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.     

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.     

Bidirectional reaction steps in metabolic networks: II. Flux estimation and statistical analysis

Metabolic carbon labelling experiments enable a large amount of extracellular fluxes and intracellular carbon isotope enrichments to be measured. Since the relation between the measured quantities and the unknown intracellular metabolic fluxes is given by bilinear balance equations, flux determination from this data set requires the numerical solution of a nonlinear inverse problem. To this end, a general algorithm for flux estimation from metabolic carbon labelling experiments based on the least squares approach is developed in this contribution and complemented by appropriate tools for statistical analysis. The linearization technique usually applied for the computation of nonlinear confidence regions is shown to be inappropriate in the case of large exchange fluxes. For this reason a sophisticated compactification transformation technique for nonlinear statistical analysis is developed. Statistical analysis is then performed by computing appropriate statistical quality measures like output sensitivities, parameter sensitivities and the parameter covariance matrix. This allows one to determine the order of magnitude of exchange fluxes in most practical situations. An application study with a large data set from lysine-producing Corynebacterium glutamicum demonstrates the power and limitations of the carbon-labelling technique. It is shown that all intracellular fluxes in central metabolism can be quantitated without assumptions on intracellular energy yields. At the same time several exchange fluxes are determined which is invaluable information for metabolic engineering. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 118-135, 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.     

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.     

Metabolite and isotopomer balancing in the analysis of metabolic cycles: I. Theory

Proper analysis of label distribution in metabolic pathway intermediates is critical for correct interpretation of experimental data and strategic experimental design. While, for example, 13C nuclear magnetic resonance (NMR) spectroscopy is usually limited to the measurement of degrees of 13C enrichment, more information about metabolic fluxes can be extracted from the fine structure of NMR spectra, or molecular weight distributions of isotopomers of metabolic intermediates (measured by gas chromatography-mass spectrometry). For this purpose, rigorous accounting for the contribution of all pathways to label distribution is required, especially contributions resulting from multiple turns of metabolic cycles. In this paper we present a mathematical model developed to analyze isotopomer distributions of tricarboxylic acid cycle (TCA) intermediates following the administration of 13C (or 14C) labeled substrates. The theory presented provides the basis to analyze 13C NMR spectra and molecular weight distributions of metabolites. In a companion paper (Park et al., 1999), the theory is applied to the analysis of several cases of biological significance. Copyright 1999 John Wiley & Sons, Inc.     

Metabolomic and mass isotopomer analysis of liver gluconeogenesis and citric acid cycle: II. Heterogeneity of metabolite labeling pattern

In this second of two companion articles, we compare the mass isotopomer distribution of metabolites of liver gluconeogenesis and citric acid cycle labeled from NaH(13)CO(3) or dimethyl [1,4-(13)C(2)]succinate. The mass isotopomer distribution of intermediates reveals the reversibility of the isocitrate dehydrogenase + aconitase reactions, even in the absence of a source of alpha-ketoglutarate. In addition, in many cases, a number of labeling incompatibilities were found as follows: (i) glucose versus triose phosphates and phosphoenolpyruvate; (ii) differences in the labeling ratios C-4/C-3 of glucose versus (glyceraldehyde 3-phosphate)/(dihydroxyacetone phosphate); and (iii) labeling of citric acid cycle intermediates in tissue versus effluent perfusate. Overall, our data show that gluconeogenic and citric acid cycle intermediates cannot be considered as sets of homogeneously labeled pools. This probably results from the zonation of hepatic metabolism and, in some cases, from differences in the labeling pattern of mitochondrial versus extramitochondrial metabolites. Our data have implications for the use of labeling patterns for the calculation of metabolic rates or fractional syntheses in liver, as well as for modeling liver intermediary metabolism.     

Triple isotopic labeling and kinetic isotope effects: exposing H-transfer steps in enzymatic systems

Kinetic isotope effect (KIE) studies can provide insight into the mechanism and kinetics of specific chemical steps in complex catalytic cascades. Recent results from hydrogen KIE measurements have examined correlations between enzyme dynamics and catalytic function, leading to a surge of studies in this area. Unfortunately, most enzymatic H-transfer reactions are not rate limiting, and the observed KIEs do not reliably reflect the intrinsic KIEs on the chemical step of interest. Given their importance to understanding the chemical step under study, accurate determination of the intrinsic KIE from the observed data is essential. In 1975, Northrop developed an elegant method to assess intrinsic KIEs from their observed values [Northrop, D. B. (1975) Steady-state analysis of kinetic isotope effects in enzymic reactions. Biochemistry 14, 2644-2651]. The Northrop method involves KIE measurements using all three hydrogen isotopes, where one of them serves as the reference isotope. This method has been successfully used with different combinations of observed KIEs over the years, but criteria for a rational choice of reference isotope have never before been experimentally determined. Here we compare different reference isotopes (and hence distinct experimental designs) using the reduction of dihydrofolate and dihydrobiopterin by two dissimilar enzymes as model reactions. A number of isotopic labeling patterns have been applied to facilitate the comparative study of reference isotopes. The results demonstrate the versatility of the Northrop method and that such experiments are limited only by synthetic techniques, availability of starting materials, and the experimental error associated with the use of distinct combinations of isotopologues.     

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.     

