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.     

Re-examination of metabolic fluxes in <Emphasis Type="Italic">Escherichia coli</Emphasis> during anaerobic fermentation of glucose using <Superscript>13</Superscript>C labeling experiments and 2-dimensional nuclear magnetic resonance (NMR) spectroscopy

Improved design of metabolic flux estimation using mixed label ¹³C labeling experiments and identifiability analysis motivated re-examination of metabolic fluxes during anaerobic fermentation in the Escherichia coli. Comprehensive metabolic flux maps were determined by using a mixture of differently labeled glucose and compared to conventional flux maps obtained using extracellular measurements and comprehensive metabolic flux maps obtained using only U-¹³C glucose as the substrate. As expected, conventional flux analysis performs poorly in comparison to ¹³C-MFA, especially in the Embden-Meyerhof-Parnas (EMP) and pentose phosphate (PP) pathways. Identifiability analysis indicated and experiments confirmed that a mixture of 10% U-^l3C glucose, 25% 1-¹³C glucose, and 65% naturally labeled glucose significantly improved the statistical quality of all calculated fluxes in the PP pathway, the EMP pathway, the anaplerotic reactions, and the tricarboxylic acid cycle. Modifying the network topology for the presence and absence of the Entner-Doudoroff pathway and the glyoxylate shunt did not affect the value or quality of estimated fluxes significantly. Extracellular measurement of formate production was necessary for the accurate estimation of the fluxes around the formate node.     

Bioreaction network topology and metabolic flux ratio analysis by biosynthetic fractional 13C labeling and two-dimensional NMR spectroscopy

Biosynthetically directed fractional 13C labeling of the proteinogenic amino acids is achieved by feeding a mixture of uniformly 13C-labeled and unlabeled carbon source compounds into a bioreaction network. Analysis of the resulting labeling pattern enables both a comprehensive characterization of the network topology and the determination of metabolic flux ratios. Attractive features with regard to routine applications are (i) an inherently small demand for 13C-labeled source compounds and (ii) the high sensitivity of two-dimensional [13C,1H]-correlation nuclear magnetic resonance spectroscopy for analysis of 13C-labeling patterns. A user-friendly program, FCAL, is available to allow rapid data analysis. This novel approach, which recently also has been employed in conjunction with metabolic flux balancing to obtain reliable estimates of in vivo fluxes, enables efficient support of metabolic engineering and biotechnology process design.     

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.     

Metabolic network fluxes in heterotrophic Arabidopsis cells: stability of the flux distribution under different oxygenation conditions

Steady-state labeling experiments with [1-¹³C]Glc were used to measure multiple metabolic fluxes through the pathways of central metabolism in a heterotrophic cell suspension culture of Arabidopsis (Arabidopsis thaliana). The protocol was based on in silico modeling to establish the optimal labeled precursor, validation of the isotopic and metabolic steady state, extensive nuclear magnetic resonance analysis of the redistribution of label into soluble metabolites, starch, and protein, and a comprehensive set of biomass measurements. Following a simple modification of the cell culture procedure, cells were grown at two oxygen concentrations, and flux maps of central metabolism were constructed on the basis of replicated experiments and rigorous statistical analysis. Increased growth rate at the higher O₂ concentration was associated with an increase in fluxes throughout the network, and this was achieved without any significant change in relative fluxes despite differences in the metabolite profile of organic acids, amino acids, and carbohydrates. The balance between biosynthesis and respiration within the tricarboxylic acid cycle was unchanged, with 38% ± 5% of carbon entering used for biosynthesis under standard O₂ conditions and 33% ± 2% under elevated O₂. These results add to the emerging picture of the stability of the central metabolic network and its capacity to respond to physiological perturbations with the minimum of rearrangement. The lack of correlation between the change in metabolite profile, which implied significant disruption of the metabolic network following the alteration in the oxygen supply, and the unchanging flux distribution highlights a potential difficulty in the interpretation of metabolomic data.     

The topology of metabolic isotope labeling networks

Background  

Metabolic Flux Analysis (MFA) based on isotope labeling experiments (ILEs) is a widely established tool for determining fluxes in metabolic pathways. Isotope labeling networks (ILNs) contain all essential information required to describe the flow of labeled material in an ILE. Whereas recent experimental progress paves the way for high-throughput MFA, large network investigations and exact statistical methods, these developments are still limited by the poor performance of computational routines used for the evaluation and design of ILEs. In this context, the global analysis of ILN topology turns out to be a clue for realizing large speedup factors in all required computational procedures.     

