Optimal Fluctuations for Nonlinear Chemical Reaction Systems with General Rate Law

TL;DR

This paper explores the nature of optimal fluctuations in nonlinear chemical reaction systems, introducing a new prehistory probability approach that links these fluctuations to time-reversed processes, with implications for understanding rare events in biochemistry.

Contribution

It proposes an alternative prehistory probability framework for analyzing optimal fluctuations in chemical reactions, extending the prehistorical approach to Langevin dynamics and revealing a connection with time reversal.

Findings

Derived evolution laws for prehistory probabilities in both non-stationary and stationary cases.

Proved law of large numbers and central limit theorem for reversed processes.

Demonstrated the relationship between optimal fluctuations and time-reversal symmetry.

Abstract

This paper investigates optimal fluctuations for chemical reaction systems with N species, M reactions, and general rate law. In the limit of large volume, large fluctuations for such models occur with overwhelming probability in the vicinity of the so-called optimal path, which is a basic consequence of the Freidlin-Wentzell theory, and is vital in biochemistry as it unveils the almost deterministic mechanism concealed behind rare noisy phenomena such as escapes from the attractive domain of a stable state and transitions between different metastable states. In this study, an alternative description for optimal fluctuations is proposed in both non-stationary and stationary settings by means of a quantity called prehistory probability in the same setting, respectively. The evolution law of each of them is derived, showing their relationship with the time reversal of a specified family of probability distributions respectively. The law of large numbers and the central limit theorem for the reversed processes are then proved. In doing so, the prehistorical approach to optimal fluctuations for Langevin dynamics is naturally generalized to the present case, thereby suggesting a strong connection between optimal fluctuations and the time reversal of the chemical reaction model.