p13CMFA: Parsimonious 13C metabolic flux analysis

Carles Foguet; Anusha Jayaraman; Silvia Marin; Vitaly A. Selivanov; Pablo Moreno; Ramon Messeguer; Pedro de Atauri; Marta Cascante

doi:10.1371/journal.pcbi.1007310

Abstract

Deciphering the mechanisms of regulation of metabolic networks subjected to perturbations, including disease states and drug-induced stress, relies on tracing metabolic fluxes. One of the most informative data to predict metabolic fluxes are ¹³C based metabolomics, which provide information about how carbons are redistributed along central carbon metabolism. Such data can be integrated using ¹³C Metabolic Flux Analysis (¹³C MFA) to provide quantitative metabolic maps of flux distributions. However, ¹³C MFA might be unable to reduce the solution space towards a unique solution either in large metabolic networks or when small sets of measurements are integrated. Here we present parsimonious ¹³C MFA (p¹³CMFA), an approach that runs a secondary optimization in the ¹³C MFA solution space to identify the solution that minimizes the total reaction flux. Furthermore, flux minimization can be weighted by gene expression measurements allowing seamless integration of gene expression data with ¹³C data. As proof of concept, we demonstrate how p¹³CMFA can be used to estimate intracellular flux distributions from ¹³C measurements and transcriptomics data. We have implemented p¹³CMFA in Iso2Flux, our in-house developed isotopic steady-state ¹³C MFA software. The source code is freely available on GitHub (https://github.com/cfoguet/iso2flux/releases/tag/0.7.2).

Author summary

¹³C Metabolic Flux Analysis (¹³C MFA) is a well-established technique that has proven to be a valuable tool in quantifying the metabolic flux profile of central carbon metabolism. When a biological system is incubated with a ¹³C-labeled substrate, ¹³C propagates to metabolites throughout the metabolic network in a flux and pathway-dependent manner. ¹³C MFA integrates measurements of ¹³C enrichment in metabolites to identify the flux distributions consistent with the measured ¹³C propagation. However, there is often a range of flux values that can lead to the observed ¹³C distribution. Indeed, either when the metabolic network is large or a small set of measurements are integrated, the range of valid solutions can be too wide to accurately estimate part of the underlying flux distribution. Here we propose to use flux minimization to select the best flux solution in the¹³C MFA solution space. Furthermore, this approach can integrate gene expression data to give greater weight to the minimization of fluxes through enzymes with low gene expression evidence in order to ensure that the selected solution is biologically relevant. The concept of using flux minimization to select the best solution is widely used in flux balance analysis, but it had never been applied in the framework of ¹³C MFA. We have termed this new approach parsimonious ¹³C MFA (p¹³CMFA).

Citation: Foguet C, Jayaraman A, Marin S, Selivanov VA, Moreno P, Messeguer R, et al. (2019) p¹³CMFA: Parsimonious ¹³C metabolic flux analysis. PLoS Comput Biol 15(9): e1007310. https://doi.org/10.1371/journal.pcbi.1007310

Editor: Vassily Hatzimanikatis, Ecole Polytechnique Fédérale de Lausanne, SWITZERLAND

Received: October 23, 2018; Accepted: August 6, 2019; Published: September 6, 2019

Copyright: © 2019 Foguet et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Information files.

Funding: This work was supported by European Commission (EC-654241, FP7-PEOPLE-264780), MINECO-European Commission FEDER funds – “Una manera de hacer Europa” (SAF2017-89673-R; SAF2015-70270-REDT), Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) – Generalitat de Catalunya (2017SGR1033) and Instituto de Salud Carlos III (CIBEREHD, CB17/04/00023). CF acknowledges the support received through “Becas de la Caixa para estudios de doctorado en universidades españolas” funded by the “La Caixa” foundation. CF also acknowledges the support from the Spanish National Bioinformatics Institute (INB-ISCIII-ES-ELIXIR). MC acknowledges the support received through the prize “ICREA Academia” for excellence in research, funded by ICREA foundation – Generalitat de Catalunya. The funders had no role in study design, data collection, and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Fluxomics is the omics field that analyses metabolic fluxes (i.e., reaction and transport rates in living cells) which are a close reflection of the metabolic phenotype. As such, quantitative tracking of metabolic fluxes is vital for deciphering the regulation mechanisms of metabolic networks subjected to perturbations, including disease states and drug-induced stress. However, unlike other omics data that can be quantified directly, the fluxome can only be estimated through an indirect interpretation of experimental data[1–3].

There are two main model-based approaches to quantifying metabolic fluxes, Flux Balance Analysis (FBA) and ¹³C Metabolic Flux Analysis (¹³C MFA). Both methods use stoichiometric, thermodynamic and experimental constraints to find the range of feasible fluxes across a metabolic network and then find the flux distributions within that space that optimize a given objective function. However, both techniques differ in the type of objective function optimized.

