Obstruction of Absolute Concentration Robustness by Conservation Laws in Non-Redundant Zero-One Networks

Abstract

Absolute concentration robustness (ACR) is a structural property of biochemical reaction networks in which a species attains the same steady-state concentration at every positive steady state, independently of initial conditions and rate constants. Existing detection methods rely on algebraic elimination and typically scale exponentially with network size. We develop a topology-based alternative for non-redundant zero-one networks of stoichiometric dimension at most two, a class that already captures enzyme catalysis, carbon-nanotube transitions, and other elementary biochemical mechanisms. Organizing our analysis around a structural index $s^*$, the number of distinct rows in the stoichiometric matrix, we obtain a complete classification of all such networks admitting non-vacuous ACR for generic rate constants. In dimension one, ACR occurs only for the elementary inflow and outflow module. In dimension two, ACR is possible if and only if $s^*\leq 3$; for $s^*=3$, the admissible networks are precisely those obtained as species refinements of consistent subnetworks of five canonical biochemical prototypes. For $s^*\geq 4$, non-vacuous ACR is impossible for any generic rate assignment.