Using isotopomer path tracing to quantify metabolic fluxes in pathway models containing reversible reactions

As a more complete picture of the genetic and enzymatic composition of cells becomes available, there is a growing need to describe how cellular regulatory elements interact with the cellular environment to affect cell physiology. One means for describing intracellular regulatory mechanisms is concurrent measurement of multiple metabolic pathways and their interactions by metabolic flux analysis. Flux of carbon through a metabolic pathway responds to all cellular regulatory systems, including changes in enzyme and substrate concentrations, enzyme activation or inhibition, and ultimately genetic control. The extent to which metabolic flux analysis can describe cellular physiology depends on the number of pathways in the model and the quality of the data. Intracellular information is obtainable from isotopic tracer experiments, the most extensive being the determination of the isotopomer distribution, or specific labeling pattern, of intracellular metabolites. We present a rapid and novel solution method that determines the flux of carbon through complex pathway models using isotopomer data. This time-consuming problem was solved with the introduction of isotopomer path tracing, which drastically reduces the number of isotopomer variables to the number of isotopomers observed experimentally. We propose a partitioned solution method that takes advantage of the nearly linear relationship between fluxes and isotopomers. Whereas the stoichiometric matrix and the isotopomer matrix are invertible, simulated annealing and the Newton-Raphson method are used for the nonlinear components. Reversible reactions are described by a new parameter, the association factor, which scales hyperbolically with the rate of metabolite exchange. Automating the solution method permits a variety of models to be compared, thus enhancing the accuracy of results. A simplified example that contains all of the complexities of a comprehensive pathway model is presented. Copyright John Wiley & Sons, Inc.     

Rapid simulation and analysis of isotopomer distributions using constraints based on enzyme mechanisms: an example from HT29 cancer cells

MOTIVATION: Addition of labeled substrates and the measurement of the subsequent distribution of the labels in isotopomers in reaction networks provide a unique method for assessing metabolic fluxes in whole cells. However, owing to insufficiency of information, attempts to quantify the fluxes often yield multiple possible sets of solutions that are consistent with a given experimental pattern of isotopomers. In the study of the pentose phosphate pathways, the need to consider isotope exchange reactions of transketolase (TK) and transaldolase (TA) (which in past analyses have often been ignored) magnifies this problem; but accounting for the interrelation between the fluxes known from biochemical studies and kinetic modeling solves it. The mathematical relationships between kinetic and equilibrium constants restrict the domain of estimated fluxes to the ones compatible not only with a given set of experimental data, but also with other biochemical information. METHOD: We present software that integrates kinetic modeling with isotopomer distribution analysis. It solves the ordinary differential equations for total concentrations (accounting for the kinetic mechanisms) as well as for all isotopomers in glycolysis and the pentose phosphate pathway (PPP). In the PPP the fluxes created in the TK and TA reactions are expressed through unitary rate constants. The algorithms that account for all the kinetic and equilbrium constant constraints are integrated with the previously developed algorithms, which have been further optimized. The most time-consuming calculations were programmed directly in assembly language; this gave an order of magnitude decrease in the computation time, thus allowing analysis of more complex systems. The software was developed as C-code linked to a program written in Mathematica (Wolfram Research, Champaign, IL), and also as a C++ program independent from Mathematica. RESULTS: Implementing constraints imposed by kinetic and equilibrium constants in the isotopomer distribution analysis in the data from the cancer cells eliminated estimates of fluxes that were inconsistent with the kinetic mechanisms of TK and TA. Fluxes measured experimentally in cells can be used to estimate better the kinetics of TK and TA as they operate in situ. Thus, our approach of integrating various methods for in situ flux analysis opens up the possibility of designing new types of experiments to probe metabolic interrelationships, including the incorporation of additional biochemical information. AVAILABILITY: Software is available freely at: http://www.bq.ub.es/bioqint/selivanov.htm CONTACT: martacascante@ub.edu     

Tandem mass spectrometry: a novel approach for metabolic flux analysis

The goal of metabolic flux analysis (MFA) is the accurate estimation of intracellular fluxes in metabolic networks. Here, we introduce a new method for MFA based on tandem mass spectrometry (MS) and stable-isotope tracer experiments. We demonstrate that tandem MS provides more labeling information than can be obtained from traditional full scan MS analysis and allows estimation of fluxes with better precision. We present a modeling framework that takes full advantage of the additional labeling information obtained from tandem MS for MFA. We show that tandem MS data can be computed for any network model, any compound and any tandem MS fragmentation using linear mapping of isotopomers. The inherent advantages of tandem MS were illustrated in two network models using simulated and literature data. Application of tandem MS increased the observability of the models and improved the precision of estimated fluxes by 2- to 5-fold compared to traditional MS analysis.     

