  Zero-one reaction networks play key roles in cell signaling such as
signalling pathways regulated by protein phosphorylation. Multistability of
zero-one networks is a key dynamics feature enabling decision-making in cells.
Since multistability (or, nondegenerate multistationarity) can be lifted from a
smaller subnetwork (low-dimensional networks with less species and fewer
reactions) to large networks, we aim to explore the multistability problem of
small zero-one networks. We prove that any zero-one network with a
one-dimensional stoichiometric subspace admits at most one (stable) positive
steady state (this steady state is also called a structural attractor), and we
completely classify all the one-dimensional zero-one networks according to if
they indeed admits a (stable) positive steady state or not. Also, we prove that
any two-dimensional zero-one network with up to three species either admits
only degenerate positive steady states, or admits at most one (stable) positive
steady state. In these proofs, we apply the theorem based on the Brouwer degree
theory and the theory of real algebraic geometry. Moreover, using the tools of
computational algebraic geometry, we provide a systematical way for detecting
the smallest zero-one networks that admit nondegenerate
multistationarity/multistability. We show that the smallest zero-one networks
that admit nondegenerate multistationarity contain three species and five
reactions, and the smallest zero-one networks that admit multistability contain
three species and six reactions.
