A review of the deterministic and diffusion approximations for stochastic chemical reaction networks

TL;DR

This review compares deterministic and diffusion approximations for stochastic chemical reaction networks, highlighting how diffusion models can capture complex behaviors missed by deterministic models and serve as valuable theoretical tools.

Contribution

It advocates for the diffusion approximation as a meaningful theoretical approach and provides explicit constructions controlling the approximation error.

Findings

Diffusion approximation captures qualitative properties missed by deterministic models.

Explicit construction of process and approximation controls trajectory distance.

Discussion of limitations and future directions for diffusion methods.

Abstract

This work reviews deterministic and diffusion approximations of the stochastic chemical reaction networks and explains their applications. We discuss the added value the diffusion approximation provides for systems with different phenomena, such as a deficiency and a bistability. It is advocated that the diffusion approximation can be considered as an alternative theoretical approach to study the reaction networks rather than a simulation shortcut. We discuss two examples in which the diffusion approximation is able to catch qualitative properties of reaction networks that the deterministic model misses. We provide an explicit construction of the original process and the diffusion approximation such that the distance between their trajectories is controlled and demonstrate this construction for the examples. We also discuss the limitations and potential directions of the developments.