The relationships between the metabolic fluxes and 13C-labeled isotopomer distribution for the flux analysis of the main metabolic pathways

It has been known that ¹³C-labeling technique is quite useful in estimating the metabolic fluxes. Although the program-based flux analysis is powerful, it is not easy to be confident with the result obtained without experiences and exhaustive trial and errors based on statistical analysis for the confidence intervals in practice. It is, therefore, quite important to grasp the relationship between the fluxes and the ¹³C-labeled isotopomer distribution to get deeper insight into the metabolic flux analysis. In the present research, it was shown explicitly how the isotopomer distribution changes with respect to the fluxes in relation to the labeling patterns of the substrate, where either labeled glucose, acetate, or pyruvate was used as a carbon source. Some of the analytical expressions were derived based on the matrix representation, and they were utilized for analysis. It was shown that the isotopomer pattern does not necessarily change uniformly with respect to fluxes, but changes in a complicated way in particular for the case of using pyruvate as a carbon source where some isotopomers do not necessarily change monotonically. It was shown to be quite important to grasp how the isotopomer pattern changes with respect to fluxes and the labeling pattern of the substrate for flux determination and the experimental design. It was also shown that the mixture of [1-¹³C] acetate and [2-¹³C] acetate should not be used from the information index point of view. Some of the experimental data were evaluated from the present approach. It was also shown that the isotopomer distribution is less sensitive to the bidirectional fluxes in the reversible pathway.     

Numerical bias estimation for mass spectrometric mass isotopomer analysis

Mass spectrometric (MS) isotopomer analysis has become a standard tool for investigating biological systems using stable isotopes. In particular, metabolic flux analysis uses mass isotopomers of metabolic products typically formed from ¹³C-labeled substrates to quantitate intracellular pathway fluxes. In the current work, we describe a model-driven method of numerical bias estimation regarding MS isotopomer analysis. Correct bias estimation is crucial for measuring statistical qualities of measurements and obtaining reliable fluxes. The model we developed for bias estimation corrects a priori unknown systematic errors unique for each individual mass isotopomer peak. For validation, we carried out both computational simulations and experimental measurements. From stochastic simulations, it was observed that carbon mass isotopomer distributions and measurement noise can be determined much more precisely only if signals are corrected for possible systematic errors. By removing the estimated background signals, the residuals resulting from experimental measurement and model expectation became consistent with normality, experimental variability was reduced, and data consistency was improved. The method is useful for obtaining systematic error-free data from ¹³C tracer experiments and can also be extended to other stable isotopes. As a result, the reliability of metabolic fluxes that are typically computed from mass isotopomer measurements is increased.     

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.     

<Superscript>13</Superscript>C metabolite profiling to compare the central metabolic flux in two yeast strains

¹³C metabolite profiling to quantify the dynamic changes of central carbon metabolites was attempted using mass isotopomer distribution analysis in two yeast strains, Saccharomyces cerevisiae and Kluyveromyces marxianus. Mass and isotopomer balances of the intermediates were examined and calculated in both yeast species and central carbon metabolic fluxes were successfully determined. Metabolic fluxes of pentose phosphate pathway in K. marxianus were 1.66 times higher than S. cerevisiae. The flux difference was also supported by relatively high abundance of partially labeled fructose 6-phosphate and 3-phosphoglycerate as well as an increased concentration of labeled L-valine in K. marxianus. Metabolic flux analysis combined with dynamic metabolite profiling has provided better understanding in the central carbon metabolic pathways of two model organisms and can be applied as a method to analyze more complicated metabolic networks in other organisms.     

