Dynamic flux balance analysis of diauxic growth in Escherichia coli

Genome-scale metabolic modeling of a clostridial co-culture for consolidated bioprocessing

An alternative consolidated bioprocessing approach is the use of a co-culture containing cellulolytic and solventogenic clostridia. It has been demonstrated that the rate of cellulose utilization in the co-culture of Clostridium acetobutylicum and Clostridium cellulolyticum is improved compared to the mono-culture of C. cellulolyticum, suggesting the presence of syntrophy between these two species. However, the metabolic interactions in the co-culture are not well understood. To understand the metabolic interactions in the co-culture, we developed a genome-scale metabolic model of C. cellulolyticum comprising of 431 genes, 621 reactions, and 603 metabolites. The C. cellulolyticum model can successfully predict the chemostat growth and byproduct secretion with cellulose as the substrate. However, a growth arrest phenomenon, which occurs in batch cultures of C. cellulolyticum at cellulose concentrations higher than 6.7 g/L, cannot be predicted by dynamic flux balance analysis due to the lack of understanding of the underlying mechanism. These genome-scale metabolic models of the pure cultures have also been integrated using a community modeling framework to develop a dynamic model of metabolic interactions in the co-culture. Co-culture simulations suggest that cellobiose inhibition cannot be the main factor that is responsible for improved cellulose utilization relative to mono-culture of C. cellulolyticum.

The virus as metabolic engineer

Recent genome-wide screens of host genetic requirements for viral infection have reemphasized the critical role of host metabolism in enabling the production of viral particles. In this review, we highlight the metabolic aspects of viral infection found in these studies, and focus on the opportunities these requirements present for metabolic engineers. In particular, the objectives and approaches that metabolic engineers use are readily comparable to the behaviors exhibited by viruses during infection. As a result, metabolic engineers have a unique perspective that could lead to novel and effective methods to combat viral infection.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Dynamic metabolic flux analysis demonstrated on cultures where the limiting substrate is changed from carbon to nitrogen and vice versa

The main requirement for metabolic flux analysis (MFA) is that the cells are in a pseudo-steady state, that there is no accumulation or depletion of intracellular metabolites. In the past, the applications of MFA were limited to the analysis of continuous cultures. This contribution introduces the concept of dynamic MFA and extends MFA so that it is applicable to transient cultures. Time series of concentration measurements are transformed into flux values. This transformation involves differentiation, which typically increases the noisiness of the data. Therefore, a noise-reducing step is needed. In this work, polynomial smoothing was used. As a test case, dynamic MFA is applied on Escherichia coli cultivations shifting from carbon limitation to nitrogen limitation and vice versa. After switching the limiting substrate from N to C, a lag phase was observed accompanied with an increase in maintenance energy requirement. This lag phase did not occur in the C- to N-limitation case.

Computational biology approaches to plant metabolism and photosynthesis: applications for corals in times of climate change and environmental stress

Knowledge of factors that are important in reef resilience helps us to understand how reef ecosystems react following major anthropogenic and environmental disturbances. The symbiotic relationship between the photosynthetic zooxanthellae algal cells and corals is that the zooxanthellae provide the coral with carbon, while the coral provides protection and access to enough light for the zooxanthellae to photosynthesise. This article reviews some recent advances in computational biology relevant to photosynthetic organisms, including Beyesian approaches to kinetics, computational methods for flux balances in metabolic processes, and determination of clades of zooxanthallae. Application of these systems will be important in the conservation of coral reefs in times of climate change and environmental stress.

Genome-scale dynamic modeling of the competition between Rhodoferax and Geobacter in anoxic subsurface environments

The advent of rapid complete genome sequencing, and the potential to capture this information in genome-scale metabolic models, provide the possibility of comprehensively modeling microbial community interactions. For example, Rhodoferax and Geobacter species are acetate-oxidizing Fe(III)-reducers that compete in anoxic subsurface environments and this competition may have an influence on the in situ bioremediation of uranium-contaminated groundwater. Therefore, genome-scale models of Geobacter sulfurreducens and Rhodoferax ferrireducens were used to evaluate how Geobacter and Rhodoferax species might compete under diverse conditions found in a uranium-contaminated aquifer in Rifle, CO. The model predicted that at the low rates of acetate flux expected under natural conditions at the site, Rhodoferax will outcompete Geobacter as long as sufficient ammonium is available. The model also predicted that when high concentrations of acetate are added during in situ bioremediation, Geobacter species would predominate, consistent with field-scale observations. This can be attributed to the higher expected growth yields of Rhodoferax and the ability of Geobacter to fix nitrogen. The modeling predicted relative proportions of Geobacter and Rhodoferax in geochemically distinct zones of the Rifle site that were comparable to those that were previously documented with molecular techniques. The model also predicted that under nitrogen fixation, higher carbon and electron fluxes would be diverted toward respiration rather than biomass formation in Geobacter, providing a potential explanation for enhanced in situ U(VI) reduction in low-ammonium zones. These results show that genome-scale modeling can be a useful tool for predicting microbial interactions in subsurface environments and shows promise for designing bioremediation strategies.

