A priori analysis of metabolic flux identifiability from (13)C-labeling data

The (13)C-labeling technique was introduced in the field of metabolic engineering as a tool for determining fluxes that could not be found using the 'classical' method of flux balancing. An a priori flux identifiability analysis is required in order to determine whether a (13)C-labeling experiment allows the identification of all the fluxes. In this article, we propose a method for identifiability analysis that is based on the recently introduced 'cumomer' concept. The method improves upon previous identifiability methods in that it provides a way of systematically reducing the metabolic network on the basis of structural elements that constitute a network and to use the implicit function theorem to analytically determine whether the fluxes in the reduced network are theoretically identifiable for various types of real measurement data. Application of the method to a realistic flux identification problem shows both the potential of the method in yielding new, interesting conclusions regarding the identifiability and its practical limitations that are caused by the fact that symbolic calculations grow fast with the dimension of the studied system.     

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.     

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.     

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.     

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.     

Rational design of 13C-labeling experiments for metabolic flux analysis in mammalian cells

ABSTRACT: BACKGROUND: 13C-Metabolic flux analysis (13C-MFA) is a standard technique to probe cellular metabolism and elucidate in vivo metabolic fluxes. 13C-Tracer selection is an important step in conducting 13C-MFA, however, current methods are restricted to trial-and-error approaches, which commonly focus on an arbitrary subset of the tracer design space. To systematically probe the complete tracer design space, especially for complex systems such as mammalian cells, there is a pressing need for new rational approaches to identify optimal tracers. RESULTS: Recently, we introduced a new framework for optimal 13C-tracer design based on elementary metabolite units (EMU) decomposition, in which a measured metabolite is decomposed into a linear combination of so-called EMU basis vectors. In this contribution, we applied the EMU method to a realistic network model of mammalian metabolism with lactate as the measured metabolite. The method was used to select optimal tracers for the two free fluxes in the system, the oxidative pentose phosphate pathway (oxPPP) flux and anaplerosis by pyruvate carboxylase (PC). Our approach was based on sensitivity analysis of EMU basis vector coefficients with respect to free fluxes. Through efficient grouping of coefficient sensitivities, simple tracer selection rules were derived for high-resolution quantification of the fluxes in the mammalian network model. The approach resulted in a significant reduction of the number of possible tracers and the feasible tracers were evaluated using numerical simulations. Two optimal, novel tracers were identified that have not been previously considered for 13C-MFA of mammalian cells, specifically [2,3,4,5,6-13C]glucose for elucidating oxPPP flux and [3,4-13C]glucose for elucidating PC flux. We demonstrate that 13C-glutamine tracers perform poorly in this system in comparison to the optimal glucose tracers. CONCLUSIONS: In this work, we have demonstrated that optimal tracer design does not need to be a pure simulation-based trial-and-error process; rather, rational insights into tracer design can be gained through the application of the EMU basis vector methodology. Using this approach, rational labeling rules can be established a priori to guide the selection of optimal 13C-tracers for high-resolution flux elucidation in complex metabolic network models.     

Visual exploration of isotope labeling networks in 3D

Isotope labeling networks (ILNs) are graphs explaining the flow of isotope labeled molecules in a metabolic network. Moreover, they are the structural backbone of metabolic flux analysis (MFA) by isotopic tracers which has been established as a standard experimental tool in fluxomics. To configure an isotope labeling experiment (ILE) for MFA, the structure of the corresponding ILN must be understood to a certain extent even by a practitioner. Graph algorithms help to analyze the network structure but produce rather abstract results. Here, the major obstruction is the high dimension of these networks and the large number of network components which, consequently, are hard to figure out manually. At the interface between theory and experiment, the three-dimensional interactive visualization tool CumoVis has been developed for exploring the network structure in a step by step manner. Navigation and orientation within ILNs are supported by exploiting the natural 3D structure of an underlying metabolite network with stacked labeled particles on top of each metabolite node. Network exploration is facilitated by rotating, zooming, forward and backward path tracing and, most important, network component reduction. All features of CumoVis are explained with an educational example and a realistic network describing carbon flow in the citric acid cycle.     

Kinetic isotope effects significantly influence intracellular metabolite 13C labeling patterns and flux determination

Rigorous mathematical modeling of carbon-labeling experiments allows estimation of fluxes through the pathways of central carbon metabolism, yielding powerful information for basic scientific studies as well as for a wide range of applications. However, the mathematical models that have been developed for flux determination from ¹³C labeling data have commonly neglected the influence of kinetic isotope effects on the distribution of ¹³C label in intracellular metabolites, as these effects have often been assumed to be inconsequential. We have used measurements of the ¹³C isotope effects on the pyruvate dehydrogenase enzyme from the literature to model isotopic fractionation at the pyruvate node and quantify the modeling errors expected to result from the assumption that isotope effects are negligible. We show that under some conditions kinetic isotope effects have a significant impact on the ¹³C labeling patterns of intracellular metabolites, and the errors associated with neglecting isotope effects in ¹³C-metabolic flux analysis models can be comparable in size to measurement errors associated with GC–MS. Thus, kinetic isotope effects must be considered in any rigorous assessment of errors in ¹³C labeling data, goodness-of-fit between model and data, confidence intervals of estimated metabolic fluxes, and statistical significance of differences between estimated metabolic flux distributions.     

