Degeneracy of Zero-one Reaction Networks

TL;DR

This paper investigates the degeneracy in two-dimensional zero-one biochemical reaction networks with up to three species, identifying all degenerate cases and linking their steady-state systems to binomial systems.

Contribution

It provides a complete classification of degenerate two-dimensional zero-one reaction networks with up to three species using an efficient algorithm.

Findings

All degenerate networks with three species are identified.

Degenerate networks' steady states are equivalent to binomial systems.

One-dimensional networks cannot degenerate.

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.