The generalized flux-minimization method and its application to metabolic networks affected by enzyme deficiencies

The flux-minimization method [Holzhütter, H.G., 2004. The principle of flux-minimization and its application to calculate stationary fluxes in metabolic networks. Eur. J. Biochem. 271, 2905-2922] has been proposed as an alternative to kinetic modeling to calculate stationary fluxes in metabolic networks. Here a generalization of this method is proposed that takes into account possible limitations of internal fluxes, e.g. due to enzyme defects or partial inhibition of enzyme activities by drugs. The generalized method consists in the minimization of an objective function which expresses the compromise that has to be made between minimization of internal fluxes on one hand and maintenance of the metabolic output required for various cellular functions on the other. Fulfillment of the latter condition is measured through a fitness function, which evaluates the relative deviation of the output fluxes from demanded target values. The method is applied to assess the metabolic consequences caused by a deficiency of enzymes involved in the metabolism of erythrocytes. The obtained results are in good agreement with those obtained on the basis of a comprehensive kinetic model.     

Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks

Background  

In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.     

Metabolic flux analysis of CHO cells at growth and non-growth phases using isotopic tracers and mass spectrometry

Chinese hamster ovary (CHO) cells are the main platform for production of biotherapeutics in the biopharmaceutical industry. However, relatively little is known about the metabolism of CHO cells in cell culture. In this work, metabolism of CHO cells was studied at the growth phase and early stationary phase using isotopic tracers and mass spectrometry. CHO cells were grown in fed-batch culture over a period of six days. On days 2 and 4, [1,2-¹³C] glucose was introduced and the labeling of intracellular metabolites was measured by gas chromatography-mass spectrometry (GC–MS) at 6, 12 and 24 h following the introduction of tracer. Intracellular metabolic fluxes were quantified from measured extracellular rates and ¹³C-labeling dynamics of intracellular metabolites using non-stationary ¹³C-metabolic flux analysis (¹³C-MFA). The flux results revealed significant rewiring of intracellular metabolic fluxes in the transition from growth to non-growth, including changes in energy metabolism, redox metabolism, oxidative pentose phosphate pathway and anaplerosis. At the exponential phase, CHO cell metabolism was characterized by a high flux of glycolysis from glucose to lactate, anaplerosis from pyruvate to oxaloacetate and from glutamate to α-ketoglutarate, and cataplerosis though malic enzyme. At the stationary phase, the flux map was characterized by a reduced flux of glycolysis, net lactate uptake, oxidative pentose phosphate pathway flux, and reduced rate of anaplerosis. The fluxes of pyruvate dehydrogenase and TCA cycle were similar at the exponential and stationary phase. The results presented here provide a solid foundation for future studies of CHO cell metabolism for applications such as cell line development and medium optimization for high-titer production of recombinant proteins.     

Sequential Metabolic Phases as a Means to Optimize Cellular Output in a Constant Environment

Temporal changes of gene expression are a well-known regulatory feature of all cells, which is commonly perceived as a strategy to adapt the proteome to varying external conditions. However, temporal (rhythmic and non-rhythmic) changes of gene expression are also observed under virtually constant external conditions. Here we hypothesize that such changes are a means to render the synthesis of the metabolic output more efficient than under conditions of constant gene activities. In order to substantiate this hypothesis, we used a flux-balance model of the cellular metabolism. The total time span spent on the production of a given set of target metabolites was split into a series of shorter time intervals (metabolic phases) during which only selected groups of metabolic genes are active. The related flux distributions were calculated under the constraint that genes can be either active or inactive whereby the amount of protein related to an active gene is only controlled by the number of active genes: the lower the number of active genes the more protein can be allocated to the enzymes carrying non-zero fluxes. This concept of a predominantly protein-limited efficiency of gene expression clearly differs from other concepts resting on the assumption of an optimal gene regulation capable of allocating to all enzymes and transporters just that fraction of protein necessary to prevent rate limitation. Applying this concept to a simplified metabolic network of the central carbon metabolism with glucose or lactate as alternative substrates, we demonstrate that switching between optimally chosen stationary flux modes comprising different sets of active genes allows producing a demanded amount of target metabolites in a significantly shorter time than by a single optimal flux mode at fixed gene activities. Our model-based findings suggest that temporal expression of metabolic genes can be advantageous even under conditions of constant external substrate supply.     

