A SageMath Package for Elementary and Sign Vectors with Applications to Chemical Reaction Networks

TL;DR

This paper introduces a SageMath package for computing elementary and sign vectors of real subspaces, with applications to analyzing conditions for complex-balanced equilibria in chemical reaction networks.

Contribution

The paper presents a new SageMath package that efficiently computes elementary and sign vectors, and applies it to chemical reaction network analysis.

Findings

The package computes elementary vectors from maximal minors.

Sign vectors help determine equilibria conditions.

Algorithmic checks for complex-balanced equilibria are demonstrated.

Abstract

We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples.