Optimal protein production by a synthetic microbial consortium: coexistence, distribution of labor, and syntrophy

The bacterium E. coli is widely used to produce recombinant proteins such as growth hormone and insulin. One inconvenience with E. coli cultures is the secretion of acetate through overflow metabolism. Acetate inhibits cell growth and represents a carbon diversion, which results in several negative effects on protein production. One way to overcome this problem is the use of a synthetic consortium of two different E. coli strains, one producing recombinant proteins and one reducing the acetate concentration. In this paper, we study a mathematical model of such a synthetic community in a chemostat where both strains are allowed to produce recombinant proteins. We give necessary and sufficient conditions for the existence of a coexistence equilibrium and show that it is unique. Based on this equilibrium, we define a multi-objective optimization problem for the maximization of two important bioprocess performance metrics, process yield and productivity. Solving numerically this problem, we find the best available trade-offs between the metrics. Under optimal operation of the mixed community, both strains must produce the protein of interest, and not only one (distribution instead of division of labor). Moreover, in this regime acetate secretion by one strain is necessary for the survival of the other (syntrophy). The results thus illustrate how complex multi-level dynamics shape the optimal production of recombinant proteins by synthetic microbial consortia.

Resource allocation accounts for the large variability of rate-yield phenotypes across bacterial strains

Different strains of a microorganism growing in the same environment display a wide variety of growth rates and growth yields. We developed a coarse-grained model to test the hypothesis that different resource allocation strategies, corresponding to different compositions of the proteome, can account for the observed rate-yield variability. The model predictions were verified by means of a database of hundreds of published rate-yield and uptake-secretion phenotypes of Escherichia coli strains grown in standard laboratory conditions. We found a very good quantitative agreement between the range of predicted and observed growth rates, growth yields, and glucose uptake and acetate secretion rates. These results support the hypothesis that resource allocation is a major explanatory factor of the observed variability of growth rates and growth yields across different bacterial strains. An interesting prediction of our model, supported by the experimental data, is that high growth rates are not necessarily accompanied by low growth yields. The resource allocation strategies enabling high-rate, high-yield growth of E. coli lead to a higher saturation of enzymes and ribosomes, and thus to a more efficient utilization of proteomic resources. Our model thus contributes to a fundamental understanding of the quantitative relationship between rate and yield in E. coli and other microorganisms. It may also be useful for the rapid screening of strains in metabolic engineering and synthetic biology.

Maturation models of fluorescent proteins are necessary for unbiased estimates of promoter activity

Fluorescent proteins (FPs) are a powerful tool to quantitatively monitor gene expression. The dynamics of a promoter and its regulation can be inferred from fluorescence data. The interpretation of fluorescent data, however, is strongly dependent on the maturation of FPs since different proteins mature in distinct ways. We propose a novel approach for analyzing fluorescent reporter data by incorporating maturation dynamics in the reconstruction of promoter activities. Our approach consists of developing and calibrating mechanistic maturation models for distinct FPs. These models are then used alongside a Bayesian approach to estimate promoter activities from fluorescence data. We demonstrate by means of targeted experiments in Escherichia coli that our approach provides robust estimates and that accounting for maturation is, in many cases, essential for the interpretation of gene expression data.

Multiomics Study of Bacterial Growth Arrest in a Synthetic Biology Application

We investigated the scalability of a previously developed growth switch based on external control of RNA polymerase expression. Our results indicate that, in liter-scale bioreactors operating in fed-batch mode, growth-arrested Escherichia coli cells are able to convert glucose to glycerol at an increased yield. A multiomics quantification of the physiology of the cells shows that, apart from acetate production, few metabolic side effects occur. However, a number of specific responses to growth slow-down and growth arrest are launched at the transcriptional level. These notably include the downregulation of genes involved in growth-associated processes, such as amino acid and nucleotide metabolism and translation. Interestingly, the transcriptional responses are buffered at the proteome level, probably due to the strong decrease of the total mRNA concentration after the diminution of transcriptional activity and the absence of growth dilution of proteins. Growth arrest thus reduces the opportunities for dynamically adjusting the proteome composition, which poses constraints on the design of biotechnological production processes but may also avoid the initiation of deleterious stress responses.

Optimal proteome allocation and the temperature dependence of microbial growth laws

Although the effect of temperature on microbial growth has been widely studied, the role of proteome allocation in bringing about temperature-induced changes remains elusive. To tackle this problem, we propose a coarse-grained model of microbial growth, including the processes of temperature-sensitive protein unfolding and chaperone-assisted (re)folding. We determine the proteome sector allocation that maximizes balanced growth rate as a function of nutrient limitation and temperature. Calibrated with quantitative proteomic data for Escherichia coli, the model allows us to clarify general principles of temperature-dependent proteome allocation and formulate generalized growth laws. The same activation energy for metabolic enzymes and ribosomes leads to an Arrhenius increase in growth rate at constant proteome composition over a large range of temperatures, whereas at extreme temperatures resources are diverted away from growth to chaperone-mediated stress responses. Our approach points at risks and possible remedies for the use of ribosome content to characterize complex ecosystems with temperature variation.