In FBA, the objective function is a set of fluxes to be minimized or maximized. These fluxes must represent a biological objective deemed desirable in the conditions of study (e.g., synthesis of biomass components for proliferating systems)[4]. A significant limitation of FBA is that the choice of objective(s) can significantly influence the predicted flux distributions.

In ¹³C MFA, the objective function is to minimize the difference between simulated and measured ¹³C enrichment in metabolites [5,6]. ¹³C enrichment is quantified in metabolic products and intermediates after incubating samples with metabolic substrates labeled with ¹³C (tracers) and provides information about how carbons are redistributed along metabolic pathways[7]. Compared to FBA, ¹³C MFA has a greater capacity to elucidate the fluxes of central carbon metabolism. However, ¹³C MFA is more complex to solve than FBA due to the non-linear nature of the ¹³C MFA objective.

A significant limitation of FBA is that there is generally a wide range of optimal flux distributions[8]. This is not usually the case with ¹³C MFA which can generally determine flux distributions with a high degree of accuracy. ¹³C MFA achieves this by integrating large sets of measured isotopologue fractions from parallel experiments with tracers optimized for different parts of the network[9–16]. However, when ¹³C MFA is used in large metabolic networks and with a limited set of measurements, it can also suffer from the same limitation as FBA and result on a wide interval of flux values for part of the metabolic network[5,17–19].

On FBA, an approach to reduce the range of optimal solutions consists in running a second optimization step on the optimal solution range. One of such methods is parsimonious FBA (pFBA)[20]. This approach, which follows the principle of parsimony or simplicity, consists on finding the optimal value of the primary objective function through FBA and then running a second optimization step where the sum of reaction fluxes is minimized while maintaining the optimal primary objective. The GIMME (and its derivative GIM3E) algorithms[21,22] are based on a similar principle as pFBA. Unlike standard pFBA, where all reactions fluxes are minimized with equal weight, GIMME integrates gene expression data to give greater weight to the minimization of fluxes through reactions catalyzed by lowly expressed enzymes. Different to FBA, for ¹³C MFA, there is currently no approach that relies on a second optimization to reduce the solution space when experimental data is insufficient to constrain the system towards a unique solution.

In addition to model-based approaches (e.g., FBA or ¹³C MFA), metabolic fluxes can also be analyzed through the direct or semidirect interpretation of ¹³C data. This approach primarily consists of predicting the contribution of a labeled substrate to the synthesis of a given metabolite (nutrient contribution) and predicting the relative activity of pathways (pathway activity analysis). Pathway activity analysis assumes that the isotopologue fractions used as a surrogate for the pathways of interest are primarily generated through them. This assumption is generally based on the assertion that the pathways of interest are the most direct way to generate such fractions from the labeled substrate used in the experiment[2,7,23–25]. Unlike ¹³C MFA, direct interpretation of ¹³C data is generally not able to quantify network-wide flux maps. Instead, it provides a series of qualitative or semiquantitative flux predictions around each analyzed metabolite. Strategies that couple direct interpretation of ¹³C data to regression and correlation analyses are widely applied to unveil the effect of an external perturbation, such as a therapeutic intervention, on central carbon metabolism[26–30].

Here we present parsimonious ¹³C MFA (p¹³CMFA), a new model-based approach to flux estimation. p¹³CMFA first minimizes the difference between experimental and simulated ¹³C enrichment in metabolites (¹³C MFA) and then applies the flux minimization principle to select the best solution among the solutions that fit experimental ¹³C data. Hence, p¹³CMFA can be used to select the best flux map in instances where experimental ¹³C measurements are not enough to fully constrain the ¹³C MFA solution space. Furthermore, the minimization can be weighted by gene expression allowing seamless integration of ¹³C with gene expression data (Fig 1).

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Fig 1. The conceptual basis of p¹³CMFA: From an infinite space of flux (v) solutions, a space of feasible solutions is obtained through the integration of stoichiometric and thermodynamic constraints (in the form of a constraint-based model) and the measured extracellular fluxes.

Applying ¹³C MFA to integrate experimental ¹³C data can further reduce the solution space to those flux distributions that are consistent with such data. Through flux minimization, p¹³CMFA can identify the optimal flux distribution that lies on the edge of the ¹³C MFA solution space. Such minimization can be weighted according to the gene expression evidence for each enzyme.

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We have implemented p¹³CMFA in Iso2Flux, our in-house developed isotopic steady-state ¹³C MFA software (https://github.com/cfoguet/iso2flux/releases/tag/0.7.2). As a proof of concept, we have applied it to the analysis of the metabolic flux distribution in HUVECs (Human umbilical vein endothelial cells) through the integration of a small set of ¹³C enrichment measurements and transcriptomics data. Furthermore, we validated the predictive capacity of p¹³CMFA using data from a published study of HTC116 cells where fluxes had been estimated with a high degree of confidence[14]. Using only a small subset of the measurements from such study, p¹³CMFA was able to achieve significantly better flux predictions than both ¹³C MFA and GIMME.

