic compartments that was proposed in (Duso and Zechner, PNAS, 2020). This paper builds on (Anderson and Howells, Bull. Math. Biol., 2023) where the special case where the compartment dynamics do not depend on their contents was studied mathematically. In particular, we demonstrate that the explosivity characterization from (Anderson and Howells, Bull. Math. Biol., 2023) fails in this setting and provide new sufficient conditions for non-explosivity and positive recurrence, under the assumption that the underlying CRN admits a linear Lyapunov function. These results extend the theoretical foundation for modeling content-mediated compartment dynamics, with implications for systems such as cell division and intracellular transport.