Simultaneous tracers and a unified model of positional and mass isotopomers for quantification of metabolic flux in liver

Computational models based on the metabolism of stable isotope tracers can yield valuable insight into the metabolic basis of disease. The complexity of these models is limited by the number of tracers and the ability to characterize tracer labeling in downstream metabolites. NMR spectroscopy is ideal for multiple tracer experiments since it precisely detects the position of tracer nuclei in molecules, but it lacks sensitivity for detecting low-concentration metabolites. GC-MS detects stable isotope mass enrichment in low-concentration metabolites, but lacks nuclei and positional specificity. We performed liver perfusions and in vivo infusions of ²H and ¹³C tracers, yielding complex glucose isotopomers that were assigned by NMR and fit to a newly developed metabolic model. Fluxes regressed from ²H and ¹³C NMR positional isotopomer enrichments served to validate GC-MS-based flux estimates obtained from the same experimental samples. NMR-derived fluxes were largely recapitulated by modeling the mass isotopomer distributions of six glucose fragment ions measured by GC-MS. Modest differences related to limited fragmentation coverage of glucose C1–C3 were identified, but fluxes such as gluconeogenesis, glycogenolysis, cataplerosis and TCA cycle flux were tightly correlated between the methods. Most importantly, modeling of GC-MS data could assign fluxes in primary mouse hepatocytes, an experiment that is impractical by ²H or ¹³C NMR.     

Metabolic flux analysis in biotechnology processes

Metabolic flux analysis (MFA) has become a fundamental tool of metabolic engineering to elucidate the metabolic state of the cell and has been applied to various biotechnological processes. In recent years, considerable technical advances have been made. Developments of analytical instruments allow us to determine ¹³C labeling distribution of intracellular metabolites with high accuracy and sensitivity. Moreover, kinetic information of intracellular label distribution during isotopic instationary enables us to calculate metabolic fluxes with shortened experimental time and decreased amount of labeled substrate. The ¹³C MFA may be one of the most promising approaches for the target estimation to improve strain performances and production processes.     

The thermodynamic meaning of metabolic exchange fluxes

Metabolic flux analysis (MFA) deals with the experimental determination of steady-state fluxes in metabolic networks. An important feature of the ¹³C MFA method is its capability to generate information on both directions of bidirectional reaction steps given by exchange fluxes. The biological interpretation of these exchange fluxes and their relation to thermodynamic properties of the respective reaction steps has never been systematically investigated. As a central result, it is shown here that for a general class of enzyme reaction mechanisms the quotients of net and exchange fluxes measured by ¹³C MFA are coupled to Gibbs energies of the reaction steps. To establish this relation the concept of apparent flux ratios of enzymatic isotope-labeling networks is introduced and some computing rules for these flux ratios are given. Application of these rules reveals a conceptional pitfall of ¹³C MFA, which is the inherent dependency of measured exchange fluxes on the chosen tracer atom. However, it is shown that this effect can be neglected for typical biochemical reaction steps under physiological conditions. In this situation, the central result can be formulated as a two-sided inequality relating fluxes, pool sizes, and standard Gibbs energies. This relation has far-reaching consequences for metabolic flux analysis, quantitative metabolomics, and network thermodynamics.     

Shewanella oneidensis MR-1 Fluxome under Various Oxygen Conditions

The central metabolic fluxes of Shewanella oneidensis MR-1 were examined under carbon-limited (aerobic) and oxygen-limited (microaerobic) chemostat conditions, using ¹³C-labeled lactate as the sole carbon source. The carbon labeling patterns of key amino acids in biomass were probed using both gas chromatography-mass spectrometry (GC-MS) and ¹³C nuclear magnetic resonance (NMR). Based on the genome annotation, a metabolic pathway model was constructed to quantify the central metabolic flux distributions. The model showed that the tricarboxylic acid (TCA) cycle is the major carbon metabolism route under both conditions. The Entner-Doudoroff and pentose phosphate pathways were utilized primarily for biomass synthesis (with a flux below 5% of the lactate uptake rate). The anaplerotic reactions (pyruvate to malate and oxaloacetate to phosphoenolpyruvate) and the glyoxylate shunt were active. Under carbon-limited conditions, a substantial amount (9% of the lactate uptake rate) of carbon entered the highly reversible serine metabolic pathway. Under microaerobic conditions, fluxes through the TCA cycle decreased and acetate production increased compared to what was found for carbon-limited conditions, and the flux from glyoxylate to glycine (serine-glyoxylate aminotransferase) became measurable. Although the flux distributions under aerobic, microaerobic, and shake flask culture conditions were different, the relative flux ratios for some central metabolic reactions did not differ significantly (in particular, between the shake flask and aerobic-chemostat groups). Hence, the central metabolism of S. oneidensis appears to be robust to environmental changes. Our study also demonstrates the merit of coupling GC-MS with ¹³C NMR for metabolic flux analysis to reduce the use of ¹³C-labeled substrates and to obtain more-accurate flux values.     

