(Springer Science and Business Media LLC, 2024-08) Gedeon, Tomáš
In this review paper we summarize a recent progress on the problem of describing range of dynamics supported by a network. We show that there is natural connection between network models consisting of collections of multivalued monotone boolean functions and ordinary differential equations models. We show how to construct such collections and use them to answer questions about prevalence of cellular phenotypes that correspond to equilibria of network models.
We consider two types of models of regulatory network dynamics: Boolean maps and systems of switching ordinary differential equations. Our goal is to construct all models in each category that are compatible with the directed signed graph that describe the network interactions. This leads to consideration of lattice of monotone Boolean functions (MBF), poset of non-degenerate MBFs, and a lattice of chains in these sets. We describe explicit inductive construction of these posets where the induction is on the number of inputs in MBF. Our results allow enumeration of potential dynamic behavior of the network for both model types, subject to practical limitation imposed by the size of the lattice of MBFs described by the Dedekind number.
