Analysis of the differentiation and of the heterogeneity within a population of Escherichia coli undergoing induced beta-galactosidase synthesis

Mechanistic aspects of IPTG (isopropylthio-β-galactoside) transport across the cytoplasmic membrane of Escherichia coli-a rate limiting step in the induction of recombinant protein expression

Coupling transcription of a cloned gene to the lac operon with induction by isopropylthio-β-galactoside (IPTG) has been a favoured approach for recombinant protein expression using Escherichia coli as a heterologous host for more than six decades. Despite a wealth of experimental data gleaned over this period, a quantitative relationship between extracellular IPTG concentration and consequent levels of recombinant protein expression remains surprisingly elusive across a broad spectrum of experimental conditions. This is because gene expression under lac operon regulation is tightly correlated with intracellular IPTG concentration due to allosteric regulation of the lac repressor protein (lacY). An in-silico mathematical model established that uptake of IPTG across the cytoplasmic membrane of E. coli by simple diffusion was negligible. Conversely, lacY mediated active transport was a rapid process, taking only some seconds for internal and external IPTG concentrations to equalize. Optimizing kcat and KM parameters by targeted mutation of the galactoside binding site in lacY could be a future strategy to improve the performance of recombinant protein expression. For example, if kcat were reduced whilst KM was increased, active transport of IPTG across the cytoplasmic membrane would be reduced, thereby lessening the metabolic burden on the cell and expediating accumulation of recombinant protein. The computational model described herein is made freely available and is amenable to optimize recombinant protein expression in other heterologous hosts.

ONE-SENTENCE SUMMARY

A computational model made freely available to optimize recombinant protein expression in Escherichia coli other heterologous hosts.

Positive feedback exists and drives the glucose-mediated repression in Escherichia coli

The expression of the lac operon of E. coli is subject to positive feedback during growth in the presence of gratuitous inducers, but its existence in the presence of lactose remains controversial. The key question in this debate is: Do the lactose enzymes, Lac permease and β-galactosidase, promote accumulation of allolactose? If so, positive feedback exists since allolactose does stimulate synthesis of the lactose enzymes. Here, we addressed the above question by developing methods for determining the intracellular allolactose concentration as well as the kinetics of enzyme induction and dilution. We show that, during lac induction in the presence of lactose, the intracellular allolactose concentration increases with the lactose enzyme level, which implies that lactose enzymes promote allolactose accumulation, and positive feedback exists. We also show that, during lac repression in the presence of lactose + glucose, the intracellular allolactose concentration decreases with the lactose enzyme levels, which suggests that, under these conditions, the positive feedback loop turns in the reverse direction. The induction and dilution rates derived from the transient data show that the positive feedback loop is reversed due to a radical shift of the steady-state induction level. This is formally identical to the mechanism driving catabolite repression in the presence of TMG + glucose.

Bistable behaviour and medium-dependent post-translational regulation of the tryptophanase operon regulatory pathway in Echerichia coli

The present work is aimed at studying the dynamic behaviour of the tryptopnanase (tna) operon, which encodes the proteins necessary to uptake and metabolise tryptophan to use it as a carbon source in the absence of glucose. To this end, we designed a micro-bioreactor capable of driving a bacterial culture to a stationary state. This allowed us to explore (at the single cell level) the tna operon steady-state dynamics under multiple culture conditions. Our experimental results suggest that the tna operon is bistable for a specific range of environmental tryptophan and glucose concentrations, and evidence that both reagents play a role on the activation of the enzyme in charge of metabolising tryptophan: tryptophanase (TnaA). Based on our experimental data and the already known regulatory mechanisms, we developed a mathematical model for the tna operon regulatory pathway. Our modelling results reinforce the claim that the tna operon is bistable, and further suggest that the activity of enzyme TnaA is regulated by the environmental levels of glucose and tryptophan via a common signalling pathway. Possible biological implications of our findings are further discussed.