Optimal tracers for parallel labeling experiments and 13C metabolic flux analysis: A new precision and synergy scoring system

¹³C-Metabolic flux analysis (¹³C-MFA) is a widely used approach in metabolic engineering for quantifying intracellular metabolic fluxes. The precision of fluxes determined by ¹³C-MFA depends largely on the choice of isotopic tracers and the specific set of labeling measurements. A recent advance in the field is the use of parallel labeling experiments for improved flux precision and accuracy. However, as of today, no systemic methods exist for identifying optimal tracers for parallel labeling experiments. In this contribution, we have addressed this problem by introducing a new scoring system and evaluating thousands of different isotopic tracer schemes. Based on this extensive analysis we have identified optimal tracers for ¹³C-MFA. The best single tracers were doubly ¹³C-labeled glucose tracers, including [1,6-¹³C]glucose, [5,6-¹³C]glucose and [1,2-¹³C]glucose, which consistently produced the highest flux precision independent of the metabolic flux map (here, 100 random flux maps were evaluated). Moreover, we demonstrate that pure glucose tracers perform better overall than mixtures of glucose tracers. For parallel labeling experiments the optimal isotopic tracers were [1,6-¹³C]glucose and [1,2-¹³C]glucose. Combined analysis of [1,6-¹³C]glucose and [1,2-¹³C]glucose labeling data improved the flux precision score by nearly 20-fold compared to widely use tracer mixture 80% [1-¹³C]glucose +20% [U-¹³C]glucose.     

Quantification and identification of isotopomer distributions of metabolites in crude cell extracts using 1H TOCSY

Isotopomer analysis is a very powerful technique for determining site enrichment with stable isotopes. Such information helps determine the relative flux through metabolic pathways. We have developed ¹H NMR detection methods to isotopomer analysis of human rhabdomyosarcoma cells grown in the presence of uniformly ¹³C-labeled glucose. We show that TOCSY can be used both to identify the isotopomer distributions in a substantial number of key compounds and to determine the site-specific enrichment with good precision. Effects of differential relaxation have been specifically addressed. We have identified and quantified isotopomer distributions in Ala, Lactate, (glycolysis markers), nucleotide riboses (pentose phosphate markers), Asp, Glu and Gln (citric acid cycle and anaplerosis markers) as well as in nucleotide pyrimidine rings. Due to the high sensitivity of proton experiments, a reasonable throughput was achieved using a cold probe on only 3–5 mg dry cell weight. This methodology can be applied to biological system using different labeled precursors to examine their metabolic phenotypes and their response to external perturbations. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Unraveling the metabolism of HEK-293 cells using lactate isotopomer analysis

HEK-293 is the most extensively used human cell line for the production of viral vectors and is gaining increasing attention for the production of recombinant proteins by transient transfection. To further improve the metabolic characterization of this cell line, we have performed cultures using ¹³C-labeled substrates and measured the resulting mass isotopomer distributions in lactate by LC/MS. Simultaneous metabolite and isotopomer balancing allowed improvement and validation of the metabolic model and quantification of key intracellular pathways. We have determined the amounts of glucose carbon channeled through the PPP, incorporated into the TCA cycle for energy production and lipids biosynthesis, as well as the cytosolic and mitochondrial malic enzyme fluxes. Our analysis also revealed that glutamine did not significantly contribute to lactate formation. An improved and quantitative understanding of the central carbon metabolism is greatly needed to pursue the rational development of engineering approaches at both the cellular and process levels.     

