Bistable behavior in a model of the lac operon in Escherichia coli with variable growth rate

Exploring gene regulation and biological processes in insects: Insights from omics data using gene regulatory network models

Gene regulatory network (GRN) comprises complicated yet intertwined gene-regulator relationships. Understanding the GRN dynamics will unravel the complexity behind the observed gene expressions. Insect gene regulation is often complicated due to their complex life cycles and diverse ecological adaptations. The main interest of this review is to have an update on the current mathematical modelling methods of GRNs to explain insect science. Several popular GRN architecture models are discussed, together with examples of applications in insect science. In the last part of this review, each model is compared from different aspects, including network scalability, computation complexity, robustness to noise and biological relevancy.

LacOp: A free web-based lac operon simulation that enhances student learning of gene regulation concepts

Here, we describe a free, web-based simulation of the lac operon, "LacOp," that is designed to enhance the learning of prokaryotic gene regulation and pathways in advanced high school and undergraduate genetics courses. This new electronic resource was created by a team of students in an advanced undergraduate course and is hosted online (http://flask-env.rnwhymamqf.us-west-2.elasticbeanstalk.com/lacop). LacOp has a simple web interface compatible with a range of devices, including smartphones. To determine whether the LacOp simulation enhances student learning from traditional instruction, we introduced the lac operon to undergraduate genetics students through a traditional classroom experience followed by use of the LacOp simulation. Students worked on their own using the included tutorial to create and test the effect of various genotypes on E. coli lactose metabolism and regulation. Upon completion of the tutorial, students showed measurable gains in conceptual understanding of the lac operon. These students also reported a generally favorable opinion of the LacOP simulation as a use of their instructional time.

Switching off: The phenotypic transition to the uninduced state of the lactose uptake pathway

The lactose uptake pathway of E. coli is a paradigmatic example of multistability in gene regulatory circuits. In the induced state of the lac pathway, the genes comprising the lac operon are transcribed, leading to the production of proteins that import and metabolize lactose. In the uninduced state, a stable repressor-DNA loop frequently blocks the transcription of the lac genes. Transitions from one phenotypic state to the other are driven by fluctuations, which arise from the random timing of the binding of ligands and proteins. This stochasticity affects transcription and translation, and ultimately molecular copy numbers. Our aim is to understand the transition from the induced to the uninduced state of the lac operon. We use a detailed computational model to show that repressor-operator binding and unbinding, fluctuations in the total number of repressors, and inducer-repressor binding and unbinding all play a role in this transition. Based on the timescales on which these processes operate, we construct a minimal model of the transition to the uninduced state and compare the results with simulations and experimental observations. The induced state turns out to be very stable, with a transition rate to the uninduced state lower than 2×10^-9 per minute. In contrast to the transition to the induced state, the transition to the uninduced state is well described in terms of a 2D diffusive system crossing a barrier, with the diffusion rates emerging from a model of repressor unbinding.

Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Fluctuating-rate model with multiple gene states

Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.

Geometric Characterization of Data Sets with Unique Reduced Gröbner Bases

Model selection based on experimental data is an important challenge in biological data science. Particularly when collecting data is expensive or time-consuming, as it is often the case with clinical trial and biomolecular experiments, the problem of selecting information-rich data becomes crucial for creating relevant models. We identify geometric properties of input data that result in an unique algebraic model, and we show that if the data form a staircase, or a so-called linear shift of a staircase, the ideal of the points has a unique reduced Gröbner basis and thus corresponds to a unique model. We use linear shifts to partition data into equivalence classes with the same basis. We demonstrate the utility of the results by applying them to a Boolean model of the well-studied lac operon in E. coli.

