Bidirectional reaction steps in metabolic networks: I. Modeling and simulation of carbon isotope labeling experiments

The extension of metabolite balancing with carbon labeling experiments, as described by Marx et al. (Biotechnol. Bioeng. 49: 11-29), results in a much more detailed stationary metabolic flux analysis. As opposed to basic metabolite flux balancing alone, this method enables both flux directions of bidirectional reaction steps to be quantitated. However, the mathematical treatment of carbon labeling systems is much more complicated, because it requires the solution of numerous balance equations that are bilinear with respect to fluxes and fractional labeling. In this study, a universal modeling framework is presented for describing the metabolite and carbon atom flux in a metabolic network. Bidirectional reaction steps are extensively treated and their impact on the system's labeling state is investigated. Various kinds of modeling assumptions, as usually made for metabolic fluxes, are expressed by linear constraint equations. A numerical algorithm for the solution of the resulting linear constrained set of nonlinear equations is developed. The numerical stability problems caused by large bidirectional fluxes are solved by a specially developed transformation method. Finally, the simulation of carbon labeling experiments is facilitated by a flexible software tool for network synthesis. An illustrative simulation study on flux identifiability from available flux and labeling measurements in the cyclic pentose phosphate pathway of a recombinant strain of Zymomonas mobilis concludes this contribution. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 101-117, 1997.     

Metabolic modeling of Saccharomyces cerevisiae using the optimal control of homeostasis: a cybernetic model definition

A model is presented to describe the observed behavior of microorganisms that aim at metabolic homeostasis while growing and adapting to their environment in an optimal way. The cellular metabolism is seen as a network with a multiple controller system with both feedback and feedforward control, i.e., a model based on a dynamic optimal metabolic control. The dynamic network consists of aggregated pathways, each having a control setpoint for the metabolic states at a given growth rate. This set of strategies of the cell forms a true cybernetic model with a minimal number of assumptions. The cellular strategies and constraints were derived from metabolic flux analysis using an identified, biochemically relevant, stoichiometry matrix derived from experimental data on the cellular composition of continuous cultures of Saccharomyces cerevisiae. Based on these data a cybernetic model was developed to study its dynamic behavior. The growth rate of the cell is determined by the structural compounds and fluxes of compounds related to central metabolism. In contrast to many other cybernetic models, the minimal model does not consist of any assumed internal kinetic parameters or interactions. This necessitates the use of a stepwise integration with an optimization of the fluxes at every time interval. Some examples of the behavior of this model are given with respect to steady states and pulse responses. This model is very suitable for describing semiquantitatively dynamics of global cellular metabolism and may form a useful framework for including structured and more detailed kinetic models.     

Dynamic modeling of the central carbon metabolism of Escherichia coli

Application of metabolic engineering principles to the rational design of microbial production processes crucially depends on the ability to describe quantitatively the systemic behavior of the central carbon metabolism to redirect carbon fluxes to the product-forming pathways. Despite the importance for several production processes, development of an essential dynamic model for central carbon metabolism of Escherichia coli has been severely hampered by the current lack of kinetic information on the dynamics of the metabolic reactions. Here we present the design and experimental validation of such a dynamic model, which, for the first time, links the sugar transport system (i.e., phosphotransferase system [PTS]) with the reactions of glycolysis and the pentose-phosphate pathway. Experimental observations of intracellular concentrations of metabolites and cometabolites at transient conditions are used to validate the structure of the model and to estimate the kinetic parameters. Further analysis of the detailed characteristics of the system offers the possibility of studying important questions regarding the stability and control of metabolic fluxes.     