Metabolic Modeling of Dynamic Brain 13C NMR Multiplet Data: Concepts and Simulations with a Two-Compartment Neuronal-Glial Model

Metabolic modeling of dynamic ¹³C labeling curves during infusion of ¹³C-labeled substrates allows quantitative measurements of metabolic rates in vivo. However metabolic modeling studies performed in the brain to date have only modeled time courses of total isotopic enrichment at individual carbon positions (positional enrichments), not taking advantage of the additional dynamic ¹³C isotopomer information available from fine-structure multiplets in ¹³C spectra. Here we introduce a new ¹³C metabolic modeling approach using the concept of bonded cumulative isotopomers, or bonded cumomers. The direct relationship between bonded cumomers and ¹³C multiplets enables fitting of the dynamic multiplet data. The potential of this new approach is demonstrated using Monte-Carlo simulations with a brain two-compartment neuronal-glial model. The precision of positional and cumomer approaches are compared for two different metabolic models (with and without glutamine dilution) and for different infusion protocols ([1,6-¹³C₂]glucose, [1,2-¹³C₂]acetate, and double infusion [1,6-¹³C₂]glucose?+?[1,2-¹³C₂]acetate). In all cases, the bonded cumomer approach gives better precision than the positional approach. In addition, of the three different infusion protocols considered here, the double infusion protocol combined with dynamic bonded cumomer modeling appears the most robust for precise determination of all fluxes in the model. The concepts and simulations introduced in the present study set the foundation for taking full advantage of the available dynamic ¹³C multiplet data in metabolic modeling.     

Enhanced glycoprotein production in HEK-293 cells expressing pyruvate carboxylase

There is an imperative need for expression systems allowing the efficient and robust manufacturing of high quality glycoproteins. In the present work, HEK-293 cells stably expressing interferon-α2b were further engineered with the insertion of the yeast pyruvate carboxylase 2 gene. In batch cultures, marked reductions in lactate and ammonia production were observed compared to the parental cell clone. Although the maximum specific growth rate remained unchanged, the altered metabolism led to a 2-fold increase in maximum cell density and 33% increase in the integral of viable cell concentration and interferon production yield. The underlying metabolic changes were further investigated using various ¹³C-labeled substrates and measuring the resulting lactate mass isotopomer distributions. Simultaneous metabolite and isotopomer balancing allowed the accurate determination of key intracellular fluxes. Such detailed and quantitative knowledge about the central carbon metabolism of the cells is instrumental to further support the development of high-yield fed-batch processes.     

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.     

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.     

Simultaneous Steady-state and Dynamic 13C NMR Can Differentiate Alternative Routes of Pyruvate Metabolism in Living Cancer Cells

Metabolic reprogramming facilitates cancer cell growth, so quantitative metabolic flux measurements could produce useful biomarkers. However, current methods to analyze flux in vivo provide either a steady-state overview of relative activities (infusion of ¹³C and analysis of extracted metabolites) or a dynamic view of a few reactions (hyperpolarized ¹³C spectroscopy). Moreover, although hyperpolarization has successfully quantified pyruvate-lactate exchanges, its ability to assess mitochondrial pyruvate metabolism is unproven in cancer. Here, we combined ¹³C hyperpolarization and isotopomer analysis to quantify multiple fates of pyruvate simultaneously. Two cancer cell lines with divergent pyruvate metabolism were incubated with thermally polarized [3-¹³C]pyruvate for several hours, then briefly exposed to hyperpolarized [1-¹³C]pyruvate during acquisition of NMR spectra using selective excitation to maximize detection of H[¹³C]O₃⁻ and [1-¹³C]lactate. Metabolites were then extracted and subjected to isotopomer analysis to determine relative rates of pathways involving [3-¹³C]pyruvate. Quantitation of hyperpolarized H[¹³C]O₃⁻ provided a single definitive metabolic rate, which was then used to convert relative rates derived from isotopomer analysis into quantitative fluxes. This revealed that H[¹³C]O₃⁻ appearance reflects activity of pyruvate dehydrogenase rather than pyruvate carboxylation followed by subsequent decarboxylation reactions. Glucose substantially altered [1-¹³C]pyruvate metabolism, enhancing exchanges with [1-¹³C]lactate and suppressing H[¹³C]O₃⁻ formation. Furthermore, inhibiting Akt, an oncogenic kinase that stimulates glycolysis, reversed these effects, indicating that metabolism of pyruvate by both LDH and pyruvate dehydrogenase is subject to the acute effects of oncogenic signaling on glycolysis. The data suggest that combining ¹³C isotopomer analyses and dynamic hyperpolarized ¹³C spectroscopy may enable quantitative flux measurements in living tumors.     