Enhanced production of heterologous proteins by a synthetic microbial community: Conditions and trade-offs

Synthetic microbial consortia have been increasingly utilized in biotechnology and experimental evidence shows that suitably engineered consortia can outperform individual species in the synthesis of valuable products. Despite significant achievements, though, a quantitative understanding of the conditions that make this possible, and of the trade-offs due to the concurrent growth of multiple species, is still limited. In this work, we contribute to filling this gap by the investigation of a known prototypical synthetic consortium. A first E. coli strain, producing a heterologous protein, is sided by a second E. coli strain engineered to scavenge toxic byproducts, thus favoring the growth of the producer at the expense of diverting part of the resources to the growth of the cleaner. The simplicity of the consortium is ideal to perform an in depth-analysis and draw conclusions of more general interest. We develop a coarse-grained mathematical model that quantitatively accounts for literature data from different key growth phenotypes. Based on this, assuming growth in chemostat, we first investigate the conditions enabling stable coexistence of both strains and the effect of the metabolic load due to heterologous protein production. In these conditions, we establish when and to what extent the consortium outperforms the producer alone in terms of productivity. Finally, we show in chemostat as well as in a fed-batch scenario that gain in productivity comes at the price of a reduced yield, reflecting at the level of the consortium resource allocation trade-offs that are well-known for individual species.

In nature, microorganisms occur in communities comprising a variety of mutually interacting species. Established through evolution, these interactions allow for the survival and growth of microorganisms in their natural environment, and give rise to complex dynamics that could not be exhibited by any of the species in isolation. The richness of microbial community dynamics has been leveraged to outperform individual species in biotechnological production processes and other processes of high societal value. Yet, in view of their complexity, natural communities are difficult to study and control. In order to overcome these issues, a rapidly growing research field concerns the rational design and engineering of synthetic microbial consortia. Despite the great potential of synthetic microbial consortia, and significant efforts devoted to their mathematical modelling and analysis, a detailed understanding of how enhanced production can be achieved, and at what cost, is still unavailable. In this work, based on a quantitative model of a prototypical synthetic microbial consortium, we determine precise conditions under which a consortium outperforms individual species in the production of a recombinant protein. Moreover, we identify the inherent trade-offs between productivity and efficiency of substrate utilization.

Acetate Metabolism and the Inhibition of Bacterial Growth by Acetate

During aerobic growth on glucose, Escherichia coli excretes acetate, a mechanism called "overflow metabolism." At high concentrations, the secreted acetate inhibits growth. Several mechanisms have been proposed for explaining this phenomenon, but a thorough analysis is hampered by the diversity of experimental conditions and strains used in these studies. Here, we describe the construction of a set of isogenic strains that remove different parts of the metabolic network involved in acetate metabolism. Analysis of these strains reveals that (i) high concentrations of acetate in the medium inhibit growth without significantly perturbing central metabolism; (ii) growth inhibition persists even when acetate assimilation is completely blocked; and (iii) regulatory interactions mediated by acetyl-phosphate play a small but significant role in growth inhibition by acetate. The major contribution to growth inhibition by acetate may originate in systemic effects like the uncoupling effect of organic acids or the perturbation of the anion composition of the cell, as previously proposed. Our data suggest, however, that under the conditions considered here, the uncoupling effect plays only a limited role.IMPORTANCE High concentrations of organic acids such as acetate inhibit growth of Escherichia coli and other bacteria. This phenomenon is of interest for understanding bacterial physiology but is also of practical relevance. Growth inhibition by organic acids underlies food preservation and causes problems during high-density fermentation in biotechnology. What causes this phenomenon? Classical explanations invoke the uncoupling effect of acetate and the establishment of an anion imbalance. Here, we propose and investigate an alternative hypothesis: the perturbation of acetate metabolism due to the inflow of excess acetate. We find that this perturbation accounts for 20% of the growth-inhibitory effect through a modification of the acetyl phosphate concentration. Moreover, we argue that our observations are not expected based on uncoupling alone.

An ensemble of mathematical models showing diauxic growth behaviour

Background

Carbon catabolite repression (CCR) controls the order in which different carbon sources are metabolised. Although this system is one of the paradigms of regulation in bacteria, the underlying mechanisms remain controversial. CCR involves the coordination of different subsystems of the cell - responsible for the uptake of carbon sources, their breakdown for the production of energy and precursors, and the conversion of the latter to biomass. The complexity of this integrated system, with regulatory mechanisms cutting across metabolism, gene expression, and signalling, has motivated important modelling efforts over the past four decades, especially in the enterobacterium Escherichia coli.