Towards kinetic modeling of genome-scale metabolic networks without sacrificing stoichiometric,thermodynamic and physiological constraints

Mathematical modeling is an essential tool for the comprehensive understanding of cell metabolism and its interactions with the environmental and process conditions. Recent developments in the construction and analysis of stoichiometric models made it possible to define limits on steady-state metabolic behavior using flux balance analysis. However, detailed information on enzyme kinetics and enzyme regulation is needed to formulate kinetic models that can accurately capture the dynamic metabolic responses. The use of mechanistic enzyme kinetics is a difficult task due to uncertainty in the kinetic properties of enzymes. Therefore, the majority of recent works considered only mass action kinetics for reactions in metabolic networks. Herein, we applied the optimization and risk analysis of complex living entities (ORACLE) framework and constructed a large-scale mechanistic kinetic model of optimally grown Escherichia coli. We investigated the complex interplay between stoichiometry, thermodynamics, and kinetics in determining the flexibility and capabilities of metabolism. Our results indicate that enzyme saturation is a necessary consideration in modeling metabolic networks and it extends the feasible ranges of metabolic fluxes and metabolite concentrations. Our results further suggest that enzymes in metabolic networks have evolved to function at different saturation states to ensure greater flexibility and robustness of cellular metabolism.     

Quantitative analysis of metabolic fluxes in Escherichia coli, using two-dimensional NMR spectroscopy and complete isotopomer models.

Complete isotopomer models that simulate distribution of label in 13C tracer experiments are applied to the quantification of metabolic fluxes in the primary carbon metabolism of E. coli under aerobic and anaerobic conditions. The concept of isotopomer mapping matrices (IMMs) is used to simplify the formulation of isotopomer mass balances by expressing all isotopomer mass balances of a metabolite pool in a single matrix equation. A numerically stable method to calculate the steady-state isotopomer distribution in metabolic networks in introduced. Net values of intracellular fluxes and the degree of reversibility of enzymatic steps are estimated by minimization of the deviations between experimental and simulated measurements. The metabolic model applied includes the Embden-Meyerhof-Parnas and the pentose phosphate pathway, the tricarboxylic acid cycle, anaplerotic reaction sequences and pathways involved in amino acid synthesis. The study clearly demonstrates the value of complete isotopomer models for maximizing the information obtainable from 13C tracer experiments. The approach applied here offers a completely general and comprehensive analysis of carbon tracer experiments where any set of experimental data on the labeling state and extracellular fluxes can be used for the quantification of metabolic fluxes in complex metabolic networks.     

Towards Kinetic Modeling of Global Metabolic Networks: Methylobacterium extorquens AM1 Growth as Validation

Here we report a systematic method for constructing a large scale kinetic metabolic model and its initial application to the modeling of central metabolism of Methylobacterium extorquens AM1, a methylotrophic and environmental important bacterium. Its central metabolic network includes formaldehyde metabolism, serine cycle, citric acid cycle, pentose phosphate pathway, gluconeogensis, PHB synthesis and acetyl-CoA conversion pathway, respiration and energy metabolism. Through a systematic and consistent procedure of finding a set of parameters in the physiological range we overcome an outstanding difficulty in large scale kinetic modeling: the requirement for a massive number of enzymatic reaction parameters. We are able to construct the kinetic model based on general biological considerations and incomplete experimental kinetic parameters. Our method consists of the following major steps: 1) using a generic enzymatic rate equation to reduce the number of enzymatic parameters to a minimum set while still preserving their characteristics; 2) using a set of steady state fluxes and metabolite concentrations in the physiological range as the expected output steady state fluxes and metabolite concentrations for the kinetic model to restrict the parametric space of enzymatic reactions; 3) choosing enzyme constants K’s and K’eqs optimized for reactions under physiological concentrations, if their experimental values are unknown; 4) for models which do not cover the entire metabolic network of the organisms, designing a dynamical exchange for the coupling between the metabolism represented in the model and the rest not included.     

Genome-scale metabolic flux analysis of Streptomyces lividans growing on a complex medium

