Individual molecules dynamics in reaction network models

TL;DR

This paper develops a computationally efficient method to approximate and analyze the dynamics of individual molecules within stochastic reaction networks, enabling detailed single-molecule and system-level insights.

Contribution

It introduces a novel approximation approach based on large volume limits for tracking individual molecules and their collective behavior in reaction networks.

Findings

Efficient approximation of single-molecule trajectories.

Explicit error bounds for the approximation.

Parallelizable simulation technique for reaction networks.

Abstract

In a stochastic reaction network setting we consider the problem of tracking the fate of individual molecules. We show that using the classical large volume limit results, we may approximate the dynamics of a single tracked molecule in a simple and computationally efficient way. We give examples on how this approach may be used to obtain various characteristics of single-molecule dynamics (for instance, the distribution of the number of infections in a single individual in the course of an epidemic or the activity time of a single enzyme molecule). Moreover, we show how to approximate the overall dynamics of species of interest in the full system with a collection of independent single-molecule trajectories, and give explicit bounds for the approximation error in terms of the reaction rates. This approximation, which is well defined for all times, leads to an efficient and fully parallelizable simulation technique for which we provide some numerical examples.