A neural network model predicts community-level signaling states in a diverse microbial community

Signal crosstalk within biological communication networks is common, and such crosstalk can have unexpected consequences for decision making in heterogeneous communities of cells. Here we examined crosstalk within a bacterial community composed of five strains of Bacillus subtilis, with each strain producing a variant of the quorum sensing peptide ComX. In isolation, each strain produced one variant of the ComX signal to induce expression of genes associated with bacterial competence. When strains were combined, a mixture of ComX variants was produced resulting in variable levels of gene expression. To examine gene regulation in mixed communities, we implemented a neural network model. Experimental quantification of asymmetric crosstalk between pairs of strains parametrized the model, enabling the accurate prediction of activity within the full five-strain network. Unlike the single strain system in which quorum sensing activated upon exceeding a threshold concentration of the signal, crosstalk within the five-strain community resulted in multiple community-level quorum sensing states, each with a unique combination of quorum sensing activation among the five strains. Quorum sensing activity of the strains within the community was influenced by the combination and ratio of strains as well as community dynamics. The community-level signaling state was altered through an external signal perturbation, and the output state depended on the timing of the perturbation. Given the ubiquity of signal crosstalk in diverse microbial communities, the application of such neural network models will increase accuracy of predicting activity within microbial consortia and enable new strategies for control and design of bacterial signaling networks.

PROKARYO: an illustrative and interactive computational model of the lactose operon in the bacterium Escherichia coli

Background

We are creating software for agent-based simulation and visualization of bio-molecular processes in bacterial and eukaryotic cells. As a first example, we have built a 3-dimensional, interactive computer model of an Escherichia coli bacterium and its associated biomolecular processes. Our illustrative model focuses on the gene regulatory processes that control the expression of genes involved in the lactose operon. Prokaryo, our agent-based cell simulator, incorporates cellular structures, such as plasma membranes and cytoplasm, as well as elements of the molecular machinery, including RNA polymerase, messenger RNA, lactose permease, and ribosomes.

Results

The dynamics of cellular ’agents’ are defined by their rules of interaction, implemented as finite state machines. The agents are embedded within a 3-dimensional virtual environment with simulated physical and electrochemical properties. The hybrid model is driven by a combination of (1) mathematical equations (DEQs) to capture higher-scale phenomena and (2) agent-based rules to implement localized interactions among a small number of molecular elements. Consequently, our model is able to capture phenomena across multiple spatial scales, from changing concentration gradients to one-on-one molecular interactions.

We use the classic gene regulatory mechanism of the lactose operon to demonstrate our model’s resolution, visual presentation, and real-time interactivity. Our agent-based model expands on a sophisticated mathematical E. coli metabolism model, through which we highlight our model’s scientific validity.

Conclusion

We believe that through illustration and interactive exploratory learning a model system like Prokaryo can enhance the general understanding and perception of biomolecular processes. Our agent-DEQ hybrid modeling approach can also be of value to conceptualize, illustrate, and—eventually—validate cell experiments in the wet lab.

Effect of Intrinsic Noise on the Phenotype of Cell Populations Featuring Solution Multiplicity: An Artificial lac Operon Network Paradigm

Heterogeneity in cell populations originates from two fundamentally different sources: the uneven distribution of intracellular content during cell division, and the stochastic fluctuations of regulatory molecules existing in small amounts. Discrete stochastic models can incorporate both sources of cell heterogeneity with sufficient accuracy in the description of an isogenic cell population; however, they lack efficiency when a systems level analysis is required, due to substantial computational requirements. In this work, we study the effect of cell heterogeneity in the behaviour of isogenic cell populations carrying the genetic network of lac operon, which exhibits solution multiplicity over a wide range of extracellular conditions. For such systems, the strategy of performing solely direct temporal solutions is a prohibitive task, since a large ensemble of initial states needs to be tested in order to drive the system--through long time simulations--to possible co-existing steady state solutions. We implement a multiscale computational framework, the so-called "equation-free" methodology, which enables the performance of numerical tasks, such as the computation of coarse steady state solutions and coarse bifurcation analysis. Dynamically stable and unstable solutions are computed and the effect of intrinsic noise on the range of bistability is efficiently investigated. The results are compared with the homogeneous model, which neglects all sources of heterogeneity, with the deterministic cell population balance model, as well as with a stochastic model neglecting the heterogeneity originating from intrinsic noise effects. We show that when the effect of intrinsic source of heterogeneity is intensified, the bistability range shifts towards higher extracellular inducer concentration values.