Results

Starting from a simple core model with only four intracellular metabolites, we develop an ensemble of model variants, all showing diauxic growth behaviour during a batch process. The model variants fall into one of the four categories: flux balance models, kinetic models with growth dilution, kinetic models with regulation, and resource allocation models. The model variants differ from one another in only a single aspect, each breaking the symmetry between the two substrate assimilation pathways in a different manner, and can be quantitatively compared using a so-called diauxic growth index. For each of the model variants, we predict the behaviour in two new experimental conditions, namely a glucose pulse for a culture growing in minimal medium with lactose and a batch culture with different initial concentrations of the components of the transport systems. When qualitatively comparing these predictions with experimental data for these two conditions, a number of models can be excluded while other model variants are still not discriminable. The best-performing model variants are based on inducer inclusion and activation of enzymatic genes by a global transcription factor, but the other proposed factors may complement these well-known regulatory mechanisms.

Conclusions

The model ensemble presented here offers a better understanding of the variety of mechanisms that have been proposed to play a role in CCR. In addition, it provides an educational resource for systems biology, as it gives an introduction to a broad range of modeling approaches in the context of a simple but biologically relevant example.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0604-8) contains supplementary material, which is available to authorized users.

Estimation of time-varying growth, uptake and excretion rates from dynamic metabolomics data

Motivation

Technological advances in metabolomics have made it possible to monitor the concentration of extracellular metabolites over time. From these data, it is possible to compute the rates of uptake and excretion of the metabolites by a growing cell population, providing precious information on the functioning of intracellular metabolism. The computation of the rate of these exchange reactions, however, is difficult to achieve in practice for a number of reasons, notably noisy measurements, correlations between the concentration profiles of the different extracellular metabolites, and discontinuties in the profiles due to sudden changes in metabolic regime.

Results

We present a method for precisely estimating time-varying uptake and excretion rates from time-series measurements of extracellular metabolite concentrations, specifically addressing all of the above issues. The estimation problem is formulated in a regularized Bayesian framework and solved by a combination of extended Kalman filtering and smoothing. The method is shown to improve upon methods based on spline smoothing of the data. Moreover, when applied to two actual datasets, the method recovers known features of overflow metabolism in Escherichia coli and Lactococcus lactis, and provides evidence for acetate uptake by L. lactis after glucose exhaustion. The results raise interesting perspectives for further work on rate estimation from measurements of intracellular metabolites.

Availability and implementation

The Matlab code for the estimation method is available for download at https://team.inria.fr/ibis/rate-estimation-software/, together with the datasets.

Supplementary information

Supplementary data are available at Bioinformatics online.

Mathematical modelling of microbes: metabolism, gene expression and growth

The growth of microorganisms involves the conversion of nutrients in the environment into biomass, mostly proteins and other macromolecules. This conversion is accomplished by networks of biochemical reactions cutting across cellular functions, such as metabolism, gene expression, transport and signalling. Mathematical modelling is a powerful tool for gaining an understanding of the functioning of this large and complex system and the role played by individual constituents and mechanisms. This requires models of microbial growth that provide an integrated view of the reaction networks and bridge the scale from individual reactions to the growth of a population. In this review, we derive a general framework for the kinetic modelling of microbial growth from basic hypotheses about the underlying reaction systems. Moreover, we show that several families of approximate models presented in the literature, notably flux balance models and coarse-grained whole-cell models, can be derived with the help of additional simplifying hypotheses. This perspective clearly brings out how apparently quite different modelling approaches are related on a deeper level, and suggests directions for further research.

Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies

Microbial physiology exhibits growth laws that relate the macromolecular composition of the cell to the growth rate. Recent work has shown that these empirical regularities can be derived from coarse-grained models of resource allocation. While these studies focus on steady-state growth, such conditions are rarely found in natural habitats, where microorganisms are continually challenged by environmental fluctuations. The aim of this paper is to extend the study of microbial growth strategies to dynamical environments, using a self-replicator model. We formulate dynamical growth maximization as an optimal control problem that can be solved using Pontryagin’s Maximum Principle. We compare this theoretical gold standard with different possible implementations of growth control in bacterial cells. We find that simple control strategies enabling growth-rate maximization at steady state are suboptimal for transitions from one growth regime to another, for example when shifting bacterial cells to a medium supporting a higher growth rate. A near-optimal control strategy in dynamical conditions is shown to require information on several, rather than a single physiological variable. Interestingly, this strategy has structural analogies with the regulation of ribosomal protein synthesis by ppGpp in the enterobacterium Escherichia coli. It involves sensing a mismatch between precursor and ribosome concentrations, as well as the adjustment of ribosome synthesis in a switch-like manner. Our results show how the capability of regulatory systems to integrate information about several physiological variables is critical for optimizing growth in a changing environment.

