Characterization of Boolean Networks with Single or Bistable States

Many biological systems, such as metabolic pathways, exhibit bistability behavior: these biological systems exhibit two distinct stable states with switching between the two stable states controlled by certain conditions. Since understanding bistability is key for understanding these biological systems, mathematical modeling of the bistability phenomenon has been at the focus of researches in quantitative and system biology. Recent study shows that Boolean networks offer relative simple mathematical models that are capable of capturing these essential information. Thus a better understanding of the Boolean networks with bistability property is desirable for both theoretical and application purposes. In this paper, we describe an algebraic condition for the number of stable states (fixed points) of a Boolean network based on its polynomial representation, and derive algorithms for a Boolean network to have a single stable state or two stable states. As an example, we also construct a Boolean network with exactly two stable states for the lac operon's $\beta$-galactosidase regulatory pathway when glucose is absent based on a delay differential equation model