Network Topology as a Driver of 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. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits 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 also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.

💡 Analysis

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. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits 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 also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.

📄 Content

arXiv:0807.3995v1 [q-bio.MN] 25 Jul 2008 NETWORK TOPOLOGY AS A DRIVER OF BISTABILITY IN THE LAC OPERON BRANDILYN STIGLER AND ALAN VELIZ-CUBA 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. We in- clude the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits 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 also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.

1. Introduction The lac operon in the bacterium Escherichia coli has been used as a model system of gene regulation since the landmark work by Jacob and Monod in 1961 [6]. Its study has led to numerous insights into sugar metabolism, including how the presence of a substrate could trigger induction of its catabolizing enzyme, yet in the presence of a preferred energy source, namely glucose, the substrate is rendered ineffective. Originally termed the “glucose effect”, catabolite repression became known as one of the mechanisms by which glucose regulates the induction of sugar-metabolizing operons. Early work on the lac operon also led to the discovery that transcription of an operon’s genes is subject to positive or negative control and that the system of genes is either inducible (inducers are needed to kick-start transcription) or repressible (corepressors are needed to stop transcription). The lac operon is one of the earliest examples of a inducible system of genes being under both positive and negative control. There are many formulations modeling the behavior and interaction of the lac genes. The first model was proposed by Goodwin two years after the discovery of the lac operon [3]. Since then there has been a steady flow of models following the advances in biological insight of the system, with the majority describing operon induction using artificial non- metabolizable compounds such as IPTG and TMG [10, 16, 17, 13]. For example, the first model to consider catabolite repression and inducer exclusion, another control mechanism of the operon by glucose, when the cells were grown in both glucose and lactose (lactose was the inducer) was presented by Wong et al. [18]. Their model consisted of up to 13 ordinary differential equations involving 65 parameters. Further Santill´an and coauthors have presented mathematical models and analysis purporting bistability (the operon is either induced or uninduced) [19, 12, 13, 11] as observed in the experiments of [9, 10]. These findings have given rise to the analogy of the lac operon acting as a biological switch [10, 5]. The first author was supported by NSF Agreement Nr. 0112050. The second author was supported by NSF grant DMS-051144. 1 2 BRANDILYN STIGLER AND ALAN VELIZ-CUBA Most mathematical formulations of the lac operon, as well as other genetic systems, are given as systems of differential equations; however, discrete modeling frameworks are receiving more attention for their use in offering global insights. In fact Albert and Othmer suggested that network topology and the type of interactions, as opposed to quantitative mathematical functions with estimated parameters, were sufficient to capture the dynamics of gene networks, which they demonstrated by constructing a Boolean model for a segment polarity network in Drosophila melanogaster [2]. Setty et al. defined a logical function for the transcription of the lac genes in terms of the proteins regulating the operon, namely CRP and LacI [14]. Although the authors initially aimed to construct a simple Boolean function to mimic the switching behavior of the operon, they discovered that AND-like and OR-like expressions could not reproduce the complexity that the lac genes exhibited. Instead they found that a logical function on 4 states (as opposed to 2 states - 0 and 1) was more biologically relevant. Mayo et al. tested and showed that this logical function was robust with respect to point mutations, that is, given the formulation in [14], the operon is still functional after point mutations [8]. To our knowledge, the model of Setty et al. is the first discrete model of the lac genes. While this is an important example of the applicability of logical functions for describing operon dynamics, one limitation is that it does not predict bistability. We propose

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