13C-Labeled Gluconate Tracing as a Direct and Accurate Method for Determining the Pentose Phosphate Pathway Split Ratio in Penicillium chrysogenum

In this study we developed a new method for accurately determining the pentose phosphate pathway (PPP) split ratio, an important metabolic parameter in the primary metabolism of a cell. This method is based on simultaneous feeding of unlabeled glucose and trace amounts of [U-¹³C]gluconate, followed by measurement of the mass isotopomers of the intracellular metabolites surrounding the 6-phosphogluconate node. The gluconate tracer method was used with a penicillin G-producing chemostat culture of the filamentous fungus Penicillium chrysogenum. For comparison, a ¹³C-labeling-based metabolic flux analysis (MFA) was performed for glycolysis and the PPP of P. chrysogenum. For the first time mass isotopomer measurements of ¹³C-labeled primary metabolites are reported for P. chrysogenum and used for a ¹³C-based MFA. Estimation of the PPP split ratio of P. chrysogenum at a growth rate of 0.02 h⁻¹ yielded comparable values for the gluconate tracer method and the ¹³C-based MFA method, 51.8% and 51.1%, respectively. A sensitivity analysis of the estimated PPP split ratios showed that the 95% confidence interval was almost threefold smaller for the gluconate tracer method than for the ¹³C-based MFA method (40.0 to 63.5% and 46.0 to 56.5%, respectively). From these results we concluded that the gluconate tracer method permits accurate determination of the PPP split ratio but provides no information about the remaining cellular metabolism, while the ¹³C-based MFA method permits estimation of multiple fluxes but provides a less accurate estimate of the PPP split ratio.     

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.     

Dynamic flux responses in riboflavin overproducing Bacillus subtilis to increasing glucose limitation in fed‐batch culture

How do intracellular fluxes respond to dynamically increasing glucose limitation when the physiology changes from strong overflow metabolism near to exclusively maintenance metabolism? Here we investigate this question in a typical industrial, glucose‐limited fed‐batch cultivation with a riboflavin overproducing Bacillus subtilis strain. To resolve dynamic flux changes, a novel approach to ¹³C flux analysis was developed that is based on recording ¹³C labeling patterns in free intracellular amino acids. Fluxes are then estimated with stationary flux ratio and iterative isotopomer balancing methods, for which a decomposition of the process into quasi‐steady states and estimation of isotopic steady state ¹³C labeling patterns was necessary. By this approach, we achieve a temporal resolution of 30–60 min that allows us to resolve the slow metabolic transients that typically occur in such cultivations. In the late process phase we found, most prominently, almost exclusive respiratory metabolism, significantly increased pentose phosphate pathway contribution and a strongly decreased futile cycle through the PEP carboxykinase. As a consequence, higher catabolic NADPH formation occurred than was necessary to satisfy the anabolic demands, suggesting a transhydrogenase‐like mechanism to close the balance of reducing equivalents. Biotechnol. Bioeng. 2010. 105: 795–804. © 2009 Wiley Periodicals, Inc.     

An improved method for statistical analysis of metabolic flux analysis using isotopomer mapping matrices with analytical expressions

Analytical expressions were derived for calculating the sensitivities of isotopomer distribution vectors, the weighted output matrix with respect to the fluxes, and the covariance matrix for the metabolic flux analysis based on isotopomers mapping matrices (IMM). These expressions allow us to implement efficient statistical analysis, avoiding the time-consuming Monte Carlo techniques for estimating the confidence interval of the fluxes. The analytical expressions are also useful in implementing a faster design of experiment, which requires repetitive computation of the covariance matrix that is not straightforward to make in practice with the numerical techniques based on the conventional IMM. The proposed method was applied for analyzing the central carbon metabolism of the mixotrophically cultivated Synechocystis sp. PCC6803, and the confidence intervals of all its fluxes were computed based on the isotopomer distribution measured using NMR and GC-MS. It was found that the best feasible mixture for labeling experiment is 70% unlabeled, 10% [U-13C] and 20% [1,2-13C2] labeled glucose to obtain the most reliable metabolic fluxes.