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.     

Insights into plant metabolic networks from steady-state metabolic flux analysis

Steady-state metabolic flux analysis (MFA) is an experimental approach that allows the measurement of multiple fluxes in the core network of primary carbon metabolism. It is based on isotopic labelling experiments, and although well established in the analysis of micro-organisms, and some mammalian systems, the extension of the method to plant cells has been challenging because of the extensive subcellular compartmentation of the metabolic network. Despite this difficulty there has been substantial progress in developing robust protocols for the analysis of heterotrophic plant metabolism by steady-state MFA, and flux maps have now been published that reflect the metabolic phenotypes of excised root tips, developing embryos and cotyledons, hairy root cultures, and cell suspensions under a variety of physiological conditions. There has been a steady improvement in the quality, extent and statistical reliability of these analyses, and new information is emerging on the performance of the plant metabolic network and the contributions of specific pathways.     

Quantification of compartmented metabolic fluxes in developing soybean embryos by employing biosynthetically directed fractional (13)C labeling, two-dimensional [(13)C, (1)H] nuclear magnetic resonance, and comprehensive isotopomer balancing

Metabolic flux quantification in plants is instrumental in the detailed understanding of metabolism but is difficult to perform on a systemic level. Toward this aim, we report the development and application of a computer-aided metabolic flux analysis tool that enables the concurrent evaluation of fluxes in several primary metabolic pathways. Labeling experiments were performed by feeding a mixture of U-(13)C Suc, naturally abundant Suc, and Gln to developing soybean (Glycine max) embryos. Two-dimensional [(13)C, (1)H] NMR spectra of seed storage protein and starch hydrolysates were acquired and yielded a labeling data set consisting of 155 (13)C isotopomer abundances. We developed a computer program to automatically calculate fluxes from this data. This program accepts a user-defined metabolic network model and incorporates recent mathematical advances toward accurate and efficient flux evaluation. Fluxes were calculated and statistical analysis was performed to obtain sds. A high flux was found through the oxidative pentose phosphate pathway (19.99 +/- 4.39 micromol d(-1) cotyledon(-1), or 104.2 carbon mol +/- 23.0 carbon mol per 100 carbon mol of Suc uptake). Separate transketolase and transaldolase fluxes could be distinguished in the plastid and the cytosol, and those in the plastid were found to be at least 6-fold higher. The backflux from triose to hexose phosphate was also found to be substantial in the plastid (21.72 +/- 5.00 micromol d(-1) cotyledon(-1), or 113.2 carbon mol +/-26.0 carbon mol per 100 carbon mol of Suc uptake). Forward and backward directions of anaplerotic fluxes could be distinguished. The glyoxylate shunt flux was found to be negligible. Such a generic flux analysis tool can serve as a quantitative tool for metabolic studies and phenotype comparisons and can be extended to other plant systems.     

COMPLETE-MFA: Complementary parallel labeling experiments technique for metabolic flux analysis

We have developed a novel approach for measuring highly accurate and precise metabolic fluxes in living cells, termed COMPLETE-MFA, short for complementary parallel labeling experiments technique for metabolic flux analysis. The COMPLETE-MFA method is based on combined analysis of multiple isotopic labeling experiments, where the synergy of using complementary tracers greatly improves the precision of estimated fluxes. In this work, we demonstrate the COMPLETE-MFA approach using all singly labeled glucose tracers, [1-¹³C], [2-¹³C], [3-¹³C], [4-¹³C], [5-¹³C], and [6-¹³C]glucose to determine precise metabolic fluxes for wild-type Escherichia coli. Cells were grown in six parallel cultures on defined medium with glucose as the only carbon source. Mass isotopomers of biomass amino acids were measured by gas chromatography–mass spectrometry (GC–MS). The data from all six experiments were then fitted simultaneously to a single flux model to determine accurate intracellular fluxes. We obtained a statistically acceptable fit with more than 300 redundant measurements. The estimated flux map is the most precise flux result obtained thus far for E. coli cells. To our knowledge, this is the first time that six isotopic labeling experiments have been successfully integrated for high-resolution ¹³C-flux analysis.     

A new framework for the estimation of control parameters in metabolic pathways using lin-log kinetics.

