Dynamic metabolic flux analysis (DMFA): a framework for determining fluxes at metabolic non-steady state

Metabolic flux analysis (MFA) is a key tool for measuring in vivo metabolic fluxes in systems at metabolic steady state. Here, we present a new method for dynamic metabolic flux analysis (DMFA) of systems that are not at metabolic steady state. The advantages of our DMFA method are: (1) time-series of metabolite concentration data can be applied directly for estimating dynamic fluxes, making data smoothing and estimation of average extracellular rates unnecessary; (2) flux estimation is achieved without integration of ODEs, or iterations; (3) characteristic metabolic phases in the fermentation data are identified automatically by the algorithm, rather than selected manually/arbitrarily. We demonstrate the application of the new DMFA framework in three example systems. First, we evaluated the performance of DMFA in a simple three-reaction model in terms of accuracy, precision and flux observability. Next, we analyzed a commercial glucose-limited fed-batch process for 1,3-propanediol production. The DMFA method accurately captured the dynamic behavior of the fed-batch fermentation and identified characteristic metabolic phases. Lastly, we demonstrate that DMFA can be used without any assumed metabolic network model for data reconciliation and detection of gross measurement errors using carbon and electron balances as constraints.     

Kinetic isotope effects significantly influence intracellular metabolite 13C labeling patterns and flux determination

Rigorous mathematical modeling of carbon-labeling experiments allows estimation of fluxes through the pathways of central carbon metabolism, yielding powerful information for basic scientific studies as well as for a wide range of applications. However, the mathematical models that have been developed for flux determination from ¹³C labeling data have commonly neglected the influence of kinetic isotope effects on the distribution of ¹³C label in intracellular metabolites, as these effects have often been assumed to be inconsequential. We have used measurements of the ¹³C isotope effects on the pyruvate dehydrogenase enzyme from the literature to model isotopic fractionation at the pyruvate node and quantify the modeling errors expected to result from the assumption that isotope effects are negligible. We show that under some conditions kinetic isotope effects have a significant impact on the ¹³C labeling patterns of intracellular metabolites, and the errors associated with neglecting isotope effects in ¹³C-metabolic flux analysis models can be comparable in size to measurement errors associated with GC–MS. Thus, kinetic isotope effects must be considered in any rigorous assessment of errors in ¹³C labeling data, goodness-of-fit between model and data, confidence intervals of estimated metabolic fluxes, and statistical significance of differences between estimated metabolic flux distributions.     

Towards dynamic metabolic flux analysis in CHO cell cultures

Chinese hamster ovary (CHO) cells are the most widely used mammalian cell line for biopharmaceutical production, with a total global market approaching $100 billion per year. In the pharmaceutical industry CHO cells are grown in fed-batch culture, where cellular metabolism is characterized by high glucose and glutamine uptake rates combined with high rates of ammonium and lactate secretion. The metabolism of CHO cells changes dramatically during a fed-batch culture as the cells adapt to a changing environment and transition from exponential growth phase to stationary phase. Thus far, it has been challenging to study metabolic flux dynamics in CHO cell cultures using conventional metabolic flux analysis techniques that were developed for systems at metabolic steady state. In this paper we review progress on flux analysis in CHO cells and techniques for dynamic metabolic flux analysis. Application of these new tools may allow identification of intracellular metabolic bottlenecks at specific stages in CHO cell cultures and eventually lead to novel strategies for improving CHO cell metabolism and optimizing biopharmaceutical process performance.     

Finding elementary flux modes in metabolic networks based on flux balance analysis and flux coupling analysis: application to the analysis of Escherichia coli metabolism

Elementary modes (EMs) are steady-state metabolic flux vectors with minimal set of active reactions. Each EM corresponds to a metabolic pathway. Therefore, studying EMs is helpful for analyzing the production of biotechnologically important metabolites. However, memory requirements for computing EMs may hamper their applicability as, in most genome-scale metabolic models, no EM can be computed due to running out of memory. In this study, we present a method for computing randomly sampled EMs. In this approach, a network reduction algorithm is used for EM computation, which is based on flux balance-based methods. We show that this approach can be used to recover the EMs in the medium- and genome-scale metabolic network models, while the EMs are sampled in an unbiased way. The applicability of such results is shown by computing “estimated” control-effective flux values in Escherichia coli metabolic network.     