Phenotypic instability in fungi

Fungi are prone to phenotypic instability, that is, the vegetative phase of these organisms, be they yeasts or molds, undergoes frequent switching between two or more behaviors, often with different morphologies, but also sometime having different physiologies without any obvious morphological outcome. In the context of industrial utilization of fungi, this can have a negative impact on the maintenance of strains and/or on their productivity. Instabilities have been shown to result from various mechanisms, either genetic or epigenetic. This chapter will review different types of instabilities and discuss some lesser-known ones, mostly in filamentous fungi, while it will direct readers to additional literature in the case of well-known phenomena such as the amyloid prions or fungal senescence. It will present in depth the "white/opaque" switch of Candida albicans and the "crippled growth" degeneration of the model fungus Podospora anserina. These are two of the most thoroughly studied epigenetic phenotypic switches. I will also discuss the "sectors" presented by many filamentous ascomycetes, for which a prion-based model exists but is not demonstrated. Finally, I will also describe intriguing examples of phenotypic instability for which an explanation has yet to be provided.

Stochastic developmental variation, an epigenetic source of phenotypic diversity with far-reaching biological consequences

This article reviews the production of different phenotypes from the same genotype in the same environment by stochastic cellular events, nonlinear mechanisms during patterning and morphogenesis, and probabilistic self-reinforcing circuitries in the adult life. These aspects of phenotypic variation are summarized under the term 'stochastic developmental variation' (SDV) in the following. In the past, SDV has been viewed primarily as a nuisance, impairing laboratory experiments, pharmaceutical testing, and true-to-type breeding. This article also emphasizes the positive biological effects of SDV and discusses implications for genotype-to-phenotype mapping, biological individuation, ecology, evolution, and applied biology. There is strong evidence from experiments with genetically identical organisms performed in narrowly standardized laboratory set-ups that SDV is a source of phenotypic variation in its own right aside from genetic variation and environmental variation. It is obviously mediated by molecular and higher-order epigenetic mechanisms. Comparison of SDV in animals, plants, fungi, protists, bacteria, archaeans, and viruses suggests that it is a ubiquitous and phylogenetically old phenomenon. In animals, it is usually smallest for morphometric traits and highest for life history traits and behaviour. SDV is thought to contribute to phenotypic diversity in all populations but is particularly relevant for asexually reproducing and genetically impoverished populations, where it generates individuality despite genetic uniformity. In each generation, SDV produces a range of phenotypes around a well-adapted target phenotype, which is interpreted as a bet-hedging strategy to cope with the unpredictability of dynamic environments. At least some manifestations of SDV are heritable, adaptable, selectable, and evolvable, and therefore, SDV may be seen as a hitherto overlooked evolution factor. SDV is also relevant for husbandry, agriculture, and medicine because most pathogens are asexuals that exploit this third source of phenotypic variation to modify infectivity and resistance to antibiotics. Since SDV affects all types of organisms and almost all aspects of life, it urgently requires more intense research and a better integration into biological thinking.

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.

Trade-offs in engineering sugar utilization pathways for titratable control

Titratable systems are common tools in metabolic engineering to tune the levels of enzymes and cellular components as part of pathway optimization. For nonmodel microorganisms with limited genetic tools, inducible sugar utilization pathways offer built-in titratable systems. However, these pathways can exhibit undesirable single-cell behaviors that hamper the uniform and tunable control of gene expression. Here, we applied mathematical modeling and single-cell measurements of l-arabinose utilization in Escherichia coli to systematically explore how sugar utilization pathways can be altered to achieve desirable inducible properties. We found that different pathway alterations, such as the removal of catabolism, constitutive expression of high-affinity or low-affinity transporters, or further deletion of the other transporters, came with trade-offs specific to each alteration. For instance, sugar catabolism improved the uniformity and linearity of the response at the cost of requiring higher sugar concentrations to induce the pathway. Within these alterations, we also found that a uniform and linear response could be achieved with a single alteration: constitutively expressing the high-affinity transporter. Equivalent modifications to the d-xylose utilization pathway yielded similar responses, demonstrating the applicability of our observations. Overall, our findings indicate that there is no ideal set of typical alterations when co-opting natural utilization pathways for titratable control and suggest design rules for manipulating these pathways to advance basic genetic studies and the metabolic engineering of microorganisms for optimized chemical production.