Microbial growth is the process by which cells sustain and reproduce themselves from available matter and energy. Strategies enabling microorganisms to optimize their growth rate have been extensively studied, but mostly in stable environments. Here, we build a coarse-grained model of microbial growth and use methods from optimal control theory to determine a resource allocation scheme that would lead to maximal biomass accumulation when the cells are dynamically shifted from one growth medium to another. We compare this optimal solution with several cellular implementations of growth control, based on the capacity of the cell to sense different physiological variables. We find that strategies maximizing growth in steady-state conditions perform quite differently in dynamical conditions. Moreover, the control strategy with performance close to the theoretical maximum exploits information of more than one physiological variable, suggesting that optimization of microbial growth in dynamical rather than steady environments requires broader sensory capacities. Interestingly, the ppGpp alarmone system in the enterobacterium Escherichia coli, known to play an important role in growth control, has structural similarities with the control strategy approaching the theoretical maximum. It senses a discrepancy between the concentrations of precursors and ribosomes, and adjusts ribosome synthesis in an on-off fashion. This suggests that E. coli is adapted for environments with intermittent, rapid changes in nutrient availability.

A synthetic growth switch based on controlled expression of RNA polymerase

The ability to control growth is essential for fundamental studies of bacterial physiology and biotechnological applications. We have engineered an Escherichia coli strain in which the transcription of a key component of the gene expression machinery, RNA polymerase, is under the control of an inducible promoter. By changing the inducer concentration in the medium, we can adjust the RNA polymerase concentration and thereby switch bacterial growth between zero and the maximal growth rate supported by the medium. We show that our synthetic growth switch functions in a medium-independent and reversible way, and we provide evidence that the switching phenotype arises from the ultrasensitive response of the growth rate to the concentration of RNA polymerase. We present an application of the growth switch in which both the wild-type E. coli strain and our modified strain are endowed with the capacity to produce glycerol when growing on glucose. Cells in which growth has been switched off continue to be metabolically active and harness the energy gain to produce glycerol at a twofold higher yield than in cells with natural control of RNA polymerase expression. Remarkably, without any further optimization, the improved yield is close to the theoretical maximum computed from a flux balance model of E. coli metabolism. The proposed synthetic growth switch is a promising tool for gaining a better understanding of bacterial physiology and for applications in synthetic biology and biotechnology.

Robust reconstruction of gene expression profiles from reporter gene data using linear inversion

MOTIVATION

Time-series observations from reporter gene experiments are commonly used for inferring and analyzing dynamical models of regulatory networks. The robust estimation of promoter activities and protein concentrations from primary data is a difficult problem due to measurement noise and the indirect relation between the measurements and quantities of biological interest.

RESULTS

We propose a general approach based on regularized linear inversion to solve a range of estimation problems in the analysis of reporter gene data, notably the inference of growth rate, promoter activity, and protein concentration profiles. We evaluate the validity of the approach using in silico simulation studies, and observe that the methods are more robust and less biased than indirect approaches usually encountered in the experimental literature based on smoothing and subsequent processing of the primary data. We apply the methods to the analysis of fluorescent reporter gene data acquired in kinetic experiments with Escherichia coli. The methods are capable of reliably reconstructing time-course profiles of growth rate, promoter activity and protein concentration from weak and noisy signals at low population volumes. Moreover, they capture critical features of those profiles, notably rapid changes in gene expression during growth transitions.

AVAILABILITY AND IMPLEMENTATION

The methods described in this article are made available as a Python package (LGPL license) and also accessible through a web interface. For more information, see https://team.inria.fr/ibis/wellinverter.

Mass spectrometry-based workflow for accurate quantification of Escherichia coli enzymes: how proteomics can play a key role in metabolic engineering

Metabolic engineering aims to design high performance microbial strains producing compounds of interest. This requires systems-level understanding; genome-scale models have therefore been developed to predict metabolic fluxes. However, multi-omics data including genomics, transcriptomics, fluxomics, and proteomics may be required to model the metabolism of potential cell factories. Recent technological advances to quantitative proteomics have made mass spectrometry-based quantitative assays an interesting alternative to more traditional immuno-affinity based approaches. This has improved specificity and multiplexing capabilities. In this study, we developed a quantification workflow to analyze enzymes involved in central metabolism in Escherichia coli (E. coli). This workflow combined full-length isotopically labeled standards with selected reaction monitoring analysis. First, full-length (15)N labeled standards were produced and calibrated to ensure accurate measurements. Liquid chromatography conditions were then optimized for reproducibility and multiplexing capabilities over a single 30-min liquid chromatography-MS analysis. This workflow was used to accurately quantify 22 enzymes involved in E. coli central metabolism in a wild-type reference strain and two derived strains, optimized for higher NADPH production. In combination with measurements of metabolic fluxes, proteomics data can be used to assess different levels of regulation, in particular enzyme abundance and catalytic rate. This provides information that can be used to design specific strains used in biotechnology. In addition, accurate measurement of absolute enzyme concentrations is key to the development of predictive kinetic models in the context of metabolic engineering.

