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arXiv · 2501.12233

When algebra twinks system biology: a conjecture on the structure of Gröbner bases in complex chemical reaction networks

Paola Ferrari·Sara Sommariva·Michele Piana·Federico Benvenuto·Matteo Varbaro
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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. Gröbner 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 Gröbner 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 Gröbner basis approach reliably captures all real positive solutions. Our computational experiments reveal the potential of Gröbner 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.

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Paola Ferrari, Sara Sommariva, Michele Piana, Federico Benvenuto, Matteo Varbaro. 2025-09-26. When algebra twinks system biology: a conjecture on the structure of Gröbner bases in complex chemical reaction networks. https://arxiv.org/abs/2501.12233

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