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.     

Determination of full 13C isotopomer distributions for metabolic flux analysis using heteronuclear spin echo difference NMR spectroscopy

13C-isotopomer labeling experiments play an increasingly important role in the analysis of intracellular metabolic fluxes for genetic engineering purposes. 13C NMR spectroscopy is a key technique in the experimental determination of isotopomer distributions. However, only subsets of isotopomers can be quantitated using this technique due to redundancies in the scalar coupling patterns and due to invisibility of the 12C isotope in NMR. Therefore, we developed and describe in this paper a 1H NMR spectroscopy method that allows to determine the complete isotopomer distribution in metabolites having a backbone consisting of up to at least four carbons. The proposed pulse sequences employ up to three alternately applied frequency-selective inversion pulses in the 13C channel. In a first application study, the complete isotopomer distribution of aspartate isolated from [1-13C]ethanol-grown Ashbya gossypii was determined. A tentative model of the central metabolism of this organism was constructed and used for metabolic flux analysis. The aspartate isotopomer NMR data played a key role in the successful determination of the flux through the peroxisomal glyoxylate pathway. The new NMR method can be highly instrumental in generating the data upon which isotopomer labeling experiments for flux analysis, that are becoming increasingly important, are based.     

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 flux analysis with a comprehensive isotopomer model in Bacillus subtilis

Fluxes in central carbon metabolism of a genetically engineered, riboflavin-producing Bacillus subtilis strain were investigated in glucose-limited chemostat cultures at low (0.11 h(-1)) and high (0.44 h(-1)) dilution rates. Using a mixture of 10% [U-(13)C] and 90% glucose labeled at natural abundance, (13)C-labeling experiments were carried out to provide additional information for metabolic flux balancing. The resulting labeling pattern in the proteinogenic amino acids were analyzed by two-dimensional [(13)C, (1)H] nuclear magnetic resonance (NMR) spectroscopy. To account rigorously for all available data from these experiments, we developed a comprehensive isotopomer model of B. subtilis central metabolism. Using this model, intracellular carbon net and exchange fluxes were estimated on the basis of validated physiological data and biomass composition in combination with 2D NMR data from 45 individual carbon atom spectra in the amino acids. Glucose catabolism proceeded primarily via glycolysis but pentose phosphate pathway fluxes increased with increasing growth rate. Moreover, significant back fluxes from the TCA cycle to the lower part of glycolysis via the gluconeogenic PEP carboxykinase were detected. The malic enzyme reaction, in contrast, was found to be inactive. A thorough statistical analysis was performed to prove the reliability of the isotopomer balance model and the obtained results. Specifically, a chi(2) test was applied to validate the model and the chi-square criterion was used to explore the sensitivity of model predictions to the experimental data.     

An improved statistical method for detecting heterotachy in nucleotide sequences

The principle of heterotachy states that the substitution rate of sites in a gene can change through time. In this article, we propose a powerful statistical test to detect sites that evolve according to the process of heterotachy. We apply this test to an alignment of 1289 eukaryotic rRNA molecules to 1) determine how widespread the phenomenon of heterotachy is in ribosomal RNA, 2) to test whether these heterotachous sites are nonrandomly distributed, that is, linked to secondary structure features of ribosomal RNA, and 3) to determine the impact of heterotachous sites on the bootstrap support of monophyletic groupings. Our study revealed that with 21 monophyletic taxa, approximately two-thirds of the sites in the considered set of sequences is heterotachous. Although the detected heterotachous sites do not appear bound to specific structural features of the small subunit rRNA, their presence is shown to have a large beneficial influence on the bootstrap support of monophyletic groups. Using extensive testing, we show that this may not be due to heterotachy itself but merely due to the increased substitution rate at the detected heterotachous sites.     

An optimized algorithm for flux estimation from isotopomer distribution in glucose metabolites

MOTIVATION: Analysis of the conversion of (13)C glucose within the metabolic network allows the evaluation of the biochemical fluxes in interconnecting metabolic pathways. Such analyses require solving hundreds of equations with respect to individual isotopomer concentrations, and this assumes applying special software even for constructing the equations. The algorithm, proposed by others could be improved. METHOD: A C-code linked to the program written in Mathematica (Wolfram Research Inc.), constructs and solves differential equations for all isotopomer concentrations, using the general enzyme characteristics (K(m), equilibrium constant, etc.). This code uses innovative algorithm of determination for the isotopomers-products, thus essentially decreasing the computation time. Feasible metabolic fluxes are provided by the parameters of enzyme kinetics found from the data fitting. RESULTS: The software effectively evaluates metabolic fluxes based on the measured isotopomer distribution, as was illustrated by the analysis of glycolysis and pentose phosphate cycle. The mechanism of transketolase and transaldolase catalysis was shown to induce a specific kind of isotopomer re-distribution, which, despite the significance of its effect, usually is not taken into account. AVAILABILITY: The software could be freely downloaded from the site: http://bq.ub.es/bioqint/label_distribution/.     

Strategies for metabolic flux analysis in plants using isotope labelling

Flux measurements through metabolic pathways generate insights into the integration of metabolism, and there is increasing interest in using such measurements to quantify the metabolic effects of mutation and genetic manipulation. Isotope labelling provides a powerful approach for measuring metabolic fluxes, and it gives rise to several distinct methods based on either dynamic or steady-state experiments. We discuss the application of these methods to photosynthetic and non-photosynthetic plant tissues, and we illustrate the different approaches with an analysis of the pathways interconverting hexose phosphates and triose phosphates. The complicating effects of the pentose phosphate pathway and the problems arising from the extensive compartmentation of plant cell metabolism are considered. The non-trivial nature of the analysis is emphasised by reference to invalid deductions in earlier work. It is concluded that steady-state isotopic labelling experiments can provide important information on the fluxes through primary metabolism in plants, and that the combination of stable isotope labelling with detection by nuclear magnetic resonance is particularly informative.     