Results

Description of the p¹³CMFA approach

p¹³CMFA consists of two consecutive optimizations: first, the optimal solution to the ¹³C MFA problem is identified (Eq 1); secondly, the weighted sum of reaction fluxes is minimized within the optimal solution space of ¹³C MFA (Eq 2).

The ¹³C MFA optimization (Eq 1) identifies the flux distribution that minimizes the difference between measured and simulated isotopologue fractions [5,7]: (Eq 1)

where,

v is a vector of flux values describing a valid steady-state flux distribution;

X_opt is the optimal value of the ¹³C MFA objective;

E_j is the experimentally quantified fraction for isotopologue j;

Y_j(v) is the simulated isotopologue fraction for isotopologue j with flux distribution v. Such simulation is performed by solving a complex non-linear system of equations built around isotopologues balances [1].

σ_j is the experimental standard deviation of the measurements of isotopologue j;

S is the stoichiometric matrix;

lb and ub are vectors defining the upper and lower bounds for flux values. Flux bounds can be used to integrate experimental flux measurements;

Either in large metabolic networks or when small sets of ¹³C measurements are integrated, the ¹³C MFA problem can be undetermined and there can be a wide range of possible solutions. Such indetermination emerges from cycles and alternative pathways in the metabolic network, which lead to many possible flux combinations that can result in the measured ¹³C label patterns. Furthermore, many of the ¹³C MFA solutions can involve large fluxes through futile cycles, which are usually artifacts of the optimization process as in vivo enzyme activities cannot support such large flux values. Therefore, to select the best solution among the many solutions that fit experimental ¹³C data, p¹³CMFA runs a second optimization where the weighted sum of fluxes is minimized (Eq 2): (Eq 2) where:

w_i is the weight given to the minimization of flux through reaction i;

T is the maximum value that the ¹³C MFA objective can deviate from its optimal value (primary objective tolerance) when fluxes are minimized;

The difference between the optimal ¹³C MFA objective function value and the objective function value when the total reaction flux is minimized can be assumed to follow a Chi2-distribution with one degree of freedom. Therefore, setting T to 3.84 gives a p¹³CMFA solution within the 95% confidence intervals of ¹³C MFA[5].

With p¹³CMFA, the activity through cycles is minimized to the minimum amount needed to account for experimental measurements. Furthermore, gene expression measurements can be integrated to give greater weight to the minimization of fluxes through reactions catalyzed by lowly expressed enzymes. Then, in instances where multiple pathways can result in similar label patterns, those pathways with stronger gene expression evidence are selected. Hence, p¹³CMFA reduces the solution space towards a unique solution without requiring a simplification of the metabolic network or additional ¹³C measurements (Fig 1).

Example of p¹³CMFA usage

As an example of a potential application of p¹³CMFA, we applied it to analyze the metabolic flux distribution in HUVECs using a publicly available dataset not large enough to make meaningful flux predictions with conventional ¹³C MFA.

In this study, available in the MetaboLights repository[31] (accession number MTBLS412), HUVECs were incubated in the presence of the tracer [1,2-¹³C₂]-glucose, and the relative abundance of ¹³C isotopologues was quantified in glycogen, ribose, lactate, and glutamate. The rates of production/consumption of glucose, glycogen, lactate, glutamate, and glutamine were also quantified. The data were integrated into a stoichiometric model of central metabolism which includes glycolysis, glycogen metabolism, pentose phosphate pathway (PPP), tricarboxylic acid (TCA) cycle, fatty acid synthesis, and energy and redox metabolism (S1 ZIP).

To predict the flux distribution using conventional ¹³C MFA, 95% confidence intervals were computed for each predicted flux value. From this analysis, the space of flux solutions consistent with the measured ¹³C enrichment was estimated. The resulting space of solution was still mostly undetermined and, in general, ¹³C MFA was unable to significantly constraint the flux ranges emerging from the stoichiometric and thermodynamic constraints and the measured extracellular fluxes (Fig 2, S1 Table). For instance, despite integrating measurements of ¹³C enrichment in ribose, it was not possible to conclude whether the oxidative branch of the pentose phosphate pathway contributed more to de novo ribose synthesis than the non-oxidative branch or vice versa.

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Fig 2. Flux spectrum, GIMME solutions, ¹³C MFA flux ranges, and p¹³CMFA solutions for some key reaction fluxes.

Flux spectrum represents the feasible flux ranges considering only the stoichiometric and thermodynamic constraints and the measured extracellular fluxes. GIMME flux values are obtained when total reaction flux is minimized weighted by gene expression without integrating ¹³C data. For ¹³C MFA, the flux values obtained after the ¹³C MFA optimization and the range of the 95% confidence intervals for such values are shown. The p¹³CMFA flux values are obtained when total reaction flux is minimized within the ¹³C MFA solution space. Fluxes are expressed in μmol·h-1·million-cells-1.

