Determining the Equivalence of Small Zero-one Reaction Networks
Yue Jiao, Xiaoxian Tang
TL;DR
This work addresses the problem of determining when two zero‑one reaction networks are equivalent with respect to their steady‑state ideals under relabeling. It introduces a high‑efficiency approach that pre‑filters networks using criteria based on the stoichiometric matrix $\\mathcal{N}$ and reactant matrix $\\mathcal{Y}$ to avoid expensive Gröbner basis computations, enabling large‑scale classification of networks by equivalence. Theoretical results provide practical tests for inconsistency, vacuity, and equivalence, which are then integrated into a comprehensive algorithm that enumerates matrices, prunes nonviable cases, and groups networks before performing basis computations. Empirically, the method classifies over three million networks for the (3,3,5) class, achieving substantial speedups and revealing that small quadratic zero‑one networks in 3D and 4D lack positive steady states for generic rate constants, implying no multistability or periodic orbits. The findings support the notion that dynamically rich small zero‑one networks must be at least cubic and demonstrate how structured algebraic checks can drastically reduce computational cost in network equivalence problems.
Abstract
Zero-one reaction networks are pivotal to cellular signaling, and establishing the equivalence of such networks represents a foundational computational challenge in the realm of chemical reaction network research. Herein, we propose a high-efficiency approach for identifying the equivalence of zero-one networks. Its efficiency stems from a set of criteria tailored to judge the equivalence of steady-state ideals derived from zero-one networks, which effectively reduces the computational cost associated with Gröbner basis calculations. Experimental results demonstrate that our proposed method can successfully categorize more than three million networks by their equivalence within a feasible timeframe. Also, our computational results for two important classes of quadratic zero-one networks (3-dimensional with 3 species, 6 reactions; 4-dimensional with 4 species, 5 reactions) show that they have no positive steady states for a generic choice of rate constants, implying these small networks generically exhibit neither multistability nor periodic orbits.