Molecular-level tradeoffs and metabolic adaptation to simultaneous stressors

Life is a dynamic process driven by the complex interplay between physical constraints and selection pressures, ranging from nutrient limitation to inhibitory substances to predators. These stressors are not mutually exclusive; microbes have faced concurrent challenges for eons. Genome-enabled systems biology approaches are adapting economic and ecological concepts like tradeoff curves and strategic resource allocation theory to analyze metabolic adaptations to simultaneous stressors. These methodologies can accurately describe and predict metabolic adaptations to concurrent stresses by considering the tradeoff between investment of limiting resources into enzymatic machinery and the resulting cellular function. The approaches represent promising links between computational biology and well-established economic and ecological methodologies for analyzing the interplay between physical constraints and microbial fitness.

Modeling with a view to target identification in metabolic engineering: a critical evaluation of the available tools

The state of the art tools for modeling metabolism, typically used in the domain of metabolic engineering, were reviewed. The tools considered are stoichiometric network analysis (elementary modes and extreme pathways), stoichiometric modeling (metabolic flux analysis, flux balance analysis, and carbon modeling), mechanistic and approximative modeling, cybernetic modeling, and multivariate statistics. In the context of metabolic engineering, one should be aware that the usefulness of these tools to optimize microbial metabolism for overproducing a target compound depends predominantly on the characteristic properties of that compound. Because of their shortcomings not all tools are suitable for every kind of optimization; issues like the dependence of the target compound's synthesis on severe (redox) constraints, the characteristics of its formation pathway, and the achievable/desired flux towards the target compound should play a role when choosing the optimization strategy.

Achieving optimal growth through product feedback inhibition in metabolism

Recent evidence suggests that the metabolism of some organisms, such as Escherichia coli, is remarkably efficient, producing close to the maximum amount of biomass per unit of nutrient consumed. This observation raises the question of what regulatory mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. We analyze several representative metabolic modules—starting from a linear pathway and advancing to a bidirectional pathway and metabolic cycle, and finally to integration of two different nutrient inputs. In each case, our mathematical analysis shows that product-feedback inhibition is not only homeostatic but also, with appropriate feedback connections, can minimize futile cycling and optimize fluxes. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pool sizes can be restricted if feedback inhibition is ultrasensitive. Indeed, the multi-layer regulation of metabolism by control of enzyme expression, enzyme covalent modification, and allostery is expected to result in such ultrasensitive feedbacks. To experimentally test whether the qualitative predictions from our analysis of feedback inhibition apply to metabolic modules beyond linear pathways, we examine the case of nitrogen assimilation in E. coli, which involves both nutrient integration and a metabolic cycle. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the network and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes in nutrient-switching experiments.

Bacteria live in remarkably diverse environments and constantly adapt to changing nutrient conditions. Recent evidence suggests that some bacteria, such as E. coli, are extraordinarily efficient in producing biomass under a variety of different nutrient conditions. This observation raises the question of what physical mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. Product-feedback inhibition is a metabolic regulatory scheme in which an end product inhibits the first dedicated step of the chain of reactions leading to its own synthesis. Our mathematical analysis of several representative metabolic modules suggests that simple feedback inhibition can indeed allow for optimal and efficient biomass production. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pools can be restricted if feedback inhibition is ultrasensitive. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the nitrogen assimilation network in E. coli and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes seen in experiments.

Transient metabolic modeling of Escherichia coli MG1655 and MG1655 DeltaackA-pta, DeltapoxB Deltapppc ppc-p37 for recombinant beta-galactosidase production

Escherichia coli is one of the most widely used hosts for the production of recombinant proteins, among other reasons because its genetics are far better characterized than those of any other microorganism. To improve the understanding of recombinant protein synthesis in E. coli, the production of a model recombinant protein, beta-galactosidase, was studied in response to the constitutive overexpression of the anaplerotic reaction afforded by PEP carboxylase. To this end, an IPTG wash-in experiment was performed starting from a well-defined steady-state condition for both the wild-type E. coli and a mutant with a defective acetate pathway and a constitutively overexpressed ppc. In order to compare the dynamics of the fluxes over time during the wash-in experiment, a method referred to as transient metabolic flux analysis, which is based on steady-state metabolic flux analysis, was used. This allowed us to track the intracellular changes/fluxes in both strains. It was observed that the flux towards fermentation products was 3.6 times lower in the ppc overexpression mutant compared to the wild-type E. coli. In the former on the other hand, the PPC flux is in general higher. In addition, the flux towards beta-galactosidase was higher (12.4 times), resulting in five times more protein activity. These results indicate that by constitutively overexpressing the anaplerotic ppc gene in E. coli, the TCA cycle intermediates are increasingly replenished. The additional supply of these protein precursors has a positive result on recombinant protein production.

Modelling the optimal timing in metabolic pathway activation-use of Pontryagin's Maximum Principle and role of the Golden section

The time course of enzyme concentrations in metabolic pathways can be predicted on the basis of the optimality criterion of minimizing the time period in which an essential product is generated. This criterion is in line with the widely accepted view that high fitness requires high pathway flux. Here, based on Pontryagin's Maximum Principle, a method is developed to solve the corresponding constrained optimal control problem in an almost exclusively analytical way and, thus, to calculate optimal enzyme profiles, when linear, irreversible rate laws are assumed. Three different problem formulations are considered and the corresponding optimization results are derived. Besides the minimization of transition time, we consider an operation time in which 90% of the substrate has been converted into product. In that case, only the enzyme at the lower end of the pathway rather than all enzymes are active in the last phase. In all cases, biphasic or multiphasic time courses are obtained. The biological meaning of the results in terms of a consecutive just-in-time expression of metabolic genes is discussed. For the special case of two-enzyme systems, the role of the Golden section in the solution is outlined.

