A spatial measure-valued model for chemical reaction networks in heterogeneous systems

TL;DR

This paper introduces a new measure-valued process to model chemical reaction networks in spatially heterogeneous systems, capturing both reaction dynamics and molecular movement, with novel asymptotic limits and a hybrid stochastic-deterministic process.

Contribution

It presents a novel measure-valued process for spatial chemical reactions, including new asymptotic limits and a hybrid stochastic-deterministic evolution model.

Findings

Derived asymptotic limits for the process

Identified a new spatial random evolution process

Established conditions for measure-valued piecewise deterministic Markov processes

Abstract

We propose a novel measure valued process which models the behaviour of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species and continuous movement of molecules in space. Reactions rates at a spatial location are proportional to the mass of different species present locally and to a location specific chemical rate, which may be a function of the local or global species mass as well. We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types. In particular, when the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.