Dynamical Properties of Discrete Reaction Networks

TL;DR

This paper provides efficient algebraic methods to characterize key dynamical properties like irreducibility and recurrence in Discrete Reaction Networks, which are applicable to stochastic models regardless of kinetics details.

Contribution

It introduces necessary and sufficient algebraic conditions, verifiable via linear programming, for analyzing irreducibility and recurrence in DRNs, bridging discrete and stochastic reaction network models.

Findings

Conditions for irreducibility and recurrence are established.

Linear programming can verify these conditions efficiently.

Results apply to large copy number and general cases.

Abstract

Reaction networks are commonly used to model the evolution of populations of species subject to transformations following an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modelling the underlying discrete nondeterministic transitions of stochastic models of reactions networks. In that sense, any proof of non-reachability in DRNs directly applies to any concrete stochastic models, independently of kinetics laws and constants. Moreover, if stochastic kinetic rates never vanish, reachability properties are equivalent in the two settings. The analysis of two global dynamical properties of DRNs is addressed: irreducibility, i.e., the ability to reach any discrete state from any other state; and recurrence, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The obtained necessary and sufficient conditions involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.