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.

Hysteresis can grant fitness in stochastically varying environment

Although the existence of multiple stable phenotypes of living organisms enables random switching between phenotypes as well as non-random history dependent switching called hysteresis, only random switching has been considered in prior experimental and theoretical models of adaptation to variable environments. This work considers the possibility that hysteresis may also evolve together with random phenotype switching to maximize population growth. In addition to allowing the possibility that switching rates between different phenotypes may depend not only on a continuous environmental input variable, but also on the phenotype itself, the present work considers an opportunity cost of the switching events. This opportunity cost arises as a result of a lag phase experimentally observed after phenotype switching and stochastic behavior of the environmental input. It is shown that stochastic environmental variation results in maximal asymptotic growth rate when organisms display hysteresis for sufficiently slowly varying environmental input. At the same time, sinusoidal input does not cause evolution of memory suggesting that the connection between the lag phase, stochastic environmental variation and evolution of hysteresis is a result of a stochastic resonance type phenomenon.

Bacterial sugar utilization gives rise to distinct single-cell behaviours

Inducible utilization pathways reflect widespread microbial strategies to uptake and consume sugars from the environment. Despite their broad importance and extensive characterization, little is known how these pathways naturally respond to their inducing sugar in individual cells. Here, we performed single-cell analyses to probe the behaviour of representative pathways in the model bacterium Escherichia coli. We observed diverse single-cell behaviours, including uniform responses (d-lactose, d-galactose, N-acetylglucosamine, N-acetylneuraminic acid), 'all-or-none' responses (d-xylose, l-rhamnose) and complex combinations thereof (l-arabinose, d-gluconate). Mathematical modelling and probing of genetically modified pathways revealed that the simple framework underlying these pathways - inducible transport and inducible catabolism - could give rise to most of these behaviours. Sugar catabolism was also an important feature, as disruption of catabolism eliminated tunable induction as well as enhanced memory of previous conditions. For instance, disruption of catabolism in pathways that respond to endogenously synthesized sugars led to full pathway induction even in the absence of exogenous sugar. Our findings demonstrate the remarkable flexibility of this simple biological framework, with direct implications for environmental adaptation and the engineering of synthetic utilization pathways as titratable expression systems and for metabolic engineering.

Regulation of the histidine utilization (hut) system in bacteria

The ability to degrade the amino acid histidine to ammonia, glutamate, and a one-carbon compound (formate or formamide) is a property that is widely distributed among bacteria. The four or five enzymatic steps of the pathway are highly conserved, and the chemistry of the reactions displays several unusual features, including the rearrangement of a portion of the histidase polypeptide chain to yield an unusual imidazole structure at the active site and the use of a tightly bound NAD molecule as an electrophile rather than a redox-active element in urocanase. Given the importance of this amino acid, it is not surprising that the degradation of histidine is tightly regulated. The study of that regulation led to three central paradigms in bacterial regulation: catabolite repression by glucose and other carbon sources, nitrogen regulation and two-component regulators in general, and autoregulation of bacterial regulators. This review focuses on three groups of organisms for which studies are most complete: the enteric bacteria, for which the regulation is best understood; the pseudomonads, for which the chemistry is best characterized; and Bacillus subtilis, for which the regulatory mechanisms are very different from those of the Gram-negative bacteria. The Hut pathway is fundamentally a catabolic pathway that allows cells to use histidine as a source of carbon, energy, and nitrogen, but other roles for the pathway are also considered briefly here.

