Individual molecules dynamics in reaction network models

Abstract

In a stochastic reaction network setting we define a subset of species as 'trackable' if we can consistently follow the fate of its individual molecules. We show that using the classical large volume limit results, we may approximate the dynamics of a single molecule of trackable species 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 trackable species 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 example.