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

Characterization of Boolean Networks with Single or Bistable States