Metabolic flux analysis in mammalian cell culture

Mammalian cell culture metabolism is characterized by glucoglutaminolysis, that is, high glucose and glutamine uptake combined with a high rate of lactate and non-essential amino acid secretion. Stress associated with acid neutralization and ammonia accumulation necessitates complex feeding schemes and limits cell densities achieved in fed-batch culture. Conventional and constraint-based metabolic flux analysis has been successfully used to study the metabolic phenotype of mammalian cells in culture, while ¹³C tracer analysis has been used to study small network models and validate assumptions of metabolism. Large-scale ¹³C metabolic flux analysis, which is required to improve confidence in the network models and their predictions, remains a major challenge. Advances in both modeling and analytical techniques are bringing this challenge within sight.     

The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks.

Cellular functions are ultimately linked to metabolic fluxes brought about by thousands of chemical reactions and transport processes. The synthesis of the underlying enzymes and membrane transporters causes the cell a certain 'effort' of energy and external resources. Considering that those cells should have had a selection advantage during natural evolution that enabled them to fulfil vital functions (such as growth, defence against toxic compounds, repair of DNA alterations, etc.) with minimal effort, one may postulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important 'target' fluxes required to accomplish a specific pattern of cellular functions, the stationary metabolic fluxes have to become a minimum. To convert this principle into a mathematical method enabling the prediction of stationary metabolic fluxes, the total flux in the network is measured by a weighted linear combination of all individual fluxes whereby the thermodynamic equilibrium constants are used as weighting factors, i.e. the more the thermodynamic equilibrium lies on the right-hand side of the reaction, the larger the weighting factor for the backward reaction. A linear programming technique is applied to minimize the total flux at fixed values of the target fluxes and under the constraint of flux balance (= steady-state conditions) with respect to all metabolites. The theoretical concept is applied to two metabolic schemes: the energy and redox metabolism of erythrocytes, and the central metabolism of Methylobacterium extorquens AM1. The flux rates predicted by the flux-minimization method exhibit significant correlations with flux rates obtained by either kinetic modelling or direct experimental determination. Larger deviations occur for segments of the network composed of redundant branches where the flux-minimization method always attributes the total flux to the thermodynamically most favourable branch. Nevertheless, compared with existing methods of structural modelling, the principle of flux minimization appears to be a promising theoretical approach to assess stationary flux rates in metabolic systems in cases where a detailed kinetic model is not yet available.     

A kinetic model of Escherichia coli core metabolism satisfying multiple sets of mutant flux data

In contrast to stoichiometric-based models, the development of large-scale kinetic models of metabolism has been hindered by the challenge of identifying kinetic parameter values and kinetic rate laws applicable to a wide range of environmental and/or genetic perturbations. The recently introduced ensemble modeling (EM) procedure provides a promising remedy to address these challenges by decomposing metabolic reactions into elementary reaction steps and incorporating all phenotypic observations, upon perturbation, in its model parameterization scheme. Here, we present a kinetic model of Escherichia coli core metabolism that satisfies the fluxomic data for wild-type and seven mutant strains by making use of the EM concepts. This model encompasses 138 reactions, 93 metabolites and 60 substrate-level regulatory interactions accounting for glycolysis/gluconeogenesis, pentose phosphate pathway, TCA cycle, major pyruvate metabolism, anaplerotic reactions and a number of reactions in other parts of the metabolism. Parameterization is performed using a formal optimization approach that minimizes the discrepancies between model predictions and flux measurements. The predicted fluxes by the model are within the uncertainty range of experimental flux data for 78% of the reactions (with measured fluxes) for both the wild-type and seven mutant strains. The remaining flux predictions are mostly within three standard deviations of reported ranges. Converting the EM-based parameters into a Michaelis–Menten equivalent formalism revealed that 35% of K_m and 77% of k_cat parameters are within uncertainty range of the literature-reported values. The predicted metabolite concentrations by the model are also within uncertainty ranges of metabolomic data for 68% of the metabolites. A leave-one-out cross-validation test to evaluate the flux prediction performance of the model showed that metabolic fluxes for the mutants located in the proximity of mutations used for training the model can be predicted more accurately. The constructed model and the parameterization procedure presented in this study pave the way for the construction of larger-scale kinetic models with more narrowly distributed parameter values as new metabolomic/fluxomic data sets are becoming available for E. coli and other organisms.     