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Nevertheless, p¹³CMFA can be applied to select the best solution in the ¹³C MFA solution space. With this aim, transcriptomic data taken from the literature[32] were used to add additional penalties to the flux through lowly expressed enzymes. Indeed, by applying p¹³CMFA, we can now conclude that, under the condition of the study, glucose is mostly directed towards lactate production except for a small part going to the TCA cycle through pyruvate dehydrogenase (Fig 2, Fig 3). Glutamine is mainly metabolized to glutamate or directed to glycolysis through the TCA cycle and phosphoenolpyruvate carboxykinase. In the PPP, the non-oxidative branch contributes to roughly 60% of the net ribose synthesis. Only the glycogen phosphorylase/glycogen synthase futile cycle is predicted to be active, while the remaining futile cycles (i.e., the hexokinase/glucose 6-phosphatase, phosphofructokinase/fructose bis-phosphatase, pyruvate carboxylase/phosphoenolpyruvate carboxykinase, and glutaminase/glutamine synthase cycles) are predicted to be inactive. Concerning redox metabolism, most of the reduced NAD+ (NADH) produced in the mitochondria is exported to the cytosol through the malate-aspartate shuttle, where it is used to reduce pyruvate to lactate.

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Fig 3. Schematic representation of the central carbon metabolism flux map obtained with p¹³CMFA.

Reaction fluxes are indicated for some key reactions in μmol·h-1·million-cells-1. Arrows indicate net flux direction, and line width is representative of flux magnitude. Reactions and metabolites of redox and energy metabolism have been omitted from this figure for clarity. 2PG: 2-Phosphoglycerate. 3PG: 3-Phosphoglycerate. AcCoA: Acetyl-CoA. aKG: α-Ketoglutarate. Asp: Aspartate. bPG13: 1,3-Bisphosphoglycerate. Cit: Citrate. DhaP: Dihydroxyacetone phosphate. Fru16bP: Fructose 1,6-bisphosphate. Fru6P: Fructose 6-phosphate. Fum: Fumarate. GaP: Glyceraldehyde-3-Phosphate. Glc: Glucose. Glc1P: Glucose 1-phosphate. Glc6P: Glucose 6-phosphate. Gln: Glutamine. Glu: Glutamate. Glucon6P: Gluconate 6-phosphate. Glygn: Glycogen. iCit: Isocitrate. Lac: Lactate. Mal: Malate. OAA: Oxaloacetate. PEP: Phosphoenolpyruvate. Pyr: Pyruvate. Rib5P: Ribose 5-phosphate. Rul5P: Ribulose 5-phosphate. Sed7P: Sedoheptulose 7-phosphate. Suc: Succinate. SucCoa: Succinyl-CoA. UDPGlc: Uridine diphosphate glucose. The subscripts e, c, and m denote the extracellular, cytosolic and mitochondrial compartments, respectively.

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To evaluate the contribution of ¹³C MFA to the p¹³CMFA solution, GIMME (i.e., flux minimization weighted by gene expression without integrating ¹³C data) was also performed (Fig 2, S1 Table). Lacking ¹³C data, GIMME does not predict any activity in the oxidative branch of the pentose phosphate pathway, nor on the glycogen phosphorylase/glycogen synthase futile cycle. Furthermore, GIMME predicts a significantly larger flux through pyruvate dehydrogenase than p¹³CMFA. Interestingly, p¹³CMFA predicts an increased activity of the TCA cycle compared to the GIMME solution. This increased activity is fueled by alternative sources of acetyl-CoA such as fatty acid oxidation or catabolism of ketogenic amino acids. Hence, p¹³CMFA is able to take advantage of measured ¹³C enrichments and predict significantly different flux maps than those derived from flux minimization alone.

Validation of the p¹³CMFA approach

To validate the p¹³CMFA method, we used data from a metabolic characterization of the colon cancer cell line HCT 116 published by Tarrado-Castellarnau et al. [14]. In this study, 25 direct flux measurements and 24 sets of isotopologue fractions, measured after incubation with either [1,2-¹³C₂]-glucose or [U-¹³C₅]-glutamine, had been integrated in the framework of ¹³C MFA. With such a large set of experimental measurements, ¹³C MFA had been able to estimate the flux through 62 reactions with a high degree of accuracy. In the same study, transcriptomics data were also collected.