Accomplishments in genome-scale in silico modeling for industrial and medical biotechnology

Driven by advancements in high-throughput biological technologies and the growing number of sequenced genomes, the construction of in silico models at the genome scale has provided powerful tools to investigate a vast array of biological systems and applications. Here, we review comprehensively the uses of such models in industrial and medical biotechnology, including biofuel generation, food production, and drug development. While the use of in silico models is still in its early stages for delivering to industry, significant initial successes have been achieved. For the cases presented here, genome-scale models predict engineering strategies to enhance properties of interest in an organism or to inhibit harmful mechanisms of pathogens. Going forward, genome-scale in silico models promise to extend their application and analysis scope to become a trans-formative tool in biotechnology.

Systems biology approaches to understanding mycobacterial survival mechanisms

The advent of high-throughput platforms for the interrogation of biological systems at the cellular and molecular level have allowed living cells to be observed and understood at a hitherto unprecedented level of detail and have enabled the construction of comprehensive, predictive in silico models. Here, we review the application of such high-throughput, systems-biological techniques to mycobacteria-specifically to the pernicious human pathogen Mycobacterium tuberculosis (MTb) and its ability to survive in human hosts. We discuss the development and application of transcriptomic, proteomic, regulomic, and metabolomic techniques for MTb as well as the development and application of genome-scale in silico models. Thus far, systems-biological approaches have largely focused on in vitro models of MTb growth; reliably extending these approaches to in vivo conditions relevant to infection is a significant challenge for the future that holds the ultimate promise of novel chemotherapeutic interventions.

Biocharts: a visual formalism for complex biological systems

We address one of the central issues in devising languages, methods and tools for the modelling and analysis of complex biological systems, that of linking high-level (e.g. intercellular) information with lower-level (e.g. intracellular) information. Adequate ways of dealing with this issue are crucial for understanding biological networks and pathways, which typically contain huge amounts of data that continue to grow as our knowledge and understanding of a system increases. Trying to comprehend such data using the standard methods currently in use is often virtually impossible. We propose a two-tier compound visual language, which we call Biocharts, that is geared towards building fully executable models of biological systems. One of the main goals of our approach is to enable biologists to actively participate in the computational modelling effort, in a natural way. The high-level part of our language is a version of statecharts, which have been shown to be extremely successful in software and systems engineering. The statecharts can be combined with any appropriately well-defined language (preferably a diagrammatic one) for specifying the low-level dynamics of the pathways and networks. We illustrate the language and our general modelling approach using the well-studied process of bacterial chemotaxis.

A systems approach to plant bioprocess optimization

A dynamic model for plant cell metabolism was used as a basis for a rational analysis of plant production potential in in vitro cultures. The model was calibrated with data from 3-L bioreactor cultures. A dynamic sensitivity analysis framework was developed to analyse the response curves of secondary metabolite production to metabolic and medium perturbations. Simulation results suggest that a straightforward engineering of cell metabolism or medium composition might only have a limited effect on productivity. To circumvent the problem of the dynamic allocation of resources between growth and production pathways, the sensitivity analysis framework was used to assess the effect of stabilizing intracellular nutrient concentrations. Simulations showed that a stabilization of intracellular glucose and nitrogen reserves could lead to a 116% increase in the specific production of secondary metabolites compared with standard culture protocol. This culture strategy was implemented experimentally using a perfusion bioreactor. To stabilize intracellular concentrations, adaptive medium feeding was performed using model mass balances and estimations. This allowed for a completely automated culture, with controlled conditions and pre-defined decision making algorithm. The proposed culture strategy leads to a 73% increase in specific production and a 129% increase in total production, as compared with a standard batch culture protocol. The sensitivity analysis on a mathematical model of plant metabolism thus allowed producing new insights on the links between intracellular nutritional management and cell productivity. The experimental implementation was also a significant improvement on current plant bioprocess strategies.

A systems biology framework for modeling metabolic enzyme inhibition of Mycobacterium tuberculosis

BACKGROUND

Because metabolism is fundamental in sustaining microbial life, drugs that target pathogen-specific metabolic enzymes and pathways can be very effective. In particular, the metabolic challenges faced by intracellular pathogens, such as Mycobacterium tuberculosis, residing in the infected host provide novel opportunities for therapeutic intervention.

RESULTS

We developed a mathematical framework to simulate the effects on the growth of a pathogen when enzymes in its metabolic pathways are inhibited. Combining detailed models of enzyme kinetics, a complete metabolic network description as modeled by flux balance analysis, and a dynamic cell population growth model, we quantitatively modeled and predicted the dose-response of the 3-nitropropionate inhibitor on the growth of M. tuberculosis in a medium whose carbon source was restricted to fatty acids, and that of the 5'-O-(N-salicylsulfamoyl) adenosine inhibitor in a medium with low-iron concentration.

CONCLUSION

The predicted results quantitatively reproduced the experimentally measured dose-response curves, ranging over three orders of magnitude in inhibitor concentration. Thus, by allowing for detailed specifications of the underlying enzymatic kinetics, metabolic reactions/constraints, and growth media, our model captured the essential chemical and biological factors that determine the effects of drug inhibition on in vitro growth of M. tuberculosis cells.