The control properties of biochemical pathways can be described by control coefficients and elasticities, as defined in the framework of metabolic control analysis. The determination of these parameters using the traditional metabolic control analysis relationships is, however, limited by experimental difficulties (e.g. realizing and measuring small changes in biological systems) and lack of appropriate mathematical procedures (e.g. when the more practical large changes are made). In this paper, the recently developed lin-log approach is proposed to avoid the above-mentioned problems and is applied to estimate control parameters from measurements obtained in steady state experiments. The lin-log approach employs approximative linear-logarithmic kinetics parameterized by elasticities and provides analytical solutions for fluxes and metabolite concentrations when large changes are made. Published flux and metabolite concentration data are used, obtained from a reconstructed section of glycolysis converting 3-phosphoglycerate to pyruvate [Giersch, C. (1995) Eur. J. Biochem. 227, 194-201]. With the lin-log approach, all data from different experiments can be combined to give realistic elasticity and flux control coefficient estimates by linear regression. Despite the large changes, a good agreement of fluxes and metabolite concentrations is obtained between the measured and calculated values according to the lin-log model. Furthermore, it is shown that the lin-log approach allows a rigorous statistical evaluation to identify the optimal reference state and the optimal model structure assumption. In conclusion, the lin-log approach addresses practical problems encountered in the traditional metabolic control analysis-based methods by introducing suitable nonlinear kinetics, thus providing a novel framework with improved procedures for the estimation of elasticities and control parameters from large perturbation experiments.     

Vacuolar compartmentation complicates the steady-state analysis of glucose metabolism and forces reappraisal of sucrose cycling in plants

Steady-state stable isotope labelling provides a method for generating flux maps of the compartmented network of central metabolism in heterotrophic plant tissues. Theoretical analysis of the contribution of the vacuole to the regeneration of glucose by endogenous processes shows that numerical fitting of isotopomeric data will only generate an accurate map of the fluxes involving intracellular glucose if information is available on the labelling of both the cytosolic and vacuolar glucose pools. In the absence of this information many of the calculated fluxes are at best unreliable or at worst indeterminate. This result suggests that the anomalously high rates of sucrose cycling and glucose resynthesis that have been reported in earlier steady-state analyses of tissues labelled with (13)C-glucose precursors may be an artefact of assuming that the labelling pattern of extracted glucose reflected the labelling of the cytosolic pool. The analysis emphasises that although subcellular information can sometimes be deduced from a steady-state analysis without recourse to subcellular fractionation, the success of this procedure depends critically on the structure of the metabolic network. It is concluded that methods need to be implemented that will allow measurement of the subcellular labelling pattern of glucose and other metabolites, as part of the routine analysis of the redistribution of label in steady-state stable isotope labelling experiments, if the true potential of network flux analysis for generating metabolic phenotypes is to be realized.     

High-throughput metabolic flux analysis based on gas chromatography-mass spectrometry derived 13C constraints

13C-constrained flux balancing analysis based on gas chromatography-mass spectrometry data is presented here as a simple and robust method for the estimation of intracellular carbon fluxes. In this approach, the underdetermined system of metabolite balances deduced from stoichiometric relations and measured extracellular rates is complemented with 13C constraints from metabolic flux ratio analysis. Fluxes in central carbon metabolism of exponentially growing Escherichia coli were estimated by 13C-constrained flux balancing from three different 13C-labeled glucose experiments. The best resolution of the network was achieved using 13C constraints derived from [U-13C]glucose and [1-13C]glucose experiments. The corresponding flux estimate was in excellent agreement with a solution that was independently obtained with a comprehensive isotopomer model. This new methodology was also demonstrated to faithfully capture the intracellular flux distribution in E. coli shake flasks and 1-ml deep-well microtiter plates. Due to its simplicity, speed, and robustness, 13C-constrained metabolic flux balancing is promising for routine and high-throughput analysis on a miniaturized scale.     

