Multistability of small zero-one reaction networks

Zero-one biochemical reaction networks play key roles in cell signalling such as signalling pathways regulated by protein phosphorylation. Multistability of reaction networks is a crucial dynamics feature enabling decision-making in cells. It is well known that multistability can be lifted from a "subnetwork" (a network with less species and fewer reactions) to large networks. So, we aim to explore the multistability problem of small zero-one networks. In this work, we prove the following main results: 1. any zero-one network with a one-dimensional stoichiometric subspace admits at most one positive steady state (it must be stable), and all the one-dimensional zero-one networks can be classified according to if they indeed admit a stable positive steady state or not; 2. any two-dimensional zero-one network with up to three species either admits only degenerate positive steady states, or admits at most one positive steady state (it must be stable); 3. the smallest zero-one networks (here, by "smallest", we mean these networks contain species as few as possible) that admit nondegenerate multistationarity/multistability contain three species and five/six reactions, and they are three dimensional. In these proofs, we use the theorems based on the Brouwer degree theory and the theory of real algebraic geometry. Moreover, applying the tools of computational real algebraic geometry, we provide a systematical way for detecting the networks that admit nondegenerate multistationarity/multistability.