A rational design approach for amino acid supplementation in hepatocyte culture

Improvement of culture media for mammalian cells is conducted via empirical adjustments, sometimes aided by statistical design methodologies. Here, we demonstrate a proof of principle for the use of constraints-based modeling to achieve enhanced performance of liver-specific functions of cultured hepatocytes during plasma exposure by adjusting amino acid supplementation and hormone levels in the medium. Flux balance analysis (FBA) is used to determine an amino acid flux profile consistent with a desired output; this is used to design an amino acid supplementation. Under conditions of no supplementation, empirical supplementation, and designed supplementation, hepatocytes were exposed to plasma and their morphology, specific cell functions (urea, albumin production) and lipid metabolism were measured. Urea production under the designed amino acid supplementation was found to be increased compared with previously reported (empirical) amino acid supplementation. Not surprisingly, the urea production attained was less than the theoretical value, indicating the existence of pathways or constraints not present in the current model. Although not an explicit design objective, albumin production was also increased by designed amino acid supplementation, suggesting a functional linkage between these outputs. In conjunction with traditional approaches to improving culture conditions, the rational design approach described herein provides a novel means to tune the metabolic outputs of cultured hepatocytes.

Systematic development of hybrid cybernetic models: application to recombinant yeast co-consuming glucose and xylose

The hybrid cybernetic modeling approach of Kim et al. (Kim et al. [2008] Biotechnol. Prog., in press) views the substrate uptake flux in microorganisms as being distributed in a regulated way among different elementary modes (EMs) of a metabolic network, which intracellular fluxes related to the uptake rates by the pseudo-steady-state approximation on intracellular metabolites. While the conceptual development has been demonstrated by Kim et al. (Kim et al. [2008] Biotechnol. Prog., in press) using a rather simple example (i.e., Escherichia coli metabolizing a single substrate), its extension to a larger scale network involving multiple substrates results in serious overparameterization (which implies an excessive number of parameters relative to the measurements available to determine them). Through the case study of recombinant Saccharomyces yeast co-consuming glucose and xylose, we present a systematic way of formulating a minimal order hybrid cybernetic model (HCM) for a general metabolic network. The overparameterization problem mostly arising from a large number of EMs is avoided using a model reduction technique developed by Song and Ramkrishna (Song and Ramkrishna [2009a] Biotechnol. Bioeng. 102(2):554-568) where an original set of EMs is condensed to a much smaller subset. Detailed discussions follow on the issue of determining the minimal set of active modes needed for the description of the simultaneous consumption of multiple substrates. The developed HCM is compared with other metabolic models: macroscopic bioreaction models (Provost et al. [2006] Bioprocess Biosyt. Eng. 29(5-6):349-366), and dynamic flux balance analysis. It is shown that the HCM outperforms the other two as validated using various sets of fermentation data. The difference among the models is more dramatic in a situation such as the sequential utilization of glucose and xylose, which is observed under realistic fermentation conditions.

Steady-state and dynamic flux balance analysis of ethanol production by Saccharomyces cerevisiae

Steady-state and dynamic flux balance analysis (DFBA) was used to investigate the effects of metabolic model complexity and parameters on ethanol production predictions for wild-type and engineered Saccharomyces cerevisiae. Three metabolic network models ranging from a single compartment representation of metabolism to a genome-scale reconstruction with seven compartments and detailed charge balancing were studied. Steady-state analysis showed that the models generated similar wild-type predictions for the biomass and ethanol yields, but for ten engineered strains the seven compartment model produced smaller ethanol yield enhancements. Simplification of the seven compartment model to two intracellular compartments produced increased ethanol yields, suggesting that reaction localisation had an impact on mutant phenotype predictions. Further analysis with the seven compartment model demonstrated that steady-state predictions can be sensitive to intracellular model parameters, with the biomass yield exhibiting high sensitivity to ATP utilisation parameters and the biomass composition. The incorporation of gene expression data through the zeroing of metabolic reactions associated with unexpressed genes was shown to produce negligible changes in steady-state predictions when the oxygen uptake rate was suitably constrained. Dynamic extensions of the single and seven compartment models were developed through the addition of glucose and oxygen uptake expressions and transient extracellular balances. While the dynamic models produced similar predictions of the optimal batch ethanol productivity for the wild type, the single compartment model produced significantly different predictions for four implementable gene insertions. A combined deletion/overexpression/insertion mutant with improved ethanol productivity capabilities was computationally identified by dynamically screening multiple combinations of the ten metabolic engineering strategies. The authors concluded that extensive compartmentalisation and detailed charge balancing can be important for reliably screening metabolic engineering strategies that rely on modification of the global redox balance and that DFBA offers the potential to identify novel mutants for enhanced metabolite production in batch and fed-batch cultures.

iBsu1103: a new genome-scale metabolic model of Bacillus subtilis based on SEED annotations

BACKGROUND

Bacillus subtilis is an organism of interest because of its extensive industrial applications, its similarity to pathogenic organisms, and its role as the model organism for Gram-positive, sporulating bacteria. In this work, we introduce a new genome-scale metabolic model of B. subtilis 168 called iBsu1103. This new model is based on the annotated B. subtilis 168 genome generated by the SEED, one of the most up-to-date and accurate annotations of B. subtilis 168 available.

