Fast reactions with non-interacting species in stochastic reaction networks

TL;DR

This paper develops a method to simplify stochastic reaction networks with non-interacting species by fast degradation, enabling easier analysis of complex biochemical systems through a two-time-scale reduction approach.

Contribution

It introduces a novel reduction technique for stochastic reaction networks with non-interacting species, based on a two-time-scale embedding and structural conditions.

Findings

Derived simplified reaction networks that approximate the original in the small-parameter limit.

Provided sufficient conditions for reduction applicability in both homogeneous and time-varying settings.

Analyzed examples demonstrating the effectiveness of the reduction method.

Abstract

We consider stochastic reaction networks modeled by continuous-time Markov chains. Such reaction networks often contain many reactions, potentially occurring at different time scales, and have unknown parameters (kinetic rates, total amounts). This makes their analysis complex. We examine stochastic reaction networks with non-interacting species that often appear in examples of interest (e.g. in the two-substrate Michaelis Menten mechanism). Non-interacting species typically appear as intermediate (or transient) chemical complexes that are depleted at a fast rate. We embed the Markov process of the reaction network into a one-parameter family under a two time-scale approach, such that molecules of non-interacting species are degraded fast. We derive simplified reaction networks where the non-interacting species are eliminated and that approximate the scaled Markov process in the limit as the parameter becomes small. Then, we derive sufficient conditions for such reductions based on the reaction network structure for both homogeneous and time-varying stochastic settings, and study examples and properties of the reduction.