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.     

Optimization of 13C isotopic tracers for metabolic flux analysis in mammalian cells

Mammalian cells consume and metabolize various substrates from their surroundings for energy generation and biomass synthesis. Glucose and glutamine, in particular, are the primary carbon sources for proliferating cancer cells. While this combination of substrates generates static labeling patterns for use in (13)C metabolic flux analysis (MFA), the inability of single tracers to effectively label all pathways poses an obstacle for comprehensive flux determination within a given experiment. To address this issue we applied a genetic algorithm to optimize mixtures of (13)C-labeled glucose and glutamine for use in MFA. We identified tracer combinations that minimized confidence intervals in an experimentally determined flux network describing central carbon metabolism in tumor cells. Additional simulations were used to determine the robustness of the [1,2-(13)C(2)]glucose/[U-(13)C(5)]glutamine tracer combination with respect to perturbations in the network. Finally, we experimentally validated the improved performance of this tracer set relative to glucose tracers alone in a cancer cell line. This versatile method allows researchers to determine the optimal tracer combination to use for a specific metabolic network, and our findings applied to cancer cells significantly enhance the ability of MFA experiments to precisely quantify fluxes in higher organisms.     

Metabolic flux analysis in Escherichia coli by integrating isotopic dynamic and isotopic stationary 13C labeling data

The novel concept of isotopic dynamic 13C metabolic flux analysis (ID-13C MFA) enables integrated analysis of isotopomer data from isotopic transient and/or isotopic stationary phase of a 13C labeling experiment, short-time experiments, and an extended range of applications of 13C MFA. In the presented work, an experimental and computational framework consisting of short-time 13C labeling, an integrated rapid sampling procedure, a LC-MS analytical method, numerical integration of the system of isotopomer differential equations, and estimation of metabolic fluxes was developed and applied to determine intracellular fluxes in glycolysis, pentose phosphate pathway (PPP), and citric acid cycle (TCA) in Escherichia coli grown in aerobic, glucose-limited chemostat culture at a dilution rate of D = 0.10 h(-1). Intracellular steady state concentrations were quantified for 12 metabolic intermediates. A total of 90 LC-MS mass isotopomers were quantified at sampling times t = 0, 91, 226, 346, 589 s and at isotopic stationary conditions. Isotopic stationarity was reached within 10 min in glycolytic and PPP metabolites. Consistent flux solutions were obtained by ID-13C MFA using isotopic dynamic and isotopic stationary 13C labeling data and by isotopic stationary 13C MFA (IS-13C MFA) using solely isotopic stationary data. It is demonstrated that integration of dynamic 13C labeling data increases the sensitivity of flux estimation, particularly at the glucose-6-phosphate branch point. The identified split ratio between glycolysis and PPP was 55%:44%. These results were confirmed by IS-13C MFA additionally using labeling data in proteinogenic amino acids (GC-MS) obtained after 5 h from sampled biomass.     

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.     

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.     

Metabolic flux analysis at ultra short time scale: isotopically non-stationary 13C labeling experiments

A novel approach to (13)C metabolic flux analysis (MFA) is presented using cytosolic metabolite pool sizes and their (13)C labeling data from an isotopically non-stationary (13)C labeling experiment (INST-CLE). The procedure is demonstrated with an E. coli wild type strain grown at fed batch conditions. The intra cellular labeling dynamics are excited by a sudden step increase of the (13)C portion in the substrate feed. Due to unchanged saturation of the substrate uptake system, the metabolic fluxes remain constant during the following sampling time period of only 16s, in which 20 samples are taken by an automated rapid sampling device immediately stopping metabolism by methanol quenching. Subsequent cell disruptive sample preparation and LC-MS/MS enabled simultaneous determination of pool sizes and mass isotopomers of intra cellular metabolites requiring detection limits in the nM range. Based on this data the new computational flux analysis tool 13CFLUX/INST is used to determine the intra cellular fluxes based on a complex carbon labeling network model. The measured data is in good agreement with the model predictions, thus proving the applicability of the new isotopically non-stationary (13)C metabolic flux analysis (INST-(13)C-MFA) concept. Moreover, it is shown that significant new information with respect to flux identifiability, non-measurable pool sizes, data consistency, or large storage pools can be taken from the novel kind of experimental data. This offers new insight into the biological operation of the metabolic network in vivo.     