Development of a mathematical model for evaluating the dynamics of normal and apoptotic Chinese hamster ovary cells

A metabolic flux based methodology was developed for modeling the metabolism of a Chinese hamster ovary cell line. The elimination of insignificant fluxes resulted in a simplified metabolic network which was the basis for modeling the significant metabolites. Employing kinetic rate expressions for growing and non-growing subpopulations, a logistic model was developed for cell growth and dynamic models were formulated to describe culture composition and monoclonal antibody (MAb) secretion. The model was validated for a range of nutrient concentrations. Good agreement was obtained between model predictions and experimental data. The ultimate goal of this study is to establish a comprehensive dynamic model which may be used for model-based optimization of the cell culture for MAb production in both batch and fed-batch systems.     

Consistent Estimation of Gibbs Energy Using Component Contributions

Standard Gibbs energies of reactions are increasingly being used in metabolic modeling for applying thermodynamic constraints on reaction rates, metabolite concentrations and kinetic parameters. The increasing scope and diversity of metabolic models has led scientists to look for genome-scale solutions that can estimate the standard Gibbs energy of all the reactions in metabolism. Group contribution methods greatly increase coverage, albeit at the price of decreased precision. We present here a way to combine the estimations of group contribution with the more accurate reactant contributions by decomposing each reaction into two parts and applying one of the methods on each of them. This method gives priority to the reactant contributions over group contributions while guaranteeing that all estimations will be consistent, i.e. will not violate the first law of thermodynamics. We show that there is a significant increase in the accuracy of our estimations compared to standard group contribution. Specifically, our cross-validation results show an 80% reduction in the median absolute residual for reactions that can be derived by reactant contributions only. We provide the full framework and source code for deriving estimates of standard reaction Gibbs energy, as well as confidence intervals, and believe this will facilitate the wide use of thermodynamic data for a better understanding of metabolism.     

Calculation errors of time-varying flux control coefficients obtained from elasticity coefficients by means of summation and connectivity theorems in metabolic control analysis

This paper investigates the accuracy of a matrix method proposed by other researchers to calculate time-varying flux control coefficients (dynamic FCCs) from elasticity coefficients by means of summation and connectivity theorems in the framework of metabolic control analysis. A mathematical model for the fed-batch penicillin V fermentation process is used as a case example for discussion. Calculated results reveal that this method produces significant calculation errors because the theorems are essentially valid only in steady state, although it may provide rough time-transient behaviors of FCCs. Strictly, therefore, dynamic FCCs should be directly calculated from the differential equations for metabolite concentrations and sensitivities.     

Optimal Bioreactor Operational Policies for the Enzymatic Hydrolysis of Sugarcane Bagasse

The consolidation of the industrial production of second-generation (2G) bioethanol relies on the improvement of the economics of the process. Within this general scope, this paper addresses one aspect that impacts the costs of the biochemical route for producing 2G bioethanol: defining optimal operational policies for the reactor running the enzymatic hydrolysis of the C6 biomass fraction. The use of fed-batch reactors is one common choice for this process, aiming at maximum yields and productivities. The optimization problem for fed-batch reactors usually consists in determining substrate feeding profiles, in order to maximize some performance index. In the present control problem, the performance index and the system dynamics are both linear with respect to the control variable (the trajectory of substrate feed flow). Simple Michaelis–Menten pseudo-homogeneous kinetic models with product inhibition were used in the dynamic modeling of a fed-bath reactor, and two feeding policies were implemented and validated in bench-scale reactors processing pre-treated sugarcane bagasse. The first approach applied classical optimal control theory. The second policy was defined with the purpose of sustaining high rates of glucose production, adding enzyme (Accellerase® 1500) and substrate simultaneously during the reaction course. A methodology is described, which used economical criteria for comparing the performance of the reactor operating in successive batches and in fed-batch modes. Fed-batch mode was less sensitive to enzyme prices than successive batches. Process intensification in the fed-batch reactor led to glucose final concentrations around 200 g/L.     

