Fixed points and multistability in monotone Boolean network models

Gene regulatory networks (GRN) control the expression levels of proteins in cells, and understanding their dynamics is key to potentially controlling disease processes. Steady states of GRNs are interpreted as cellular phenotypes, and the first step in understanding GRN dynamics is describing the collection of steady states the network can support in different conditions. We consider a collection of all monotone Boolean function models compatible with a given GRN, and ask which steady states are supported by most models. We find that for networks with no negative loops, there is an explicit hierarchy in the prevalence of individual steady states, as well as the prevalence of bistability and multistability. The key insight that we use is that monotone Boolean models supporting a given equilibrium are a product of prime ideals and prime filters of the lattices of monotone Boolean functions. To illustrate our result, we show that in the EMT network associated with cancer metastasis, the most common equilibria correspond to epithelial (E) and mesenchymal (M) states, and the bistability between them is the most common bistability among all network-compatible monotone Boolean models. Author summaryCells adjust their behavior in response to external inputs via networks of genes that regulate each others expression, until they arrive at a new steady state. Each interacting network of genes can behave in different ways that depend on internal and external cellular conditions. In this paper we consider, for a given network, an entire collection of particular type of models (monotone Boolean models) that represent all different ways that network can behave. Then, for any given state a network can potentially be in, we describe all monotone Boolean models that have that state as a steady state. We consider those states that are supported by more models to more likely represent the states that the network will be in. We apply our approach to EMT network that is important in cancer metastasis. We show that the most common steady states are those correspond to epithelial and mesenchymal states, and that the bistability between these two states is the most common bistability. This confirms the experimental results that these are the most common states of the EMT network.