Deficiency, Kinetic Invertibility, and Catalysis in Stochastic Chemical Reaction Networks

TL;DR

This paper investigates the conditions under which the dual master equation of stochastic chemical reaction networks satisfies the law of mass-action, revealing that deficiency-zero networks uniquely allow for kinetic invertibility, and explores the catalytic properties related to deficiency.

Contribution

It establishes a topological criterion based on deficiency for the invertibility of the dual master equation in stochastic chemical networks and analyzes catalytic networks' deficiency properties.

Findings

Dual master equation satisfies mass-action only for deficiency-zero networks.

Most networks are non-invertible due to deficiency.

Catalytic networks are not deficiency-zero when driven out of equilibrium.

Abstract

Stochastic chemical processes are described by the chemical master equation satisfying the law of mass-action. We first ask whether the dual master equation, which has the same steady state as the chemical master equation, but with inverted reaction currents, satisfies the law of mass-action, namely, still describes a chemical process. We prove that the answer depends on the topological property of the underlying chemical reaction network known as deficiency. The answer is yes only for deficiency-zero networks. It is no for all other networks, implying that their steady-state currents cannot be inverted by controlling the kinetic constants of the reactions. Hence, the network deficiency imposes a form of non-invertibility to the chemical dynamics. We then ask whether catalytic chemical networks are deficiency-zero. We prove that the answer is no when they are driven out of equilibrium due to the exchange of some species with the environment.