Deterministic and stochastic simulation and analysis of biochemical reaction networks the lactose operon example

A brief introduction to mathematical modeling of biochemical regulatory reaction networks is presented. Both deterministic and stochastic modeling techniques are covered with examples from enzyme kinetics, coupled reaction networks with oscillatory dynamics and bistability. The Yildirim-Mackey model for lactose operon is used as an example to discuss and show how deterministic and stochastic methods can be used to investigate various aspects of this bacterial circuit.

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.

Epigenetics of the yeast galactose genetic switch

The transcriptional activation of enzymes involved in galactose utilization (GAL genes) in Saccharomyces cerevisiae is regulated by a complex interplay between three regulatory proteins encoded by GAL4 (transcriptional activator), GAL3 (signal transducer) and GAL80 (repressor). The relative concentrations of the signal transducer and the repressor are maintained by autoregulation. Cells disabled for autoregulation exhibit phenotypes distinctly different from that of the wild type cells, enabling us to explore the biological significance of autoregulation. The redundancy in signal transduction due to the presence of GAL1 (alternate signal transducer) also makes it a suitable model to understand the phenomenon of epigenetics. In this article we review some of the recent attempts made to understand the importance of epigenetics in the establishment of cellular and transcriptional memory.

Comparison of deterministic and stochastic models of the lac operon genetic network

The lac operon has been a paradigm for genetic regulation with positive feedback, and several modeling studies have described its dynamics at various levels of detail. However, it has not yet been analyzed how stochasticity can enrich the system's behavior, creating effects that are not observed in the deterministic case. To address this problem we use a comparative approach. We develop a reaction network for the dynamics of the lac operon genetic switch and derive corresponding deterministic and stochastic models that incorporate biological details. We then analyze the effects of key biomolecular mechanisms, such as promoter strength and binding affinities, on the behavior of the models. No assumptions or approximations are made when building the models other than those utilized in the reaction network. Thus, we are able to carry out a meaningful comparison between the predictions of the two models to demonstrate genuine effects of stochasticity. Such a comparison reveals that in the presence of stochasticity, certain biomolecular mechanisms can profoundly influence the region where the system exhibits bistability, a key characteristic of the lac operon dynamics. For these cases, the temporal asymptotic behavior of the deterministic model remains unchanged, indicating a role of stochasticity in modulating the behavior of the system.

Bistability, epigenetics, and bet-hedging in bacteria

Clonal populations of microbial cells often show a high degree of phenotypic variability under homogeneous conditions. Stochastic fluctuations in the cellular components that determine cellular states can cause two distinct subpopulations, a property called bistability. Phenotypic heterogeneity can be readily obtained by interlinking multiple gene regulatory pathways, effectively resulting in a genetic logic-AND gate. Although switching between states can occur within the cells' lifetime, cells can also pass their cellular state over to the next generation by a mechanism known as epigenetic inheritance and thus perpetuate the phenotypic state. Importantly, heterogeneous populations can demonstrate increased fitness compared with homogeneous populations. This suggests that microbial cells employ bet-hedging strategies to maximize survival. Here, we discuss the possible roles of interlinked bistable networks, epigenetic inheritance, and bet-hedging in bacteria.

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.

Predicting the asymmetric response of a genetic switch to noise

We present a simple analytical tool which gives an approximate insight into the stationary behavior of nonlinear systems undergoing the influence of a weak and rapid noise from one dominating source, e.g. the kinetic equations describing a genetic switch with the concentration of one substrate fluctuating around a constant mean. The proposed method allows for predicting the asymmetric response of the genetic switch to noise, arising from the noise-induced shift of stationary states. The method has been tested on an example model of the lac operon regulatory network: a reduced Yildirim-Mackey model with fluctuating extracellular lactose concentration. We calculate analytically the shift of the system's stationary states in the presence of noise. The results of the analytical calculation are in excellent agreement with the results of numerical simulation of the noisy system. The simulation results suggest that the structure of the kinetics of the underlying biochemical reactions protects the bistability of the lactose utilization mechanism from environmental fluctuations. We show that, in the consequence of the noise-induced shift of stationary states, the presence of fluctuations stabilizes the behavior of the system in a selective way: Although the extrinsic noise facilitates, to some extent, switching off the lactose metabolism, the same noise prevents it from switching on.