Analysis of Escherichia coli anaplerotic metabolism and its regulation mechanisms from the metabolic responses to altered dilution rates and phosphoenolpyruvate carboxykinase knockout

The gluconeogenic phosphoenolpyruvate (PEP) carboxykinase is active in Escherichia coli during its growth on glucose. The present study investigated the influence of growth rates and PEP carboxykinase knockout on the anaplerotic fluxes in E. coli. The intracellular fluxes were determined using the complementary methods of flux ratio analysis and metabolic flux analysis based on [U-(13)C(6)]glucose labeling experiments and 2D nuclear magnetic resonance (NMR) spectroscopy of cellular amino acids and glycerol. Significant activity of PEP carboxykinase was identified in wild-type E. coli, and the ATP dissipation for the futile cycling via this reaction accounted for up to 8.2% of the total energy flux. Flux analysis of pck deletion mutant revealed that abolishment of PEP carboxykinase activity resulted in a remarkably reduced flux through the anaplerotic PEP carboxylase and the activation of the glyoxylate shunt, with 23% of isocitrate found being channeled in the glyoxylate shunt. The changes in intracellular metabolite concentrations and specific enzyme activities associated with different growth rates and pck deletion, were also determined. Combining the measurement data of in vivo fluxes, metabolite concentrations and enzyme activities, the in vivo regulations of PEP carboxykinase flux, PEP carboxylation, and glyoxylate shunt in E. coli are discussed.     

Identification of optimal measurement sets for complete flux elucidation in metabolic flux analysis experiments

Metabolic flux analysis (MFA) methods use external flux and isotopic measurements to quantify the magnitude of metabolic flows in metabolic networks. A key question in this analysis is choosing a set of measurements that is capable of yielding a unique flux distribution (identifiability). In this article, we introduce an optimization-based framework that uses incidence structure analysis to determine the smallest (or most cost-effective) set of measurements leading to complete flux elucidation. This approach relies on an integer linear programming formulation OptMeas that allows for the measurement of external fluxes and the complete (or partial) enumeration of the isotope forms of metabolites without requiring any of these to be chosen in advance. We subsequently query and refine the measurement sets suggested by OptMeas for identifiability and optimality. OptMeas is first tested on small to medium-size demonstration examples. It is subsequently applied to a large-scale E. coli isotopomer mapping model with more than 17,000 isotopomers. A number of additional measurements are identified leading to maximum flux elucidation in an amorphadiene producing E. coli strain.     

Computational tools for isotopically instationary 13C labeling experiments under metabolic steady state conditions

(13)C metabolic flux analysis (MFA) has become an important and powerful tool for the quantitative analysis of metabolic networks in the framework of metabolic engineering. Isotopically instationary (13)C MFA under metabolic stationary conditions is a promising refinement of classical stationary MFA. It accounts for the experimental requirements of non-steady-state cultures as well as for the shortening of the experimental duration. This contribution extends all computational methods developed for classical stationary (13)C MFA to the instationary situation by using high-performance computing methods. The developed tools allow for the simulation of instationary carbon labeling experiments (CLEs), sensitivity calculation with respect to unknown parameters, fitting of the model to the measured data, statistical identifiability analysis and an optimal experimental design facility. To explore the potential of the new approach all these tools are applied to the central metabolism of Escherichia coli. The achieved results are compared to the outcome of the stationary counterpart, especially focusing on statistical properties. This demonstrates the specific strengths of the instationary method. A new ranking method is proposed making both an a priori and an a posteriori design of the sampling times available. It will be shown that although still not all fluxes are identifiable, the quality of flux estimates can be strongly improved in the instationary case. Moreover, statements about the size of some immeasurable pool sizes can be made.     

Identification of metabolic fluxes in hepatic cells from transient 13C-labeling experiments: Part II. Flux estimation

This contribution addresses the identification of metabolic fluxes and metabolite concentrations in mammalian cells from transient (13)C-labeling experiments. Whilst part I describes experimental set-up and acquisition of required metabolite and (13)C-labeling data, part II focuses on setting up network models and the estimation of intracellular fluxes. Metabolic fluxes were determined in glycolysis, pentose-phosphate pathway (PPP), and citric acid cycle (TCA) in a hepatoma cell line grown in aerobic batch cultures. In glycolytic and PPP metabolite pools isotopic stationarity was observed within 30 min, whereas in the TCA cycle the labeling redistribution did not reach isotopic steady state even within 180 min. In silico labeling dynamics were in accordance with in vivo (13)C-labeling data. Split ratio between glycolysis and PPP was 57%:43%; intracellular glucose concentration was estimated at 101.6 nmol per 10(6) cells. In contrast to isotopic stationary (13)C-flux analysis, transient (13)C-flux analysis can also be applied to industrially relevant mammalian cell fed-batch and batch cultures.     