SBML qualitative models: a model representation format and infrastructure to foster interactions between qualitative modelling formalisms and tools

BACKGROUND

Qualitative frameworks, especially those based on the logical discrete formalism, are increasingly used to model regulatory and signalling networks. A major advantage of these frameworks is that they do not require precise quantitative data, and that they are well-suited for studies of large networks. While numerous groups have developed specific computational tools that provide original methods to analyse qualitative models, a standard format to exchange qualitative models has been missing.

RESULTS

We present the Systems Biology Markup Language (SBML) Qualitative Models Package ("qual"), an extension of the SBML Level 3 standard designed for computer representation of qualitative models of biological networks. We demonstrate the interoperability of models via SBML qual through the analysis of a specific signalling network by three independent software tools. Furthermore, the collective effort to define the SBML qual format paved the way for the development of LogicalModel, an open-source model library, which will facilitate the adoption of the format as well as the collaborative development of algorithms to analyse qualitative models.

CONCLUSIONS

SBML qual allows the exchange of qualitative models among a number of complementary software tools. SBML qual has the potential to promote collaborative work on the development of novel computational approaches, as well as on the specification and the analysis of comprehensive qualitative models of regulatory and signalling networks.

Shared control of gene expression in bacteria by transcription factors and global physiology of the cell

A simple, parameterless mathematical model, in combination with real-time monitoring of promoter activities, shows how control of gene expression in bacteria is shared between transcription factors and global physiological effects.

We present an approach based on a simple, paramaterless mathematical model to analyze the control of gene expression by transcription factors and the global physiological state of the cell.

We illustrate the strength of this approach by means of time-resolved measurements of the transcriptional activities of genes in a central regulatory circuit in Escherichia coli.

We conclude that global physiological effects rather than transcription factors dominate the control of gene expression during a growth transition.

Our results call for a reappraisal of the role of transcription factors, which may be most appropriately viewed as complementing and finetuning global control exerted by the physiological state of the cell.

Gene expression is controlled by the joint effect of (i) the global physiological state of the cell, in particular the activity of the gene expression machinery, and (ii) DNA-binding transcription factors and other specific regulators. We present a model-based approach to distinguish between these two effects using time-resolved measurements of promoter activities. We demonstrate the strength of the approach by analyzing a circuit involved in the regulation of carbon metabolism in E. coli. Our results show that the transcriptional response of the network is controlled by the physiological state of the cell and the signaling metabolite cyclic AMP (cAMP). The absence of a strong regulatory effect of transcription factors suggests that they are not the main coordinators of gene expression changes during growth transitions, but rather that they complement the effect of global physiological control mechanisms. This change of perspective has important consequences for the interpretation of transcriptome data and the design of biological networks in biotechnology and synthetic biology.

A genome-wide screen for identifying all regulators of a target gene

We have developed a new screening methodology for identifying all genes that control the expression of a target gene through genetic or metabolic interactions. The screen combines mutant libraries with luciferase reporter constructs, whose expression can be monitored in vivo and over time in different environmental conditions. We apply the method to identify the genes that control the expression of the gene acs, encoding the acetyl coenzyme A synthetase, in Escherichia coli. We confirm most of the known genetic regulators, including CRP-cAMP, IHF and components of the phosphotransferase system. In addition, we identify new regulatory interactions, many of which involve metabolic intermediates or metabolic sensing, such as the genes pgi, pfkA, sucB and lpdA, encoding enzymes in glycolysis and the TCA cycle. Some of these novel interactions were validated by quantitative reverse transcriptase-polymerase chain reaction. More generally, we observe that a large number of mutants directly or indirectly influence acs expression, an effect confirmed for a second promoter, sdhC. The method is applicable to any promoter fused to a luminescent reporter gene in combination with a deletion mutant library.

Towards multiscale plant models: integrating cellular networks

One of the ambitions of 'crop systems biology' is to combine information from molecular biology with a broader view of plant development and growth. In the context of modeling, this calls for a multiscale perspective that focuses on the interplay between cellular and macroscopic studies. With this in mind, in this review we aim to draw attention to a panel of approaches that were developed in the context of systems biology and are used for analyzing and describing the behavior of cellular networks. Ultimately, insights obtained from these methods can be exploited to refine the description of plant processes, leading to integrated plant-cellular models.

Towards multiscale plant models: integrating cellular networks

One of the ambitions of 'crop systems biology' is to combine information from molecular biology with a broader view of plant development and growth. In the context of modeling, this calls for a multiscale perspective that focuses on the interplay between cellular and macroscopic studies. With this in mind, in this review we aim to draw attention to a panel of approaches that were developed in the context of systems biology and are used for analyzing and describing the behavior of cellular networks. Ultimately, insights obtained from these methods can be exploited to refine the description of plant processes, leading to integrated plant-cellular models.

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only 4 out of 31 reactions, and 37 out of 100 parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.

