Dynamics and bistability in a reduced model of the lac operon (original) (raw)
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Biophysical Journal, 2003
A mathematical model for the regulation of induction in the lac operon in Escherichia coli is presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; b-galactosidase, which is involved in the conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We have paid particular attention to the estimation of the parameters in the model. We have tested our model against two sets of b-galactosidase activity versus time data, as well as a set of data on b-galactosidase activity during periodic phosphate feeding. In all three cases we find excellent agreement between the data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation in the model dynamics.
Origin of Bistability in the lac Operon
Biophysical Journal, 2007
Multistability is an emergent dynamic property that has been invoked to explain multiple coexisting biological states. In this work, we investigate the origin of bistability in the lac operon. To do this, we develop a mathematical model for the regulatory pathway in this system and compare the model predictions with other experimental results in which a nonmetabolizable inducer was employed. We investigate the effect of lactose metabolism using this model, and show that it greatly modifies the bistable region in the external lactose (Le) versus external glucose (Ge) parameter space. The model also predicts that lactose metabolism can cause bistability to disappear for very low Ge. We have also carried out stochastic numerical simulations of the model for several values of Ge and Le. Our results indicate that bistability can help guarantee that Escherichia coli consumes glucose and lactose in the most efficient possible way. Namely, the lac operon is induced only when there is almost no glucose in the growing medium, but if Le is high, the operon induction level increases abruptly when the levels of glucose in the environment decrease to very low values. We demonstrate that this behavior could not be obtained without bistability if the stability of the induced and uninduced states is to be preserved. Finally, we point out that the present methods and results may be useful to study the emergence of multistability in biological systems other than the lac operon.
Bistable behavior in a model of the lac operon in Escherichia coli with variable growth rate
Biophysical journal, 2008
This work is a continuation from another study previously published in this journal. Both the former and the present works are dedicated to investigating the bistable behavior of the lac operon in Escherichia coli from a mathematical modeling point of view. In the previous article, we developed a detailed mathematical model that accounts for all of the known regulatory mechanisms in this system, and studied the effect of inducing the operon with lactose instead of an artificial inducer. In this article, the model is improved to account, in a more detailed way, for the interaction of the repressor molecules with the three lac operators. A recently discovered cooperative interaction between the CAP molecule (an activator of the lactose operon) and Operator 3 (which influences DNA folding) is also included in this new version of the model. The growth rate dependence on the rate of energy entering the bacteria (in the form of transported glucose molecules and of metabolized lactose molecules) is also considered. A large number of numerical experiments is carried out with this improved model. The results are discussed in regard to the bistable behavior of the lactose operon. Special attention is paid to the effect that a variable growth rate has on the system dynamics.
Bistability and Nonmonotonic Induction of the lac Operon in the Natural Lactose Uptake System
Biophysical Journal, 2017
The E. coli lac operon is regulated by a positive feedback loop whose potential to generate an all-or-none response in single cells has been a paradigm for bistable gene expression. However, so far bistable lac induction has only been observed using gratuitous inducers raising the question about the biological relevance of bistable lac induction in the natural setting with lactose as the inducer. In fact, the existing experimental evidence points to a graded rather than an all-or-none response in the natural lactose uptake system. In contrast, predictions based on computational models of the lactose uptake pathway remain controversial. While some argue in favor of bistability others argue against it. Here, we reinvestigate lac operon expression in single cells using a combined experimental / modeling approach. To this end we parametrize a well-supported mathematical model using transient measurements of LacZ activity upon induction with different amounts of lactose. The resulting model predicts a monostable induction curve for the wildtype system, but indicates that overexpression of the LacI repressor would drive the system into the bistable regime. Both predictions were confirmed experimentally supporting the view that the wildtype lac induction circuit generates a graded response rather than bistability. More interestingly, we find that the lac induction curve exhibits a pronounced maximum at intermediate lactose concentrations. Supported by our data a model-based analysis suggests that the non-monotonic response results from saturation of the LacI repressor at low inducer concentrations and dilution of Lac enzymes due to an increased growth rate beyond the saturation point. We speculate that the observed maximum in the lac expression level helps to save cellular resources by limiting Lac enzyme expression at high inducer concentrations.
Influence of catabolite repression and inducer exclusion on the bistable behavior of the lac operon
Biophysical journal, 2004
A mathematical model of the lac operon which includes all of the known regulatory mechanisms, including external-glucose-dependent catabolite repression and inducer exclusion, as well as the time delays inherent to transcription and translation, is presented. With this model we investigate the influence of external glucose, by means of catabolite repression and the regulation of lactose uptake, on the bistable behavior of this system.
Bistable behavior of the lac operon in E. coli when induced with a mixture of lactose and TMG
Frontiers in physiology, 2010
In this work we investigate multistability in the lac operon of Escherichia coli when it is induced by a mixture of lactose and the non-metabolizable thiomethyl galactoside (TMG). In accordance with previously published experimental results and computer simulations, our simulations predict that: (1) when the system is induced by TMG, the system shows a discernible bistable behavior while, (2) when the system is induced by lactose, bistability does not disappear but excessively high concentrations of lactose would be required to observe it. Finally, our simulation results predict that when a mixture of lactose and TMG is used, the bistability region in the extracellular glucose concentration vs. extracellular lactose concentration parameter space changes in such a way that the model predictions regarding bistability could be tested experimentally. These experiments could help to solve a recent controversy regarding the existence of bistability in the lac operon under natural conditions.
TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES, 2016
This paper presents the results of a theoretical and numerical study on the analysis of bistable behavior of the most studied gene regulatory network, the lac operon, in terms of the model parameters. The boundedness of the state variables for the considered model are demonstrated, the parameter values providing the existence of the multiple equilibria and thus the bistable behavior are determined, and a local stability analysis of the equilibria is performed. The parameter region yielding the existence of multiple equilibria is determined in an algebraic way based on discriminants. The model given in the state equation form is defined by the ordinary differential equations with the rational right-hand sides constituted within Hill and Michaelis-Menten approaches based on enzyme kinetics. The presented method can also be used in the parametric studies of other gene regulatory and metabolic networks given by state equations with rational right hand sides.
Modeling Gene Expression: Lac operon
2021 43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), 2021
Gene regulation is an essential process for cell development, having a profound effect in dictating cell functions. Bacterial genes are often regulated through inducible systems like the Lac operon which plays an important role in cell metabolism. An accurate model of its regulation can reveal the dynamics of gene expression. In this paper, a mathematical model of this system is constructed by focusing on regulation by the Lac repressor. The results show, as expected, that the concentration of lactose approaches zero while glucose concentration approaches the initial concentration of lactose by the action of β-galactosidase, expressed by the Lac operon. Addition of PD control improves stability of the system, with the phase margin increasing from 45°t o 90°. Modeling the dynamics of gene expression in inducible operons like Lac operon can be essential for its applications in the production of recombinant proteins and its potential usage in gene therapy. I.
Boolean Models Can Explain Bistability in the lac Operon
Journal of Computational Biology, 2011
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.