Boolean Models Can Explain Bistability in the lac Operon

Abstract

The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. Our model includes the two main glucose control mechanisms of catabolite repression and inducer exclusion. We show that this Boolean model is capable of predicting the ON and OFF steady states and bistability. Further, we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model exhibits the same dynamics. This work suggests that the key to model qualitative dynamics of gene systems is the topology of the network and Boolean models are well suited for this purpose.

The Boolean Kinetics of signal transduction

“…Details of the ODEs and rate equations are given in the supplementary material. The model of the lac operon was based upon [8,9] and [11]. The model of MAP kinase signalling is based upon that of Brightman and Fell [21], and the model of PI3K signalling was adapted from [22].…”

“…Veliz-Cuba and Stigler [9] showed that Boolean models could explain bistability (i.e. the existence of stable ON or OFF states, but not intermediate states) in the lac operon.…”

Coupled cell networks: Boolean perspective

“…For example, Francis Robert [10] defines 1 * 1 = 0, whereas other authors (for example [18]) consider 1 * 1 to be 1. Except of adding and multiplying variables, we are also allowed to add 1, which is equivalent to negation.…”

“…Large networks are hard to analyze mathematically, both from discrete and from continuous point of view [15], [17]. Usually in such cases a model reduction technique is applied [13], [17], [18], [19]. The authors of [1] base their model reduction strategy intended for CCNs on cell coloring.…”

“…Most known Boolean models in systems biology are characterized by cooperating species, each of which is governed by a different set of rules [18], [26]. Models based on the idea of identical entities governed by identical sets of rules are quite common in ecology, however these models are usually not Boolean [27].…”