Bistability of the lac operon during growth of Escherichia coli on lactose and lactose+glucose

The lac operon of Escherichia coli can exhibit bistability. Early studies showed that bistability occurs during growth on TMG/succinate and lactose+glucose, but not during growth on lactose. More recently, studies with lacGFP-transfected cells show bistability during growth on TMG/succinate, but not during growth on lactose and lactose+glucose. In the literature, these results are invariably attributed to variations in the destabilizing effect of the positive feedback generated by induction. Specifically, during growth on TMG/succinate, lac induction generates strong positive feedback because the permease stimulates the accumulation of intracellular TMG, which in turn, promotes the synthesis of even more permease. This positive feedback is attenuated during growth on lactose because hydrolysis of intracellular lactose by beta-galactosidase suppresses the stimulatory effect of the permease. It is attenuated even more during growth on lactose + glucose because glucose inhibits the uptake of lactose. But it is clear that the stabilizing effect of dilution also changes dramatically as a function of the medium composition. For instance, during growth on TMG/succinate, the dilution rate of lac permease is proportional to its activity, e, because the specific growth rate is independent of e (it is completely determined by the concentration of succinate). However, during growth on lactose, the dilution rate of the permease is proportional to e2 because the specific growth rate is proportional to the specific lactose uptake rate, which in turn, proportional to e. We show that: (a) This dependence on e2 creates such a strong stabilizing effect that bistability is virtually impossible during growth on lactose, even in the face of the intense positive feedback generated by induction. (b) This stabilizing effect is weakened during growth on lactose+glucose because the specific growth rate on glucose is independent of e, so that the dilution rate once again contains a term that is proportional to e. These results imply that the lac operon is much more prone to bistability if the medium contains carbon sources that cannot be metabolized by the lac enzymes, e.g., succinate during growth on TMG/succinate and glucose during growth on lactose+glucose. We discuss the experimental data in the light of these results.

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

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.

Origin of bistability in the lac Operon

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.

Phenotypic variation in bacteria: the role of feedback regulation

To survive in rapidly changing environmental conditions, bacteria have evolved a diverse set of regulatory pathways that govern various adaptive responses. Recent research has reinforced the notion that bacteria use feedback-based circuitry to generate population heterogeneity in natural situations. Using artificial gene networks, it has been shown that a relatively simple 'wiring' of a bacterial genetic system can generate two or more stable subpopulations within an overall genetically homogeneous population. This review discusses the ubiquity of these processes throughout nature, as well as the presumed molecular mechanisms responsible for the heterogeneity observed in a selection of bacterial species.

L’épigénétique comme aspect de la postgénomique

A cell transmits to its progeny the activity level of many of its genes, not just their sequence. Just like the sequence may vary through a mutation, the gene activity level may change through an "epimutation" (an epigenetic modification) which is heritable and does not entail any concomitant genetic alteration. An epimutation can have important phenotypic consequences, that eventually survive to the loss of the environmental conditions that triggered it. For instance, epimutations are responsible for the divergence between a neuron and an epithelial cell that both come from the same egg and contain the same genome complement. This phenotypic difference is much larger than the one between the neurons from two animal species with dissimilar genotypes, thereby underlining the importance of epimutations. Tradition opposes the genetic and epigenetic visions, the latter being often adequated to the DNA methylation phenomenon. However, epimutations display a rich spectrum of modes that can all fit in a unique reference system based on correlated chemical, spatial and temporal scales. This reference system allows the integration of purely genetic mutations at one of its ends, thus paving the way to a new, gradual vision that encompasses the genome and the epigenome. At the other end can be found two types of epimutations that are both wide-ranging in space and rapid in producing phenotypic alterations. Firstly, long-range rearrangements of the three-dimensional structure of the chromosome may influence gene expression in an heritable fashion. Such rearrangements seem to result from the collective dynamics of DNA-related activities, particularly transcription. Lastly, heritable regulatory states, e.g. a differentiated state that results from tipping a regulatory "toggle switch", involve components that are distributed throughout the nucleus or the cytoplasm, and possibly all the way to cell confines.

Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks

The biochemical reaction networks include elementary reactions differing by many orders of magnitude in the numbers of molecules involved. The kinetics of reactions involving small numbers of molecules can be studied by exact stochastic simulation. This approach is not practical for the simulation of metabolic processes because of the computational cost of accounting for individual molecular collisions. We present the "maximal time step method," a novel approach combining the Gibson and Bruck algorithm with the Gillespie tau-leap method. This algorithm allows stochastic simulation of systems composed of both intensive metabolic reactions and regulatory processes involving small numbers of molecules. The method is applied to the simulation of glucose, lactose, and glycerol metabolism in Escherichia coli. The gene expression, signal transduction, transport, and enzymatic activities are modeled simultaneously. We show that random fluctuations in gene expression can propagate to the level of metabolic processes. In the cells switching from glucose to a mixture of lactose and glycerol, random delays in transcription initiation determine whether lactose or glycerol operon is induced. In a small fraction of cells severe decrease in metabolic activity may also occur. Both effects are epigenetically inherited by the progeny of the cell in which the random delay in transcription initiation occurred.

Modeling operon dynamics: the tryptophan and lactose operons as paradigms

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.

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 beta-galactosidase involvement and simplify it by ignoring permease dynamics (assuming a constant permease concentration). To test the behavior of the reduced model, three existing sets of data on beta-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.

Modeling network dynamics: the lac operon, a case study

We use the lac operon in Escherichia coli as a prototype system to illustrate the current state, applicability, and limitations of modeling the dynamics of cellular networks. We integrate three different levels of description (molecular, cellular, and that of cell population) into a single model, which seems to capture many experimental aspects of the system.

Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data

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; beta-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 beta-galactosidase activity versus time data, as well as a set of data on beta-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.

Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior

A biological introduction serves to remind us that differentiation is an epigenetic process, that multistationarity can account for epigenetic differences, including those involved in cell differentiation, and that positive feedback circuits are a necessary condition for multistationarity and, by inference, for differentiation. The core of the paper is comprised of a formal description of feedback circuits and unions of disjoint circuits. We introduce the concepts of full-circuit (a circuit or union of disjoint circuits which involves all the variables of the system), and of ambiguous circuit (a circuit whose sign depends on the location in phase space). We describe the partition of phase space (a) according to the signs of the ambiguous circuits, and (b) according to the signs of the eigenvalues or their real part. We introduce a normalization of the system versus one of the circuits; in two variables, this permits an entirely general description in terms of a common diagram in the "circuit space." The paper ends with general statements concerning the requirements for multistationarity, stable periodicity, and deterministic chaos. (c) 2001 American Institute of Physics.

Multistability: a major means of differentiation and evolution in biological systems

Very simple biochemical systems regulated at the level of gene expression or protein function are capable of complex dynamic behaviour. Among the various patterns of regulation associated with non-linear kinetics, multistability, which corresponds to a true switch between alternate steady states, allows a graded signal to be turned into a discontinuous evolution of the system along several possible distinct pathways, which can be either reversible or irreversible. Multistability plays a significant role in some of the basic processes of life. It might account for maintenance of phenotypic differences in the absence of genetic or environmental differences, as has been demonstrated experimentally for the regulation of the lactose operon in Escherichia coli. Cell differentiation might also be explained as multistability.

