When algebra twinks system biology: a conjecture on the structure of Gr\"obner bases in complex chemical reaction networks

TL;DR

This paper explores the use of Gr"obner bases to systematically find all positive steady states in complex chemical reaction networks, especially those with pairwise interactions, demonstrating their effectiveness over traditional numerical methods.

Contribution

It proposes a conjecture that CRNs with pairwise interactions have Gr"obner bases with a near-triangular structure, enabling comprehensive solution enumeration.

Findings

Gr"obner bases reliably capture all positive solutions in example networks

The approach overcomes limitations of local numerical methods

Potential for broad application in biochemical system analysis

Abstract

We address the challenge of identifying all real positive steady states in chemical reaction networks (CRNs) governed by mass-action kinetics. Traditional numerical methods often require specific initial guesses and may fail to find all the solutions in systems exhibiting multistability. Gr\"obner bases offer an algebraic framework that systematically transforms polynomial equations into simpler forms, facilitating comprehensive solution enumeration. In this work, we propose a conjecture that CRNs with at most pairwise interactions yield Gr\"obner bases possessing a near-"triangular" structure, under appropriate assumptions. We illustrate this phenomenon using examples from a gene regulatory network and the Wnt signaling pathway, where the Gr\"obner basis approach reliably captures all real positive solutions. Our computational experiments reveal the potential of Gr\"obner bases to overcome limitations of local numerical methods for finding the steady states of complex biological systems, making them a powerful tool for understanding dynamical processes across diverse biochemical models.