From this large data set, we selected a partial data set consisting of 7 experimental flux measurements (the rates of uptake/secretion of glucose, lactate, glutamine, glutamate and, oxygen and the rate of protein and glycogen synthesis) and 4 sets of isotopologue fractions (isotopologue fractions in ribose, lactate, glutamate and glycogen measured after incubation with 1,2-¹³C₂]-glucose). Those are the sets of isotopologues and fluxes that were analyzed in the HUVECs case study with the addition of the rate of protein synthesis and oxygen consumption which Tarrado-Castellarnau et al. described as key determinants of the metabolic phenotype of HCT 116 cells. The partial data set was used to apply pFBA, GIMME, ¹³C MFA and p¹³CMFA in the framework of the metabolic network defined by Tarrado-Castellarnau et al. [14] (S2 Zip). p¹³CMFA was applied both with and without integrating gene expression data (p¹³CMFA+ge and p¹³CMFA-ge, respectively). Two complementary metrics, Pearson’s correlation and Euclidian distance, were used to evaluate the similarity between the predicted flux distributions and the flux maps estimated by Tarrado-Castellarnau et al. using the full dataset[14] (Fig 4, S2 Table). The results show that p¹³CMFA-ge yields a significantly more accurate flux prediction than both pFBA (i.e., flux minimization without integrating ¹³C data), and ¹³C MFA. Interestingly, while integrating gene expression significantly enhances the accuracy of p¹³CMFA (p¹³CMFA+ge compared to p¹³CMFA-ge), such effect is less marked than the effect of adding gene expression to standard flux minimization (GIMME compared to pFBA). This is due to the fact that p¹³CMFA-ge flux predictions have already a remarkable level of accuracy; hence, less information can be gained by adding transcriptomics data. Nevertheless, even if GIMME achieves flux predictions of similar accuracy to p¹³CMFA-ge, p¹³CMFA+ge results on flux predictions that are significantly more accurate than those obtained with GIMME. Hence, in instances were only a limited number of ¹³C measurements are available, p¹³CMFA is a valid method for obtaining accurate flux estimations, regardless of the availability of gene expression data.

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Fig 4. Comparison of the predictive power of ¹³C MFA, pFBA, GIMME, and p¹³CMFA.

A: Pearson’s correlation coefficients between the reference flux distribution and the flux maps obtained from ¹³C MFA (optimal solution), pFBA, GIMME, and p¹³CMFA. p¹³CMFA was applied both with and without integrating gene expression data (p¹³CMFA+ge and p¹³CMFA-ge, respectively). The statistical significance of the difference between correlation coefficients was evaluated using the Fisher r-to-z transformation[33]. B: Euclidian distances between the reference flux distribution and the flux maps obtained from ¹³C MFA (optimal solution), pFBA, GIMME, and p¹³CMFA.

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Discussion

¹³C MFA is a well-established technique and has proven to be an extremely valuable tool in quantifying metabolic fluxes[9–18]. However, to fully determine fluxes through a large metabolic network, parallel labeling experiments must be performed and ¹³C propagation must be quantified in many metabolites in the network[19]. Indeed, when applying ¹³C MFA either with a small set of experimental data or with a large metabolic network, part of the ¹³C MFA solution space can be too wide to draw meaningful conclusions about the underlying flux distribution. This solution space can be reduced by removing degrees of freedom from the system, for instance, by removing reactions from the network or making reactions irreversible. However, this can introduce an arbitrary bias in the resulting flux distribution.

Here we describe p¹³CMFA, a new approach for ¹³C data integration which can overcome these limitations of ¹³C MFA and estimate a realistic solution within an undetermined ¹³C MFA solution space. This solution will be the flux distribution within the ¹³C MFA solution space that minimizes the weighted sum of reaction fluxes. Thus, it will be the most enzymatically efficient solution. In that regard, p¹³CMFA is partially based on a similar principle as pathway activity analysis (i.e., the assumption that specific fractions of isotopologues are primarily generated through the simplest combinations of pathways). However, unlike pathway activity analysis, p¹³CMFA is able to integrate all quantified isotopologue fractions and flux measurements (e.g. rates of metabolite uptake and secretion) to generate network-wide flux maps consistent with such data. Furthermore, p¹³CMFA is highly flexible; for instance, here we show that it can be used to seamlessly integrate gene expression data by giving higher weight to the minimization of the fluxes through lowly expressed enzymes.

As a proof of concept, we exemplified how p¹³CMFA can be used to estimate flux distributions integrating only limited sets of ¹³C measurements in a test case where traditional ¹³C MFA was unable to provide a narrow solution space. Furthermore, we demonstrated that, when a limited set of measurements are integrated, p¹³CMFA can yield more accurate flux predictions than both ¹³C MFA and GIMME.

p¹³C MFA does not aim to be a replacement of ¹³C MFA; instead, it seeks to supplement it by identifying the more straightforward solution in parts of the network that cannot be uniquely determined. In that regard, it can be used to quantitatively study flux distributions in instances where not enough information can be obtained with conventional ¹³C MFA. Nor does it aim to replace the direct interpretation of ¹³C data. The latter is still a suitable technique when the goal of the analysis is to compare the relative activity of well-established pathways across conditions or quantify substrate contributions rather than to generate complete flux maps.