Optimization of regulatory architectures in metabolic reaction networks

Successful biotechnological applications, such as amino acid production, have demonstrated significant improvement in bioprocess performance by genetic modifications of metabolic control architectures and enzyme expression levels. However, the stoichiometric complexity of metabolic pathways, along with their strongly nonlinear nature and regulatory coupling, necessitates the use of structured kinetic models to direct experimental applications and aid in quantitative understanding of cellular bioprocesses. A novel optimization problem is introduced here, the objective of which is to identify changes in the regulatory characteristics of pertinent enzymes and in their cellular content which should be implemented to optimize a particular metabolic process. The mathematical representation of the metabolic reaction networks used is the S-system representation, which at steady state is characterized by linear equations. Exploiting the linearity of the representation, we formulated the optimization problem as a mixed-integer linear programming (MILP) problem. This formulation allows the consideration of a regulatory superstructure that contains all alternative regulatory structures that can be considered for a given pathway. The proposed approach is developed and illustrated using a simple linear pathway. Application of the framework on a complicated pathway-namely, the xanthine monophosphate (XMP) and guanosine monophosphate (GMP) synthesis pathway-identified the modification of the regulatory architecture that, along with changes in enzyme expression levels, can increase the XMP and GMP concentration by over 114 times the reference value, which is 50 times more than could be achieved by changes in enzyme expression levels only. (c) 1996 John Wiley & Sons, Inc.     

Modular decomposition of metabolic reaction networks based on flux analysis and pathway projection

MOTIVATION: The rational decomposition of biochemical networks into sub-structures has emerged as a useful approach to study the design of these complex systems. A biochemical network is characterized by an inhomogeneous connectivity distribution, which gives rise to several organizational features, including modularity. To what extent the connectivity-based modules reflect the functional organization of the network remains to be further explored. In this work, we examine the influence of physiological perturbations on the modular organization of cellular metabolism. RESULTS: Modules were characterized for two model systems, liver and adipocyte primary metabolism, by applying an algorithm for top-down partition of directed graphs with non-uniform edge weights. The weights were set by the engagement of the corresponding reactions as expressed by the flux distribution. For the base case of the fasted rat liver, three modules were found, carrying out the following biochemical transformations: ketone body production, glucose synthesis and transamination. This basic organization was further modified when different flux distributions were applied that describe the liver's metabolic response to whole body inflammation. For the fully mature adipocyte, only a single module was observed, integrating all of the major pathways needed for lipid storage. Weaker levels of integration between the pathways were found for the early stages of adipocyte differentiation. Our results underscore the inhomogeneous distribution of both connectivity and connection strengths, and suggest that global activity data such as the flux distribution can be used to study the organizational flexibility of cellular metabolism. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

The activity reaction core and plasticity of metabolic networks

Understanding the system-level adaptive changes taking place in an organism in response to variations in the environment is a key issue of contemporary biology. Current modeling approaches, such as constraint-based flux-balance analysis, have proved highly successful in analyzing the capabilities of cellular metabolism, including its capacity to predict deletion phenotypes, the ability to calculate the relative flux values of metabolic reactions, and the capability to identify properties of optimal growth states. Here, we use flux-balance analysis to thoroughly assess the activity of Escherichia coli, Helicobacter pylori, and Saccharomyces cerevisiae metabolism in 30,000 diverse simulated environments. We identify a set of metabolic reactions forming a connected metabolic core that carry non-zero fluxes under all growth conditions, and whose flux variations are highly correlated. Furthermore, we find that the enzymes catalyzing the core reactions display a considerably higher fraction of phenotypic essentiality and evolutionary conservation than those catalyzing noncore reactions. Cellular metabolism is characterized by a large number of species-specific conditionally active reactions organized around an evolutionary conserved, but always active, metabolic core. Finally, we find that most current antibiotics interfering with bacterial metabolism target the core enzymes, indicating that our findings may have important implications for antimicrobial drug-target discovery.     

Zonation of labeling of lipogenic acetyl-CoA across the liver: implications for studies of lipogenesis by mass isotopomer analysis

Measurement of fractional lipogenesis by condensation polymerization methods assumes constant enrichment of lipogenic acetyl-CoA in all hepatocytes. mass isotopomer distribution analysis (MIDA) and isotopomer spectral analysis (ISA) represent such methods and are based on the combinatorial analyses of mass isotopomer distributions (MIDs) of fatty acids and sterols. We previously showed that the concentration and enrichment of [13C]acetate decrease markedly across the dog liver because of the simultaneous uptake and production of acetate. To test for zonation of the enrichment of lipogenic acetyl-CoA, conscious dogs, prefitted with transhepatic catheters, were infused with glucose and [1,2-13C2]acetate in a branch of the portal vein. Analyses of MIDs of fatty acids and sterols isolated from liver, bile, and plasma very low density lipoprotein by a variant of ISA designed to detect gradients in precursor enrichment revealed marked zonation of enrichment of lipogenic acetyl-CoA. As control experiments where no zonation of acetyl-CoA enrichment would be expected, isolated rat livers were perfused with 10 mm [1,2-13C2]acetate. The ISA analyses of MIDs of fatty acids and sterols from liver and bile still revealed a zonation of acetyl-CoA enrichment. We conclude that zonation of hepatic acetyl-CoA enrichment occurs under a variety of animal models and physiological conditions. Failure to consider gradients of precursor enrichment can lead to underestimations of fractional lipogenesis calculated from the mass isotopomer distributions. The degree of such underestimation was modeled in vitro, and the data are reported in the companion paper (Bederman, I. R., Kasumov, T., Reszko, A. E., David, F., Brunengraber, H., and Kelleher, J. K. (2004) J. Biol. Chem. 279, 43217-43226).