Determining the bistability parameter ranges of artificially induced lac operon using the root locus method

This paper employs the root locus method to conduct a detailed investigation of the parameter regions that ensure bistability in a well-studied gene regulatory network namely, lac operon of Escherichia coli (E. coli). In contrast to previous works, the parametric bistability conditions observed in this study constitute a complete set of necessary and sufficient conditions. These conditions were derived by applying the root locus method to the polynomial equilibrium equation of the lac operon model to determine the parameter values yielding the multiple real roots necessary for bistability. The lac operon model used was defined as an ordinary differential equation system in a state equation form with a rational right hand side, and it was compatible with the Hill and Michaelis-Menten approaches of enzyme kinetics used to describe biochemical reactions that govern lactose metabolism. The developed root locus method can be used to study the steady-state behavior of any type of convergent biological system model based on mass action kinetics. This method provides a solution to the problem of analyzing gene regulatory networks under parameter uncertainties because the root locus method considers the model parameters as variable, rather than fixed. The obtained bistability ranges for the lac operon model parameters have the potential to elucidate the appearance of bistability for E. coli cells in in vivo experiments, and they could also be used to design robust hysteretic switches in synthetic biology.

The utility of simple mathematical models in understanding gene regulatory dynamics

In this review, we survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behaviour of simple bacterial operons starting with the initial work of the 1960's. We concentrate on the simplest of situations, discussing both repressible and inducible systems and then turning to concrete examples related to the biology of the lactose and tryptophan operons. We conclude with a brief discussion of the role of both extrinsic noise and so-called intrinsic noise in the form of translational and/or transcriptional bursting.

Coarse-grained analysis of stochastically simulated cell populations with a positive feedback genetic network architecture

Among the different computational approaches modelling the dynamics of isogenic cell populations, discrete stochastic models can describe with sufficient accuracy the evolution of small size populations. However, for a systematic and efficient study of their long-time behaviour over a wide range of parameter values, the performance of solely direct temporal simulations requires significantly high computational time. In addition, when the dynamics of the cell populations exhibit non-trivial bistable behaviour, such an analysis becomes a prohibitive task, since a large ensemble of initial states need to be tested for the quest of possibly co-existing steady state solutions. In this work, we study cell populations which carry the lac operon network exhibiting solution multiplicity over a wide range of extracellular conditions (inducer concentration). By adopting ideas from the so-called "equation-free" methodology, we perform systems-level analysis, which includes numerical tasks such as the computation of coarse steady state solutions, coarse bifurcation analysis, as well as coarse stability analysis. Dynamically stable and unstable macroscopic (population level) steady state solutions are computed by means of bifurcation analysis utilising short bursts of fine-scale simulations, and the range of bistability is determined for different sizes of cell populations. The results are compared with the deterministic cell population balance model, which is valid for large populations, and we demonstrate the increased effect of stochasticity in small size populations with asymmetric partitioning mechanisms.

The effect of LacI autoregulation on the performance of the lactose utilization system in Escherichia coli

The lactose operon of Escherichia coli is a paradigm system for quantitative understanding of gene regulation in prokaryotes. Yet, none of the many mathematical models built so far to study the dynamics of this system considered the fact that the Lac repressor regulates its own transcription by forming a transcriptional roadblock at the O3 operator site. Here we study the effect of autoregulation on intracellular LacI levels and also show that cAMP-CRP binding does not affect the efficiency of autoregulation. We built a mathematical model to study the role of LacI autoregulation in the lactose utilization system. Previously, it has been argued that negative autoregulation can significantly reduce noise as well as increase the speed of response. We show that the particular molecular mechanism, a transcriptional roadblock, used to achieve self-repression in the lac system does neither. Instead, LacI autoregulation balances two opposing states, one that allows quicker response to smaller pulses of external lactose, and the other that minimizes production costs in the absence of lactose.

On the relationship of steady states of continuous and discrete models arising from biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.