Incorporating metabolic flux ratios into constraint-based flux analysis by using artificial metabolites and converging ratio determinants

One of the well-established approaches for the quantitative characterization of large-scale underdetermined metabolic network is constraint-based flux analysis, which quantifies intracellular metabolic fluxes to characterize the metabolic status. The system is typically underdetermined, and thus usually is solved by linear programming with the measured external fluxes as constraints. Thus, the intracellular flux distribution calculated may not represent the true values. (13)C-constrained flux analysis allows more accurate determination of internal fluxes, but is currently limited to relatively small metabolic networks due to the requirement of complicated mathematical formulation and limited parameters available. Here, we report a strategy of employing such partial information obtained from the (13)C-labeling experiments as additional constraints during the constraint-based flux analysis. A new methodology employing artificial metabolites and converging ratio determinants (CRDs) was developed for improving constraint-based flux analysis. The CRDs were determined based on the metabolic flux ratios obtained from (13)C-labeling experiments, and were incorporated into the mass balance equations for the artificial metabolites. These new mass balance equations were used as additional constraints during the constraint-based flux analysis with genome-scale E. coli metabolic model, which allowed more accurate determination of intracellular metabolic fluxes.     

Quantitative approaches for analysing fluxes through plant metabolic networks using NMR and stable isotope labelling

The quantitative analysis of metabolic networks is a prerequisite for understanding the integration and regulation of plant metabolism and for devising rational approaches for manipulating resource allocation in plants. The analysis of steady state stable isotope labelling experiments using nuclear magnetic resonance (NMR) spectroscopy has developed into a powerful method for determining these fluxes in micro-organisms and its application to heterotrophic plant metabolism is increasing. After an introductory discussion of the well known role of stable isotopes in pathway delineation, the review considers their application to metabolic flux analysis in plants. These applications are divided into two groups – small scale analyses of fluxes through particular pathways and large scale analyses of multiple fluxes through metabolic networks – and the problems caused by the complexity of intermediary metabolism in plants are discussed. It is concluded that metabolic flux analysis provides a powerful method for defining the metabolic phenotype of wild type, mutant and transgenic plants and that its development should be pursued.     

OpenFLUX: efficient modelling software for 13C-based metabolic flux analysis

Background  

The quantitative analysis of metabolic fluxes, i.e., in vivo activities of intracellular enzymes and pathways, provides key information on biological systems in systems biology and metabolic engineering. It is based on a comprehensive approach combining (i) tracer cultivation on ¹³C substrates, (ii) ¹³C labelling analysis by mass spectrometry and (iii) mathematical modelling for experimental design, data processing, flux calculation and statistics. Whereas the cultivation and the analytical part is fairly advanced, a lack of appropriate modelling software solutions for all modelling aspects in flux studies is limiting the application of metabolic flux analysis.     

Determination of confidence intervals of metabolic fluxes estimated from stable isotope measurements

Metabolic fluxes, estimated from stable isotope studies, provide a key to quantifying physiology in fields ranging from metabolic engineering to the analysis of human metabolic diseases. A serious drawback of the flux estimation method in current use is that it does not produce confidence limits for the estimated fluxes. Without this information it is difficult to interpret flux results and expand the physiological significance of flux studies. To address this shortcoming we derived analytical expressions of flux sensitivities with respect to isotope measurements and measurement errors. These tools allow the determination of local statistical properties of fluxes and relative importance of measurements. Furthermore, we developed an efficient algorithm to determine accurate flux confidence intervals and demonstrated that confidence intervals obtained with this method closely approximate true flux uncertainty. In contrast, confidence intervals approximated from local estimates of standard deviations are inappropriate due to inherent system nonlinearities. We applied these methods to analyze the statistical significance and confidence of estimated gluconeogenesis fluxes from human studies with [U-13C]glucose as tracer and found true limits for flux estimation in specific human isotopic protocols.     

Determination of metabolic fluxes in a non-steady-state system

Estimation of fluxes through metabolic networks from redistribution patterns of (13)C has become a well developed technique in recent years. However, the approach is currently limited to systems at metabolic steady-state; dynamic changes in metabolic fluxes cannot be assessed. This is a major impediment to understanding the behaviour of metabolic networks, because steady-state is not always experimentally achievable and a great deal of information about the control hierarchy of the network can be derived from the analysis of flux dynamics. To address this issue, we have developed a method for estimating non-steady-state fluxes based on the mass-balance of mass isotopomers. This approach allows multiple mass-balance equations to be written for the change in labelling of a given metabolite pool and thereby permits over-determination of fluxes. We demonstrate how linear regression methods can be used to estimate non-steady-state fluxes from these mass balance equations. The approach can be used to calculate fluxes from both mass isotopomer and positional isotopomer labelling information and thus has general applicability to data generated from common spectrometry- or NMR-based analytical platforms. The approach is applied to a GC-MS time-series dataset of (13)C-labelling of metabolites in a heterotrophic Arabidopsis cell suspension culture. Threonine biosynthesis is used to demonstrate that non-steady-state fluxes can be successfully estimated from such data while organic acid metabolism is used to highlight some common issues that can complicate flux estimation. These include multiple pools of the same metabolite that label at different rates and carbon skeleton rearrangements.     