Design of substrate label for steady state flux measurements in plant systems using the metabolic network of Brassica napus embryos

Steady state metabolic flux analysis using (13)C labeled substrates is of growing importance in plant physiology and metabolic engineering. The quality of the flux estimates in (13)C metabolic flux analysis depend on the: (i) network structure; (ii) flux values; (iii) design of the labeling substrate; and (iv) label measurements performed. Whereas the first two parameters are facts of nature, the latter two are to some extent controlled by the experimenter, yet they have received little attention in most plant studies. Using the metabolic flux map of developing Brassica napus (Rapeseed) embryos, this study explores the value of optimal substrate label designs obtained with different statistical criteria and addresses the applicability of different optimal designs to biological questions. The results demonstrate the value of optimizing the choice of labeled substrates and show that substrate combinations commonly used in bacterial studies can be far from optimal for mapping fluxes in plant systems. The value of performing additional experiments and the inclusion of measurements is also evaluated.     

Integrated 13C-metabolic flux analysis of 14 parallel labeling experiments in Escherichia coli

The use of parallel labeling experiments for ¹³C metabolic flux analysis (¹³C-MFA) has emerged in recent years as the new gold standard in fluxomics. The methodology has been termed COMPLETE-MFA, short for complementary parallel labeling experiments technique for metabolic flux analysis. In this contribution, we have tested the limits of COMPLETE-MFA by demonstrating integrated analysis of 14 parallel labeling experiments with Escherichia coli. An effort on such a massive scale has never been attempted before. In addition to several widely used isotopic tracers such as [1,2-¹³C]glucose and mixtures of [1-¹³C]glucose and [U-¹³C]glucose, four novel tracers were applied in this study: [2,3-¹³C]glucose, [4,5,6-¹³C]glucose, [2,3,4,5,6-¹³C]glucose and a mixture of [1-¹³C]glucose and [4,5,6-¹³C]glucose. This allowed us for the first time to compare the performance of a large number of isotopic tracers. Overall, there was no single best tracer for the entire E. coli metabolic network model. Tracers that produced well-resolved fluxes in the upper part of metabolism (glycolysis and pentose phosphate pathways) showed poor performance for fluxes in the lower part of metabolism (TCA cycle and anaplerotic reactions), and vice versa. The best tracer for upper metabolism was 80% [1-¹³C]glucose+20% [U-¹³C]glucose, while [4,5,6-¹³C]glucose and [5-¹³C]glucose both produced optimal flux resolution in the lower part of metabolism. COMPLETE-MFA improved both flux precision and flux observability, i.e. more independent fluxes were resolved with smaller confidence intervals, especially exchange fluxes. Overall, this study demonstrates that COMPLETE-MFA is a powerful approach for improving flux measurements and that this methodology should be considered in future studies that require very high flux resolution.     

Improvements in metabolic flux analysis using carbon bond labeling experiments: bondomer balancing and Boolean function mapping

The biosynthetically directed fractional (13)C labeling method for metabolic flux evaluation relies on performing a 2-D [(13)C, (1)H] NMR experiment on extracts from organisms cultured on a uniformly labeled carbon substrate. This article focuses on improvements in the interpretation of data obtained from such an experiment by employing the concept of bondomers. Bondomers take into account the natural abundance of (13)C; therefore many bondomers in a real network are zero, and can be precluded a priori--thus resulting in fewer balances. Using this method, we obtained a set of linear equations which can be solved to obtain analytical formulas for NMR-measurable quantities in terms of fluxes in glycolysis and the pentose phosphate pathways. For a specific case of this network with four degrees of freedom, a priori identifiability of the fluxes was shown possible for any set of fluxes. For a more general case with five degrees of freedom, the fluxes were shown identifiable for a representative set of fluxes. Minimal sets of measurements which best identify the fluxes are listed. Furthermore, we have delineated Boolean function mapping, a new method to iteratively simulate bondomer abundances or efficiently convert carbon skeleton rearrangement information to mapping matrices. The efficiency of this method is expected to be valuable while analyzing metabolic networks which are not completely known (such as in plant metabolism) or while implementing iterative bondomer balancing methods.     