Genetic network analyzer: a tool for the qualitative modeling and simulation of bacterial regulatory networks

Genetic Network Analyzer (GNA) is a tool for the qualitative modeling and simulation of gene regulatory networks, based on so-called piecewise-linear differential equation models. We describe the use of this tool in the context of the modeling of bacterial regulatory networks, notably the network of global regulators controlling the adaptation of Escherichia coli to carbon starvation conditions. We show how the modeler, by means of GNA, can define a regulatory network, build a model of the network, determine the steady states of the system, perform a qualitative simulation of the network dynamics, and analyze the simulation results using model-checking tools. The example illustrates the interest of qualitative approaches for the analysis of the dynamics of bacterial regulatory networks.

Importance of metabolic coupling for the dynamics of gene expression following a diauxic shift in Escherichia coli

Gene regulatory networks consist of direct interactions, but also include indirect interactions mediated by metabolism. We investigate to which extent these indirect interactions arising from metabolic coupling influence the dynamics of the system. To this end, we build a qualitative model of the gene regulatory network controlling carbon assimilation in Escherichia coli, and use this model to study the changes in gene expression following a diauxic shift from glucose to acetate. In particular, we compare the relative variation in the steady-state concentrations of enzymes and transcription regulators during growth on glucose and acetate, as well as the dynamic response of gene expression to the exhaustion of glucose and the subsequent assimilation of acetate. We find significant differences between the dynamics of the system in the absence and presence of metabolic coupling. This shows that interactions arising from metabolic coupling cannot be ignored when studying the dynamics of gene regulatory networks.

The logic layout of the TOL network of Pseudomonas putida pWW0 plasmid stems from a metabolic amplifier motif (MAM) that optimizes biodegradation of m-xylene

Background

The genetic network of the TOL plasmid pWW0 of the soil bacterium Pseudomonas putida mt-2 for catabolism of m-xylene is an archetypal model for environmental biodegradation of aromatic pollutants. Although nearly every metabolic and transcriptional component of this regulatory system is known to an extraordinary molecular detail, the complexity of its architecture is still perplexing. To gain an insight into the inner layout of this network a logic model of the TOL system was implemented, simulated and experimentally validated. This analysis made sense of the specific regulatory topology out on the basis of an unprecedented network motif around which the entire genetic circuit for m-xylene catabolism gravitates.

Results

The most salient feature of the whole TOL regulatory network is the control exerted by two distinct but still intertwined regulators (XylR and XylS) on expression of two separated catabolic operons (upper and lower) for catabolism of m-xylene. Following model reduction, a minimal modular circuit composed by five basic variables appeared to suffice for fully describing the operation of the entire system. In silico simulation of the effect of various perturbations were compared with experimental data in which specific portions of the network were activated with selected inducers: m-xylene, o-xylene, 3-methylbenzylalcohol and 3-methylbenzoate. The results accredited the ability of the model to faithfully describe network dynamics. This analysis revealed that the entire regulatory structure of the TOL system enables the action an unprecedented metabolic amplifier motif (MAM). This motif synchronizes expression of the upper and lower portions of a very long metabolic system when cells face the head pathway substrate, m-xylene.

Conclusion

Logic modeling of the TOL circuit accounted for the intricate regulatory topology of this otherwise simple metabolic device. The found MAM appears to ensure a simultaneous expression of the upper and lower segments of the m-xylene catabolic route that would be difficult to bring about with a standard substrate-responsive single promoter. Furthermore, it is plausible that the MAM helps to avoid biochemical conflicts between competing plasmid-encoded and chromosomally-encoded pathways in this bacterium.

Identification of metabolic network models from incomplete high-throughput datasets

Motivation: High-throughput measurement techniques for metabolism and gene expression provide a wealth of information for the identification of metabolic network models. Yet, missing observations scattered over the dataset restrict the number of effectively available datapoints and make classical regression techniques inaccurate or inapplicable. Thorough exploitation of the data by identification techniques that explicitly cope with missing observations is therefore of major importance.

Results: We develop a maximum-likelihood approach for the estimation of unknown parameters of metabolic network models that relies on the integration of statistical priors to compensate for the missing data. In the context of the linlog metabolic modeling framework, we implement the identification method by an Expectation-Maximization (EM) algorithm and by a simpler direct numerical optimization method. We evaluate performance of our methods by comparison to existing approaches, and show that our EM method provides the best results over a variety of simulated scenarios. We then apply the EM algorithm to a real problem, the identification of a model for the Escherichia coli central carbon metabolism, based on challenging experimental data from the literature. This leads to promising results and allows us to highlight critical identification issues.

Contact:sara.berthoumieux@inria.fr; eugenio.cinquemani@inria.fr

Supplementary information:Supplementary data are available at Bioinformatics online.