Mathematical modeling of gene expression: a guide for the perplexed biologist

The detailed analysis of transcriptional networks holds a key for understanding central biological processes, and interest in this field has exploded due to new large-scale data acquisition techniques. Mathematical modeling can provide essential insights, but the diversity of modeling approaches can be a daunting prospect to investigators new to this area. For those interested in beginning a transcriptional mathematical modeling project, we provide here an overview of major types of models and their applications to transcriptional networks. In this discussion of recent literature on thermodynamic, Boolean, and differential equation models, we focus on considerations critical for choosing and validating a modeling approach that will be useful for quantitative understanding of biological systems.

Boolean models can explain bistability in the lac operon

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.

Design of the lac gene circuit revisited

The lactose (lac) operon of Escherichia coli serves as the paradigm for gene regulation, not only for bacteria, but also for all biological systems from simple phage to humans. The details of the systems may differ, but the key conceptual framework remains, and the original system continues to reveal deeper insights with continued experimental and theoretical study. Nearly as long lasting in impact as the pivotal work of Jacob and Monod is the classic experiment of Novick and Weiner in which they demonstrated all-or-none gene expression in response to an artificial inducer. These results are often cited in claims that normal gene expression is in fact a discontinuous bistable phenomenon. In this paper, I review several levels of analysis of the lac system and introduce another perspective based on the construction of the system design space. These represent variations on a theme, based on a simply stated design principle, that captures the key qualitative features of the system in a largely mechanism-independent fashion. Moreover, this principle can be readily interpreted in terms of specific mechanisms to make predictions regarding monostable vs. bistable behavior. The regions of design space representing bifurcations are compared with the corresponding regions identified through bifurcation analysis. I present evidence based on biological considerations as well as modeling and analysis to suggest that induction of the lac system in its natural setting is a monostable continuously graded phenomenon. Nevertheless, it must be acknowledged that the lac stability question remains unsettled, and it undoubtedly will remain so until there are definitive experimental results.

Noise contributions in an inducible genetic switch: a whole-cell simulation study

Stochastic expression of genes produces heterogeneity in clonal populations of bacteria under identical conditions. We analyze and compare the behavior of the inducible lac genetic switch using well-stirred and spatially resolved simulations for Escherichia coli cells modeled under fast and slow-growth conditions. Our new kinetic model describing the switching of the lac operon from one phenotype to the other incorporates parameters obtained from recently published in vivo single-molecule fluorescence experiments along with in vitro rate constants. For the well-stirred system, investigation of the intrinsic noise in the circuit as a function of the inducer concentration and in the presence/absence of the feedback mechanism reveals that the noise peaks near the switching threshold. Applying maximum likelihood estimation, we show that the analytic two-state model of gene expression can be used to extract stochastic rates from the simulation data. The simulations also provide mRNA–protein probability landscapes, which demonstrate that switching is the result of crossing both mRNA and protein thresholds. Using cryoelectron tomography of an E. coli cell and data from proteomics studies, we construct spatial in vivo models of cells and quantify the noise contributions and effects on repressor rebinding due to cell structure and crowding in the cytoplasm. Compared to systems without spatial heterogeneity, the model for the fast-growth cells predicts a slight decrease in the overall noise and an increase in the repressors rebinding rate due to anomalous subdiffusion. The tomograms for E. coli grown under slow-growth conditions identify the positions of the ribosomes and the condensed nucleoid. The smaller slow-growth cells have increased mRNA localization and a larger internal inducer concentration, leading to a significant decrease in the lifetime of the repressor–operator complex and an increase in the frequency of transcriptional bursts.