Modeling isotopomer distributions in biochemical networks using isotopomer mapping matrices

Within the last decades NMR spectroscopy has undergone tremendous development and has become a powerful analytical tool for the investigation of intracellular flux distributions in biochemical networks using (13)C-labeled substrates. Not only are the experiments much easier to conduct than experiments employing radioactive tracer elements, but NMR spectroscopy also provides additional information on the labeling pattern of the metabolites. Whereas the maximum amount of information obtainable with (14)C-labeled substrates is the fractional enrichment in the individual carbon atom positions, NMR spectroscopy can also provide information on the degree of labeling at neighboring carbon atom positions by analyzing multiplet patterns in NMR spectra or using 2-dimensional NMR spectra. It is possible to quantify the mole fractions of molecules that show a specific labeling pattern, i.e., information of the isotopomer distribution in metabolite pools can be obtained. The isotopomer distribution is the maximum amount of information that in theory can be obtained from (13)C-tracer studies. The wealth of information contained in NMR spectra frequently leads to overdetermined algebraic systems. Consequently, fluxes must be estimated by nonlinear least squares analysis, in which experimental labeling data is compared with simulated steady state isotopomer distributions. Hence, mathematical models are required to compute the steady state isotopomer distribution as a function of a given set of steady state fluxes. Because 2(n) possible labeling patterns exist in a molecule of n carbon atoms, and each pattern corresponds to a separate state in the isotopomer model, these models are inherently complex. Model complexity, so far, has restricted usage of isotopomer information to relatively small metabolic networks. A general methodology for the formulation of isotopomer models is described. The model complexity of isotopomer models is reduced to that of classical metabolic models by expressing the 2(n) isotopomer mass balances of a metabolite pool in a single matrix equation. Using this approach an isotopomer model has been implemented that describes label distribution in primary carbon metabolism, i.e., in a metabolic network including the Embden-Meyerhof-Parnas and pentose phosphate pathway, the tricarboxylic acid cycle, and selected anaplerotic reaction sequences. The model calculates the steady state label distribution in all metabolite pools as a function of the steady state fluxes and is applied to demonstrate the effect of selected anaplerotic fluxes on the labeling pattern of the pathway intermediates. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55:831-840, 1997.     

Metabolic isotopomer labeling systems. Part II: structural flux identifiability analysis

Metabolic flux analysis using carbon labeling experiments (CLEs) is an important tool in metabolic engineering where the intracellular fluxes have to be computed from the measured extracellular fluxes and the partially measured distribution of 13C labeling within the intracellular metabolite pools. The relation between unknown fluxes and measurements is described by an isotopomer labeling system (ILS) (see Part I [Math. Biosci. 169 (2001) 173]). Part II deals with the structural flux identifiability of measured ILSs in the steady state. The central question is whether the measured data contains sufficient information to determine the unknown intracellular fluxes. This question has to be decided a priori, i.e. before the CLE is carried out. In structural identifiability analysis the measurements are assumed to be noise-free. A general theory of structural flux identifiability for measured ILSs is presented and several algorithms are developed to solve the identifiability problem. In the particular case of maximal measurement information, a symbolical algorithm is presented that decides the identifiability question by means of linear methods. Several upper bounds of the number of identifiable fluxes are derived, and the influence of the chosen inputs is evaluated. By introducing integer arithmetic this algorithm can even be applied to large networks. For the general case of arbitrary measurement information, identifiability is decided by a local criterion. A new algorithm based on integer arithmetic enables an a priori local identifiability analysis to be performed for networks of arbitrary size. All algorithms have been implemented and flux identifiability is investigated for the network of the central metabolic pathways of a microorganism. Moreover, several small examples are worked out to illustrate the influence of input metabolite labeling and the paradox of information loss due to network simplification.     