Metabolic analysis of the synthesis of high levels of intracellular human SOD in Saccharomyces cerevisiae rhSOD 2060 411 SGA122

The synthesis of human superoxide dismutase (SOD) in batch cultures of a Saccharomyces cerevisiae strain using a glucose-limited minimal medium was studied through metabolic flux analysis. A stoichiometric model was built, which included 78 reactions, according to metabolic pathways operative in these strains during respirofermentative and oxidative metabolism. It allowed calculation of the distribution of metabolic fluxes during diauxic growth on glucose and ethanol. Fermentation profiles and metabolic fluxes were analyzed at different phases of diauxic growth for the recombinant strain (P+) and for its wild type (P-). The synthesis of SOD by the strain P+ resulted in a decrease in specific growth rate of 34 and 54% (growth on glucose and ethanol respectively) in comparison to the wild type. Both strains exhibited similar flux of glucose consumption and ethanol synthesis but important differences in carbon distribution with biomass/substrate yields and ATP production 50% higher in P-. A higher contribution of fermentative metabolism, with 64% of the energy produced at the phosphorylation level, was observed during SOD production. The flux of precursors to amino acids and nucleotides was higher in the recombinant strain, in agreement with the higher total RNA and protein levels. Lower specific growth rates in strain P+ appear to be related to the decrease in the rate of synthesis of nonrecombinant protein, as well as a decrease in the activities of the pentose phosphate (PP) pathway and TCA cycle. A very different way of entry into the stationary phase was observed for each strain: in the wild-type strain most metabolic fluxes decreased and fluxes related to energy reserve synthesis increased, while in the P+ strain the flux of 22 reactions (including PP pathway and amino acids biosynthesis) related to SOD production increased their fluxes. Changes in SOD production rates at different physiological states appear to be related to the differences in building blocks availability between respirofermentative and oxidative metabolism. Using the present expression system, ideal conditions for SOD synthesis are represented by either active growth during respirofermentative metabolism or transition from a growing to a nongrowing state. An increase in SOD flux could be achieved using an expression system nonassociated to growth and potentially eliminating part of the metabolic burden.     

Strategies for investigating the plant metabolic network with steady-state metabolic flux analysis: lessons from an Arabidopsis cell culture and other systems

Steady-state (13)C metabolic flux analysis (MFA) is currently the experimental method of choice for generating flux maps of the compartmented network of primary metabolism in heterotrophic and mixotrophic plant tissues. While statistically robust protocols for the application of steady-state MFA to plant tissues have been developed by several research groups, the implementation of the method is still far from routine. The effort required to produce a flux map is more than justified by the information that it contains about the metabolic phenotype of the system, but it remains the case that steady-state MFA is both analytically and computationally demanding. This article provides an overview of principles that underpin the implementation of steady-state MFA, focusing on the definition of the metabolic network responsible for redistribution of the label, experimental considerations relating to data collection, the modelling process that allows a set of metabolic fluxes to be deduced from the labelling data, and the interpretation of flux maps. The article draws on published studies of Arabidopsis cell cultures and other systems, including developing oilseeds, with the aim of providing practical guidance and strategies for handling the issues that arise when applying steady-state MFA to the complex metabolic networks encountered in plants.     

Interpretation of metabolic flux maps by limitation potentials and constrained limitation sensitivities

Two new concepts, "Limitation Potential" and "Constraint Limitation Sensitivity" are introduced that use definitions derived from metabolic flux analysis (MFA) and metabolic network analysis (MNA). They are applied to interpret a measured flux distribution in the context of all possible flux distributions and thus combine MFA with MNA. The proposed measures are used to quantify and compare the influence of intracellular fluxes on the production yield. The methods are purely based on the stoichiometry of the network and constraints that are given from irreversible fluxes. In contrast to metabolic control analysis (MCA), within this approach no information about the kinetic mechanisms are needed. A limitation potential (LP) is defined as the reduction of the reachable (theoretical) maximum by a measured flux. Measured fluxes that strongly narrow the reachable maximum are assumed to be limiting as the network has no ability to counterbalance the restriction due to the observed flux. In a second step, the sensitivity of the reduced maximum is regarded. This measure provides information about the necessitated changes to reach higher yields. The methods are applied to interpret the capabilities of a network based on measured fluxes for a L-phenylalanine producer. The strain was examined by a series of experiments and three flux maps of the production phase are analyzed. It can be shown that the reachable yield is drastically reduced by the measured efflux into the TCA cycle, while the oxidative pentose-phosphate pathway only plays a secondary role on the reachable maximum.     

Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix epsilon, and the vector e of enzyme concentrations fulfil in optimal states the relations c(T)e = e and epsilon(T)e = 0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients.     