Constraint-based metabolic modeling comprises various excellent tools to assess experimentally observed phenotypic behavior of micro-organisms in terms of intracellular metabolic fluxes. In combination with genome-scale metabolic networks, micro-organisms can be investigated in much more detail and under more complex environmental conditions. Although complex media are ubiquitously applied in industrial fermentations and are often a prerequisite for high protein secretion yields, such multi-component conditions are seldom investigated using genome-scale flux analysis. In this paper, a systematic and integrative approach is presented to determine metabolic fluxes in Streptomyces lividans TK24 grown on a nutritious and complex medium. Genome-scale flux balance analysis and randomized sampling of the solution space are combined to extract maximum information from exometabolome profiles. It is shown that biomass maximization cannot predict the observed metabolite production pattern as such. Although this cellular objective commonly applies to batch fermentation data, both input and output constraints are required to reproduce the measured biomass production rate. Rich media hence not necessarily lead to maximum biomass growth. To eventually identify a unique intracellular flux vector, a hierarchical optimization of cellular objectives is adopted. Out of various tested secondary objectives, maximization of the ATP yield per flux unit returns the closest agreement with the maximum frequency in flux histograms. This unique flux estimation is hence considered as a reasonable approximation for the biological fluxes. Flux maps for different growth phases show no active oxidative part of the pentose phosphate pathway, but NADPH generation in the TCA cycle and NADPH transdehydrogenase activity are most important in fulfilling the NADPH balance. Amino acids contribute to biomass growth by augmenting the pool of available amino acids and by boosting the TCA cycle, particularly when using glutamate and aspartate. Depletion of glutamate and aspartate causes a distinct shift in fluxes of the central carbon and nitrogen metabolism. In the current work, hurdles encountered in flux analysis at a genome-scale level are addressed using hierarchical flux balance analysis and uniform sampling of the constrained solution space. This general framework can now be adopted in further studies of S. lividans, e.g., as a host for heterologous protein production.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

Impact of Stoichiometry Representation on Simulation of Genotype-Phenotype Relationships in Metabolic Networks

Genome-scale metabolic networks provide a comprehensive structural framework for modeling genotype-phenotype relationships through flux simulations. The solution space for the metabolic flux state of the cell is typically very large and optimization-based approaches are often necessary for predicting the active metabolic state under specific environmental conditions. The objective function to be used in such optimization algorithms is directly linked with the biological hypothesis underlying the model and therefore it is one of the most relevant parameters for successful modeling. Although linear combination of selected fluxes is widely used for formulating metabolic objective functions, we show that the resulting optimization problem is sensitive towards stoichiometry representation of the metabolic network. This undesirable sensitivity leads to different simulation results when using numerically different but biochemically equivalent stoichiometry representations and thereby makes biological interpretation intrinsically subjective and ambiguous. We hereby propose a new method, Minimization of Metabolites Balance (MiMBl), which decouples the artifacts of stoichiometry representation from the formulation of the desired objective functions, by casting objective functions using metabolite turnovers rather than fluxes. By simulating perturbed metabolic networks, we demonstrate that the use of stoichiometry representation independent algorithms is fundamental for unambiguously linking modeling results with biological interpretation. For example, MiMBl allowed us to expand the scope of metabolic modeling in elucidating the mechanistic basis of several genetic interactions in Saccharomyces cerevisiae.     

OptForce: An Optimization Procedure for Identifying All Genetic Manipulations Leading to Targeted Overproductions

Computational procedures for predicting metabolic interventions leading to the overproduction of biochemicals in microbial strains are widely in use. However, these methods rely on surrogate biological objectives (e.g., maximize growth rate or minimize metabolic adjustments) and do not make use of flux measurements often available for the wild-type strain. In this work, we introduce the OptForce procedure that identifies all possible engineering interventions by classifying reactions in the metabolic model depending upon whether their flux values must increase, decrease or become equal to zero to meet a pre-specified overproduction target. We hierarchically apply this classification rule for pairs, triples, quadruples, etc. of reactions. This leads to the identification of a sufficient and non-redundant set of fluxes that must change (i.e., MUST set) to meet a pre-specified overproduction target. Starting with this set we subsequently extract a minimal set of fluxes that must actively be forced through genetic manipulations (i.e., FORCE set) to ensure that all fluxes in the network are consistent with the overproduction objective. We demonstrate our OptForce framework for succinate production in Escherichia coli using the most recent in silico E. coli model, iAF1260. The method not only recapitulates existing engineering strategies but also reveals non-intuitive ones that boost succinate production by performing coordinated changes on pathways distant from the last steps of succinate synthesis.     