FiatFlux – a software for metabolic flux analysis from <Superscript>13</Superscript>C-glucose experiments

Background  

Quantitative knowledge of intracellular fluxes is important for a comprehensive characterization of metabolic networks and their functional operation. In contrast to direct assessment of metabolite concentrations, in vivo metabolite fluxes must be inferred indirectly from measurable quantities in ¹³C experiments. The required experience, the complicated network models, large and heterogeneous data sets, and the time-consuming set-up of highly controlled experimental conditions largely restricted metabolic flux analysis to few expert groups. A conceptual simplification of flux analysis is the analytical determination of metabolic flux ratios exclusively from MS data, which can then be used in a second step to estimate absolute in vivo fluxes.     

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.     

Using isotopomer path tracing to quantify metabolic fluxes in pathway models containing reversible reactions

As a more complete picture of the genetic and enzymatic composition of cells becomes available, there is a growing need to describe how cellular regulatory elements interact with the cellular environment to affect cell physiology. One means for describing intracellular regulatory mechanisms is concurrent measurement of multiple metabolic pathways and their interactions by metabolic flux analysis. Flux of carbon through a metabolic pathway responds to all cellular regulatory systems, including changes in enzyme and substrate concentrations, enzyme activation or inhibition, and ultimately genetic control. The extent to which metabolic flux analysis can describe cellular physiology depends on the number of pathways in the model and the quality of the data. Intracellular information is obtainable from isotopic tracer experiments, the most extensive being the determination of the isotopomer distribution, or specific labeling pattern, of intracellular metabolites. We present a rapid and novel solution method that determines the flux of carbon through complex pathway models using isotopomer data. This time-consuming problem was solved with the introduction of isotopomer path tracing, which drastically reduces the number of isotopomer variables to the number of isotopomers observed experimentally. We propose a partitioned solution method that takes advantage of the nearly linear relationship between fluxes and isotopomers. Whereas the stoichiometric matrix and the isotopomer matrix are invertible, simulated annealing and the Newton-Raphson method are used for the nonlinear components. Reversible reactions are described by a new parameter, the association factor, which scales hyperbolically with the rate of metabolite exchange. Automating the solution method permits a variety of models to be compared, thus enhancing the accuracy of results. A simplified example that contains all of the complexities of a comprehensive pathway model is presented. Copyright John Wiley & Sons, Inc.     

13C tracer experiments and metabolite balancing for metabolic flux analysis: comparing two approaches

Conventional metabolic flux analysis uses the information gained from determination of measurable fluxes and a steady-state assumption for intracellular metabolites to calculate the metabolic fluxes in a given metabolic network. The determination of intracellular fluxes depends heavily on the correctness of the assumed stoichiometry including the presence of all reactions with a noticeable impact on the model metabolite balances. Determination of fluxes in complex metabolic networks often requires the inclusion of NADH and NADPH balances, which are subject to controversial debate. Transhydrogenation reactions that transfer reduction equivalents from NADH to NADPH or vice versa can usually not be included in the stoichiometric model, because they result in singularities in the stoichiometric matrix. However, it is the NADPH balance that, to a large extent, determines the calculated flux through the pentose phosphate pathway. Hence, wrong assumptions on the presence or activity of transhydrogenation reactions will result in wrong estimations of the intracellular flux distribution. Using 13C tracer experiments and NMR analysis, flux analysis can be performed on the basis of only well established stoichiometric equations and measurements of the labeling state of intracellular metabolites. Neither NADH/NADPH balancing nor assumptions on energy yields need to be included to determine the intracellular fluxes. Because metabolite balancing methods and the use of 13C labeling measurements are two different approaches to the determination of intracellular fluxes, both methods can be used to verify each other or to discuss the origin and significance of deviations in the results. Flux analysis based entirely on metabolite balancing and flux analysis, including labeling information, have been performed independently for a wild-type strain of Aspergillus oryzae producing alpha-amylase. Two different nitrogen sources, NH4+ and NO3-, have been used to investigate the influence of the NADPH requirements on the intracellular flux distribution. The two different approaches to the calculation of fluxes are compared and deviations in the results are discussed. Copyright 1998 John Wiley & Sons, Inc.