A Linear Programming Approach to Dynamical Equivalence, Linear Conjugacy, and the Deficiency One Theorem

TL;DR

This paper introduces a mixed-integer linear programming method to identify whether a chemical reaction system can be represented by a network satisfying the Deficiency One Theorem, aiding in the analysis of steady states.

Contribution

It develops a computational framework to find dynamically equivalent or linearly conjugate networks that meet the Deficiency One Theorem criteria, extending previous work on weakly reversible, deficiency-zero systems.

Findings

Successfully determines network representations satisfying the Deficiency One Theorem.

Extends computational methods to broader classes of reaction networks.

Facilitates analysis of steady states in complex chemical systems.

Abstract

The well-known Deficiency One Theorem gives structural conditions on a chemical reaction network under which, for any set of parameter values, the steady states of the corresponding mass action system may be easily characterized. It is also known, however, that mass action systems are not uniquely associated with reaction networks and that some representations may satisfy the Deficiency One Theorem while others may not. In this paper we present a mixed-integer linear programming framework capable of determining whether a given mass action system has a dynamically equivalent or linearly conjugate representation which has an underlying network satisfying the Deficiency One Theorem. This extends recent computational work determining linearly conjugate systems which are weakly reversible and have a deficiency of zero.