Metabolic fluxes and metabolic engineering

Metabolic engineering is the directed improvement of cellular properties through the modification of specific biochemical reactions or the introduction of new ones, with the use of recombinant DNA technology. As such, metabolic engineering emphasizes metabolic pathway integration and relies on metabolic fluxes as determinants of cell physiology and measures of metabolic control. The combination of analytical methods to quantify fluxes and their control with molecular biological techniques to implement genetic modifications is the essence of metabolic engineering. Strategies for metabolic flux determination are reviewed in this paper and it is shown how metabolic fluxes can be used in the systematic elucidation of metabolic control in the framework of reaction grouping and top-down metabolic control analysis.     

Uniform sampling of steady-state flux spaces: means to design experiments and to interpret enzymopathies

Reconstruction of genome-scale metabolic networks is now possible using multiple different data types. Constraint-based modeling is an approach to interrogate capabilities of reconstructed networks by constraining possible cellular behavior through the imposition of physicochemical laws. As a result, a steady-state flux space is defined that contains all possible functional states of the network. Uniform random sampling of the steady-state flux space allows for the unbiased appraisal of its contents. Monte Carlo sampling of the steady-state flux space of the reconstructed human red blood cell metabolic network under simulated physiologic conditions yielded the following key results: 1), probability distributions for the values of individual metabolic fluxes showed a wide variety of shapes that could not have been inferred without computation; 2), pairwise correlation coefficients were calculated between all fluxes, determining the level of independence between the measurement of any two fluxes, and identifying highly correlated reaction sets; and 3), the network-wide effects of the change in one (or a few) variables (i.e., a simulated enzymopathy or fixing a flux range based on measurements) were computed. Mathematical models provide the most compact and informative representation of a hypothesis of how a cell works. Thus, understanding model predictions clearly is vital to driving forward the iterative model-building procedure that is at the heart of systems biology. Taken together, the Monte Carlo sampling procedure provides a broadening of the constraint-based approach by allowing for the unbiased and detailed assessment of the impact of the applied physicochemical constraints on a reconstructed network.     

An integrated analysis of enzyme activities, cofactor pools and metabolic fluxes in baculovirus-infected Spodoptera frugiperda Sf9 cells

The scarcity of fundamental knowledge on the baculovirus-host cell interaction is a major drawback for the improvement of bioprocesses through Metabolic Engineering. After the first hours post-infection, the virus takes over the control of cellular machinery, leading to the repression of host gene expression and imposing a high metabolic burden to insect cells. Nevertheless, there is a lack of detailed data on the metabolic responses to infection, which are ultimately responsible for system productivity performance. In this work, a further insight into the central metabolism of Sf9 cells is achieved by a combined analysis of enzyme activities, cellular cofactors (ATP and NAD(P)(+)/NAD(P)H) and metabolic fluxes. Hexokinase and isocitrate dehydrogenase were identified as feasible limiting steps of metabolism; carbon and nitrogen metabolism enzymes were differentially regulated during batch cultures. Moreover, alterations occurring after infection demonstrated the importance of maintaining the energetic state of the cells for baculovirus replication, since ATP accumulated in a MOI-dependent way, and the glutamate dehydrogenase anaplerotic pathway was greatly activated. Altogether, cellular de-energization and stress responses are relevant factors in the metabolic burden imposed by infection. The implications for the improvement of bioprocess performance are discussed.     

Identification of metabolic engineering targets for the enhancement of 1,4-butanediol production in recombinant E. coli using large-scale kinetic models

Rational metabolic engineering methods are increasingly employed in designing the commercially viable processes for the production of chemicals relevant to pharmaceutical, biotechnology, and food and beverage industries. With the growing availability of omics data and of methodologies capable to integrate the available data into models, mathematical modeling and computational analysis are becoming important in designing recombinant cellular organisms and optimizing cell performance with respect to desired criteria. In this contribution, we used the computational framework ORACLE (Optimization and Risk Analysis of Complex Living Entities) to analyze the physiology of recombinant Escherichia coli producing 1,4-butanediol (BDO) and to identify potential strategies for improved production of BDO. The framework allowed us to integrate data across multiple levels and to construct a population of large-scale kinetic models despite the lack of available information about kinetic properties of every enzyme in the metabolic pathways. We analyzed these models and we found that the enzymes that primarily control the fluxes leading to BDO production are part of central glycolysis, the lower branch of tricarboxylic acid (TCA) cycle and the novel BDO production route. Interestingly, among the enzymes between the glucose uptake and the BDO pathway, the enzymes belonging to the lower branch of TCA cycle have been identified as the most important for improving BDO production and yield. We also quantified the effects of changes of the target enzymes on other intracellular states like energy charge, cofactor levels, redox state, cellular growth, and byproduct formation. Independent earlier experiments on this strain confirmed that the computationally obtained conclusions are consistent with the experimentally tested designs, and the findings of the present studies can provide guidance for future work on strain improvement. Overall, these studies demonstrate the potential and effectiveness of ORACLE for the accelerated design of microbial cell factories.     

