Bidirectional reaction steps in metabolic networks: III. Explicit solution and analysis of isotopomer labeling systems

The last few years have brought tremendous progress in experimental methods for metabolic flux determination by carbon-labeling experiments. A significant enlargement of the available measurement data set has been achieved, especially when isotopomer fractions within intracellular metabolite pools are quantitated. This information can be used to improve the statistical quality of flux estimates. Furthermore, several assumptions on bidirectional intracellular reaction steps that were hitherto indispensable may now become obsolete. To make full use of the complete measurement information a general mathematical model for isotopomer systems is established in this contribution. Then, by introducing the important new concept of cumomers and cumomer fractions, it is shown that the arising nonlinear isotopomer balance equations can be solved analytically in all cases. In particular, the solution of the metabolite flux balances and the positional carbon-labeling balances presented in part I of this series turn out to be just the first two steps of the general solution procedure for isotopomer balances. A detailed analysis of the isotopomer network structure then opens up new insights into the intrinsic structure of isotopomer systems. In particular, it turns out that isotopomer systems are not as complex as they appear at first glance. This enables some far-reaching conclusions to be drawn on the information potential of isotopomer experiments with respect to flux identification. Finally, some illustrative examples are examined to show that an information increase is not guaranteed when isotopomer measurements are used in addition to positional enrichment data.     

GC-MS analysis of amino acids rapidly provides rich information for isotopomer balancing

Gas chromatography-mass spectrometry (GC-MS) is a rapid method that provides rich information on isotopomer distributions for metabolic flux analysis. First, we established a fast and reliable experimental protocol for GC-MS analysis of amino acids from total biomass hydrolyzates, and common experimental pitfalls are discussed. Second, a suitable interface for the use of mass isotopomer data is presented. For this purpose, a general, matrix-based correction procedure that accounts for naturally occurring isotopes is introduced. Simulated and experimentally determined mass distributions of unlabeled amino acids showed a deviation of less than 3% for 89% of the analyzed fragments. Third, to investigate general properties of GC-MS-based isotopomer balancing, altered flux ratios through glycolysis and pentose phosphate pathway and/or exchange fluxes were simulated. Different fluxes were found to exert specific and significant influence on the mass isotopomer distributions, thus indicating that GC-MS data contain valuable information for metabolic flux analysis. Fourth, comparison of different methods revealed that GC-MS analysis provides the largest number of independent constraints on amino acid isotopomer abundance, followed by correlation spectroscopy and fractional enrichment analysis by nuclear magnetic resonance.     

Mass isotopomer analysis: theoretical and practical considerations.

A theory of mass isotopomer analysis based on the well-known principle of isotope dilution mass spectrometry is reviewed. An algorithm for the determination of isotope incorporation into a metabolic substrate from a labeled precursor using mass isotopomer analysis is presented. The steps include the determination of the contribution of the derivatization reagent to the observed spectrum of the derivatized substrate and the correction of contribution from 13C natural abundance using multiple linear regression analysis. Examples of the application of this theory to determine the spectrum of the trimethylsilyl derivative of the 'pure unlabeled' or mononuclidic cholesterol, and the calculation of mass isotopomer distribution in cholesterol due to tracer incorporation using this 'pure unlabeled' spectrum, are also provided.     

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.     

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.     

Metabolic isotopomer labeling systems. Part I: global dynamic behavior

In the last few years metabolic flux analysis (MFA) using carbon labeling experiments (CLE) has become a major diagnostic tool in metabolic engineering. The mathematical centerpiece of MFA is the solution of isotopomer labeling systems (ILS). An ILS is a high-dimensional nonlinear differential equation system that describes the distribution of isotopomers over a metabolic network during a carbon labeling experiment. This contribution presents a global analysis of the dynamic behavior of general ILSs. It is proven that an ILS is globally stable under very weak conditions that are always satisfied in practice. In particular it is shown that in some sense ILSs are a nonlinear extension to the classical theory of compartmental systems. The central stability condition for compartmental systems, i.e., the non-existence of traps in linear compartmental networks, is also the major stability condition for ILSs. As an important side result of the proof, it is shown that ILSs can be transformed to a cascade of linear systems with time-dependent inhomogeneous terms. This cascade structure has considerable consequences for the development of efficient numerical algorithms for the solution of ILSs and thus for MFA.     

Bidirectional reaction steps in metabolic networks: IV. Optimal design of isotopomer labeling experiments

This article generalizes the statistical tools for the evaluation of carbon-labeling experiments that have been developed for the case of positional enrichment systems in part II of this series to the general case of isotopomer systems. For this purpose, a new generalized measurement equation is introduced that can describe all kinds of measured data, like positional enrichments, relative (13)C nuclear magnetic resonance ((13)C NMR) multiplet intensities, or mass isotopomer fractions produced with mass spectroscopy (MS) instruments. Then, to facilitate the specification of the various measurement procedures available, a new flexible textual notation is introduced from which the complicated generalized measurement equations are generated automatically. Based on these measurement equations, a statistically optimal flux estimator is established and parameter covariance matrices for the flux estimation are computed. Having implemented these tools, different kinds of labeling experiments can be compared by using statistical quality measures. A general framework for the optimal design of carbon-labeling experiments is established on the basis of this method. As an example it is applied to the Corynebacterium network from part II extended by various NMR and MS measurements. In particular, the positional enrichment, multiplet, or mass isotopomer measurements with the greatest information content for flux estimation are computed (measurement design) and various differently labeled input substrates are compared with respect to flux estimation (input design). It is examined in detail how the measurement procedure influences the estimation quality of specific fluxes like the pentose phosphate pathway influx.     