Model reduction using piecewise-linear approximations preserves dynamic properties of the carbon starvation response in Escherichia coli

The adaptation of the bacterium Escherichia coli to carbon starvation is controlled by a large network of biochemical reactions involving genes, mRNAs, proteins, and signalling molecules. The dynamics of these networks is difficult to analyze, notably due to a lack of quantitative information on parameter values. To overcome these limitations, model reduction approaches based on quasi-steady-state (QSS) and piecewise-linear (PL) approximations have been proposed, resulting in models that are easier to handle mathematically and computationally. These approximations are not supposed to affect the capability of the model to account for essential dynamical properties of the system, but the validity of this assumption has not been systematically tested. In this paper, we carry out such a study by evaluating a large and complex PL model of the carbon starvation response in E. coli using an ensemble approach. The results show that, in comparison with conventional nonlinear models, the PL approximations generally preserve the dynamics of the carbon starvation response network, although with some deviations concerning notably the quantitative precision of the model predictions. This encourages the application of PL models to the qualitative analysis of bacterial regulatory networks, in situations where the reference time scale is that of protein synthesis and degradation.

Efficient parameter search for qualitative models of regulatory networks using symbolic model checking

Motivation: Investigating the relation between the structure and behavior of complex biological networks often involves posing the question if the hypothesized structure of a regulatory network is consistent with the observed behavior, or if a proposed structure can generate a desired behavior.

Results: The above questions can be cast into a parameter search problem for qualitative models of regulatory networks. We develop a method based on symbolic model checking that avoids enumerating all possible parametrizations, and show that this method performs well on real biological problems, using the IRMA synthetic network and benchmark datasets. We test the consistency between IRMA and time-series expression profiles, and search for parameter modifications that would make the external control of the system behavior more robust.

Availability: GNA and the IRMA model are available at http://ibis.inrialpes.fr/

Contact:gregory.batt@inria.fr

Supplementary information:Supplementary data are available at Bioinformatics online.

Experimental and computational validation of models of fluorescent and luminescent reporter genes in bacteria

BACKGROUND

Fluorescent and luminescent reporter genes have become popular tools for the real-time monitoring of gene expression in living cells. However, mathematical models are necessary for extracting biologically meaningful quantities from the primary data.

RESULTS

We present a rigorous method for deriving relative protein synthesis rates (mRNA concentrations) and protein concentrations by means of kinetic models of gene expression. We experimentally and computationally validate this approach in the case of the protein Fis, a global regulator of transcription in Escherichia coli. We show that the mRNA and protein concentration profiles predicted from the models agree quite well with direct measurements obtained by Northern and Western blots, respectively. Moreover, we present computational procedures for taking into account systematic biases like the folding time of the fluorescent reporter protein and differences in the half-lives of reporter and host gene products. The results show that large differences in protein half-lives, more than mRNA half-lives, may be critical for the interpretation of reporter gene data in the analysis of the dynamics of regulatory systems.

CONCLUSIONS

The paper contributes to the development of sound methods for the interpretation of reporter gene data, notably in the context of the reconstruction and validation of models of regulatory networks. The results have wide applicability for the analysis of gene expression in bacteria and may be extended to higher organisms.

A service-oriented architecture for integrating the modeling and formal verification of genetic regulatory networks

Background

The study of biological networks has led to the development of increasingly large and detailed models. Computer tools are essential for the simulation of the dynamical behavior of the networks from the model. However, as the size of the models grows, it becomes infeasible to manually verify the predictions against experimental data or identify interesting features in a large number of simulation traces. Formal verification based on temporal logic and model checking provides promising methods to automate and scale the analysis of the models. However, a framework that tightly integrates modeling and simulation tools with model checkers is currently missing, on both the conceptual and the implementational level.

Results

We have developed a generic and modular web service, based on a service-oriented architecture, for integrating the modeling and formal verification of genetic regulatory networks. The architecture has been implemented in the context of the qualitative modeling and simulation tool GNA and the model checkers NUSMV and CADP. GNA has been extended with a verification module for the specification and checking of biological properties. The verification module also allows the display and visual inspection of the verification results.

Conclusions

The practical use of the proposed web service is illustrated by means of a scenario involving the analysis of a qualitative model of the carbon starvation response in E. coli. The service-oriented architecture allows modelers to define the model and proceed with the specification and formal verification of the biological properties by means of a unified graphical user interface. This guarantees a transparent access to formal verification technology for modelers of genetic regulatory networks.

Structural identification of piecewise-linear models of genetic regulatory networks

We present a method for the structural identification of genetic regulatory networks (GRNs), based on the use of a class of Piecewise-Linear (PL) models. These models consist of a set of decoupled linear models describing the different modes of operation of the GRN and discrete switches between the modes accounting for the nonlinear character of gene regulation. They thus form a compromise between the mathematical simplicity of linear models and the biological expressiveness of nonlinear models. The input of the PL identification method consists of time-series measurements of concentrations of gene products. As output it produces estimates of the modes of operation of the GRN, as well as all possible minimal combinations of threshold concentrations of the gene products accounting for switches between the modes of operation. The applicability of the PL identification method has been evaluated using simulated data obtained from a model of the carbon starvation response in the bacterium Escherichia coli. This has allowed us to systematically test the performance of the method under different data characteristics, notably variations in the noise level and the sampling density.

