Modeling Gene Expression: Lac operon (original) (raw)
Related papers
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.
A Novel Cellular-Level Numerical Model of the Lac Operon in E. Coli
Proceedings of the World Congress …, 2008
Based on cellular biology and mathematic modeling, the present paper put forth a flow chart to depict the metabolic pathway of the lac operon structural gene expression of an Escherichia coli (E. coli) cell, and further introduces eight difference equations of the pathway, a mathematic model, for cellular-level computation. In detail, the model can compute the molecular activities and regular patterns of the mRNA, repressor, inducer, β-galactosidase, lactose, and Adenosine 5'-triphosphate (ATP) of an E. coli cell, which was not all available by traditional means and ways. The model itself and its state-variables construct a dynamics, and then numerically describe the gene expression and growth of the cell supplied with mere lactose. The model also established a basis to quantitatively predict activities of the cell. The agreement between the outcomes of the model and traditional biological experiment justified the rightness and achievement of the model.
Dynamic model of gene regulation for the lac operon
Journal of Physics: Conference Series, 2011
Gene regulatory network is a collection of DNA which interact with each other and with other matter in the cell. The lac operon is an example of a relatively simple genetic network and is one of the best-studied structures in the Escherichia coli bacteria. In this work we consider a deterministic model of the lac operon with a noise term, representing the stochastic nature of the regulation. The model is written in terms of a system of simultaneous first order differential equations with delays. We investigate an analytical and numerical solution and analyse the range of values for the parameters corresponding to a stable solution.
Dynamics and bistability in a reduced model of the lac operon
It is known that the lac operon regulatory pathway is capable of showing bistable behavior. This is an important complex feature, arising from the nonlinearity of the involved mechanisms, which is essential to understand the dynamic behavior of this molecular regulatory system. To find which of the mechanisms involved in the regulation of the lac operon is the origin of bistability, we take a previously published model which accounts for the dynamics of mRNA, lactose, allolactose, permease and -galactosidase involvement and simplify it by ignoring permease dynamics ͑assuming a constant permease concentration͒. To test the behavior of the reduced model, three existing sets of data on -galactosidase levels as a function of time are simulated and we obtain a reasonable agreement between the data and the model predictions. The steady states of the reduced model were numerically and analytically analyzed and it was shown that it may indeed display bistability, depending on the extracellular lactose concentration and growth rate.
Generalized net model of the lac operon in bacterium E. coli
2012 6th IEEE INTERNATIONAL CONFERENCE INTELLIGENT SYSTEMS, 2012
In this paper a Generalized Net (GN) model of the regulation of the lac operon in E. coli bacterium is presented. Prokaryotes, such the bacterium E. coli, have an efficient mechanism for metabolizing lactose. Three proteins that are important in lactose metabolism are all encoded in a single expressible unit of DNA, called the lac operon. The GN model, presented in this paper, describes the use of lactose as an energy source from E. coli. The net presents the turn on of several genes in the lac operon, which are required for lactose metabolism.
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.
Analysis of the Lactose metabolism in E. coli using sum-of-squares decomposition
2005
We provide a system-theoretic analysis of the mathematical model of lactose induction in E.coli which predicts the level of lactose induction into the cell for specified values of external lactose. Depending on the levels of external lactose and other parameters, the Lac operon is known to have a low steady state in which it is said to be turned off and high steady state where it is said to be turned on. Furthermore, the model has been shown experimentally to exhibit a bi-stable behavior. Using ideas from Lyapunov stability theory and sumof-squares decomposition, we characterize the reachable state space for different sets of initial conditions, calculating estimates of the regions of attraction of the biologically relevant equilibria of this system. The changes in the basins of attraction with changes in model parameters can be used to provide biological insight. Specifically, we explain the crucial role played by a small basal transcription rate in the Lac operon. We show that if the basal rate is below a threshold, the region of attraction of the low steady state grows significantly, indicating that system is trapped in the (off) mode, showing the importance of the basal rate of transcription.
Modeling operon dynamics: the tryptophan and lactose operons as paradigms
2004
Understanding the regulation of gene control networks and their ensuing dynamics will be a critical component in the understanding of the mountain of genomic data being currently collected. This paper reviews recent mathematical modeling work on the tryptophan and lactose operons which are, respectively, the classical paradigms for repressible and inducible operons. To cite this article: MC Mackey et al., CR Biologies 327 (2004).
Biotechnology Progress, 1997
A mathematical model of the lactose (lac) operon was developed to study diauxic growth on glucose and lactose. The model includes catabolite repression, inducer exclusion, lactose hydrolysis to glucose and galactose, and synthesis and degradation of allolactose. Two models for catabolite repression were tested: (i) cyclic AMP (cAMP) synthesis inversely correlated with the external glucose concentration and (ii) synthesis inversely correlated with the glucose transport rate. No significant differences in the two models were observed. In addition to synthesis, degradation and secretion of cAMP were also included in the model. Two models for the phosphorylation of the glucose produced from lactose hydrolysis were also tested: (i) phosphorylation by intracellular hexokinase and (ii) secretion of glucose and subsequent phosphorylation upon transport back into the cell. The latter model resulted in weak catabolite repression when the glucose produced from lactose was transported out of the cell, whereas the former model showed no catabolite repression during growth on lactose. Parameter sensitivity analysis indicates the importance of key parameters to lac operon expression and cell growth: the lactose and allolactose transformation rates by -galactosidase and the glucose concentrations that affect catabolite repression and inducer exclusion. Large values of the allolactose hydrolysis rate resulted in low concentrations of allolactose, low-level expression of the lac operon, and slow growth due to limited import and metabolism of lactose; small values resulted in a high concentration of allolactose, high-level expression of the lac operon, and slow growth due to a limiting concentration of glucose 6-phosphate formed from allolactose. Changes in the rates of all -galactosidase-catalyzed reactions showed similar behavior, but had more drastic effects on the growth rate. Changes in the glucose concentration that inhibited lactose transport could extend or contract the diauxic growth period during growth in the presence of glucose and lactose. Moreover, changes in the glucose concentration that affected catabolite repression affected the cAMP levels and lac operon expression, but had a lesser effect on the growth rate.
Finite state abstraction of a stochastic model of the lactose regulation system of Escherichia coli
Proceedings of the 45th IEEE Conference on Decision and Control, 2006
This paper focuses on the lactose regulation system in Escherichia coli bacteria, one of the most extensively studied examples of positive feedback in a naturally occurring gene network. State-of-the-art nonlinear dynamical system models predict a bi-stability phenomenon that is confirmed in experiments. However, such deterministic models fail to explain experimental observations of spontaneous transition between the two stable states in the system and the simultaneous occurrence of both steady states in a population of cells. In this paper, we propose a stochastic model that explains this phenomenon. Furthermore, we also extract a coarser two-state continuous-time Markov chain as a higher level abstraction of this model, and show that macroscopic properties are retained in the abstraction.