¹³C data has been widely used to assist in drug discovery. In this regard, tracer analysis coupled with regression and correlation analyses is frequently used to characterize drug response [26–29]. Such approach uses regression and correlation statistics with binary, numeric and visual analysis to integrate drug dosage, time points, as well as all necessary biological variables in order to diagnose disturbed stable isotope labeled matrices[29]. p¹³CMFA could further expand the role of ¹³C in drug discovery by allowing the integration of ¹³C and transcriptomic data in the framework of genome-scale metabolic models. In the framework of such models, drug targets are identified by systematically simulating the effect of reactions or genes knock out to cell function[34]. This is usually attained by applying the ROOM[35] or MOMA[36] algorithms, which take a unique flux solution as input (wild-type flux distribution) to predict the most likely effect of a gene KO. Hence, p¹³CMFA results could be potentially used as ROOM/MOMA inputs allowing to take full advantage of the flux information derived from both ¹³C and transcriptomics data to predict new drug targets. With atom mappings now available on a genome-scale[37], the main obstacle to applying p¹³CMFA at a genome-scale is the high computational complexity of solving the resulting non-linear problem which increases with the size of the network. Hence, the next challenge for p¹³CMFA will be optimizing its implementation for genome-scale networks.

Methods

Flux spectrum

The flux spectrum[38] (i.e., the feasible range of fluxes for a given set of stoichiometric, thermodynamic and flux boundary constraints) was determined using flux variability analysis [8]. Under this approach, each flux is minimized (Eq 3) and maximized (Eq 4) subject to constraints to find the minimum () and maximum () feasible values for each flux: (Eq 3) (Eq 4)

¹³C MFA confidence intervals

The ¹³C MFA solution space is estimated by computing the confidence intervals for each flux. Such intervals are obtained by maximizing (Eq 5) and minimizing (Eq 6) each flux subject to constraints[5]. (Eq 5) (Eq 6) where,

vmin_i: is the lower bound of the confidence interval for flux i with tolerance T;

vmax_i: is the upper bound of the confidence interval for flux i with tolerance T;

Provided that the same primary objective tolerance (T) is used in computing both the p¹³CMFA solution and the ¹³C MFA confidence intervals, the p¹³CMFA solution will always fall within the boundaries of ¹³C MFA confidence intervals (vmin_i≤v_i≤vmax_i).

GIMME and pFBA

To apply GIMME and pFBA, the sum of fluxes is minimized subject only to network stoichiometry and flux boundaries (Eq 7).

(Eq 7)

In GIMME, flux minimization weights are derived from gene expression measurements, whereas in pFBA all reactions are given the same minimization weight[20,22].

Transcriptomic analysis

Transcriptomic data of HUVECs and HCT 116 cells published by Weigand et al.[32] and Tarrado-Castellarnau[14] et al., respectively, were obtained from the Gene Expression Omnibus repository[39]. A Robust Multichip Analysis gene-level normalization[40] was performed with the Oligo package for R[41].

Using gene-protein-reaction rules, normalized transcript intensities were mapped to each enzyme-catalyzed reaction or protein-facilitated transport process. The weight given to the minimization of fluxes was assigned according to the following equation: (Eq 8) where,

ge_i is the gene expression value assigned to reaction i;

Th is the gene expression threshold. Fluxes through reactions with gene expression levels below this threshold are given additional minimization weight;

Using the same criteria as GIM3E[22], Th was set at the maximum gene expression value found in the set of genes mapped to the metabolic network (Eq 9): (Eq 9)

Using this threshold, the information gained from integrating available gene expression measurements is maximized. Other Th values were tested in the validation case study[14] and using the maximum gene expression as the threshold was found to yield the most accurate flux predictions (S3 Table).

p¹³CMFA implementation

p¹³CMFA was implemented in Iso2Flux, our in-house developed ¹³C MFA software (https://github.com/cfoguet/iso2flux/releases/tag/0.7.2).

Iso2Flux computes steady-state flux distributions as the product of the null space of the stoichiometric matrix and the vector of free fluxes. Reversible reactions are split into forward and reverse reactions. For each reversible reaction, a turnover variable (t_i) is introduced defining the flux that is common to the forward (v_i^f) and reverse (v_i^r) reactions. These variables are used to assign values to the fluxes of the forward and reverse reactions as a function of the steady-state net flux (v_i).

(Eq 10)

(Eq 11)

Iso2flux uses the Elementary Metabolite Unit (EMU) framework[1] to build the ¹³C propagation model. This framework is based on a highly efficient decomposition method that identifies the minimum amount of isotopologue transitions required to simulate the experimentally quantified isotopologues according to the defined carbon propagation rules. The isotopologue transitions are grouped into decoupled systems based on isotopologue size. Balance equations are built around each isotopologue fraction under the assumption of isotopic steady state (S1 Fig). Using the steady-state flux distribution as an input, systems of equations around isotopologues balances are solved sequentially starting with the smallest isotopologue size [1] using the fsolve function of the SciPy library (https://scipy.org/scipylib/index.html). Solving such system predicts the isotopologue distribution associated with a given steady-state flux distribution (Y_j(v)).