Kinetic studies of beta-galactosidase induction

The kinetics of beta-galactosidase induction in E. coli ML 3 have been studied. Following addition of inducer, the rate of enzyme synthesis accelerates from the uninduced to a steady-state rate. At saturating concentration of inducer the time constant (T(c)) for this process is 2.5 to 3 minutes. With decreasing inducer concentration (I), increasing time constants are observed. I/I + K' approximates I/T(c). The steady-state rate of beta-galactosidase synthesis is approximated by I(2)/I(2) + K(2). K' and K have been estimated for IPTG and TMG. The kinetics of beta-galactosidase production after the removal of inducer by dilution or after the addition of glucose have been investigated. A transition time of 2.5 to 3 minutes is observed before enzyme synthesis slows or stops. These results are consistent with the hypothesis that the enzyme-forming unit is unstable.

A complex control circuit. Regulation of immunity in temperate bacteriophages

Temperate bacteriophages can display in a stable way two essentially different behaviours. In the immune state, a gene (cI) produces a repressor which prevents expression of all the other viral genes; in the non-immune state the typically viral functions are expressed. The choice between the two pathways and the establishment of one of them have much in common with cell determination and differentiation. This choice depends on a complex control system, in fact one of the most intricate nets of regulation known in some detail. Our paper provides a formal description and partial analysis of this regulatory net. It is shown that even for relatively simple known models, this kind of analysis uncovers predictions which had previously remained hidden. Some of these predictions were checked experimentally. The experimental part chiefly deals with the efficiency of lysogenization by thermoinducible lambda phage carrying mutations in one or more of the regulatory genes, N, cro and cII. Although N- mutations are widely known for preventing efficient integration, and both N- and cII mutations for preventing efficient establishment of immunity, it is shown that, as predicted by a simple model, both N- and cII- phage efficiently lysogenize at low temperature if they are in addition cro-. In contrast with lambda N- cro+, lambda N- cro- is not propagated as a plasmid at low temperature, precisely because it establishes immunity too efficiently. Genetic control circuits are described in terms of sets of logic equations, which relate the state of expression of genes or of chemical reactions (functions) to input (genetic and environmental) variables and to the presence of gene and reaction products (internal, or memorization varibles). From the set of equations, one derives a matrix which shows the stable stationary states (if any) of the system, and from which one can derive the pathways (temporal sequences of states) consistent with the model. This kind of analysis is complementary to the more widely used analysis based on differential equations; it allows one to analyze in less detail more complex systems. The language might be used as well, mutatis mutandis, in fields very different from genetics. The last part of the discussion deals with the role of positive feedback loops in our specific problem (establishment and maintenance of immunity in temperate bacteriophages) and in developmental genetics in general. As a generalization of an old idea, it is suggested that cell determination (for a given character) depends on a set of genes whose interaction constitutes a positive feedback loop. Such a system has two stable stationary states: which one is chosen will usually depend on additional controls grafted on the loop.

Lag in adaptation to lactose as a probe to the timing of permease incorporation into the cell membrane

If bacteria are incapable of forming and incorporating proteins into the cytoplasmic membranes in all phases of the cell cycle, then not all cells from an asynchronous culture should be capable of growth when switched to a new carbon and energy source whose metabolism requires new membrane function. The transfer of an inducible culture to low lactose provides such a situation since the cells cannot grow unless galactoside permease can function to concentrate the lactose internally. From such experiments, it was concluded that the Y gene product of the lac operon is synthesized, incorporated, and can start functioning in active transport, at any time throughout the bulk of the cell cycle. Not only were the lags before growth re-ensued much shorter than would be expected if the membrane transport capability could only be developed in a small portion of the cycle, but brief pulses of a gratuitous inducer shortened the lags much further. Three types of Escherichia coli ML 30 culture were studied: cells that had exhausted the limiting glucose; cells taken directly from glucose-limited chemostats; and a washed suspension of highly catabolite repressed cells from cultures grown in high levels of glucose and gluconate. The growth studies reported here were performed on-line with a minicomputer. They represent at least an order of magnitude increase in accuracy in estimating growth parameters over previous instrumentation.