Parallel labeling experiments with [U-(13)C]glucose validate E. coli metabolic network model for (13)C metabolic flux analysis

(13)C-metabolic flux analysis (MFA) is a widely used method for measuring intracellular metabolic fluxes in living cells. (13)C MFA relies on several key assumptions: (1) the assumed metabolic network model is complete, in that it accounts for all significant enzymatic and transport reactions; (2) (13)C-labeling measurements are accurate and precise; and (3) enzymes and transporters do not discriminate between (12)C- and (13)C-labeled metabolites. In this study, we tested these inherent assumptions of (13)C MFA for wild-type E. coli by parallel labeling experiments with [U-(13)C]glucose as tracer. Cells were grown in six parallel cultures in custom-constructed mini-bioreactors, starting from the same inoculum, on medium containing different mixtures of natural glucose and fully labeled [U-(13)C]glucose, ranging from 0% to 100% [U-(13)C]glucose. Macroscopic growth characteristics of E. coli showed no observable kinetic isotope effect. The cells grew equally well on natural glucose, 100% [U-(13)C]glucose, and mixtures thereof. (13)C MFA was then used to determine intracellular metabolic fluxes for several metabolic network models: an initial network model from literature; and extended network models that accounted for potential dilution effects of isotopic labeling. The initial network model did not give statistically acceptable fits and produced inconsistent flux results for the parallel labeling experiments. In contrast, an extended network model that accounted for dilution of intracellular CO(2) by exchange with extracellular CO(2) produced statistically acceptable fits, and the estimated metabolic fluxes were consistent for the parallel cultures. This study illustrates the importance of model validation for (13)C MFA. We show that an incomplete network model can produce statistically unacceptable fits, as determined by a chi-square test for goodness-of-fit, and return biased metabolic fluxes. The validated metabolic network model for E. coli from this study can be used in future investigations for unbiased metabolic flux measurements.     

Tandem mass spectrometry: a novel approach for metabolic flux analysis

The goal of metabolic flux analysis (MFA) is the accurate estimation of intracellular fluxes in metabolic networks. Here, we introduce a new method for MFA based on tandem mass spectrometry (MS) and stable-isotope tracer experiments. We demonstrate that tandem MS provides more labeling information than can be obtained from traditional full scan MS analysis and allows estimation of fluxes with better precision. We present a modeling framework that takes full advantage of the additional labeling information obtained from tandem MS for MFA. We show that tandem MS data can be computed for any network model, any compound and any tandem MS fragmentation using linear mapping of isotopomers. The inherent advantages of tandem MS were illustrated in two network models using simulated and literature data. Application of tandem MS increased the observability of the models and improved the precision of estimated fluxes by 2- to 5-fold compared to traditional MS analysis.     

Efficient Modeling of MS/MS Data for Metabolic Flux Analysis

Metabolic flux analysis (MFA) is a widely used method for quantifying intracellular metabolic fluxes. It works by feeding cells with isotopic labeled nutrients, measuring metabolite isotopic labeling, and computationally interpreting the measured labeling data to estimate flux. Tandem mass-spectrometry (MS/MS) has been shown to be useful for MFA, providing positional isotopic labeling data. Specifically, MS/MS enables the measurement of a metabolite tandem mass-isotopomer distribution, representing the abundance in which certain parent and product fragments of a metabolite have different number of labeled atoms. However, a major limitation in using MFA with MS/MS data is the lack of a computationally efficient method for simulating such isotopic labeling data. Here, we describe the tandemer approach for efficiently computing metabolite tandem mass-isotopomer distributions in a metabolic network, given an estimation of metabolic fluxes. This approach can be used by MFA to find optimal metabolic fluxes, whose induced metabolite labeling patterns match tandem mass-isotopomer distributions measured by MS/MS. The tandemer approach is applied to simulate MS/MS data in a small-scale metabolic network model of mammalian methionine metabolism and in a large-scale metabolic network model of E. coli. It is shown to significantly improve the running time by between two to three orders of magnitude compared to the state-of-the-art, cumomers approach. We expect the tandemer approach to promote broader usage of MS/MS technology in metabolic flux analysis. Implementation is freely available at www.cs.technion.ac.il/~tomersh/methods.html