The self-adaptive differential evolution (SADE) algorithm from PyGMO (Python Parallel Global Multiobjective Optimizer, https://github.com/esa/pagmo2) was used to find the optimal solution of the ¹³C MFA (Eq 1) and p¹³CMFA (Eq 2) problems. SADE was parallelized using the generalized island-model paradigm. Under such implementation, SADE is run in parallel in different CPU processes (islands). After a given number of SADE iterations (generations), the best solutions (individuals) in each SADE process (island) are shared to parallel SADE processes (migrate to adjacent islands). To prevent bias from the starting solutions (starting populations), the islands are seeded through random sampling of all variables. Free fluxes variables are sampled using the optGpSampler implemented into COBRApy[42,43]. Turnover variables are sampled using the random.uniform function built into python. The algorithm was run with 7 islands, each with a population of 60, and with migrations between islands set to occur every 400 generations. For the analyzed ¹³C MFA and p¹³CMFA problems, repeated iterations of the algorithm were shown to reliably converge towards the same minimal objective function value.

Accommodating large metabolite pools

At the beginning of a ¹³C experiment, all internal metabolites are unlabeled (m0). Progressively, these products are enriched in ¹³C, with the subsequent decrease in m0. Isotopic steady state is quickly reached for small pools of metabolites but not necessarily for larger pools such as those of fatty acids, glycogen or metabolites present in large concentrations in the external medium[44]. For these larger pools, unlabeled isotopologues m0 are oversized and might not quickly decrease to the theoretical value that should be reached at steady-state.

However, it is possible to represent the effect of large pools in the framework of steady-state ¹³C MFA through the addition of a virtual reaction. This reaction replaces labeled isotopologues by unlabeled isotopologues in metabolites with large pools. With p¹³CMFA, the flux through this virtual reaction can be minimized. Effectively, this allows correcting steady-state ¹³C simulations for large pools while identifying the solutions that require the minimum amount of correction.

Evaluating the significance of the difference between correlation coefficients

The statistical significance of the difference between correlation coefficients was evaluated using the Fisher r-to-z transformation[33]. Following this approach, Pearson’s correlation coefficients (r) can be converted to a z-score (r’): (Eq 12)

The variance of z (Sz) will depend only on the sample size (n): (Eq 13)

From Eq 12 and Eq 13, the significance of the difference between two correlation coefficients (r1 and r2) can be evaluated by computing the z score corresponding to such difference (Eq 14) and its associated p-value.

(Eq 14)

Experimental methods

Human Umbilical Vein Endothelial Cells (HUVECs-pooled, Lonza) were maintained on 1% gelatin-coated flasks at 37°C in a humidified atmosphere of 5% CO₂ and 95% air in MCDB131 (Gibco) medium, supplemented with the recommended quantity of endothelial growth medium (EGM) SingleQuots (Lonza), 10% fetal bovine serum (FBS) (Gibco), 2 mM glutamine (Gibco) and 0.1% Streptomycin (100 μg/mL)/Penicillin (100 units/mL) (S/P) (Gibco). 1 × 106 HUVECs were seeded in 1% gelatin-coated cell culture plates for 6h, and then the maintenance medium was replaced with the MCDB131 basal medium, supplemented with 2% FBS, 2 mM glutamine and 0.1% S/P and cells were incubated overnight for nutrient deprivation. After nutrient deprivation, the medium was replaced with a restricted medium containing MCDB131 medium supplemented with 2% FBS, 2 mM glutamine and 0.1% S/P with 10 mM of 50% [1,2-¹³C₂]-glucose (Sigma-Aldrich) and cells were incubated for 40h in a humidified atmosphere with 5% CO₂ and 1% O₂ at 37°C. Both at the beginning (t = 0h) and the end (t = 40h) of incubation, media and pellets were collected. On the one hand, media and cell pellets were used for analyzing isotopologue abundances for glucose, lactate, glutamate, RNA ribose and glycogen. Raw data are publicly available in the MetaboLights repository at http://www.ebi.ac.uk/metabolights [31], with accession number MTBLS412. Isolation, derivatization and analysis details are described in MetaboLights. Glucose, lactate, glutamate, and glutamine concentrations were determined in media samples for estimation of secretion or uptake rates of these metabolites using spectrophotometric methods[45]. Also, the net rate of glycogen re-utilization into glucose was estimated by quantifying glycogen content at initial and final time points using [U-¹³C-D₇]-glucose as recovery standard[46]. All biochemical data were normalized by cell number, and by incubation time (h). The resulting rates–expressed in micromoles of metabolite consumed/produced/transformed per hour per million cells (μmol·h-1·million-cells-1)–were 0.463, 0.099, 0.050 and 1.169 for glucose uptake, glutamine uptake, glutamate secretion, and lactate secretion, respectively, and a net transformation of glycogen of 0.000175.