RESULTS

The iBsu1103 model includes 1,437 reactions associated with 1,103 genes, making it the most complete model of B. subtilis available. The model also includes Gibbs free energy change (DeltarG' degrees ) values for 1,403 (97%) of the model reactions estimated by using the group contribution method. These data were used with an improved reaction reversibility prediction method to identify 653 (45%) irreversible reactions in the model. The model was validated against an experimental dataset consisting of 1,500 distinct conditions and was optimized by using an improved model optimization method to increase model accuracy from 89.7% to 93.1%.

CONCLUSIONS

Basing the iBsu1103 model on the annotations generated by the SEED significantly improved the model completeness and accuracy compared with the most recent previously published model. The enhanced accuracy of the iBsu1103 model also demonstrates the efficacy of the improved reaction directionality prediction method in accurately identifying irreversible reactions in the B. subtilis metabolism. The proposed improved model optimization methodology was also demonstrated to be effective in minimally adjusting model content to improve model accuracy.

Recent developments in parameter estimation and structure identification of biochemical and genomic systems

The organization, regulation and dynamical responses of biological systems are in many cases too complex to allow intuitive predictions and require the support of mathematical modeling for quantitative assessments and a reliable understanding of system functioning. All steps of constructing mathematical models for biological systems are challenging, but arguably the most difficult task among them is the estimation of model parameters and the identification of the structure and regulation of the underlying biological networks. Recent advancements in modern high-throughput techniques have been allowing the generation of time series data that characterize the dynamics of genomic, proteomic, metabolic, and physiological responses and enable us, at least in principle, to tackle estimation and identification tasks using 'top-down' or 'inverse' approaches. While the rewards of a successful inverse estimation or identification are great, the process of extracting structural and regulatory information is technically difficult. The challenges can generally be categorized into four areas, namely, issues related to the data, the model, the mathematical structure of the system, and the optimization and support algorithms. Many recent articles have addressed inverse problems within the modeling framework of Biochemical Systems Theory (BST). BST was chosen for these tasks because of its unique structural flexibility and the fact that the structure and regulation of a biological system are mapped essentially one-to-one onto the parameters of the describing model. The proposed methods mainly focused on various optimization algorithms, but also on support techniques, including methods for circumventing the time consuming numerical integration of systems of differential equations, smoothing overly noisy data, estimating slopes of time series, reducing the complexity of the inference task, and constraining the parameter search space. Other methods targeted issues of data preprocessing, detection and amelioration of model redundancy, and model-free or model-based structure identification. The total number of proposed methods and their applications has by now exceeded one hundred, which makes it difficult for the newcomer, as well as the expert, to gain a comprehensive overview of available algorithmic options and limitations. To facilitate the entry into the field of inverse modeling within BST and related modeling areas, the article presented here reviews the field and proposes an operational 'work-flow' that guides the user through the estimation process, identifies possibly problematic steps, and suggests corresponding solutions based on the specific characteristics of the various available algorithms. The article concludes with a discussion of the present state of the art and with a description of open questions.

A hybrid model of anaerobic E. coli GJT001: combination of elementary flux modes and cybernetic variables

Flux balance analysis (FBA) in combination with the decomposition of metabolic networks into elementary modes has provided a route to modeling cellular metabolism. It is dependent, however, on the availability of external fluxes such as substrate uptake or growth rate before estimates can become available of intracellular fluxes. The framework classically does not allow modeling of metabolic regulation or the formulation of dynamic models except through dynamic measurement of external fluxes. The cybernetic modeling approach of Ramkrishna and coworkers provides a dynamic framework for modeling metabolic systems because of its focus on describing regulatory processes based on cybernetic arguments and hence has the capacity to describe both external and internal fluxes. In this article, we explore the alternative of developing hybrid models combining cybernetic models for the external fluxes with the flux balance approach for estimation of the internal fluxes. The approach has the merit of the simplicity of the early cybernetic models and hence computationally facile while also providing detailed information on intracellular fluxes. The hybrid model of this article is based on elementary mode decomposition of the metabolic network. The uptake rates for the various elementary modes are combined using global cybernetic variables based on maximizing substrate uptake rates. Estimation of intracellular metabolism is based on its stoichiometric coupling with the external fluxes under the assumption of (pseudo-) steady state conditions. The set of parameters of the hybrid model was estimated with the aid of nonlinear optimization routine, by fitting simulations with dynamic experimental data on concentrations of biomass, substrate, and fermentation products. The hybrid model estimations were tested with FBA (based on measured substrate uptake rate) for two different metabolic networks (one is a reduced network which fixes ATP contribution to the biomass and maintenance requirement of ATP, and the other network is a more complex network which has a separate reaction for maintenance.) for the same experiment involving anaerobic growth of E. coli GJT001. The hybrid model estimated glucose consumption and all fermentation byproducts to better than 10%. The FBA makes similar estimations of fermentation products, however, with the exception of succinate. The simulation results show that the global cybernetic variables alone can regulate the metabolic reactions obtaining a very satisfactory fit to the measured fermentation byproducts. In view of the hybrid model's ability to predict biomass growth and fermentation byproducts of anaerobic E. coli GJT001, this reduced order model offers a computationally efficient alternative to more detailed models of metabolism and hence useful for the simulation of bioreactors.

