# Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks

## TL;DR

This paper explores various forms of detailed balance in stochastic chemical reaction networks, establishing their relationships, implications for global potential functions, and providing conditions for stochastic detailed balance with applications to high-dimensional systems.

## Contribution

It clarifies the equivalence and differences among four types of detailed balance conditions and introduces a new sufficient condition for stochastic detailed balance in complex networks.

## Key findings

- Four types of detailed balance are equivalent under certain conditions.
- Local detailed balance implies the existence of a global potential function.
- A new sufficient condition for stochastic detailed balance is proposed, enabling the construction of multistable high-dimensional networks.

## Abstract

Detailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays the central role in the large deviation theory and the Freidlin-Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.