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.     

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     

A Peptide-Based Method for 13C Metabolic Flux Analysis in Microbial Communities

The study of intracellular metabolic fluxes and inter-species metabolite exchange for microbial communities is of crucial importance to understand and predict their behaviour. The most authoritative method of measuring intracellular fluxes, ¹³C Metabolic Flux Analysis (¹³C MFA), uses the labeling pattern obtained from metabolites (typically amino acids) during ¹³C labeling experiments to derive intracellular fluxes. However, these metabolite labeling patterns cannot easily be obtained for each of the members of the community. Here we propose a new type of ¹³C MFA that infers fluxes based on peptide labeling, instead of amino acid labeling. The advantage of this method resides in the fact that the peptide sequence can be used to identify the microbial species it originates from and, simultaneously, the peptide labeling can be used to infer intracellular metabolic fluxes. Peptide identity and labeling patterns can be obtained in a high-throughput manner from modern proteomics techniques. We show that, using this method, it is theoretically possible to recover intracellular metabolic fluxes in the same way as through the standard amino acid based ¹³C MFA, and quantify the amount of information lost as a consequence of using peptides instead of amino acids. We show that by using a relatively small number of peptides we can counter this information loss. We computationally tested this method with a well-characterized simple microbial community consisting of two species.     

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.     

Experimental design principles for isotopically instationary 13C labeling experiments

13C metabolic flux analysis (MFA) is a well-established tool in Metabolic Engineering that found numerous applications in recent years. However, one strong limitation of the current method is the requirement of an-at least approximate-isotopic stationary state at sampling time. This requirement leads to a principle lower limit for the duration of a 13C labeling experiment. A new methodological development is based on repeated sampling during the instationary transient of the 13C labeling dynamics. The statistical and computational treatment of such instationary experiments is a completely new terrain. The computational effort is very high because large differential equations have to be solved and, moreover, the intracellular pool sizes play a significant role. For this reason, the present contribution works out principles and strategies for the experimental design of instationary experiments based on a simple example network. Hereby, the potential of isotopically instationary experiments is investigated in detail. Various statistical results on instationary flux identifiability are presented and possible pitfalls of experimental design are discussed. Finally, a framework for almost optimal experimental design of isotopically instationary experiments is proposed which provides a practical guideline for the analysis of large-scale networks.     

Selection of tracers for 13C-metabolic flux analysis using elementary metabolite units (EMU) basis vector methodology

Metabolic flux analysis (MFA) is a powerful technique for elucidating in vivo fluxes in microbial and mammalian systems. A key step in (13)C-MFA is the selection of an appropriate isotopic tracer to observe fluxes in a proposed network model. Despite the importance of MFA in metabolic engineering and beyond, current approaches for tracer experiment design are still largely based on trial-and-error. The lack of a rational methodology for selecting isotopic tracers prevents MFA from achieving its full potential. Here, we introduce a new technique for tracer experiment design based on the concept of elementary metabolite unit (EMU) basis vectors. We demonstrate that any metabolite in a network model can be expressed as a linear combination of so-called EMU basis vectors, where the corresponding coefficients indicate the fractional contribution of the EMU basis vector to the product metabolite. The strength of this approach is the decoupling of substrate labeling, i.e. the EMU basis vectors, from the dependence on free fluxes, i.e. the coefficients. In this work, we demonstrate that flux observability inherently depends on the number of independent EMU basis vectors and the sensitivities of coefficients with respect to free fluxes. Specifically, the number of independent EMU basis vectors places hard limits on how many free fluxes can be determined in a model. This constraint is used as a guide for selecting feasible substrate labeling. In three example models, we demonstrate that by maximizing the number of independent EMU basis vectors the observability of a system is improved. Inspection of sensitivities of coefficients with respect to free fluxes provides additional constraints for proper selection of tracers. The present contribution provides a fresh perspective on an important topic in metabolic engineering, and gives practical guidelines and design principles for a priori selection of isotopic tracers for (13)C-MFA studies.     

Kinetic flux profiling for quantitation of cellular metabolic fluxes

This protocol enables quantitation of metabolic fluxes in cultured cells. Measurements are based on the kinetics of cellular incorporation of stable isotope from nutrient into downstream metabolites. At multiple time points, after cells are rapidly switched from unlabeled to isotope-labeled nutrient, metabolism is quenched, metabolites are extracted and the extract is analyzed by chromatography-mass spectrometry. Resulting plots of unlabeled compound versus time follow variants of exponential decay, with the flux equal to the decay rate multiplied by the intracellular metabolite concentration. Because labeling is typically fast (t(1/2)相似文献