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.     

Mass spectrometry for metabolic flux analysis

Mass spectrometry in combination with tracer experiments based on 13C substrates can serve as a powerful tool for the modeling and analysis of intracellular fluxes and the investigation of biochemical networks. The theoretical background for the application of mass spectrometry to metabolic flux analysis is discussed. Mass spectrometry methods are especially useful to determine mass distribution of metabolites. Additional information gained from fragmentation of metabolites, e.g., by electron impact ionization, allows further localization of labeling positions, up to complete resolution of isotopomer pools. To effectively handle mass distributions in simulation experiments, a matrix based general methodology is formulated. The natural isotope distribution of carbon, oxygen, hydrogen and nitrogen in the target metabolites is considered by introduction of correction matrices. It is shown by simulation results for the central carbon metabolism that neglecting natural isotope distributions causes significant errors in intracellular flux distributions. By varying relative fluxes into pentosephosphate pathway and pyruvate carboxylation reaction, marked changes in the mass distributions of metabolites result, which are illustrated for pyruvate, oxaloacetate, and alpha-ketoglutarate. In addition mass distributions of metabolites are significantly influenced over a broad range by the degree of reversibility of transaldolase and transketolase reactions in the pentosephosphate pathway. The mass distribution of metabolites is very sensitive towards intracellular flux patterns and can be measured with high accuracy by routine mass spectrometry methods. Copyright 1999 John Wiley & Sons, Inc.     

Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness

Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional “shooting” methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.     

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.     

Generalized n-dimensional biomechanical field analysis using statistical parametric mapping

A variety of biomechanical data are sampled from smooth n-dimensional spatiotemporal fields. These data are usually analyzed discretely, by extracting summary metrics from particular points or regions in the continuum. It has been shown that, in certain situations, such schemes can compromise the spatiotemporal integrity of the original fields. An alternative methodology called statistical parametric mapping (SPM), designed specifically for continuous field analysis, constructs statistical images that lie in the original, biomechanically meaningful sampling space. The current paper demonstrates how SPM can be used to analyze both experimental and simulated biomechanical field data of arbitrary spatiotemporal dimensionality. Firstly, 0-, 1-, 2-, and 3-dimensional spatiotemporal datasets derived from a pedobarographic experiment were analyzed using a common linear model to emphasize that SPM procedures are (practically) identical irrespective of the data's physical dimensionality. Secondly two probabilistic finite element simulation studies were conducted, examining heel pad stress and femoral strain fields, respectively, to demonstrate how SPM can be used to probe the significance of field-wide simulation results in the presence of uncontrollable or induced modeling uncertainty. Results were biomechanically intuitive and suggest that SPM may be suitable for a wide variety of mechanical field applications. SPM's main theoretical advantage is that it avoids problems associated with a priori assumptions regarding the spatiotemporal foci of field signals. SPM's main practical advantage is that a unified framework, encapsulated by a single linear equation, affords comprehensive statistical analyses of smooth scalar fields in arbitrarily bounded n-dimensional spaces.     

An efficient resampling method for assessing genome-wide statistical significance in mapping quantitative trait Loci

Assessing genome-wide statistical significance is an important and difficult problem in multipoint linkage analysis. Due to multiple tests on the same genome, the usual pointwise significance level based on the chi-square approximation is inappropriate. Permutation is widely used to determine genome-wide significance. Theoretical approximations are available for simple experimental crosses. In this article, we propose a resampling procedure to assess the significance of genome-wide QTL mapping for experimental crosses. The proposed method is computationally much less intensive than the permutation procedure (in the order of 10(2) or higher) and is applicable to complex breeding designs and sophisticated genetic models that cannot be handled by the permutation and theoretical methods. The usefulness of the proposed method is demonstrated through simulation studies and an application to a Drosophila backcross.     

A ratiometric imaging method for mapping ion flux densities

A heterogeneous distribution of ion channels on the cell surface is a prerequisite for several cellular functions. Thus, there has been considerable interest in methods allowing the mapping of ion channel distributions. Here we report on a novel ratiometric imaging technique appropriate to measure spatially resolved ion flux signals by using ion sensitive dyes. However, given that certain relevant cell properties like the surface to volume ratio may exhibit significant spatial heterogeneities, the local influx signal cannot be interpreted as a measure of the local open channel concentration or flux density. To overcome this problem, we suggest an internal normalization procedure, which, in analogy to, but clearly distinct from, well-established ratioing techniques, eliminates effects which would otherwise obscure the desired result. Ratioing is performed on flux signals from a given cell, triggered by two different, subsequent stimuli. If the two stimuli address different ion channels, the flux density distribution caused by two channel types can be determined relative to each other. In cases where one of the stimuli triggers a spatially homogeneous flux signal, ratioing yields an ion flux density map for a given channel type. Thus distribution patterns of ion channels active during a given stimulus may be derived.     

A possibilistic framework for constraint-based metabolic flux analysis

Background  

Constraint-based models allow the calculation of the metabolic flux states that can be exhibited by cells, standing out as a powerful analytical tool, but they do not determine which of these are likely to be existing under given circumstances. Typical methods to perform these predictions are (a) flux balance analysis, which is based on the assumption that cell behaviour is optimal, and (b) metabolic flux analysis, which combines the model with experimental measurements.