An algorithm for linear metabolic pathway alignment

Metabolic pathway alignment represents one of the most powerful tools for comparative analysis of metabolism. It involves recognition of metabolites common to a set of functionally-related metabolic pathways, interpretation of biological evolution processes and determination of alternative metabolic pathways. Moreover, it is of assistance in function prediction and metabolism modeling. Although research on genomic sequence alignment is extensive, the problem of aligning metabolic pathways has received less attention. We are motivated to develop an algorithm of metabolic pathway alignment to reveal the similarities between metabolic pathways. A new definition of the metabolic pathway is introduced. The algorithm has been implemented into the PathAligner system; its web-based interface is available at http://bibiserv.techfak.uni-bielefeld.de/pathaligner/.     

13C metabolic flux analysis of three divergent extremely thermophilic bacteria: Geobacillus sp. LC300, Thermus thermophilus HB8, and Rhodothermus marinus DSM 4252

Thermophilic organisms are being increasingly investigated and applied in metabolic engineering and biotechnology. The distinct metabolic and physiological characteristics of thermophiles, including broad substrate range and high uptake rates, coupled with recent advances in genetic tool development, present unique opportunities for strain engineering. However, poor understanding of the cellular physiology and metabolism of thermophiles has limited the application of systems biology and metabolic engineering tools to these organisms. To address this concern, we applied high resolution ¹³C metabolic flux analysis to quantify fluxes for three divergent extremely thermophilic bacteria from separate phyla: Geobacillus sp. LC300, Thermus thermophilus HB8, and Rhodothermus marinus DSM 4252. We performed 18 parallel labeling experiments, using all singly labeled glucose tracers for each strain, reconstructed and validated metabolic network models, measured biomass composition, and quantified precise metabolic fluxes for each organism. In the process, we resolved many uncertainties regarding gaps in pathway reconstructions and elucidated how these organisms maintain redox balance and generate energy. Overall, we found that the metabolisms of the three thermophiles were highly distinct, suggesting that adaptation to growth at high temperatures did not favor any particular set of metabolic pathways. All three strains relied heavily on glycolysis and TCA cycle to generate key cellular precursors and cofactors. None of the investigated organisms utilized the Entner-Doudoroff pathway and only one strain had an active oxidative pentose phosphate pathway. Taken together, the results from this study provide a solid foundation for future model building and engineering efforts with these and related thermophiles.     

From annotated genomes to metabolic flux models and kinetic parameter fitting

Significant advances in system-level modeling of cellular behavior can be achieved based on constraints derived from genomic information and on optimality hypotheses. For steady-state models of metabolic networks, mass conservation and reaction stoichiometry impose linear constraints on metabolic fluxes. Different objectives, such as maximization of growth rate or minimization of flux distance from a reference state, can be tested in different organisms and conditions. In particular, we have suggested that the metabolic properties of mutant bacterial strains are best described by an algorithm that performs a minimization of metabolic adjustment (MOMA) upon gene deletion. The increasing availability of many annotated genomes paves the way for a systematic application of these flux balance methods to a large variety of organisms. However, such a high throughput goal crucially depends on our capacity to build metabolic flux models in a fully automated fashion. Here we describe a pipeline for generating models from annotated genomes and discuss the current obstacles to full automation. In addition, we propose a framework for the integration of flux modeling results and high throughput proteomic data, which can potentially help in the inference of whole-cell kinetic parameters.