Coupling a genome-scale metabolic model with a reactive transport model to describe in situ uranium bioremediation

The increasing availability of the genome sequences of microorganisms involved in important bioremediation processes makes it feasible to consider developing genome-scale models that can aid in predicting the likely outcome of potential subsurface bioremediation strategies. Previous studies of the in situ bioremediation of uranium-contaminated groundwater have demonstrated that Geobacter species are often the dominant members of the groundwater community during active bioremediation and the primary organisms catalysing U(VI) reduction. Therefore, a genome-scale, constraint-based model of the metabolism of Geobacter sulfurreducens was coupled with the reactive transport model HYDROGEOCHEM in an attempt to model in situ uranium bioremediation. In order to simplify the modelling, the influence of only three growth factors was considered: acetate, the electron donor added to stimulate U(VI) reduction; Fe(III), the electron acceptor primarily supporting growth of Geobacter; and ammonium, a key nutrient. The constraint-based model predicted that growth yields of Geobacter varied significantly based on the availability of these three growth factors and that there are minimum thresholds of acetate and Fe(III) below which growth and activity are not possible. This contrasts with typical, empirical microbial models that assume fixed growth yields and the possibility for complete metabolism of the substrates. The coupled genome-scale and reactive transport model predicted acetate concentrations and U(VI) reduction rates in a field trial of in situ uranium bioremediation that were comparable to the predictions of a calibrated conventional model, but without the need for empirical calibration, other than specifying the initial biomass of Geobacter. These results suggest that coupling genome-scale metabolic models with reactive transport models may be a good approach to developing models that can be truly predictive, without empirical calibration, for evaluating the probable response of subsurface microorganisms to possible bioremediation approaches prior to implementation.

Photosynthetic metabolism of C3 plants shows highly cooperative regulation under changing environments: a systems biological analysis

We studied the robustness of photosynthetic metabolism in the chloroplasts of C(3) plants under drought stress and at high CO(2) concentration conditions by using a method called Minimization of Metabolic Adjustment Dynamic Flux Balance Analysis (M_DFBA). Photosynthetic metabolism in the chloroplasts of C(3) plants applies highly cooperative regulation to minimize the fluctuation of metabolite concentration profiles in the face of transient perturbations. Our work suggests that highly cooperative regulation assures the robustness of the biological system and that there is closer cooperation under perturbation conditions than under normal conditions. This results in minimizing fluctuations in the profiles of metabolite concentrations, which is the key to maintaining a system's function. Our methods help in understanding such phenomena and the mechanisms of robustness for complex metabolic networks in dynamic processes.

Flux balance analysis: interrogating genome-scale metabolic networks

Flux balance analysis (FBA) is a computational method to analyze reconstructions of biochemical networks. FBA requires the formulation of a biochemical network in a precise mathematical framework called a stoichiometric matrix. An objective function is defined (e.g., growth rate) toward which the system is assumed to be optimized. In this chapter, we present the methodology, theory, and common pitfalls of the application of FBA.

Gene regulation in continuous cultures: a unified theory for bacteria and yeasts

During batch growth on mixtures of two growth-limiting substrates, microbes consume the substrates either sequentially (diauxie) or simultaneously. The ubiquity of these growth patterns suggests that they may be driven by a universal mechanism common to all microbial species. Recently, we showed that a minimal model accounting only for enzyme induction and dilution, the two processes that occur in all microbes, explains the phenotypes observed in batch cultures of various wild-type and mutant/recombinant cells (Narang and Pilyugin in J. Theor. Biol. 244:326-348, 2007). Here, we examine the extension of the minimal model to continuous cultures. We show that: (1) Several enzymatic trends, attributed entirely to cross-regulatory mechanisms, such as catabolite repression and inducer exclusion, can be quantitatively explained by enzyme dilution. (2) The bifurcation diagram of the minimal model for continuous cultures, which classifies the substrate consumption pattern at any given dilution rate and feed concentrations, provides a precise explanation for the empirically observed correlations between the growth patterns in batch and continuous cultures. (3) Numerical simulations of the model are in excellent agreement with the data. The model captures the variation of the steady state substrate concentrations, cell densities, and enzyme levels during the single- and mixed-substrate growth of bacteria and yeasts at various dilution rates and feed concentrations. This variation is well approximated by simple analytical expressions that furnish deep physical insights. (4) Since the minimal model describes the behavior of the cells in the absence of cross-regulatory mechanisms, it provides a rigorous framework for quantifying the effect of these mechanisms. We illustrate this by analyzing several data sets from the literature.

HEMET: mathematical model of biochemical pathways for simulation and prediction of HEpatocyte METabolism

Many computer studies and models have been developed in order to simulate cell biochemical pathways. The difficulty of integrating all the biochemical reactions that occur in a cell in a single model is the main reason for the poor results in the prediction and simulation of cell behaviour under different chemical and physical stimuli. In this paper we have translated biochemical reactions into differential equations for the development of modular model of metabolism of a hepatocyte cultured in static and standard conditions (in a plastic multiwell placed in an incubator at 37 degrees C with 5% of CO(2)). Using biochemical equations and energetic considerations a set of non-linear differential equations has been derived and implemented in Simulink. This set of equations mimics some of the principal metabolic pathways of biomolecules present in the culture medium. The software platform developed is subdivided into separate modules, each one describing a different metabolic pathway; they constitute a library which can be used for developing new modules and models to project, predict and validate cell behaviour in vitro.

