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 and, hence, 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.
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The law of mass-action is satisfied for elementary reactions in ideal dilute solutions, that is, chemical species are noninteracting and there is a species, called the solvent, which is not involved in the chemical reactions and is much more abundant than all the other species. If reactions are not elementary or the chemical species interact, reaction rates might not satisfy the law of mass-action. In this work, we consider only elementary reactions in ideal dilute solutions.
The kinetic non-invertibility of catalytic CRNs implies that there are no kinetic constants that can generate the dual process, where the net steady-state currents of all reactions are inverted. However, this does not exclude that the net steady-state currents of some specific reactions (e.g., the reaction producing a desired species) can be inverted in catalytic CRNs by controlling the kinetic constants.
