The Ubiquity of Three Steady States: Minimal Multistable Zero-One Reaction Networks

Abstract

Biochemical networks with unit stoichiometry arise naturally in receptor-ligand binding and multisite phosphorylation systems. A central question is to identify which networks admit multistability. Because this property can be inherited from smaller subnetworks to larger ones, it is natural to seek the smallest networks with it. In this paper, we completely determine all minimal quadratic zero-one networks that exhibit multistability. Building on recent progress that has narrowed the search to the zero-one networks with 3 species, 6 reactions, and dimension 3, say (3, 6, 3) family. We develop a computational pipeline to provide a complete characterization of multistability within this class. The primary theoretical contribution is a structural simplification showing that for (3, m, 3) quadratic zero-one networks, the Jacobian determinants at consecutive positive steady states have opposite signs, and exactly half of these states are stable. This reduces multistationarity detection and stability verification to a local sign-checking problem, eliminating the need for boundary or asymptotic analysis. Applying this result, we identify 373 quadratic zero-one networks that exhibit bistability (two stable and one unstable positive steady states). Strikingly, no (3, 6, 3) quadratic zero-one network admits more than 3 positive steady states--a sharp bound that falls well below the theoretical BKK bound of 5 and the Bezout bound of 8. Among the 375 networks that admit exactly 3 positive steady states, 373 are multistable and only 2 are not. These minimal networks provide essential test cases for understanding how bistability emerges in cell signaling without requiring higher-order molecular collisions.