Degeneracy of Zero-one Reaction Networks
Xiaoxian Tang, Yihan Wang, Jiandong Zhang
TL;DR
This work studies degeneracy in zero-one reaction networks, focusing on two-dimensional, three-species cases. It introduces an extreme-ray–based transformation of the Jacobian to derive a test polynomial B(p,λ) that detects degeneracy, and applies it to exhaustively classify all degenerate networks in the target class. The key finding is that all 3152 degenerate networks have steady-state systems equivalent to binomial systems, revealing a unifying structural pattern at the degenerate boundary. The results offer a computationally efficient framework for identifying degenerate regimes and hint at deeper binomial-structure commonalities in small zero-one networks.
Abstract
Zero-one biochemical reaction networks are widely recognized for their importance in analyzing signal transduction and cellular decision-making processes. Degenerate networks reveal non-standard behaviors and mark the boundary where classical methods fail. Their analysis is key to understanding exceptional dynamical phenomena in biochemical systems. Therefore, we focus on investigating the degeneracy of zero-one reaction networks. It is known that one-dimensional zero-one networks cannot degenerate. In this work, we identify all degenerate two-dimensional zero-one reaction networks with up to three species by an efficient algorithm. By analyzing the structure of these networks, we arrive at the following conclusion: if a two-dimensional zero-one reaction network with three species is degenerate, then its steady-state system is equivalent to a binomial system.