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.     

Determination of metabolic fluxes in a non-steady-state system

Estimation of fluxes through metabolic networks from redistribution patterns of (13)C has become a well developed technique in recent years. However, the approach is currently limited to systems at metabolic steady-state; dynamic changes in metabolic fluxes cannot be assessed. This is a major impediment to understanding the behaviour of metabolic networks, because steady-state is not always experimentally achievable and a great deal of information about the control hierarchy of the network can be derived from the analysis of flux dynamics. To address this issue, we have developed a method for estimating non-steady-state fluxes based on the mass-balance of mass isotopomers. This approach allows multiple mass-balance equations to be written for the change in labelling of a given metabolite pool and thereby permits over-determination of fluxes. We demonstrate how linear regression methods can be used to estimate non-steady-state fluxes from these mass balance equations. The approach can be used to calculate fluxes from both mass isotopomer and positional isotopomer labelling information and thus has general applicability to data generated from common spectrometry- or NMR-based analytical platforms. The approach is applied to a GC-MS time-series dataset of (13)C-labelling of metabolites in a heterotrophic Arabidopsis cell suspension culture. Threonine biosynthesis is used to demonstrate that non-steady-state fluxes can be successfully estimated from such data while organic acid metabolism is used to highlight some common issues that can complicate flux estimation. These include multiple pools of the same metabolite that label at different rates and carbon skeleton rearrangements.     

Empirical modelling of lake water-temperature relationships: a comparison of approaches

1.  As a result of the role that temperature plays in many aquatic processes, good predictive models of annual maximum near-surface lake water temperature across large spatial scales are needed, particularly given concerns regarding climate change. Comparisons of suitable modelling approaches are required to determine their relative merit and suitability for providing good predictions of current conditions. We developed models predicting annual maximum near-surface lake water temperatures for lakes across Canada using four statistical approaches: multiple regression, regression tree, artificial neural networks and Bayesian multiple regression.
2.  Annual maximum near-surface (from 0 to 2 m) lake water-temperature data were obtained for more than 13 000 lakes and were matched to geographic, climatic, lake morphology, physical habitat and water chemistry data. We modelled 2348 lakes and three subsets thereof encompassing different spatial scales and predictor variables to identify the relative importance of these variables at predicting lake temperature.
3.  Although artificial neural networks were marginally better for three of the four data sets, multiple regression was considered to provide the best solution based on the combination of model performance and computational complexity. Climatic variables and date of sampling were the most important variables for predicting water temperature in our models.
4.  Lake morphology did not play a substantial role in predicting lake temperature across any of the spatial scales. Maximum near-surface temperatures for Canadian lakes appeared to be dominated by large-scale climatic and geographic patterns, rather than lake-specific variables, such as lake morphology and water chemistry.     

FiatFlux – a software for metabolic flux analysis from <Superscript>13</Superscript>C-glucose experiments

Background  

Quantitative knowledge of intracellular fluxes is important for a comprehensive characterization of metabolic networks and their functional operation. In contrast to direct assessment of metabolite concentrations, in vivo metabolite fluxes must be inferred indirectly from measurable quantities in ¹³C experiments. The required experience, the complicated network models, large and heterogeneous data sets, and the time-consuming set-up of highly controlled experimental conditions largely restricted metabolic flux analysis to few expert groups. A conceptual simplification of flux analysis is the analytical determination of metabolic flux ratios exclusively from MS data, which can then be used in a second step to estimate absolute in vivo fluxes.     

Uniform sampling of steady-state flux spaces: means to design experiments and to interpret enzymopathies