Quantification of central metabolic fluxes in the facultative methylotroph methylobacterium extorquens AM1 using 13C-label tracing and mass spectrometry

The metabolic fluxes of central carbon metabolism were measured in chemostat-grown cultures of Methylobacterium extorquens AM1 with methanol as the sole organic carbon and energy source and growth-limiting substrate. Label tracing experiments were carried out using 70% (13)C-methanol in the feed, and the steady-state mass isotopomer distributions of amino acids derived from total cell protein were measured by gas chromatography coupled to mass spectrometry. Fluxes were calculated from the isotopomer distribution data using an isotopomer balance model and evolutionary error minimization algorithm. The combination of labeled methanol with unlabeled CO(2), which enters central metabolism in two different reactions, provided the discriminatory power necessary to allow quantification of the unknown fluxes within a reasonably small confidence interval. In wild-type M. extorquens AM1, no measurable flux was detected through pyruvate dehydrogenase or malic enzyme, and very little flux through alpha-ketoglutarate dehydrogenase (1.4% of total carbon). In contrast, the alpha-ketoglutarate dehydrogenase flux was 25.5% of total carbon in the regulatory mutant strain phaR, while the pyruvate dehydrogenase and malic enzyme fluxes remained insignificant. The success of this technique with growth on C(1) compounds suggests that it can be applied to help characterize the effects of other regulatory mutations, and serve as a diagnostic tool in the metabolic engineering of methylotrophic bacteria.     

Reconstruction and flux analysis of coupling between metabolic pathways of astrocytes and neurons: application to cerebral hypoxia

Background

It is a daunting task to identify all the metabolic pathways of brain energy metabolism and develop a dynamic simulation environment that will cover a time scale ranging from seconds to hours. To simplify this task and make it more practicable, we undertook stoichiometric modeling of brain energy metabolism with the major aim of including the main interacting pathways in and between astrocytes and neurons.

Model

The constructed model includes central metabolism (glycolysis, pentose phosphate pathway, TCA cycle), lipid metabolism, reactive oxygen species (ROS) detoxification, amino acid metabolism (synthesis and catabolism), the well-known glutamate-glutamine cycle, other coupling reactions between astrocytes and neurons, and neurotransmitter metabolism. This is, to our knowledge, the most comprehensive attempt at stoichiometric modeling of brain metabolism to date in terms of its coverage of a wide range of metabolic pathways. We then attempted to model the basal physiological behaviour and hypoxic behaviour of the brain cells where astrocytes and neurons are tightly coupled.

Results

The reconstructed stoichiometric reaction model included 217 reactions (184 internal, 33 exchange) and 216 metabolites (183 internal, 33 external) distributed in and between astrocytes and neurons. Flux balance analysis (FBA) techniques were applied to the reconstructed model to elucidate the underlying cellular principles of neuron-astrocyte coupling. Simulation of resting conditions under the constraints of maximization of glutamate/glutamine/GABA cycle fluxes between the two cell types with subsequent minimization of Euclidean norm of fluxes resulted in a flux distribution in accordance with literature-based findings. As a further validation of our model, the effect of oxygen deprivation (hypoxia) on fluxes was simulated using an FBA-derivative approach, known as minimization of metabolic adjustment (MOMA). The results show the power of the constructed model to simulate disease behaviour on the flux level, and its potential to analyze cellular metabolic behaviour in silico.

Conclusion

The predictive power of the constructed model for the key flux distributions, especially central carbon metabolism and glutamate-glutamine cycle fluxes, and its application to hypoxia is promising. The resultant acceptable predictions strengthen the power of such stoichiometric models in the analysis of mammalian cell metabolism.     

Analysis of flux estimates based on (13)C-labelling experiments.

Modelling of the fluxes in central metabolism can be performed by combining labelling experiments with metabolite balancing. Using this approach, multiple samples from a cultivation of Saccharomyces cerevisiae in metabolic and isotopic steady state were analysed, and the metabolic fluxes in central metabolism were estimated. In the various samples, the estimates of the central metabolic pathways, the tricarboxylic acid cycle, the oxidative pentose phosphate pathway and the anaplerotic pathway, showed an unprecedented reproducibility. The high reproducibility was obtained with fractional labellings of individual carbon atoms as the calculational base, illustrating that the more complex modelling using isotopomers is not necessarily superior with respect to reproducibility of the flux estimates. Based on these results some general difficulties in flux estimation are discussed.     

