Two-scale large deviations for chemical reaction kinetics through second quantization path integral

TL;DR

This paper develops a formal framework using second-quantization path integrals to analyze large deviations in two-scale chemical reaction systems, revealing different stochastic behaviors across regimes.

Contribution

It introduces a novel second-quantization path integral approach to derive large deviation principles for multi-scale chemical reactions, extending classical results.

Findings

Identifies three distinct large deviation regimes with the same mean-field limit.

Shows different chemical Langevin approximations in each regime.

Provides insights into single-molecule kinetics and multi-scale systems.

Abstract

Motivated by the study of rare events for a typical genetic switching model in systems biology, in this paper we aim to establish the general two-scale large deviations for chemical reaction systems. We build a formal approach to explicitly obtain the large deviation rate functionals for the considered two-scale processes based upon the second-quantization path integral technique. We get three important types of large deviation results when the underlying two times scales are in three different regimes. This is realized by singular perturbation analysis to the rate functionals obtained by path integral. We find that the three regimes possess the same deterministic mean-field limit but completely different chemical Langevin approximations. The obtained results are natural extensions of the classical large volume limit for chemical reactions. We also discuss its implication on the single-molecule Michaelis-Menten kinetics. Our framework and results can be applied to understand general multi-scale systems including diffusion processes.