Discreteness-induced Transition in Catalytic Reaction Networks

TL;DR

This paper reports a transition in catalytic reaction networks caused by small molecule numbers, leading to large fluctuations and deviations from continuum behavior, explained by reaction termination due to molecule deficiency.

Contribution

It introduces a novel transition phenomenon in catalytic networks driven by molecule discreteness, with a quantitative analysis of critical molecule numbers and universal scaling laws.

Findings

Transition characterized by large fluctuations and species extinction.

Critical molecule number depends on species and reaction counts.

Universal scaling of reaction rates with system parameters.

Abstract

Drastic change in dynamics and statistics in a chemical reaction system, induced by smallness in the molecule number, is reported. Through stochastic simulations for random catalytic reaction networks, transition to a novel state is observed with the decrease in the total molecule number N, characterized by: i) large fluctuations in chemical concentrations as a result of intermittent switching over several states with extinction of some molecule species and ii) strong deviation of time averaged distribution of chemical concentrations from that expected in the continuum limit, i.e., $N \to \infty$. The origin of transition is explained by the deficiency of molecule leading to termination of some reactions. The critical number of molecules for the transition is obtained as a function of the number of molecules species M and that of reaction paths K, while total reaction rates, scaled properly, are shown to follow a universal form as a function of NK/M.