Inclusion of maintenance energy improves the intracellular flux predictions of CHO

Chinese hamster ovary (CHO) cells are the leading platform for the production of biopharmaceuticals with human-like glycosylation. The standard practice for cell line generation relies on trial and error approaches such as adaptive evolution and high-throughput screening, which typically take several months. Metabolic modeling could aid in designing better producer cell lines and thus shorten development times. The genome-scale metabolic model (GSMM) of CHO can accurately predict growth rates. However, in order to predict rational engineering strategies it also needs to accurately predict intracellular fluxes. In this work we evaluated the agreement between the fluxes predicted by parsimonious flux balance analysis (pFBA) using the CHO GSMM and a wide range of ¹³C metabolic flux data from literature. While glycolytic fluxes were predicted relatively well, the fluxes of tricarboxylic acid (TCA) cycle were vastly underestimated due to too low energy demand. Inclusion of computationally estimated maintenance energy significantly improved the overall accuracy of intracellular flux predictions. Maintenance energy was therefore determined experimentally by running continuous cultures at different growth rates and evaluating their respective energy consumption. The experimentally and computationally determined maintenance energy were in good agreement. Additionally, we compared alternative objective functions (minimization of uptake rates of seven nonessential metabolites) to the biomass objective. While the predictions of the uptake rates were quite inaccurate for most objectives, the predictions of the intracellular fluxes were comparable to the biomass objective function.     

Topology of the global regulatory network of carbon limitation in Escherichia coli

One fundamental shortcoming of biotechnological processes operating under carbon-limiting conditions is the high-energy demand (maintenance) of the cells. Although the function of the central carbon metabolism in supplying precursors and energy for biosynthesis has been thoroughly characterized, its regulation and dynamic behaviour during carbon-limited growth has not yet been revealed. The current work demonstrates a time series of metabolic flux distributions during fed-batch cultivation of Escherichia coli K-12 W3110 applying a constant feed rate. The fluxes in glycolysis, pentose phosphate pathway and biosynthesis fell significantly, whereas TCA cycle fluxes remained constant. The flux redistribution resulted in an enhanced energy generation in the TCA cycle and consequently, in a 20% lower biomass yield. The intracellular alarmones ppGpp and cAMP accumulated in large quantities after the onset of nutrient limitation, subsequently declining to basal levels. The network topology of the regulation of the central metabolic pathways was identified so that the observed metabolic and regulatory behaviour can be described. This provides novel aspects of global regulation of the metabolism by the cra, crp and relA/spoT modulons. The work constitutes an important step towards dynamic mathematical modelling of regulation and metabolism, which is needed for the rational optimization of biotechnological processes.     

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.     

Analysis of optimal phenotypic space using elementary modes as applied to Corynebacterium glutamicum

Background  

Quantification of the metabolic network of an organism offers insights into possible ways of developing mutant strain for better productivity of an extracellular metabolite. The first step in this quantification is the enumeration of stoichiometries of all reactions occurring in a metabolic network. The structural details of the network in combination with experimentally observed accumulation rates of external metabolites can yield flux distribution at steady state. One such methodology for quantification is the use of elementary modes, which are minimal set of enzymes connecting external metabolites. Here, we have used a linear objective function subject to elementary modes as constraint to determine the fluxes in the metabolic network of Corynebacterium glutamicum. The feasible phenotypic space was evaluated at various combinations of oxygen and ammonia uptake rates.     

A hybrid kinetic and constraint-based model of leaf metabolism allows predictions of metabolic fluxes in different environments

While flux balance analysis (FBA) provides a framework for predicting steady-state leaf metabolic network fluxes, it does not readily capture the response to environmental variables without being coupled to other modelling formulations. To address this, we coupled an FBA model of 903 reactions of soybean (Glycine max) leaf metabolism with e-photosynthesis, a dynamic model that captures the kinetics of 126 reactions of photosynthesis and associated chloroplast carbon metabolism. Successful coupling was achieved in an iterative formulation in which fluxes from e-photosynthesis were used to constrain the FBA model and then, in turn, fluxes computed from the FBA model used to update parameters in e-photosynthesis. This process was repeated until common fluxes in the two models converged. Coupling did not hamper the ability of the kinetic module to accurately predict the carbon assimilation rate, photosystem II electron flux, and starch accumulation of field-grown soybean at two CO₂ concentrations. The coupled model also allowed accurate predictions of additional parameters such as nocturnal respiration, as well as analysis of the effect of light intensity and elevated CO₂ on leaf metabolism. Predictions included an unexpected decrease in the rate of export of sucrose from the leaf at high light, due to altered starch–sucrose partitioning, and altered daytime flux modes in the tricarboxylic acid cycle at elevated CO₂. Mitochondrial fluxes were notably different between growing and mature leaves, with greater anaplerotic, tricarboxylic acid cycle and mitochondrial ATP synthase fluxes predicted in the former, primarily to provide carbon skeletons and energy for protein synthesis.