Positive Steady-State Varieties of Small Chemical Reaction Networks

TL;DR

This paper classifies positive steady-state varieties of small chemical reaction networks using algebraic geometry, providing a foundation for analyzing larger networks and simplifying previous criteria.

Contribution

It offers a systematic classification of positive steady-state varieties for 2-species, 2-reaction networks, simplifying existing criteria and aiding future research.

Findings

Complete classification of steady-state varieties for small networks

Simplified criteria for analyzing steady states

Foundation for studying larger chemical reaction networks

Abstract

Chemical reaction network theory is a field of applied mathematics concerned with modeling chemical systems, and can be used in other contexts such as in systems biology to study cellular signaling pathways or epidemiology to study the effect of human interaction on the spread of disease. In this paper, we seek to understand a chemical reaction network's equilibrium points through the lens of algebraic geometry by computing the positive part of the steady-state variety defined by polynomial equations arising from the assumption of mass-action kinetics. We provide a systematic classification of positive steady-state varieties produced by 2-species, 2-reaction networks, grounded in combinatorial and algebraic properties. While some (restricted) techniques exist to fully understand the ideal defining the positive steady-state variety, this computation presents a significant challenge in general. Our classification theorems provide a simplification of previous criteria, and aim to provide a foundation for future analysis of larger networks.