INCA 2.0: A tool for integrated,dynamic modeling of NMR- and MS-based isotopomer measurements and rigorous metabolic flux analysis

Metabolic flux analysis (MFA) combines experimental measurements and computational modeling to determine biochemical reaction rates in live biological systems. Advancements in analytical instrumentation, such as nuclear magnetic resonance (NMR) spectroscopy and mass spectrometry (MS), have facilitated chemical separation and quantification of isotopically enriched metabolites. However, no software packages have been previously described that can integrate isotopomer measurements from both MS and NMR analytical platforms and have the flexibility to estimate metabolic fluxes from either isotopic steady-state or dynamic labeling experiments. By applying physiologically relevant cardiac and hepatic metabolic models to assess NMR isotopomer measurements, we herein test and validate new modeling capabilities of our enhanced flux analysis software tool, INCA 2.0. We demonstrate that INCA 2.0 can simulate and regress steady-state ¹³C NMR datasets from perfused hearts with an accuracy comparable to other established flux assessment tools. Furthermore, by simulating the infusion of three different ¹³C acetate tracers, we show that MFA based on dynamic ¹³C NMR measurements can more precisely resolve cardiac fluxes compared to isotopically steady-state flux analysis. Finally, we show that estimation of hepatic fluxes using combined ¹³C NMR and MS datasets improves the precision of estimated fluxes by up to 50%. Overall, our results illustrate how the recently added NMR data modeling capabilities of INCA 2.0 can enable entirely new experimental designs that lead to improved flux resolution and can be applied to a wide range of biological systems and measurement time courses.     

Metabolic flux analysis in a nonstationary system: fed-batch fermentation of a high yielding strain of E. coli producing 1,3-propanediol

Metabolic fluxes estimated from stable-isotope studies provide a key to understanding cell physiology and regulation of metabolism. A limitation of the classical method for metabolic flux analysis (MFA) is the requirement for isotopic steady state. To extend the scope of flux determination from stationary to nonstationary systems, we present a novel modeling strategy that combines key ideas from isotopomer spectral analysis (ISA) and stationary MFA. Isotopic transients of the precursor pool and the sampled products are described by two parameters, D and G parameters, respectively, which are incorporated into the flux model. The G value is the fraction of labeled product in the sample, and the D value is the fractional contribution of the feed for the production of labeled products. We illustrate the novel modeling strategy with a nonstationary system that closely resembles industrial production conditions, i.e. fed-batch fermentation of Escherichia coli that produces 1,3-propanediol (PDO). Metabolic fluxes and the D and G parameters were estimated by fitting labeling distributions of biomass amino acids measured by GC/MS to a model of E. coli metabolism. We obtained highly consistent fits from the data with 82 redundant measurements. Metabolic fluxes were estimated for 20 time points during course of the fermentation. As such we established, for the first time, detailed time profiles of in vivo fluxes. We found that intracellular fluxes changed significantly during the fed-batch. The intracellular flux associated with PDO pathway increased by 10%. Concurrently, we observed a decrease in the split ratio between glycolysis and pentose phosphate pathway from 70/30 to 50/50 as a function of time. The TCA cycle flux, on the other hand, remained constant throughout the fermentation. Furthermore, our flux results provided additional insight in support of the assumed genotype of the organism.     

Dynamic model for CHO cell engineering

Industrial CHO cell fed-batch processes have progressed significantly over the past decade, with recombinant protein titer consistently reaching the gram per liter level. Such improvements have largely resulted from separate advances in process and cell line development. Model-based selection of targets for metabolic engineering in CHO cells is confounded by the dynamic nature of the fed-batch process. In this work, we use a dynamic model of CHO cell metabolism to simultaneously identify both process and cell modifications that improve antibody production. Model simulations explored ca. 9200 combinations of process variables (shift temperature, shift day, seed density, and harvest day) and knockdowns (8 metabolic enzymes). A comprehensive examination of a simulated solution space showed that optimal gene knockdown clearly depends on the process parameters such as temperature shift day, shift temperature, and seed density. Knockdown of enzymes related to lactate production were the most beneficial; however, depending on the process conditions, modulating such enzymes yielded varying productivities, ranging from a reduction in final titer to greater than 2-fold improvement.     