Supporting information

S1 Fig. Example of isotopologue balance equations in a toy metabolic network.

In this toy metabolic network, two mono-carbon metabolites (C_a and C_b) are condensed into a bi-carbon metabolite (C_a-C_b) through a reaction with a flux v₁. Metabolite C_a-C_b is removed from the system at a rate of v₂. For each metabolite, isotopologue fractions (M_x) are defined as the relative abundance of the metabolite with x number of ¹³C substitutions. Isotopologue balances for metabolite C_a-C_b are indicated. Under the assumption of isotopic steady state (i.e., isotopologue fractions are constant in time) and given v₁ and v₂, and a set of isotopologue fractions for C_a and C_b (assumed a constant input), the system can be solved to identify the steady-state isotopologue fractions for metabolite C_a-C_b.

https://doi.org/10.1371/journal.pcbi.1007310.s001

(TIF)

S1 Table. Flux spectrum, GIMME, ¹³C MFA and p¹³CMFA flux solutions for all net reaction fluxes in the HUVECs case study.

Fluxes are expressed in μmol·h-1·million-cells-1.

https://doi.org/10.1371/journal.pcbi.1007310.s002

(XLSX)

S2 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set using ¹³C MFA, pFBA, GIMME, and p¹³CMFA.

Fluxes are indicated in μmol·h-1·million-cells-1.

https://doi.org/10.1371/journal.pcbi.1007310.s003

(XLSX)

S3 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set with p¹³CMFA using different gene expression percentiles as thresholds for adding additional weight to flux minimization.

Fluxes are indicated in μmol·h-1·million-cells-1.

https://doi.org/10.1371/journal.pcbi.1007310.s004

(XLSX)

S1 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HUVECs case study.

The files are inputs for running p¹³CMFA on Iso2Flux.

https://doi.org/10.1371/journal.pcbi.1007310.s005

(ZIP)

S2 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HCT 116 cells case study.

The files are inputs for running p¹³CMFA on Iso2Flux.

https://doi.org/10.1371/journal.pcbi.1007310.s006

(ZIP)

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p¹³CMFA: Parsimonious ¹³C metabolic flux analysis

p¹³CMFA: Parsimonious ¹³C metabolic flux analysis

This is an uncorrected proof.

Figures

Abstract

Author summary

Introduction

Results

Description of the p¹³CMFA approach

Example of p¹³CMFA usage

Validation of the p¹³CMFA approach

Discussion

Methods

Flux spectrum

¹³C MFA confidence intervals

GIMME and pFBA

Transcriptomic analysis

p¹³CMFA implementation

Accommodating large metabolite pools

Evaluating the significance of the difference between correlation coefficients

Experimental methods

Supporting information

S1 Fig. Example of isotopologue balance equations in a toy metabolic network.

S1 Table. Flux spectrum, GIMME, ¹³C MFA and p¹³CMFA flux solutions for all net reaction fluxes in the HUVECs case study.

S2 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set using ¹³C MFA, pFBA, GIMME, and p¹³CMFA.

S3 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set with p¹³CMFA using different gene expression percentiles as thresholds for adding additional weight to flux minimization.

S1 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HUVECs case study.

S2 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HCT 116 cells case study.

References

This is an uncorrected proof.

Figures

Abstract

Author summary

Introduction

Results

Description of the p13CMFA approach

Example of p13CMFA usage

Validation of the p13CMFA approach

Discussion

Methods

Flux spectrum

13C MFA confidence intervals

GIMME and pFBA

Transcriptomic analysis

p13CMFA implementation

Accommodating large metabolite pools

Evaluating the significance of the difference between correlation coefficients

Experimental methods

Supporting information

S1 Fig. Example of isotopologue balance equations in a toy metabolic network.

S1 Table. Flux spectrum, GIMME, 13C MFA and p13CMFA flux solutions for all net reaction fluxes in the HUVECs case study.

S2 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set using 13C MFA, pFBA, GIMME, and p13CMFA.

S3 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set with p13CMFA using different gene expression percentiles as thresholds for adding additional weight to flux minimization.

S1 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HUVECs case study.

S2 ZIP. Files describing the metabolic network, carbon propagation rules, and experimental data used for the HCT 116 cells case study.

References

Description of the p¹³CMFA approach

Example of p¹³CMFA usage

Validation of the p¹³CMFA approach

¹³C MFA confidence intervals

p¹³CMFA implementation

S1 Table. Flux spectrum, GIMME, ¹³C MFA and p¹³CMFA flux solutions for all net reaction fluxes in the HUVECs case study.

S2 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set using ¹³C MFA, pFBA, GIMME, and p¹³CMFA.

S3 Table. Comparison between the reference flux map in HCT 116 cells and the flux maps computed from the partial data set with p¹³CMFA using different gene expression percentiles as thresholds for adding additional weight to flux minimization.