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

We address the challenge of identifying all real positive steady states in chemical reaction networks (CRNs) governed by mass-action kinetics. Gröbner bases offer an algebraic framework that systematically transforms polynomial equations into simpler forms, facilitating comprehensive enumeration of solutions. 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, and we establish this triangularity rigorously under a strengthened set of hypotheses. We illustrate the 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.