Optimization of Fed-Batch Saccharomyces cerevisiae Fermentation Using Dynamic Flux Balance Models

We developed a dynamic flux balance model for fed-batch Saccharomyces cerevisiae fermentation that couples a detailed steady-state description of primary carbon metabolism with dynamic mass balances on key extracellular species. Model-based dynamic optimization is performed to determine fed-batch operating policies that maximize ethanol productivity and/or ethanol yield on glucose. The initial volume and glucose concentrations, the feed flow rate and dissolved oxygen concentration profiles, and the final batch time are treated as decision variables in the dynamic optimization problem. Optimal solutions are generated to analyze the tradeoff between maximal productivity and yield objectives. We find that for both cases the prediction of a microaerobic region is significant. The optimization results are sensitive to network model parameters for the growth associated maintenance and P/O ratio. The results of our computational study motivate continued development of dynamic flux balance models and further exploration of their application to productivity optimization in biochemical reactors.

Integrating metabolic, transcriptional regulatory and signal transduction models in Escherichia coli

MOTIVATION

The effort to build a whole-cell model requires the development of new modeling approaches, and in particular, the integration of models for different types of processes, each of which may be best described using different representation. Flux-balance analysis (FBA) has been useful for large-scale analysis of metabolic networks, and methods have been developed to incorporate transcriptional regulation (regulatory FBA, or rFBA). Of current interest is the integration of these approaches with detailed models based on ordinary differential equations (ODEs).

RESULTS

We developed an approach to modeling the dynamic behavior of metabolic, regulatory and signaling networks by combining FBA with regulatory Boolean logic, and ordinary differential equations. We use this approach (called integrated FBA, or iFBA) to create an integrated model of Escherichia coli which combines a flux-balance-based, central carbon metabolic and transcriptional regulatory model with an ODE-based, detailed model of carbohydrate uptake control. We compare the predicted Escherichia coli wild-type and single gene perturbation phenotypes for diauxic growth on glucose/lactose and glucose/glucose-6-phosphate with that of the individual models. We find that iFBA encapsulates the dynamics of three internal metabolites and three transporters inadequately predicted by rFBA. Furthermore, we find that iFBA predicts different and more accurate phenotypes than the ODE model for 85 of 334 single gene perturbation simulations, as well for the wild-type simulations. We conclude that iFBA is a significant improvement over the individual rFBA and ODE modeling paradigms.

AVAILABILITY

All MATLAB files used in this study are available at http://www.simtk.org/home/ifba/.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Optimization in computational systems biology

Optimization aims to make a system or design as effective or functional as possible. Mathematical optimization methods are widely used in engineering, economics and science. This commentary is focused on applications of mathematical optimization in computational systems biology. Examples are given where optimization methods are used for topics ranging from model building and optimal experimental design to metabolic engineering and synthetic biology. Finally, several perspectives for future research are outlined.

Dynamic analysis of integrated signaling, metabolic, and regulatory networks

Extracellular cues affect signaling, metabolic, and regulatory processes to elicit cellular responses. Although intracellular signaling, metabolic, and regulatory networks are highly integrated, previous analyses have largely focused on independent processes (e.g., metabolism) without considering the interplay that exists among them. However, there is evidence that many diseases arise from multifunctional components with roles throughout signaling, metabolic, and regulatory networks. Therefore, in this study, we propose a flux balance analysis (FBA)–based strategy, referred to as integrated dynamic FBA (idFBA), that dynamically simulates cellular phenotypes arising from integrated networks. The idFBA framework requires an integrated stoichiometric reconstruction of signaling, metabolic, and regulatory processes. It assumes quasi-steady-state conditions for “fast” reactions and incorporates “slow” reactions into the stoichiometric formalism in a time-delayed manner. To assess the efficacy of idFBA, we developed a prototypic integrated system comprising signaling, metabolic, and regulatory processes with network features characteristic of actual systems and incorporating kinetic parameters based on typical time scales observed in literature. idFBA was applied to the prototypic system, which was evaluated for different environments and gene regulatory rules. In addition, we applied the idFBA framework in a similar manner to a representative module of the single-cell eukaryotic organism Saccharomyces cerevisiae. Ultimately, idFBA facilitated quantitative, dynamic analysis of systemic effects of extracellular cues on cellular phenotypes and generated comparable time-course predictions when contrasted with an equivalent kinetic model. Since idFBA solves a linear programming problem and does not require an exhaustive list of detailed kinetic parameters, it may be efficiently scaled to integrated intracellular systems that incorporate signaling, metabolic, and regulatory processes at the genome scale, such as the S. cerevisiae system presented here.

Cellular systems comprise many diverse components and component interactions spanning signal transduction, transcriptional regulation, and metabolism. Although signaling, metabolic, and regulatory activities are often investigated independently of one another, there is growing evidence that considerable interplay occurs among them, and that the malfunctioning of this interplay is associated with disease. The computational analysis of integrated networks has been challenging because of the varying time scales involved as well as the sheer magnitude of such systems (e.g., the numbers of rate constants involved). To this end, we developed a novel computational framework called integrated dynamic flux balance analysis (idFBA) that generates quantitative, dynamic predictions of species concentrations spanning signaling, regulatory, and metabolic processes. idFBA extends an existing approach called flux balance analysis (FBA) in that it couples “fast” and “slow” reactions, thereby facilitating the study of whole-cell phenotypes and not just sub-cellular network properties. We applied this framework to a prototypic integrated system derived from literature as well as a representative integrated yeast module (the high-osmolarity glycerol [HOG] pathway) and generated time-course predictions that matched with available experimental data. By extending this framework to larger-scale systems, phenotypic profiles of whole-cell systems could be attained expeditiously.