Reconstruction of genome-scale metabolic networks is now possible using multiple different data types. Constraint-based modeling is an approach to interrogate capabilities of reconstructed networks by constraining possible cellular behavior through the imposition of physicochemical laws. As a result, a steady-state flux space is defined that contains all possible functional states of the network. Uniform random sampling of the steady-state flux space allows for the unbiased appraisal of its contents. Monte Carlo sampling of the steady-state flux space of the reconstructed human red blood cell metabolic network under simulated physiologic conditions yielded the following key results: 1), probability distributions for the values of individual metabolic fluxes showed a wide variety of shapes that could not have been inferred without computation; 2), pairwise correlation coefficients were calculated between all fluxes, determining the level of independence between the measurement of any two fluxes, and identifying highly correlated reaction sets; and 3), the network-wide effects of the change in one (or a few) variables (i.e., a simulated enzymopathy or fixing a flux range based on measurements) were computed. Mathematical models provide the most compact and informative representation of a hypothesis of how a cell works. Thus, understanding model predictions clearly is vital to driving forward the iterative model-building procedure that is at the heart of systems biology. Taken together, the Monte Carlo sampling procedure provides a broadening of the constraint-based approach by allowing for the unbiased and detailed assessment of the impact of the applied physicochemical constraints on a reconstructed network.     

Quantitative determination of the main glucose metabolic fluxes in human erythrocytes by 13C- and 1H-MR spectroscopy.

Information displayed by homonuclear and heteronuclear spin-coupling patterns in 13C- and 1H-MR spectra allowed us to identify the major lactate isotopomers produced either from [1-(13)C]-glucose or from [2-(13)C]-glucose by human erythrocytes. Relative concentrations of detectable isotopomers were determined by integrating the corresponding MR signals. The interpretation of these data in terms of the fractional glucose metabolised through glycolysis and pentose phosphate pathway was performed by a computer simulation of the metabolism that took into account metabolic schemes pertaining to glycolysis and to the F-type of pentose phosphate pathway. The simulation was organised in a way to anticipate the populations of the isotopomers produced from any precursor at a priori established metabolic steady state. By the simulation, isotopomer populations were determined according to different values of pentose cycle, defined as the flux of glyceraldehyde 3-phosphate originating from pentose phosphate pathway at unitary glucose uptake. The populations of the isotopomers originating from [2-(13)C]-glucose were described by polynomials, and ratios between the polynomials were used in conjunction with 13C- and 1H-MR data to determine pentose cycle values. The knowledge of glucose uptake and of pentose cycle value allowed us to perform accurate measurement of the pentose phosphate pathway flux, of the hexokinase and phosphofructokinase fluxes as well as, indirectly, of the carbon dioxide production.     

Cumulative bondomers: a new concept in flux analysis from 2D [13C,1H] COSY NMR data

A well-established way of determining metabolic fluxes is to measure 2D [(13)C,(1)H] COSY NMR spectra of components of biomass grown on uniformly (13)C-labeled carbon sources. When using the entire set of measured data to simultaneously determine all fluxes in a proposed metabolic network model, the (13)C-labeling distribution in all measured compounds has to be simulated. This requires very large sets of isotopomer or cumomer balances. This article introduces the new concept of bondomers; entities that only vary in the numbers and positions of C-C bonds that have remained intact since the medium substrate molecule entered the metabolism. Bondomers are shown to have many analogies to isotopomers. One of these is that bondomers can be transformed to cumulative bondomers, just like isotopomers can be transformed to cumomers. Similarly to cumomers, cumulative bondomers allow an analytical solution of the entire set of balances describing a metabolic network. The main difference is that cumulative bondomer models are considerably smaller than corresponding cumomer models. This saves computational time, allows easier identifiability analysis, and yields new insights in the information content of 2D [(13)C,(1)H] COSY NMR data. We illustrate the theoretical concepts by means of a realistic example of the glycolytic and pentose phosphate pathways. The combinations of 2D [(13)C,(1)H] COSY NMR data that allow identification of all metabolic fluxes in these pathways are analyzed, and it is found that the NMR data contain less information than was previously expected.     

Characterization of ecosystem responses to climatic controls using artificial neural networks

Understanding and modeling ecosystem responses to their climatic controls is one of the major challenges for predicting the effects of global change. Usually, the responses are implemented in models as parameterized functional relationships of a fixed type. In contrast, the inductive approach presented here based on artificial neural networks (ANNs) allows the relationships to be extracted directly from the data. It has been developed to explore large, fragmentary, noisy, and multidimensional datasets, such as the carbon fluxes measured at the ecosystem level with the eddy covariance technique. To illustrate this, our approach has been systematically applied to the daytime carbon flux dataset of the deciduous broadleaf forest Hainich in Germany. The total explainable variability of the half‐hourly carbon fluxes from the driving climatic variables was 93.1%, showing the excellent data mining capability of the ANNs. Total photosynthetic photon flux density was identified as the dominant control of the daytime response, followed by the diffuse radiation. The vapor pressure deficit was the most important nonradiative control. From the ANNs, we were also able to deduce and visualize the dependencies and sensitivities of the response to its climatic controls. With respect to diffuse radiation, the daytime carbon response showed no saturation and the light use efficiency was three times greater for diffuse compared with direct radiation. However, with less potential radiation reaching the forest, the overall effect of diffuse radiation was slightly negative. The optimum uptake of carbon occurred at diffuse fractions between 30% and 40%. By identifying the hierarchy of the climatic controls of the ecosystem response as well as their multidimensional functional relationships, our inductive approach offers a direct interface to the data. This provides instant insight in the underlying ecosystem physiology and links the observational relationships to their representation in the modeling world.     

Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.     

Analysis of flux estimates based on (13)C-labelling experiments.

Modelling of the fluxes in central metabolism can be performed by combining labelling experiments with metabolite balancing. Using this approach, multiple samples from a cultivation of Saccharomyces cerevisiae in metabolic and isotopic steady state were analysed, and the metabolic fluxes in central metabolism were estimated. In the various samples, the estimates of the central metabolic pathways, the tricarboxylic acid cycle, the oxidative pentose phosphate pathway and the anaplerotic pathway, showed an unprecedented reproducibility. The high reproducibility was obtained with fractional labellings of individual carbon atoms as the calculational base, illustrating that the more complex modelling using isotopomers is not necessarily superior with respect to reproducibility of the flux estimates. Based on these results some general difficulties in flux estimation are discussed.     

Elementary metabolite units (EMU): a novel framework for modeling isotopic distributions

Metabolic flux analysis (MFA) has emerged as a tool of great significance for metabolic engineering and mammalian physiology. An important limitation of MFA, as carried out via stable isotope labeling and GC/MS and nuclear magnetic resonance (NMR) measurements, is the large number of isotopomer or cumomer equations that need to be solved, especially when multiple isotopic tracers are used for the labeling of the system. This restriction reduces the ability of MFA to fully utilize the power of multiple isotopic tracers in elucidating the physiology of realistic situations comprising complex bioreaction networks. Here, we present a novel framework for the modeling of isotopic labeling systems that significantly reduces the number of system variables without any loss of information. The elementary metabolite unit (EMU) framework is based on a highly efficient decomposition method that identifies the minimum amount of information needed to simulate isotopic labeling within a reaction network using the knowledge of atomic transitions occurring in the network reactions. The functional units generated by the decomposition algorithm, called EMUs, form the new basis for generating system equations that describe the relationship between fluxes and stable isotope measurements. Isotopomer abundances simulated using the EMU framework are identical to those obtained using the isotopomer and cumomer methods, however, require significantly less computation time. For a typical (13)C-labeling system the total number of equations that needs to be solved is reduced by one order-of-magnitude (100s EMUs vs. 1000s isotopomers). As such, the EMU framework is most efficient for the analysis of labeling by multiple isotopic tracers. For example, analysis of the gluconeogenesis pathway with (2)H, (13)C, and (18)O tracers requires only 354 EMUs, compared to more than two million isotopomers.     

Systems-level analysis of genome-scalein silico metabolic models using MetaFluxNet

The systems-level analysis of microbes with myriad of heterologous data generated by omics technologies has been applied to improve our understanding of cellular function and physiology and consequently to enhance production of various bioproducts. At the heart of this revolution residesin silico genome-scale metabolic model. In order to fully exploit the power of genome-scale model, a systematic approach employing user-friendly software is required. Metabolic flux analysis of genome-scale metabolic network is becoming widely employed to quantify the flux distribution and validate model-driven hypotheses. Here we describe the development of an upgraded MetaFluxNet which allows (1) construction of metabolic models connected to metabolic databases, (2) calculation of fluxes by metabolic flux analysis, (3) comparative flux analysis with flux-profile visualization, (4) the use of metabolic flux analysis markup language to enable models to be exchanged efficiently, and (5) the exporting of data from constraints-based flux analysis into various formats. MetaFluxNet also allows cellular physiology to be predicted and strategies for strain improvement to be developed from genome-based information on flux distributions. This integrated software environment promises to enhance our understanding on metabolic network at a whole organism level and to establish novel strategies for improving the properties of organisms for various biotechnological applications.     