On the analysis of substrate cycles in large metabolic systems

The simultaneous operation of paired, opposing reactions (substrate cycles) or parallel reactions (dual pathways) with seeming wastage of ATP is widespread in cellular metabolism. Analysis of such “futile” pathways has hitherto been limited to loci with only two or three interconnecting fluxes. We introduce here a method that allows straightforward analysis of more complex systems. The method involves the linear superposition of “fundamental” modes, one or more of which may be energetically wasteful. Decomposition of a flux pattern into such modes allows computation of the amount of free energy “wasted” at any locus. Appropriate normalizations of energy wastage yield a number of indices useful for assessing the energetic impact of futile pathways on the cell and for comparing the degree of regulation of substrate cycles or dual pathways under different metabolic conditions. This approach is applied to steady-state flux data obtained in the protozoanTetrahymena pyriformis and in isolated rat hepatocytes under a variety of conditions.     

High-resolution analysis of metabolic cycles in the intertidal mussel Mytilus californianus

Inhabitants of the marine rocky intertidal live in an environment that alternates between aquatic and terrestrial due to the rise and fall of the tide. The tide creates a cyclical availability of oxygen with animals having access to oxygenated water during episodes of submergence, while access to oxygen is restricted during aerial emergence. Here we performed liquid chromatography and gas chromatography-mass spectrometry enabled metabolomic profiling of gill samples isolated from the California ribbed mussel, Mytilus californianus, to investigate how metabolism is orchestrated in this variable environment. We created a simulated intertidal environment in which mussels were acclimated to alternating high and low tides of 6 h duration, and samples were taken every 2 h for 72 h to capture reproducible changes in metabolite levels over six high and six low tides. We quantified 169 named metabolites of which 24 metabolites cycled significantly with a 12-h period that was linked to the tidal cycle. These data confirmed the presence of alternating phases of fermentation and aerobic metabolism and highlight a role for carnitine-conjugated metabolites during the anaerobic phase of this cycle. Mussels at low tide accumulated eight carnitine-conjugated metabolites, arising from the degradation of fatty acids, branched-chain amino acids, and mitochondrial β-oxidation end products. The data also implicate sphingosine as a potential signaling molecule during aerial emergence. These findings identify new levels of metabolic control whose role in intertidal adaptation remains to be elucidated.     

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.     

Metabolite damage and repair in metabolic engineering design

The necessarily sharp focus of metabolic engineering and metabolic synthetic biology on pathways and their fluxes has tended to divert attention from the damaging enzymatic and chemical side-reactions that pathway metabolites can undergo. Although historically overlooked and underappreciated, such metabolite damage reactions are now known to occur throughout metabolism and to generate (formerly enigmatic) peaks detected in metabolomics datasets. It is also now known that metabolite damage is often countered by dedicated repair enzymes that undo or prevent it. Metabolite damage and repair are highly relevant to engineered pathway design: metabolite damage reactions can reduce flux rates and product yields, and repair enzymes can provide robust, host-independent solutions. Herein, after introducing the core principles of metabolite damage and repair, we use case histories to document how damage and repair processes affect efficient operation of engineered pathways – particularly those that are heterologous, non-natural, or cell-free. We then review how metabolite damage reactions can be predicted, how repair reactions can be prospected, and how metabolite damage and repair can be built into genome-scale metabolic models. Lastly, we propose a versatile ‘plug and play’ set of well-characterized metabolite repair enzymes to solve metabolite damage problems known or likely to occur in metabolic engineering and synthetic biology projects.     

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.     

Application of a metabolic balancing technique to the analysis of microbial fermentation data

A general method for the development of fermentation models, based on elemental and metabolic balances, is illustrated with three examples from the literature. Physiological parameters such as the (maximal) yield on ATP, the energetic maintenance coefficient, the P/O ratio and others are estimated by fitting model equations to experimental data. Further, phenomenological relations concerning kinetics of product formation and limiting enzyme activities are assessed. The results are compared with the conclusions of the original articles, and differences due to the application of improved models are discussed.     

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.     

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.     

On the stability of metabolic cycles

We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.     

In vivo analysis of metabolic dynamics in Saccharomyces cerevisiae : I. Experimental observations

The goal of this work was to obtain rapid sampling technique to measure transient metabolites in vivo. First, a pulse of glucose was added to a culture of the yeast Saccharomyces cerevisiae growing aerobically under glucose limitation. Next, samples were removed at 2 to 5 s intervals and quenched using methods that depend on the metabolite measured. Extracellular glucose, excreted products, as well as glycolytic intermediates (G6P, F6P, FBP, GAP, 3-PG, PEP, Pyr) and cometabolites (ATP, ADP, AMP, NAD(+), NADH) were measured using enzymatic or HPLC methods. Significant differences between the adenine nucleotide concentrations in the cytoplasm and mitochondria indicated the importance of compartmentation for the regulation of the glycolysis. Changes in the intra- and extracellular levels of metabolites confirmed that glycolysis is regulated on a time scale of seconds. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 305-316, 1997.