The regulatory network of E. coli metabolism as a Boolean dynamical system exhibits both homeostasis and flexibility of response

Background

Elucidating the architecture and dynamics of large scale genetic regulatory networks of cells is an important goal in systems biology. We study the system level dynamical properties of the genetic network of Escherichia coli that regulates its metabolism, and show how its design leads to biologically useful cellular properties. Our study uses the database (Covert et al., Nature 2004) containing 583 genes and 96 external metabolites which describes not only the network connections but also the Boolean rule at each gene node that controls the switching on or off of the gene as a function of its inputs.

Results

We have studied how the attractors of the Boolean dynamical system constructed from this database depend on the initial condition of the genes and on various environmental conditions corresponding to buffered minimal media. We find that the system exhibits homeostasis in that its attractors, that turn out to be fixed points or low period cycles, are highly insensitive to initial conditions or perturbations of gene configurations for any given fixed environment. At the same time the attractors show a wide variation when external media are varied implying that the system mounts a highly flexible response to changed environmental conditions. The regulatory dynamics acts to enhance the cellular growth rate under changed media.

Conclusion

Our study shows that the reconstructed genetic network regulating metabolism in E. coli is hierarchical, modular, and largely acyclic, with environmental variables controlling the root of the hierarchy. This architecture makes the cell highly robust to perturbations of gene configurations as well as highly responsive to environmental changes. The twin properties of homeostasis and response flexibility are achieved by this dynamical system even though it is not close to the edge of chaos.

Using a logical model to predict the growth of yeast

BACKGROUND

A logical model of the known metabolic processes in S. cerevisiae was constructed from iFF708, an existing Flux Balance Analysis (FBA) model, and augmented with information from the KEGG online pathway database. The use of predicate logic as the knowledge representation for modelling enables an explicit representation of the structure of the metabolic network, and enables logical inference techniques to be used for model identification/improvement.

RESULTS

Compared to the FBA model, the logical model has information on an additional 263 putative genes and 247 additional reactions. The correctness of this model was evaluated by comparison with iND750 (an updated FBA model closely related to iFF708) by evaluating the performance of both models on predicting empirical minimal medium growth data/essential gene listings.

CONCLUSION

ROC analysis and other statistical studies revealed that use of the simpler logical form and larger coverage results in no significant degradation of performance compared to iND750.

Experimental demonstration of ecological character displacement

Background

The evolutionary consequences of competition are of great interest to researchers studying sympatric speciation, adaptive radiation, species coexistence and ecological assembly. Competition's role in driving evolutionary change in phenotypic distributions, and thus causing ecological character displacement, has been inferred from biogeographical data and measurements of divergent selection on a focal species in the presence of competitors. However, direct experimental demonstrations of character displacement due to competition are rare.

Results

We demonstrate a causal role for competition in ecological character displacement. Using populations of the bacterium Escherichia coli that have adaptively diversified into ecotypes exploiting different carbon resources, we show that when interspecific competition is relaxed, phenotypic distributions converge. When we reinstate competition, phenotypic distributions diverge.

Conclusion

This accordion-like dynamic provides direct experimental evidence that competition for resources can cause evolutionary shifts in resource-related characters.

Predicting biological system objectives de novo from internal state measurements

Background

Optimization theory has been applied to complex biological systems to interrogate network properties and develop and refine metabolic engineering strategies. For example, methods are emerging to engineer cells to optimally produce byproducts of commercial value, such as bioethanol, as well as molecular compounds for disease therapy. Flux balance analysis (FBA) is an optimization framework that aids in this interrogation by generating predictions of optimal flux distributions in cellular networks. Critical features of FBA are the definition of a biologically relevant objective function (e.g., maximizing the rate of synthesis of biomass, a unit of measurement of cellular growth) and the subsequent application of linear programming (LP) to identify fluxes through a reaction network. Despite the success of FBA, a central remaining challenge is the definition of a network objective with biological meaning.

Results

We present a novel method called Biological Objective Solution Search (BOSS) for the inference of an objective function of a biological system from its underlying network stoichiometry as well as experimentally-measured state variables. Specifically, BOSS identifies a system objective by defining a putative stoichiometric "objective reaction," adding this reaction to the existing set of stoichiometric constraints arising from known interactions within a network, and maximizing the putative objective reaction via LP, all the while minimizing the difference between the resultant in silico flux distribution and available experimental (e.g., isotopomer) flux data. This new approach allows for discovery of objectives with previously unknown stoichiometry, thus extending the biological relevance from earlier methods. We verify our approach on the well-characterized central metabolic network of Saccharomyces cerevisiae.

Conclusion

We illustrate how BOSS offers insight into the functional organization of biochemical networks, facilitating the interrogation of cellular design principles and development of cellular engineering applications. Furthermore, we describe how growth is the best-fit objective function for the yeast metabolic network given experimentally-measured fluxes.