Incorporating metabolic flux ratios into constraint-based flux analysis by using artificial metabolites and converging ratio determinants

One of the well-established approaches for the quantitative characterization of large-scale underdetermined metabolic network is constraint-based flux analysis, which quantifies intracellular metabolic fluxes to characterize the metabolic status. The system is typically underdetermined, and thus usually is solved by linear programming with the measured external fluxes as constraints. Thus, the intracellular flux distribution calculated may not represent the true values. (13)C-constrained flux analysis allows more accurate determination of internal fluxes, but is currently limited to relatively small metabolic networks due to the requirement of complicated mathematical formulation and limited parameters available. Here, we report a strategy of employing such partial information obtained from the (13)C-labeling experiments as additional constraints during the constraint-based flux analysis. A new methodology employing artificial metabolites and converging ratio determinants (CRDs) was developed for improving constraint-based flux analysis. The CRDs were determined based on the metabolic flux ratios obtained from (13)C-labeling experiments, and were incorporated into the mass balance equations for the artificial metabolites. These new mass balance equations were used as additional constraints during the constraint-based flux analysis with genome-scale E. coli metabolic model, which allowed more accurate determination of intracellular metabolic fluxes.     

Metabolic flux and metabolic network analysis of Penicillium chrysogenum using 2D [13C, 1H] COSY NMR measurements and cumulative bondomer simulation

At present two alternative methods are available for analyzing the fluxes in a metabolic network: (1) combining measurements of net conversion rates with a set of metabolite balances including the cofactor balances, or (2) leaving out the cofactor balances and fitting the resulting free fluxes to measured (13)C-labeling data. In this study these two approaches are applied to the fluxes in the glycolysis and pentose phosphate pathway of Penicillium chrysogenum growing on either ammonia or nitrate as the nitrogen source, which is expected to give different pentose phosphate pathway fluxes. The presented flux analyses are based on extensive sets of 2D [(13)C, (1)H] COSY data. A new concept is applied for simulation of this type of (13)C-labeling data: cumulative bondomer modeling. The outcomes of the (13)C-labeling based flux analysis substantially differ from those of the pure metabolite balancing approach. The fluxes that are determined using (13)C-labeling data are shown to be highly dependent on the chosen metabolic network. Extending the traditional nonoxidative pentose phosphate pathway with additional transketolase and transaldolase reactions, extending the glycolysis with a fructose 6-phosphate aldolase/dihydroxyacetone kinase reaction sequence or adding a phosphoenolpyruvate carboxykinase reaction to the model considerably improves the fit of the measured and the simulated NMR data. The results obtained using the extended version of the nonoxidative pentose phosphate pathway model show that the transketolase and transaldolase reactions need not be assumed reversible to get a good fit of the (13)C-labeling data. Strict statistical testing of the outcomes of (13)C-labeling based flux analysis using realistic measurement errors is demonstrated to be of prime importance for verifying the assumed metabolic model.     

13C-metabolic flux analysis of Actinobacillus succinogenes fermentative metabolism at different NaHCO3 and H2 concentrations

Actinobacillus succinogenes naturally produces high concentrations of succinate, a potential intermediary feedstock for bulk chemical productions. A. succinogenes responds to high CO(2) and H(2) concentrations by producing more succinate and by producing less formate, acetate, and ethanol. To determine how intermediary fluxes in A. succinogenes respond to CO(2) and H(2) perturbations, (13)C-metabolic flux analysis was performed in batch cultures at two different NaHCO(3) concentrations, with and without H(2), using a substrate mixture of [1-(13)C]glucose, [U-(13)C]glucose, and unlabeled NaHCO(3). The resulting amino acid, organic acid, and glycogen isotopomers were analyzed by gas chromatography-mass spectrometry and NMR. In all conditions, exchange flux was observed through malic enzyme and/or oxaloacetate decarboxylase. The presence of an exchange flux between oxaloacetate, malate, and pyruvate indicates that, in addition to phosphoenolpyruvate, oxaloacetate, and malate, pyruvate is a fourth node for flux distribution between succinate and alternative fermentation products. High NaHCO(3) concentrations decreased the amount of flux shunted by C(4)-decarboxylating activities from the succinate-producing C(4) pathway to the formate-, acetate-, and ethanol-producing C(3) pathway. In addition, pyruvate carboxylating flux increased in response to high NaHCO(3) concentrations. C(3)-pathway dehydrogenase fluxes increased or decreased appropriately in response to the different redox demands imposed by the different NaHCO(3) and H(2) concentrations. Overall, these metabolic flux changes allowed A. succinogenes to maintain a constant growth rate and biomass yield in all conditions. These results are discussed with respect to A. succinogenes' physiology and to metabolic engineering strategies to increase the flux to succinate.