An on-line physiological state recognition system for the lysine fermentation process based on a metabolic reaction model

A metabolic reaction model was developed for the lysine fermentation process by Corynebacterium glutamicum AJ-3462 to estimate the physiological state of the cells-that is, the growth and production activity, and the flux distribution of metabolites-from on-line measurable rates only. First, the extended Kalman filter was applied to eliminate noise in the measured rates. Then, using the metabolic reaction model, the lysine production rate and flux distribution were calculated. The estimation results allowed the physiological state of lysine production to be recognized, and an appropriate measure corresponding to the estimated state, such as intermittent addition of glucose and/or leucine, to be taken to maintain a high level of lysine productivity in batch culture. Finally, application of the recognition system enabled lysine to be produced from glucose at a higher yield than that from glucose- or leucine-limited exponential fed-batch cultures. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 170-181, 1997.     

Adjusting for publication bias in meta-analysis via inverse probability weighting using clinical trial registries

Publication bias is a major concern in conducting systematic reviews and meta-analyses. Various sensitivity analysis or bias-correction methods have been developed based on selection models, and they have some advantages over the widely used trim-and-fill bias-correction method. However, likelihood methods based on selection models may have difficulty in obtaining precise estimates and reasonable confidence intervals, or require a rather complicated sensitivity analysis process. Herein, we develop a simple publication bias adjustment method by utilizing the information on conducted but still unpublished trials from clinical trial registries. We introduce an estimating equation for parameter estimation in the selection function by regarding the publication bias issue as a missing data problem under the missing not at random assumption. With the estimated selection function, we introduce the inverse probability weighting (IPW) method to estimate the overall mean across studies. Furthermore, the IPW versions of heterogeneity measures such as the between-study variance and the I² measure are proposed. We propose methods to construct confidence intervals based on asymptotic normal approximation as well as on parametric bootstrap. Through numerical experiments, we observed that the estimators successfully eliminated bias, and the confidence intervals had empirical coverage probabilities close to the nominal level. On the other hand, the confidence interval based on asymptotic normal approximation is much wider in some scenarios than the bootstrap confidence interval. Therefore, the latter is recommended for practical use.     

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.     

Prediction of microbial growth rate versus biomass yield by a metabolic network with kinetic parameters

Identifying the factors that determine microbial growth rate under various environmental and genetic conditions is a major challenge of systems biology. While current genome-scale metabolic modeling approaches enable us to successfully predict a variety of metabolic phenotypes, including maximal biomass yield, the prediction of actual growth rate is a long standing goal. This gap stems from strictly relying on data regarding reaction stoichiometry and directionality, without accounting for enzyme kinetic considerations. Here we present a novel metabolic network-based approach, MetabOlic Modeling with ENzyme kineTics (MOMENT), which predicts metabolic flux rate and growth rate by utilizing prior data on enzyme turnover rates and enzyme molecular weights, without requiring measurements of nutrient uptake rates. The method is based on an identified design principle of metabolism in which enzymes catalyzing high flux reactions across different media tend to be more efficient in terms of having higher turnover numbers. Extending upon previous attempts to utilize kinetic data in genome-scale metabolic modeling, our approach takes into account the requirement for specific enzyme concentrations for catalyzing predicted metabolic flux rates, considering isozymes, protein complexes, and multi-functional enzymes. MOMENT is shown to significantly improve the prediction accuracy of various metabolic phenotypes in E. coli, including intracellular flux rates and changes in gene expression levels under different growth rates. Most importantly, MOMENT is shown to predict growth rates of E. coli under a diverse set of media that are correlated with experimental measurements, markedly improving upon existing state-of-the art stoichiometric modeling approaches. These results support the view that a physiological bound on cellular enzyme concentrations is a key factor that determines microbial growth rate.