Expressing genes in a bacterial cell is noisy and random. A colony of bacteria grown from a single cell can show remarkable differences in the copy number per cell of a given protein after only a few generations. In this work we use computer simulations to study the variation in how individual cells in a population express a set of genes in response to an environmental signal. The modeled system is the lac genetic switch that Escherichia coli uses to find, collect, and process lactose sugar from the environment. The noise inherent in the genetic circuit controlling the cell's response determines how similar the cells are to each other and we study how the different components of the circuit affect this noise. Furthermore, an estimated 30–50% of the cell volume is taken up by a wide variety of large biomolecules. To study the response of the circuit caused by crowding, we simulate the circuit inside a three-dimensional model of an E. coli cell built using data from cryoelectron tomography reconstructions of a single cell and proteomics data. Correctly including random effects of molecular crowding will be critical to developing fully dynamic models of living cells.

Biological physics in México: Review and new challenges

Biological and physical sciences possess a long-standing tradition of cooperativity as separate but related subfields of science. For some time, this cooperativity has been limited by their obvious differences in methods and views. Biological physics has recently experienced a kind of revival (or better a rebirth) due to the growth of molecular research on animate matter. New avenues for research have been opened for both theoretical and experimental physicists. Nevertheless, in order to better travel for such paths, the contemporary biological physicist should be armed with a set of specialized tools and methods but also with a new attitude toward multidisciplinarity. In this review article, we intend to somehow summarize what has been done in the past (in particular, as an example we will take a closer look at the Mexican case), to show some examples of fruitful investigations in the biological physics area and also to set a proposal of new curricula for physics students and professionals interested in applying their science to get a better understanding of the physical basis of biological function.

Bistable behavior of the lac operon in E. coli when induced with a mixture of lactose and TMG

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.

Filling kinetic gaps: dynamic modeling of metabolism where detailed kinetic information is lacking

BACKGROUND

Integrative analysis between dynamical modeling of metabolic networks and data obtained from high throughput technology represents a worthy effort toward a holistic understanding of the link among phenotype and dynamical response. Even though the theoretical foundation for modeling metabolic network has been extensively treated elsewhere, the lack of kinetic information has limited the analysis in most of the cases. To overcome this constraint, we present and illustrate a new statistical approach that has two purposes: integrate high throughput data and survey the general dynamical mechanisms emerging for a slightly perturbed metabolic network.

METHODOLOGY/PRINCIPAL FINDINGS

This paper presents a statistic framework capable to study how and how fast the metabolites participating in a perturbed metabolic network reach a steady-state. Instead of requiring accurate kinetic information, this approach uses high throughput metabolome technology to define a feasible kinetic library, which constitutes the base for identifying, statistical and dynamical properties during the relaxation. For the sake of illustration we have applied this approach to the human Red blood cell metabolism (hRBC) and its capacity to predict temporal phenomena was evaluated. Remarkable, the main dynamical properties obtained from a detailed kinetic model in hRBC were recovered by our statistical approach. Furthermore, robust properties in time scale and metabolite organization were identify and one concluded that they are a consequence of the combined performance of redundancies and variability in metabolite participation.

CONCLUSIONS/SIGNIFICANCE

In this work we present an approach that integrates high throughput metabolome data to define the dynamic behavior of a slightly perturbed metabolic network where kinetic information is lacking. Having information of metabolite concentrations at steady-state, this method has significant relevance due its potential scope to analyze others genome scale metabolic reconstructions. Thus, I expect this approach will significantly contribute to explore the relationship between dynamic and physiology in other metabolic reconstructions, particularly those whose kinetic information is practically nulls. For instances, I envisage that this approach can be useful in genomic medicine or pharmacogenomics, where the estimation of time scales and the identification of metabolite organization may be crucial to characterize and identify (dis)functional stages.

Quantitative approaches to the study of bistability in the lac operon of Escherichia coli

In this paper, the history and importance of the lac operon in the development of molecular and systems biology are briefly reviewed. We start by presenting a description of the regulatory mechanisms in this operon, taking into account the most recent discoveries. Then we offer a survey of the history of the lac operon, including the discovery of its main elements and the subsequent influence on the development of molecular and systems biology. Next the bistable behaviour of the operon is discussed, both with respect to its discovery and its molecular origin. A review of the literature in which this bistable phenomenon has been studied from a mathematical modelling viewpoint is then given. We conclude with some brief remarks.