Search for steady states of piecewise-linear differential equation models of genetic regulatory networks

Analysis of the attractors of a genetic regulatory network gives a good indication of the possible functional modes of the system. In this paper we are concerned with the problem of finding all steady states of genetic regulatory networks described by piecewise-linear differential equation (PLDE) models. We show that the problem is NP-hard and translate it into a propositional satisfiability (SAT) problem. This allows the use of existing, efficient SAT solvers and has enabled the development of a steady state search module of the computer tool Genetic Network Analyzer (GNA). The practical use of this module is demonstrated by means of the analysis of a number of relatively small bacterial regulatory networks as well as randomly generated networks of several hundreds of genes.

Strategies for dealing with incomplete information in the modeling of molecular interaction networks

Modelers of molecular interaction networks encounter the paradoxical situation that while large amounts of data are available, these are often insufficient for the formulation and analysis of mathematical models describing the network dynamics. In particular, information on the reaction mechanisms and numerical values of kinetic parameters are usually not available for all but a few well-studied model systems. In this article we review two strategies that have been proposed for dealing with incomplete information in the study of molecular interaction networks: parameter sensitivity analysis and model simplification. These strategies are based on the biologically justified intuition that essential properties of the system dynamics are robust against moderate changes in the value of kinetic parameters or even in the rate laws describing the interactions. Although advanced measurement techniques can be expected to relieve the problem of incomplete information to some extent, the strategies discussed in this article will retain their interest as tools providing an initial characterization of essential properties of the network dynamics.

Qualitative simulation of the carbon starvation response in Escherichia coli

In case of nutritional stress, like carbon starvation, Escherichia coli cells abandon their exponential-growth state to enter a more resistant, non-growth state called stationary phase. This growth-phase transition is controlled by a genetic regulatory network integrating various environmental signals. Although E. coli is a paradigm of the bacterial world, it is little understood how its response to carbon starvation conditions emerges from the interactions between the different components of the regulatory network. Using a qualitative method that is able to overcome the current lack of quantitative data on kinetic parameters and molecular concentrations, we model the carbon starvation response network and simulate the response of E. coli cells to carbon deprivation. This allows us to identify essential features of the transition between exponential and stationary phase and to make new predictions on the qualitative system behavior following a carbon upshift.

Control of Bacteriophage Mu Lysogenic Repression

The transposable and temperate phage Mu infects Escherichia coli where it can enter the lytic life-cycle or reside as a repressed and integrated prophage. The repressor protein Rep is the key element in the lysis-lysogeny decision. We have analyzed the fate of Rep in different mutants by Western blotting under two conditions that can induce a lysogen: high temperature and stationary phase. We show that, unexpectedly, Rep accumulates under all conditions where the prophage is completely derepressed, and that this accumulation is ClpX-dependent. An analysis of the degradation kinetics shows that Rep is a target of two protease systems: inactivation of either the clpP or lon gene results in a stabilization of Rep. Such a reaction scheme explains the counterintuitive observation that derepression is correlated with high repressor concentration. We conclude that under all conditions of phage induction the repressor is sequestered in a non-active form. A quantitative simulation accounts for our experimental data. It provides a model that captures the essential features of Mu induction and explains some of the mechanisms by which the physiological signals affecting the lysis-lysogeny decision converge onto Rep.

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.

Validation of qualitative models of genetic regulatory networks by model checking: analysis of the nutritional stress response in Escherichia coli

MOTIVATION

The modeling and simulation of genetic regulatory networks have created the need for tools for model validation. The main challenges of model validation are the achievement of a match between the precision of model predictions and experimental data, as well as the efficient and reliable comparison of the predictions and observations.

RESULTS

We present an approach towards the validation of models of genetic regulatory networks addressing the above challenges. It combines a method for qualitative modeling and simulation with techniques for model checking, and is supported by a new version of the computer tool Genetic Network Analyzer (GNA). The model-validation approach has been applied to the analysis of the network controlling the nutritional stress response in Escherichia coli.

AVAILABILITY

GNA and the model of the stress response network are available at http://www-helix.inrialpes.fr/gna.

Genetic Network Analyzer: qualitative simulation of genetic regulatory networks

MOTIVATION

The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for the development of appropriate computer tools for the modeling and simulation of gene regulation processes.

RESULTS

We present Genetic Network Analyzer (GNA), a computer tool for the modeling and simulation of genetic regulatory networks. The tool is based on a qualitative simulation method that employs coarse-grained models of regulatory networks. The use of GNA is illustrated by a case study of the network of genes and interactions regulating the initiation of sporulation in Bacillus subtilis.

AVAILABILITY

GNA and the model of the sporulation network are available at http://www-helix.inrialpes.fr/gna.

Modeling and simulation of genetic regulatory systems: a literature review

In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.